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+{"text":"\\section{Introduction}\n\nThis article concerns embeddability conditions for pairs of groups from the family of \\emph{symmetric R. Thompson groups} \n$\\{V_m(G)\\}$.  The group $V_m(G)$ is the group $V_m(G)=\\langle V_m\\cup\nG\\rangle$, where $V_m\\leq \\operatorname{Aut}(\\mathfrak{C}_m)$ is the Higman-Thompson group\ndenoted $G_{m,1}$ by Higman in \\cite{HI}, acting on the Cantor space $\\mathfrak{C}_m:= \\{0,1,\\ldots, m-1\\}^\\omega$, while $G$ is a particular faithful representation of a finite group $\\widetilde{G}\\leq \\operatorname{Sym}(m)$ in $\\operatorname{Aut}(\\mathfrak{C}_m)$. \n\n\nThe groups $\\{V_m(G)\\}$ have developed as groups of interest for a variety of reasons.  Firstly, they were singled out as natural groups of interest in \\cite{NekraCuntz} and \\cite{Roever}, and they arise naturally as a fundamental subfamily of Hughes' $\\mathcal{F}\\mathcal{S}\\mathcal{S}$ groups \\cite{Hughes}.  The paper \\cite{BDJ} shows that for $n\\geq2$, $V_m\\cong V_m(G)$ if and only if $\\widetilde{G}$ is semiregular (the nontrivial elements of $\\widetilde{G}$ have no fixed points), and also, that for $m>3$ there exists $\\widetilde{G},\\widetilde{H}\\in\\operatorname{Sym}(m)$ with $\\widetilde{G}\\cong\\widetilde{H}$ but where the induced groups $V_m(G)$ and $V_m(H)$ are not isomorphic (the orbit structure of the actions of the elements of the groups $\\widetilde{G}$ and  $\\widetilde{H}$ impacts the isomorphism types of the groups $V_m(G)$ and $V_m(H)$).  In another direction, in \\cite{FarleyCoCF} Farley shows the symmetric R. Thompson groups are CoCF groups (see \\cite{HRRT} for the definition of CoCF groups).  Thus, if one can show that some group in the family $\\{V_m(G)\\}$ fails to embed in $V=V_2$, then Lehnert's conjecture will be shown to be false (see \\cite{LehnertDiss,LehnertSchweitzer, BMN}).\n\n\nWe investigate conditions on $m\\leq n$, $G\\leq \\operatorname{Sym}(m)$, and $H\\leq \\operatorname{Sym}(n)$ that guarantee the existence of embeddings between the groups $V_m(G)$ and $V_n(H)$ (we now drop the ``tilde'' notation on the groups $G$ and $H$ when thinking of them as subgroups of $\\operatorname{Sym}(m)$ and $\\operatorname{Sym}(n)$, respectively).  Thus, this note can be thought of as a continuation of the investigations in \\cite{BDJ}, and is partly inspired by the work of Birget (in \\cite{B}, he gives a method to embed $V_2$ into $V_m$ for $2\\leq m$ (embeddings in the other direction have been known since Higman's book \\cite{HI})), and partly by considering some of the questions alluded to in the previous paragraph.  In this context, our two embedding results depend on the direction of the embedding ($V_m(G)\\rightarrowtail V_n(H)$ or $V_n(H)\\rightarrowtail V_m(G)$), and our constructed embeddings require in both cases the Higman condition $n\\equiv 1\\mod(m-1)$.\n\nThe embedding $V_m(G)\\rightarrowtail V_n(H)$ is algebraic in nature, inspired by the embedding of Birget from $V_2$ into $V_n$ in \\cite{B}, while the embedding $V_n(H)\\rightarrowtail V_m(G)$ uses a topological conjugacy by rational group elements (see \\cite{GNS}).\n\nWe can now state and discuss our main results.\n\n\\begin{theorem}\\label{thm:MT1}\nLet $n,m \\geq 2$ be natural numbers such that $m<n$, and let $G \\leq \\operatorname{Sym}(m)$, $H \\leq \\operatorname{Sym}(n)$. Suppose that:\n\\begin{enumerate}\n\\item There exists an prefix code $\\widetilde{A}$ of $\\mathfrak{C}_m$ such that $\\vert \\widetilde{A} \\vert = n$,\n\\item the group $\\mathcal{R}_{G}(\\widetilde{A})$ is well defined, and \n\\item $\\mathcal{R}_{G}(\\widetilde{A})$ and $H$ are cyclically isomorphic.\n\\end{enumerate} Then $V_n(H)$ embeds in $V_m(G)$.\n\\end{theorem}\nFor this first result, observe that we require a group $\\mathcal{R}(G)$ to be well defined.  This group (when it exists) can be thought of in a natural way as a subgroup of $\\operatorname{Sym}(n)$.  We also need $\\mathcal{R}(G)$ to be cyclically isomorphic to $H$.  This is equivalent to the existence of an $n$-element complete prefix code $\\widetilde{A}$ in $\\mathcal{A}_m^*$ which is preserved by the action of the iterated permutations of $G$ on $\\mathcal{A}_m^*$ and where the cycle structure of the action of $G$ on $\\widetilde{A}$ is isomorphic (element-for-element) to the cycle structure of the action of $H$ on $\\mathcal{A}_{n}$.\n\nThus, we have the following general observation.\n\\begin{observation}\nIn practical terms, natural applications of Theorem \\ref{thm:MT1} occur by choosing $m$ and a prefix code that is closed under the action of iterated  permutations from some $G\\leq \\operatorname{Sym}(m)$.  Then, one immediately obtains a cyclically isomorphic group $H$ in $\\operatorname{Sym}(n)$.\n\\end{observation}\n\n\\begin{theorem}\\label{thm:MT2}\nLet $n,m \\geq 2$ be natural numbers such that $n = k(m-1)+ m$ for some $k \\geq 1$, and let $G \\leq \\operatorname{Sym}(m)$. Let $H = G_{ext} \\leq \\operatorname{Sym}(n)$ the extended symmetric group of $H$, whose elements act as the elements of $H$ on the first $m$ elements, and act as the identity on the remaining $n-m$.\n\nThen $V_m(G)$ embeds in $V_n(H)$.\n\\end{theorem}\nThus, to achieve such an embedding, the main obstruction is that $m$ and $n$ satisfy the Higman Condition, i.e.:\n\\begin{corollary} Let $m\\leq n$ so that $n\\equiv m\\mod (m-1)$, and let $G\\leq \\operatorname{Sym}(m)$.  Then, there exists $H\\leq \\operatorname{Sym}(n)$ so that $V_m(G)\\rightarrowtail V_n(H)$.\n\\end{corollary}\n\nAs mentioned above, for given $m$ and $G$ semiregular, the paper \\cite{BDJ} shows $V_m(G)\\cong V_m$, while Higman's book \\cite{HI} gives an embedding of $V_m$ into R. Thompson's group $V=V_2$.  The semiregularity condition above has to do with local groups of germs of the action of $V_m(G)$ on $\\mathfrak{C}_m$ (see \\cite{BDJ,BL}).  Our topological embedding preserves the local groups of germs, while the algebraic embeddings (for $G$ non-trivial) produce complicated local groups of germs.  Thus, it is impossible to chain our families of embeddings together to get an embedding from $V_m(G)$ into $V_2$ when $G$ is not semi-regular.\n\nWe therefore ask the following question:\n\\begin{question}\nDoes there exist $m\\in \\mathbb{N}$ and $G\\in \\operatorname{Sym}(m)$ so that $G$ is not semiregular, but where there is an embedding from $V_m(G)$ into $V_2$?\n\\end{question}\n\n{\\flushleft{\\it Acknowledgements:}}\\\\\nThe authors would like to thank Javier Aramayona, Jim Belk, and Matthew G. Brin for numerous and enjoyable conversations around the material in this note.\n\n\n\\section{Symmetric Thompson's groups}\n\nIn this section, we introduce symmetric Thompson's groups, giving an easy way to express its elements as tables (this corresponds to a generalised construction of Higman used by Scott and R\\\"over in the creation of their extensions of $V$ (\\cite{ScottI,ScottII,ScottIII,Roever}).   \n\n\\subsection{Tables} For natural $n>1$, let $\\mathfrak{C}_n$ be the $n$-adic Cantor set, which is constructed inductively as follows: $\\mathfrak{C}_n^1$ corresponds to first subdividing $\\mathfrak{C}_n^0= [0,1]$ into $2n-1$ closed intervals of equal length (so, sharing endpoints with neighbours), numbered $1, \\ldots, 2n-1$ from left to right, and then taking the collection of odd-numbered sub-intervals. Next, $\\mathfrak{C}_n^2$ is obtained from $\\mathfrak{C}_n^1$ by applying the same procedure to each of the intervals forming $\\mathfrak{C}_n^1$, and so on.  Then, $C_n$ is the limit of this process, so that \n\\[\n\\mathfrak{C}_n=\\cap_i \\mathfrak{C}_n^i.\n\\]  Now, let $\\mathcal{A}_n = \\{0,\\dots,n-1\\}$ and give it the discrete topology. It is easy to build a direct homemorphism from the space $\\mathcal{A}_n^\\mathbb{N}$ equipped with the product topology to $\\mathfrak{C}_n$, so every element $\\zeta \\in \\mathfrak{C}_n$ can be expressed as an infinite word $\\zeta = w_1w_2\\dots $, where $w_i \\in \\mathcal{A}_n$.  It is a classical result of Brouwer from \\cite{BRO} that all of the spaces in the set $\\{\\mathfrak{C}_n\\}$ are abstractly homeomorphic to each other.\n\nWe denote by $\\mathcal{A}_n^*$ the set of finite words in $\\mathcal{A}_n$. The empty word $\\varepsilon$ is also in $\\mathcal{A}_n^*$.\n\n\\begin{definition}[Concatenation]\nLet $u = u_1u_2\\dots u_k, u_i \\in \\mathcal{A}_n$ be a finite word and $v \\in \\mathcal{A}_n^* \\cup \\mathcal{A}_n^{\\mathbb{N}}$ with $v=v_1v_2\\ldots$ (where for all valid indices $i$ we have  $v_i\\in \\mathcal{A}_n$) . The \\textit{concatenation} of $u$ with $v$ is the (finite or infinite) word: $$u \\vert\\!\\vert v = u_1u_2\\dots u_kv_1v_2\\ldots.$$ \n\\end{definition}\n\nWith concatenation being a fundamental operation, we will often just write the concatenation of two strings without the formal concatenation operator, that is, we might write $u\\vert\\!\\vert v$ as simply $uv$, reserving the formal use of ``$\\vert\\!\\vert$'' for situations where we wish to stress that a concatenation is occuring.\n\n\n\\begin{definition}[Prefix order]\nLet $u \\in \\mathcal{A}_n^*$  and $v \\in \\mathcal{A}_n^{*}\\cup \\mathcal{A}_n^{\\mathbb{N}}$.  We say that $u$ is a \\textit{prefix} of $v$ ($u \\leq_{pref} v$) if $v = u\\vert\\!\\vert w$, for some $w \\in\\mathcal{A}_n^{*}\\cup \\mathcal{A}_n^{\\mathbb{N}}$.\n\\end{definition}\n\nNote that this property is transitive for finite length words: If $u \\leq_{pref} v$ and $v \\leq_{pref} w$ then $u \\leq_{pref} w$. In addition, $u \\leq_{pref} u$, as $\\varepsilon \\in \\mathcal{A}_n^*$.  That is, $\\leq_{pref}$ provides a partial order on $\\mathcal{A}_n^{*}$.\n\n\\begin{definition}[Prefix code]\nLet $S$ be a finite set of words in $\\mathcal{A}_n^*$. Then $S$ is an \\textit{prefix code} of $\\mathfrak{C}_n$ if for every infinite word $\\zeta \\in \\mathcal{A}_n^{\\mathbb{N}}$ there exists one and only one word $s \\in S$ such that $s \\leq_{pref} \\zeta$. (Specifically, a prefix code is a complete anti-chain for the partial order $\\leq_{pref}$.)\n\\end{definition}\n\nFor convenience, we will use the following notation: let $\\sigma \\in \\operatorname{Sym}(n)$ be an element of the symmetric group of $n$ elements. Given any word $\\zeta = z_1z_2z_3 \\dots \\in \\mathcal{A}_n^\\mathbb{*} \\cup \\mathcal{A}_n^\\mathbb{N}$ we define $\\sigma(\\zeta) = \\sigma(z_1)\\sigma(z_2)\\sigma(z_3)\\dots \\in \\mathcal{A}_n^\\mathbb{*} \\cup \\mathcal{A}_n^\\mathbb{N}$. Let $\\sigma_i \\in H \\leq \\operatorname{Sym}(n)$.   (Note that we are using left actions here, so if $\\sigma,\\tau\\in \\operatorname{Sym}(n)$ then the product $\\tau\\sigma$ means employ the permutation $\\sigma$ first, and then employ $\\tau$).\n\nWith the above notation, an element of $V_n(H)$ is a homeomorphism of $\\mathfrak{C}_n$  that can be (non-uniquely) described by a \\textit{table} as follows:\n$$v = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k\\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix},$$\nwhere $p_i,q_i \\in \\mathcal{A}_n^*$, $\\sigma_i, \\tau_i \\in H$ and such that the sets $P = \\{p_i\\}_{i=1}^k$ and $Q = \\{q_i\\}_{i=1}^k$ are prefix codes of $\\mathfrak{C}_n$. We say that $k \\geq 1$ is the \\textit{length} of the table. The homeomorphism of $\\mathfrak{C}_n$ induced can be defined as follows: for every infinite word $\\zeta$ such that $p_i\\leq_{pref}\\zeta$, that is $\\zeta = p_i \\vert\\!\\vert u$ for some $u \\in \\mathcal{A}_n^\\mathbb{N}$, we have \n$$v: p_i \\vert\\!\\vert \\sigma_i(u) \\rightarrow q_i \\vert\\!\\vert \\tau_i(u).$$\nThere are infinitely many tables which induce the same homeomorphism of $\\mathfrak{C}_n$. We proceed to define the four basic moves we can perform on a table in order to obtain an equivalent one (the four basic moves naturally split as two essential sorts of moves, together with their inverse (or ``near-inverse'') moves). \n\nThe first basic move is \\textit{expansion}: for a given prefix code $$P = \\{p_1, \\dots, p_i , \\dots , p_k\\},$$ we can consider $$\\widetilde{P} = \\{p_1, \\dots, p_i0, \\dots, p_i(n-1) , \\dots , p_k\\}$$ by expanding the word $p_i$. This expansion not only occurs in $P$, as the image of $p_i$ must be also expanded. So we have $$\\widetilde{Q} = \\{q_1, \\dots, q_i0, \\dots, q_i(n-1) , \\dots  ,q_k\\}.$$ It is easy to see that both $\\widetilde{P}$ and $\\widetilde{Q}$ are also prefix codes. Then:\n$$\\begin{bmatrix}\np_1 & \\cdots& p_i & \\cdots & p_k \\\\\n\\sigma_1 & \\cdots& \\sigma_i & \\cdots & \\sigma_k \\\\\nq_1 & \\cdots& q_i & \\cdots & q_k \\\\\n\\tau_1 & \\cdots& \\tau_i & \\cdots & \\tau_k\n\\end{bmatrix} \\equiv \n\\begin{bmatrix}\np_1 & \\cdots& p_i \\sigma_i(0) & \\cdots & p_i \\sigma_i(n-1) & \\cdots & p_k \\\\\n\\sigma_1 & \\cdots& \\sigma_i & \\cdots& \\sigma_i & \\cdots & \\sigma_k \\\\\nq_1 & \\cdots& q_i \\tau_i(0) & \\cdots & q_i \\tau_i(n-1) & \\cdots & q_k \\\\\n\\tau_1 & \\cdots& \\tau_i & \\cdots& \\tau_i & \\cdots & \\tau_k \\\\\n\\end{bmatrix}.$$  One can always perform an expansion, but not all tables look like the result of an expansion.  Naturally, the inverse of an expansion (when it is defined) is called a \\emph{reduction}.\n\nThe second move we can perform on a table is \\textit{pushing down} (resp. \\textit{pushing up}) the action of all $\\sigma_i$ such that $\\sigma_i = Id$ for every $i \\in \\{1, \\dots, k\\}$ (resp. $\\tau_i = Id$ for every $i \\in \\{1, \\dots, k\\}$):\n\\begin{align*}\n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k \n\\end{bmatrix} &\\equiv \n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\nId & Id & \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_k \\\\[2pt]\n\\tau_1\\sigma_1^{-1} & \\tau_2\\sigma_2^{-1} & \\cdots & \\tau_k\\sigma_k^{-1}\n\\end{bmatrix} \\\\ &\\equiv \n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\[2pt]\n\\sigma_1\\tau_1^{-1} & \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1} \\\\[2pt]\nq_1 & q_2 & \\cdots & q_k \\\\\nId & Id & \\cdots & Id\\\\\n\\end{bmatrix}.\n\\end{align*}\n\nIt is not hard to see that a table gives a well-defined homeomorphism of the appropriate Cantor space, and if two tables are related by a finite sequence of our four moves then they represent the same homeomorphism.  The reader can also check that if a homeomorphism of an appropriate Cantor space is represented by two tables, then in fact these tables are in the same equivalence class under our four basic moves on tables.  Thus, we can just consider our group elements to be the equivalence classes of tables with the aforementioned relations. \n\nThe {composition} of two different elements $u,v\\in V_n(H)$ is easy to compute using the equivalences. Let $u$, $v \\in V_n(H)$, such that $u$ takes the prefix code $P$ to the prefix code $Q$ (resp. $v$ takes $P'$ to $Q'$). We need to find a prefix code $S$ such that, for every element $s \\in S$, there exists one element $q \\in Q$ and one element $p' \\in P'$ such that $q\\leq_{pref} s$ and $p'\\leq_{pref} s$. This can always be done by expanding $P'$ and $Q$ until we obtain the same prefix code $S$. Thus, without loss of generality:\n$$u = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\ns_1 & s_2 & \\cdots & s_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix}, \\quad v = \\begin{bmatrix}\ns_1 & s_2 & \\cdots & s_k \\\\\n\\sigma'_1 & \\sigma'_2 & \\cdots & \\sigma'_k \\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\\n\\tau'_1 & \\tau'_2 & \\cdots & \\tau'_k\n\\end{bmatrix}.$$\n\nFinally, we push up the action of $u$ and push down the action of $v$:\n\\begin{align*}\nu&  = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1\\tau_1^{-1} & \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1}\\\\\ns_1 & s_2 & \\cdots & s_k \\\\\nId & Id & \\cdots & Id\n\\end{bmatrix},\\\\ v & = \\begin{bmatrix}\ns_1 & s_2 & \\cdots & s_k \\\\\nId & Id & \\cdots & Id\\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\[2pt]\n\\tau'_1(\\sigma'_1)^{-1} & \\tau'_2(\\sigma'_2)^{-1} & \\cdots & \\tau'_k(\\sigma'_k)^{-1}\n\\end{bmatrix},\n\\end{align*}\nso \n$$v \\circ u  = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1\\tau_1^{-1}& \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1}\\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\[2pt]\n\\tau'_1(\\sigma'_1)^{-1} & \\tau'_2(\\sigma'_2)^{-1} & \\cdots & \\tau'_k(\\sigma'_k)^{-1}\n\\end{bmatrix}.$$\n\nWe sum up the previous discussion in the following proposition:\n\\begin{proposition}\n$V_n(H)$ is a group with the composition.\n\\end{proposition}\n\n\n\\section{Topological Embeddings}\n\nIn this section, we present topological embeddings between symmetric Thompson's groups. The key idea is, given any group $V_n(H)$, to translate the action of an element $\\sigma \\in H$ into a permutation $\\widetilde{\\sigma}$ of the elements of some prefix code of $\\mathfrak{C}_m$. Therefore, $\\widetilde{\\sigma} \\in V_m(G)$ for some $G$.\n\nOur method will be first to understand when actions on prefix codes over smaller alphabets can represent embeddings of permutations on larger alphabets which commute with our core operations of expansion and contraction of prefix codes.  With that understanding in hand, we can then build the desired embedding from a group $V_n(H)$ to a group $V_m(G)$ for $m\\leq n$.\n\nWe first establish some useful definitions.\n\n\\subsection{The Root Group \\texorpdfstring{$\\boldsymbol{\\mathcal{R}_{G}(S)}$}{RGS}}\n\nGiven a linear order $\\leq$ on $\\mathcal{A}_n$ (we choose $0<1<\\ldots<n-1$), there is an induced standard dictionary order:\n\\begin{definition}[Dictionary order]\nLet $\\mathcal{A}_n = \\{a_0, \\dots, a_{n-1}\\}$ be an alphabet with linear order $\\leq$. We define the \\textit{dictionary order} on $\\mathcal{A}_n^*$ as follows. Let $u,v \\in \\mathcal{A}_n^*$, then $u \\leq_{dict} v$ if and only if:\n\\begin{enumerate}\n\\item $u \\leq_{pref} v$,\n\\item $u \\not\\leq_{pref} v$ and there exist $p, s, t \\in \\mathcal{A}_n^*$ and $\\alpha, \\beta \\in \\mathcal{A}_n$ such that $u = p\\alpha s, v = p \\beta t$, and $\\alpha < \\beta$.\n\\end{enumerate} \n\\end{definition}\n\nLet $2\\leq m <n\\in\\mathbb{N}$ be fixed, and let $G \\leq \\operatorname{Sym}(m)$. We assume for the construction Higman's condition: $n = k(m-1)+ 1$ for some $k \\geq 0$ (so, $n\\equiv 1\\mod (m-1)$).  Then, we can find $S = \\{s_0, \\dots, s_{n-1}\\}$ an ordered prefix code of $\\mathfrak{C}_m$ of length $n$ (using the dictionary order).   Observe for now that if $\\sigma(S) = S \\ \\forall \\sigma \\in G$, then $\\sigma: S \\rightarrow S$ will induce the desired rearrangement $\\widetilde{\\sigma}$ of the elements of $S$, that is, $\\widetilde{\\sigma} \\in \\operatorname{Sym}_S\\cong \\operatorname{Sym}(n)$. \n\nWe will be interested in the set $\\mathcal{T}$ of triples $(m,n,G)$\nwhere $m\\leq n$, $n\\equiv 1\\mod(m-1)$, and $G\\in \\operatorname{Sym}(m)$.  We will say a triple $(m,n,G)\\in \\mathcal{T}$ is \\emph{satisfiable} if there is a prefix code $S\\subset X_m^*$ with $|S|=n$ and where for each $\\sigma\\in G$ the action of $\\sigma$ on $X_m^*$ preserves $S$.  In this case we say \\emph{$S$ is a solution for the triple $(m,n,G)$.}\n\n\\begin{definition}[Root group]\nGiven a triple $(m,n,G)\\in\\mathcal{T}$ and a solution $S$, \nwe define the \\textit{root group} $\\mathcal{R}_{G}(S)$ to be the group of permutations of $S$ induced by the action of $G$ on $S$.\n\\end{definition}\n\nIt is the case that not every triple $(m,n,G)\\in\\mathcal{T}$ admits a solution $S$.  However, when it does admit a solution $S$, it is immediate that $\\mathcal{R}_{G}(S)$ is isomorphic to $G$.\n\n\\begin{definition}[Cycle type]\nLet $\\sigma \\in \\operatorname{Sym}(n)$. The \\textit{cycle type c} of $\\sigma$ is the multiset (a set, but allowing multiple elements that are equal) of lengths of the cycles in the cycle decomposition of $\\sigma$. We say that two subgroups $H, H' \\in \\operatorname{Sym}(n)$ are \\textit{cyclically isomorphic} if there exists an isomorphism $\\psi: H \\rightarrow H'$ which preserves the cycle type of every permutation, that is, $c(\\sigma) = c(\\psi(\\sigma))$ for all $\\sigma \\in H$. \n\\end{definition}\n\n\\begin{remark}\n Subgroups $H$ and $H'$ are cyclically isomorphic if and only if they are conjugate in $\\operatorname{Sym}(n)$, but our focus is on cycle structure and that is why we are using the language we have chosen. (N.B., there exist exotic automorphisms of $\\operatorname{Sym}(6)$ which do not arise by conjugation, but these automorphisms change the cycle structure of some elements of order two.) \n\\end{remark}\n\n\\subsection{The induced group \\texorpdfstring{$\\boldsymbol{\\mathcal{G}}$}{G}} \n\nWe proceed to define the group $\\mathcal{G} < V_m(G)$ such that $V_n(H)$ is isomorphic to $\\mathcal{G}$. \n\nLet $V_m(G)$ and $V_n(H)$ be two symmetric Thompson's groups such that $m$ and $n$ fulfill Higman's condition. Let $\\widetilde{\\mathcal{A}}_m = \\{a_0,a_1,\\dots,a_{n-1}\\}$ be an ordered prefix code of $\\mathfrak{C}_m$ of length $n$, such that $\\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m)$ is well defined. We may think of  $\\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m)$ as a subgroup of $\\operatorname{Sym}(n)$ by using the bijection from $\\widetilde{\\mathcal{A}}_m$ to $\\mathcal{A}_n$ induced by the lexicographic ordering of $\\widetilde{\\mathcal{A}}_m$.  Thus, we can define $\\mathcal{G}$ as the set of equivalence classes of tables of the form:\n$$v = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix},$$\nwhere $\\sigma_i,\\tau_i \\in \\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m) \\ \\forall i$ and the prefix codes $P$ and $Q$ consist of words in the alphabet $\\widetilde{\\mathcal{A}}_m$, that is, every $p_i$ and $q_i$ is a nonempty concatenation of elements of $\\widetilde{\\mathcal{A}}_m$. \n\nIt is straightforward to check that $\\mathcal{G}$ is a group, since the concatenation of two words in $\\widetilde{\\mathcal{A}}_m^*$ is another word in $\\widetilde{\\mathcal{A}}_m^*$. Tables of $\\mathcal{G}$ are well defined by expansions and pushings, and the action of an element $\\sigma \\in \\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m)$ on a word in $\\mathcal{\\widetilde{\\mathcal{A}}}_m$ gives another word in $\\mathcal{\\widetilde{\\mathcal{A}}}_m$ by the definition of root group, so the composition of any two elements in $\\mathcal{G}$ gives another element of $\\mathcal{G}$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:MT1}]\nLet $V_m(G)$ and $V_n(H)$ two symmetric Thompson's groups. Let $\\mathcal{A}_n = \\{0,1,\\dots,n-1\\}$ and $\\widetilde{\\mathcal{A}}_m = \\{a_0,a_1,\\dots,a_{n-1}\\}$ such that $\\mathcal{R}_G(\\widetilde{\\mathcal{A}}_m)$ is well defined. We define the following \\textit{translating map}:\n$$ \\begin{array}{cccc} \\widetilde{t}: & \\mathcal{A}_n &\\longrightarrow & \\widetilde{\\mathcal{A}}_m \\\\\n& i & \\longrightarrow &  a_i,\n\\end{array}$$\nand $$\\begin{array}{cccc} t: & H &  \\longrightarrow &  \\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m) \\\\\n& \\sigma & \\longrightarrow & \\widetilde{\\sigma}\\end{array},$$ \nwhere $t$ is the isomorphism between $H$ and $\\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m)$. Note that $t$ and $\\widetilde{t}$ have the following property:\n$$\\widetilde{t}(\\sigma(i))  \n = a_{\\sigma(i)}\n = \\widetilde{\\sigma}(a_i)\n = t(\\sigma)(a_i)\n = t(\\sigma)(\\widetilde{t}(i)) , \\forall \\sigma \\in H, \\forall i \\in \\mathcal{A}_n,$$\nas $\\widetilde{\\sigma}(a_i) = a_{\\sigma(i)}, \\forall \\sigma \\in H, \\forall i \\in \\mathcal{A}_n$, since $\\mathcal{R}_{G}(\\widetilde{\\mathcal{A}}_m)$ and $H$ are cyclically isomorphic.\n\nOur embedding is as follows:\n$$v = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix} \\overset{\\iota}{\\longrightarrow} \\begin{bmatrix}\n\\widetilde{t}(p_1) & \\widetilde{t}(p_2) & \\cdots & \\widetilde{t}(p_k) \\\\\nt(\\sigma_1) & t(\\sigma_2) & \\cdots & t(\\sigma_k) \\\\\n\\widetilde{t}(q_1) & \\widetilde{t}(q_2) & \\cdots & \\widetilde{t}(q_k) \\\\\nt(\\tau_1) & t(\\tau_2) & \\cdots &t(\\tau_k)\n\\end{bmatrix}.$$\nWe see that $\\iota$ commutes with expansions and pushings, that is, $\\iota(\\operatorname{exp}(v)) = \\operatorname{exp}(\\iota(v))$ and $\\iota(\\operatorname{push}(v)) = \\operatorname{push}(\\iota(v)) \\ \\forall v \\in V_n(H)$:\n\n\\begin{align*}\n\\operatorname{push}(v) & = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\nId & Id & \\cdots & Id \\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1\\sigma_1^{-1} & \\tau_2\\sigma_2^{-1} & \\cdots & \\tau_k\\sigma_k^{-1}\n\\end{bmatrix}, \\\\ \n\\iota(\\operatorname{push}(v)) & = \\begin{bmatrix}\n\\widetilde{t}(p_1) & \\widetilde{t}(p_2) & \\cdots & \\widetilde{t}(p_k) \\\\\nId & Id & \\cdots & Id \\\\\n\\widetilde{t}(q_1) & \\widetilde{t}(q_2) & \\cdots & \\widetilde{t}(q_k) \\\\\nt(\\tau_1\\sigma_1^{-1}) & t(\\tau_2\\sigma_2^{-1}) & \\cdots & t(\\tau_k\\sigma_k^{-1})\n\\end{bmatrix}, \\\\\n\\operatorname{push}(\\iota(v)) &  = \\begin{bmatrix}\n\\widetilde{t}(p_1) & \\widetilde{t}(p_2) & \\cdots & \\widetilde{t}(p_k) \\\\\nId & Id & \\cdots & Id \\\\\n\\widetilde{t}(q_1) & \\widetilde{t}(q_2) & \\cdots & \\widetilde{t}(q_k) \\\\\nt(\\tau_1)t(\\sigma_1^{-1}) & t(\\tau_2)t(\\sigma_2^{-1}) & \\cdots & t(\\tau_k)t(\\sigma_k^{-1})\n\\end{bmatrix}.\n\\end{align*}\nAs $t$ is an isomorphism of groups, the commutativity follows. On the other hand, suppose that we expand the prefix code $P = \\{p_1, \\dots, p_k\\}$ on $p_i$ (we argue below that $\\iota$ commutes with expanding, but our argument uses  the pushed version of $v$: it is easy to see that this is sufficient):\n$$\\operatorname{exp}(v) = \\begin{bmatrix}\np_1  & \\cdots & p_i0 & \\cdots & p_i(n-1)  & \\cdots &  p_k \\\\\nId  & \\cdots & Id & \\cdots & Id & \\cdots & Id\\\\\nq_1  & \\cdots & q_i\\vert\\!\\vert\\tau_i\\sigma_i^{-1}(0) & \\cdots & q_i\\vert\\!\\vert\\tau_i\\sigma_i^{-1}(n-1)  & \\cdots & q_k \\\\\n\\tau_1\\sigma_1^{-1} & \\cdots & \\tau_i\\sigma_i^{-1} & \\cdots & \\tau_i\\sigma_i^{-1} & \\cdots & \\tau_k\\sigma_k^{-1}\n\\end{bmatrix}.$$\nThe table for $\\iota(\\operatorname{exp}(v))$ is: \n$$ \\begin{bmatrix}\n\\widetilde{t}(p_1) & \\cdots & \\widetilde{t}(p_i0) & \\cdots & \\widetilde{t}(p_i(n-1))  & \\cdots & \\widetilde{t}(p_k) \\\\\nId  & \\cdots & Id & \\cdots & Id & \\cdots & Id\\\\\n\\widetilde{t}(q_1)  & \\cdots & \\widetilde{t}(q_i\\vert\\!\\vert\\tau_i\\sigma_i^{-1}(0)) & \\cdots & \\widetilde{t}(q_i\\vert\\!\\vert\\tau_i\\sigma_i^{-1}(n-1))  & \\cdots & \\widetilde{t}(q_k) \\\\\nt(\\tau_1\\sigma_1^{-1})  & \\cdots & t(\\tau_i\\sigma_i^{-1}) & \\cdots & t(\\tau_i\\sigma_i^{-1}) & \\cdots & t(\\tau_k\\sigma_k^{-1})\n\\end{bmatrix},$$\nFinally, the table for $\\operatorname{exp}(\\iota(v))$ is: \n$$\\begin{bmatrix}\n\\widetilde{t}(p_1)  & \\cdots & \\widetilde{t}(p_i)a_0 & \\cdots & \\widetilde{t}(p_i)a_{n-1}  & \\cdots & \\widetilde{t}(p_k) \\\\\nId  & \\cdots & Id & \\cdots & Id & \\cdots & Id\\\\\n\\widetilde{t}(q_1) & \\cdots & \\widetilde{t}(q_i)\\vert\\!\\vert t(\\tau_i\\sigma_i^{-1})(a_0) & \\cdots &\\widetilde{t}(q_i)\\vert\\!\\vert t(\\tau_i\\sigma_i^{-1})(a_{n-1})  & \\cdots & \\widetilde{t}(q_k) \\\\\nt(\\tau_1\\sigma_1^{-1})  & \\cdots & t(\\tau_i\\sigma_i^{-1}) & \\cdots & t(\\tau_i\\sigma_i^{-1}) & \\cdots & t(\\tau_k\\sigma_k^{-1})\n\\end{bmatrix},$$\nBoth tables $\\operatorname{exp}(\\iota(v)) = \\iota(\\operatorname{exp}(v))$ are equal as:\n$$ \\widetilde{t}(p_i\\vert\\!\\vert j) = \\widetilde{t}(p_i)\\vert\\!\\vert \\widetilde{t}(j) = \\widetilde{t}(p_i)\\vert\\!\\vert \\widetilde{t}(a_j),$$\n$$\\widetilde{t}(q_i\\vert\\!\\vert\\tau_i\\sigma_i^{-1}(j)) = \\widetilde{t}(q_i) \\vert\\!\\vert\\widetilde{t}(\\tau_i\\sigma_i^{-1}(j)) = \\widetilde{t}(q_i) \\vert\\!\\vert t(\\tau_i\\sigma_i^{-1})(\\widetilde{t}(j)) = \\widetilde{t}(q_i) \\vert\\!\\vert t(\\tau_i\\sigma_i^{-1})(a_j).$$\n\n(We have used the concat symbol ``$\\vert\\!\\vert$'' in these tables and the immediate discussion after, wherever we think it makes terms easier to read.  In our later explanations we will generally refrain from using it.)\n\nTo finish the proof, we need to show that $\\iota(v) \\circ \\iota(u) = \\iota(v \\circ u)$. This follows easily from the fact that $\\iota$ commutes with expansions and pushings. Without loss of generality, consider:\n$$u = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\nId & Id & \\cdots & Id \\\\\ns_1 & s_2 & \\cdots & s_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix}, \\quad \\iota(u) = \\begin{bmatrix}\n\\widetilde{t}(p_1) & \\widetilde{t}(p_2) & \\cdots & \\widetilde{t}(p_k) \\\\\nId & Id & \\cdots & Id \\\\\n\\widetilde{t}(s_1) & \\widetilde{t}(s_2) & \\cdots & \\widetilde{t}(s_k) \\\\\nt(\\tau_1) & t(\\tau_2) & \\cdots &t(\\tau_k)\n\\end{bmatrix}, $$\n\n$$v = \\begin{bmatrix}\ns_1 & s_2 & \\cdots & s_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k \\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\\n\\tau'_1 & \\tau'_2 & \\cdots & \\tau'_k\n\\end{bmatrix}, \\quad \\iota(v) = \\begin{bmatrix}\n\\widetilde{t}(s_1) & \\widetilde{t}(s_2) & \\cdots & \\widetilde{t}(s_k) \\\\\nt(\\tau_1) & t(\\tau_2) & \\cdots &t(\\tau_k)\\\\\n\\widetilde{t}(q'_1) & \\widetilde{t}(q'_2) & \\cdots & \\widetilde{t}(q'_k) \\\\\nt(\\tau'_1) & t(\\tau'_2) & \\cdots &t(\\tau'_k)\n\\end{bmatrix}. $$\nThus: \n$$\\iota(v) \\circ \\iota(u) = \\iota(v \\circ u) = \n\\begin{bmatrix}\n\\widetilde{t}(p_1) & \\widetilde{t}(p_2) & \\cdots & \\widetilde{t}(p_k) \\\\\nId & Id & \\cdots & Id \\\\\n\\widetilde{t}(q'_1) & \\widetilde{t}(q'_2) & \\cdots & \\widetilde{t}(q'_k) \\\\\nt(\\tau'_1) & t(\\tau'_2) & \\cdots &t(\\tau'_k)\n\\end{bmatrix}. $$\n\\end{proof}\n\n\n\\section{Algebraic Embeddings}\nIn this section we present some algebraic embeddings $V_m(G)\\rightarrowtail V_n(H)$ between symmetric Thompson's groups (recall here $n-m=k(m-1)$ for some positive $k$, $G\\leq \\operatorname{Sym}(m)$ and with $H$ a particular extended version of $G$ in $\\operatorname{Sym}(n)$). \n\nWe call these embeddings ``algebraic'' as they do not arise via topological conjugacy.  In particular, these embeddings do not preserve the orbit lengths of the points of $\\mathfrak{C}_n$ (when $n>m$, which is our primary case of interest).\n\n\\subsection{Successors} Here, we give the key idea for our algebraic embeddings, which relies on extending an idea of Birget into our context.\n\nThe \\textit{successor} of an element, expressed as a table, was defined in \\cite{B} in order to embed $V_2(Id)$ in $V_n(Id)$, for all $n \\geq 2$. We generalise Birget's definition.\n\n\n\n\\begin{definition}[Set of prefixes]\\label{spref}\\cite{B}\nLet $P \\subset \\mathcal{A}_n^*$ be a prefix code of $\\mathfrak{C}_n$. We define the \\textit{set of prefixes of $P$}, $\\operatorname{spref}(P)$ as follows:\n$$ \\operatorname{spref}(P) = \\{w \\in \\mathcal{A}_n^* : \\exists p \\in P,\\, w <_{pref} p \\}.$$\nIn other words, $\\operatorname{spref}(P)$ is the set of strict prefixes of the elements of $P$.\n\\end{definition}\n\nWe are embedding a symmetric Thompson's group on alphabet $\\mathcal{A}_m$ into a symmetric Thompson's group on alphabet $\\mathcal{A}_n$, where $m\\leq n$.  For this, we will assume $\\mathcal{A}_m\\subseteq \\mathcal{A}_n$.  And in particular we set $\\mathcal{A}_m=\\{a_0,a_1,\\ldots,a_{m-1}\\}$ and $\\mathcal{A}_n=\\mathcal{A}_m\\cup\\{a_m,a_{m+1},\\ldots,a_{n-1}\\}$ with symbols with distinct indices being distinct (so that $|\\mathcal{A}_n|=n$).\n\n\nIn what follows, we take a prefix code $P\\subset a_{m-1}\\vert\\!\\vert \\mathcal{A}_m^*$ (so each element of $P$ begins with the letter $a_{m-1}$) and transform it to a new prefix code $\\succ{P}\\subset \\mathcal{A}_n^*$ by appending letters from the set $\\{a_{m},a_{m+1},\\ldots,a_n\\}$. \n\n\n\\begin{definition}[Successor]\\label{successor} Let $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be a prefix code (complete, were we to remove the initial prefix letter $a_{m-1}$, so that $|P|\\equiv 1\\mod (m-1)$)  with $\\vert P \\vert = l \\geq 1$, and let $\\{p_1, \\dots, p_l\\}$ be the ordered list of all the elements of $P$, using the \\emph{reverse} dictionary order. \n\nWe build a new prefix code $\\succ{P}$ inductively using our ordered list $(p_1,p_2,\\ldots,p_l)$.\n\nLet $k$ be the smallest non-negative integer so that $n-m=k(m-1)$ (this $k$ will exist when $m$ and $n$ satisfy Higman's condition, which we require to build our embeddings).\n\nWe define (inductively) nested sets $P_{s,i}$, where $s$ will grow from $1$ to $l$, and for each value of $s$, we will have $i$ grow from $1$ to $k$.\n\nSet $\\mathcal{A}_{m,n}:= \\{a_m,a_{m+1},\\ldots,a_{n-1}\\}$.  For every $p_s \\in P$, and $i\\in \\{1,2,\\ldots, k\\}$ the \\textit{$i$-th successor} $(p_s)'_i$ of $p_s$ is the element of $\\operatorname{spref}(P)\\vert\\!\\vert \\mathcal{A}_{m,n}$ defined as follows, assuming that \n\\[P_{s,i-1} = \\left\\{\\begin{matrix}(p_1)'_1,(p_1)'_{2},\\ldots,(p_1)'_{k},\\\\(p_2)'_1,(p_2)'_{2},\\ldots,(p_2)'_{k},\\\\\n\\vdots\\\\\n(p_s)'_{1},(p_s)'_{2},\\ldots,(p_s)'_{i-1}\n\\end{matrix}\\right\\}\\]\nhas already been defined, we set:\n\n\\begin{align*}\n(p_s)'_i = \\min \\{ & xa_j \\in \\operatorname{spref}(P)\\vert\\!\\vert\\mathcal{A}_{m,n}: p_s <_{dict} xa_j  \\ \\mbox{and} \\ xa_j \\not\\in P_{s,i-1}\\},\n\\end{align*}\nwhere $\\min$ uses the dictionary order in $\\{a_0, \\dots, a_{n-1}\\}$.\n\\end{definition}\n\n\n\\begin{example}\nSuppose $m=3$ and $n=5$, so that $k=1$.  In the definition above, $a_{m-1}=2$.  So, consider the set $P=\\{20,210,211,212,22\\}$.  Now, $k=1$ and $\\operatorname{spref}(P)=\\{\\varepsilon,2,21\\}$.  We obtain\n\\[\n\\begin{array}{lll}\n p_1=22&\\quad&(p_1)'_1=23\\\\\n p_2=212&\\quad&(p_2)'_1=213\\\\ \n p_3=211&\\quad& (p_3)'_1=214\\\\\n p_4=210&\\quad& (p_4)'_1=24\\\\\n p_5=20&\\quad&(p_5)'_1=3.\n\\end{array}\n\\]\n\n\\end{example}\n\n\\begin{remark}The three constants, $n,m,k$ are not arbitrary, as the system of successors needs to be well defined.  If every element has $k$ successors, then: $$n-m = k(m-1), \\, k \\geq 0,$$\nwhich is Higman's condition.\\end{remark}\n\\begin{proof}\n If we expand an element $p_i \\in P$, we need to assign successors to each element $p_ia_j$ for every $0 \\leq j \\leq m-1$. In particular, the number of successors $k$ of every leaf does not vary, and each element $p_ia_r$ for every $m \\leq r \\leq n-1$ needs to be the successor of some element in $\\widetilde{P} = (P \\backslash \\{p_i\\}) \\cup \\{p_ia_0, \\dots,p_ia_{m-1}\\}$. Then $\\widetilde{P}$ has $m-1$ more elements than $P$ and there are $n-m$ new elements $p_ia_r$ for $m \\leq r \\leq n-1$. Thus, we need $m-1$ to evenly divide $n-m$, and $k$ is the factor of this division.\n\\end{proof}\n\nWe proceed to prove the following lemma, essential for the proof of Theorem \\ref{thm:MT2}: \n\n\\begin{lemma}\\label{lem:successor}Suppose $m\\leq n$ are naturals so that there is $k$ natural with $n-m=k(m-1)$. Suppose $l$ is a positive integer congruent to $m$ modulo $m-1$. Let $S= \\{ a_m , \\dots, a_{n-1}\\}$ and let $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be an $l$-element prefix code, ordered as $p_l <_{dict} p_{l-1}<_{dict}\\dots <_{dict} p_1$.  Let $i$ with $1\\leq i\\leq l$.  Then, the successors $(p_i)'_1$, $(p_i)'_2$, $\\ldots$, $(p_i)'_k$ are well defined, and furthermore, the expansion in which we replace $P$ by $\\widetilde{P} = (P \\backslash \\{p_i\\} ) \\cup p_i\\{a_0, \\dots, a_{m-1}\\}$ has successors $(p_ia_j)'_i$ uniquely determined as follows: \n$$\\begin{array}{lll}\n(p_ia_{m-1})'_1 & = p_ia_{m}\\\\\n& \\vdots \\\\ (p_ia_{m-1})'_k & = p_ia_{m+k-1}\\\\\n(p_ia_{m-2})'_1 & = p_ia_{m+k}\\\\\n& \\vdots \\\\ (p_ia_{m-2})'_k & = p_ia_{m+2k-1}\\\\\n& \\vdots \\\\\n(p_ia_{1})'_1 & = p_ia_{m+(m-2)k}\\\\ \n& \\vdots \\\\\n(p_ia_{1})'_k & = p_ia_{m+(m-1)k-1} = p_ia_n\\\\\n(p_ia_{0})'_1 & = (p_i)'_1\\\\ \n& \\vdots \\\\\n(p_ia_{0})'_k & = p_ia_{m+(m-1)k} = (p_i)'_k.\\\\\n\\end{array}$$\n\\end{lemma}\n\n\\begin{proof}\nWe prove the two statements by induction on $l$.  \n\n{\\flushleft {\\it Base Case ($l=1$):}}\\\\\nIf $l=1$ then $P=\\{a_{m-1}\\}$.  We have $\\operatorname{spref}{P}=\\{\\varepsilon\\}$.  It then follows that the $k$ successors are, the set $\\{a_m,a_{m+1},\\ldots,a_{m+k-1}\\}$, noting that these are given in order and are the results of the inductive definition of the $k$ successors of $a_{m-1}$.  Thus we have in the base case that the successors are well defined.  We need to verify the existence of well defined successors for an expansion of $P=\\{a_{m-1}\\}$.  In this case, $P$ admits only one expansion, which is precisely the set $\\widetilde{P}=\\{a_{m-1}a_{m-1},a_{m-1}a_{m-2},\\ldots,a_{m-1}a_0\\}.$  We have $\\operatorname{spref}({\\widetilde{P}})=\\{\\varepsilon,a_{m-1}\\}$ and we have $$\\begin{array}{lll}\n(a_{m-1}a_{m-1})'_1 & = a_{m-1}a_{m}\\\\\n& \\vdots \\\\ (a_{m-1}a_{m-1})'_k & = a_{m-1}a_{m+k-1}\\\\\n(a_{m-1}a_{m-2})'_1 & = a_{m-1}a_{m+k}\\\\\n& \\vdots \\\\ (a_{m-1}a_{m-2})'_k & = a_{m-1}a_{m+2k-1}\\\\\n& \\vdots \\\\\n(a_{m-1}a_{1})'_1 & = a_{m-1}a_{m+(m-2)k}\\\\ \n& \\vdots \\\\\n(a_{m-1}a_{1})'_k & = a_{m-1}a_{m+(m-1)k-1} = a_{m-1}a_n\\\\\n(a_{m-1}a_{0})'_1 & = (a_{m-1})'_1\\\\ \n& \\vdots \\\\\n(a_{m-1}a_{0})'_k & = a_{m-1}a_{m+(m-1)k} = (a_{m-1})'_k.\\\\\n\\end{array}$$\nWe can directly observe these successors are well defined and distinct.  Thus, the statement is true for $l=1$.\n\n\n{\\flushleft {\\it Inductive Case ($l>1$):}}\\\\\nNow let us assume that $\\widetilde{P}$ is a result of $v$ expansions from the one-element prefix code $\\{a_{m-1}\\}$, for some $v\\geq 1$, where for any prefix code resulting from $u$ expansions from $\\{a_{m-1}\\}$ for $0\\leq u<v$ the statement of the lemma holds.  We will show that the successors of our expansion $\\widetilde{P}$ are well defined.\nIn particular, if $P$ is the prefix code (of size $l$) arising from doing only the first $t-1$ expansions from $\\{a_{m-1}\\}$ towards the prefix code $\\widetilde{P}$, and $p_i$ is the element of $P$ which is being replaced by and $m$-fold expansion to create $\\widetilde{P}$, then by induction, the successors $(p_i)'_1$, $(p_i)'_2$, $\\ldots$, $(p_i)'_k$ of $p_i\\in P$ are well defined.\n\nNote that $P_{i-1,k} = \\widetilde{P}_{i-1,k}$, as the involved subsets of both $P$ and $\\widetilde{P}$ are equal. Thus the first successor to assign is $(p_ia_{m-1})'_1$. Suppose that $(p_ia_{m-1})'_1 <_{dict} p_ia_m$, then $(p_ia_{m-1})'_1 = pa_t$ for some $p \\in \\operatorname{spref}(P): p <_{dict} p_i$ and $a_t \\in \\{a_m, \\dots,a_{n-1} \\}$. Thus, before expanding $P$, $pa_t$ is one of the successors of some $p_r \\in P$. \n\nIf $p_r <_{dict} p_i$, then the set of successors of $p_i$ is defined before the set of successors of $p_r$. Because $p_i$ is expanded, the set of $k$ successors of $p_ia_{m-1}$ is equal to the set of successors of $p_i$, and this set does not contain $pa_t$, which is a contradiction. On the other hand, if $p_r >_{dict} p_i$, then $pa_t \\in P_{i-1,k} = \\widetilde{P}_{i-1,k}$, which is also a contradiction. Thus $(p_ia_{m-1})'_1 = p_ia_m$. We can use a similar argument for all $(p_ia_{m-1})'_1  \\dots (p_ia_{1})'_k$. \n\nFor $p_ia_0$, all successors of the form $p_ia_s,\\, a_s \\in \\{a_m ,\\dots, a_{n-1}\\}$, have already been assigned.  Thus, the remaining $k$ successors are precisely the $k$ successors of $p_i$, taken in order.\n\\end{proof}\n\n\\begin{remark}\\label{rem:successor}\n Birget in \\cite{B} gives a formula for the $i$-th successor of an element, for the case of $m=2$.  The statement of Lemma \\ref{lem:successor} above shows the natural generalisation of that formula holds when we have the Higman Condition (as we must for successors to be well defined).  The resulting formula is given as follows:\n \nLet $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be a prefix code with $\\vert P \\vert \\geq 2$, such that the elements of $P$ are ordered in reverse dictionary order. Then every element of $w \\in P$ can be written uniquely in the form $ua_ia_0^t$, where $u\\in \\{a_0 , \\dots, a_{m-1}\\}^*$ and $t \\geq 0$. The $i$-th successor of $w$ is:\n$$(w)'_i = (ua_ja_0^t)'_i = ua_{m-1+(m-1-j)k + i}.$$ \nWe stress that this formula is only valid if $P$ is ordered in reverse dictionary order.\n\\end{remark} \n\n\\subsection{The algebraic embedding}\nWe proceed to define the algebraic embedding of $V_m(G)$ in $V_n(H) = V_n(G_{ext})$. Let $g \\in V_m(G)$, given by the following table:\n\\begin{align*}\ng & = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_l \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_l\\\\\nq_1 & q_2 & \\cdots & q_l \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_l\n\\end{bmatrix}\n\\end{align*}\nWe define the embedding $\\iota(g)$ below.  The resulting tables are large, and our notation requires some explanation.  The idea of the embedding is to use the identity map initially, and at $a_{m+k}$ and later letters, but under the address $a_{m-1}$ we place the prefix code $p_1$ to $p_l$, and we also require action under the successors.  The first row then has entries following the ordered list given here (wrapped at natural locations due to page length constraints):\n\\[\n\\begin{matrix}\na_0,a_1,\\ldots,a_{m-2},\\\\\na_{m-1}p_1,a_{m-1}p_2,\\ldots,a_{m-1}p_l,\\\\\n(a_{m-1}p_1)'_1,(a_{m-1}p_2)'_1,  \\ldots, (a_{m-1}p_l)'_1,\\\\\n(a_{m-1}p_1)'_2,(a_{m-1}p_2)'_2,  \\ldots, (a_{m-1}p_l)'_2,\\\\\n\\dots\\\\\n(a_{m-1}p_1)'_k,(a_{m-1}p_2)'_k,  \\ldots, (a_{m-1}p_l)'_k,\\\\\na_{m+k},a_{m+k+1},\\ldots,a_{n-1}.\n\\end{matrix}\n\\]\nWe use vertical bars ``$\\vert$'' in our table at the same locations that we placed line-wraps in the row detailed above, for clarity of grouping.  The element $\\iota(g)$ is now given by the following table:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & \\cdots  \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n \\color{white} =  \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots  & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &  \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & Id & \\cdots & Id \\\\\n\\cdots  & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nNote that the set of successors of $P = \\{p_1, \\dots, p_l\\}$ are assigned supposing that $p_l <_{dict} \\dots <_{dict} p_1$. Therefore, the set of successors of $Q = \\{q_1, \\dots, q_l\\}$ is assigned following the order $q_l \\rightarrow \\dots \\rightarrow q_1$, which does not need to follow the dictionary order on $Q$. \n\nIndeed, the first and third rows of $\\iota(g)$ are both prefix codes of $\\mathfrak{C}_n$. On the one hand, suppose that the number of columns of $g$ is $l = m + d(m-1)$ for some $d \\geq 0$. It follows that the number of columns of $\\iota(g)$ whose elements of the first row start with $a_{m-1}$ is $n+ d(n-1)$ (observe that the last $k$ terms from the successor substitution will not begin with $a_{m-1}$). On the other hand, as the number of columns of $g$ is $(m + d(m-1))$, and we assign $k$ successors to every column, we have $(m+d(m-1))(k+1)$ columns on $\\iota(g)$. As we have Higman's Condition, the reader can verify that $(m+d(m-1))(k+1) = n+ d(n-1) +k$.  From this, we see firstly that $(n-1)\\vert k$, but more importantly, this embedding\/successor operation does not place any constraints on the  number of expansions $d$ that were used to create the original prefix code for the domain of $g$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:MT2}]\nIf we push down the action of every $\\sigma_i$, we have:\n\\begin{align*}\n\\operatorname{push}(g) & = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_l \\\\\nId & Id & \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_l \\\\\n\\tau_1\\sigma_1^{-1}  & \\tau_2\\sigma_2^{-1} & \\cdots & \\tau_l\\sigma_l^{-1} \\\\\n\\end{bmatrix}  \n\\end{align*}\nThus the table for $\\iota(\\operatorname{push}(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\[2pt]\nId & \\cdots & Id & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & \\cdots  \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n \\color{white} =  \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots  & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &  Id &\\cdots & Id & \\vert & Id & \\cdots & Id \\\\\n\\cdots  & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\[2pt]\n\\cdots & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & Id & \\cdots & Id \\\\\n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\nOn the other hand the table for $\\operatorname{push}(\\iota(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\[2pt]\nId & \\cdots & Id & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & \\cdots  \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n  \\color{white} =  \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots  & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &  Id &\\cdots & Id & \\vert & Id & \\cdots & Id \\\\\n\\cdots  & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\[2pt]\n\\cdots & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nRecall from the statement of Theorem \\ref{thm:MT2} that for an element $\\tau\\in\\operatorname{Sym}(m)$, the extended version $\\tau'$ of $\\tau$ in $\\operatorname{Sym}(n)$ is that element of $\\operatorname{Sym}(n)$ which agrees with $\\tau$ on the set $\\mathcal{A}_m$ and acts as the identity on the points of $\\mathcal{A}_{m,n}$ in $\\mathcal{A}_n$.  Thus, both tables are equal, as $(\\tau_i\\sigma_i^{-1})' = (\\tau'_i)(\\sigma'_i)^{-1},\\, \\forall i \\in \\{i,\\dots,l\\}$. If we expand $g$ on $p_i$:\n\n$$\\operatorname{exp}(g) = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_ia_0 & \\cdots & p_ia_{m-1} & \\cdots  &  p_l \\\\\nId & Id & \\cdots & Id & \\cdots & Id& \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_i \\tau_i(a_0) & \\cdots & q_i\\tau_i(a_{m-1}) & \\cdots  &  q_l \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_i & \\cdots & \\tau_i& \\cdots & \\tau_l\n\\end{bmatrix}$$\nThen the table for $\\iota(\\operatorname{exp}(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_ia_0 & \\cdots & a_{m-1}p_ia_{m-1} & \\cdots & a_{m-1}p_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\cdots & Id &\\cdots  & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_i\\tau_i(a_0) & \\cdots & a_{m-1}q_i\\tau_i(a_{m-1}) & \\cdots & a_{m-1}q_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_i  &\\cdots & \\tau'_i  &\\cdots & \\tau'_l & \\vert & \\cdots  \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\\color{white} =  \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots  & \\vert  & \\cdots  & (a_{m-1}p_ia_0)'_j & \\cdots & (a_{m-1}p_ia_{m-1})'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots  & \\vert  & \\cdots  & Id & \\cdots & Id & \\cdots & \\vert & Id &\\cdots & Id \\\\\n\\cdots   & \\vert  & \\cdots  & (a_{m-1}q_i\\tau_i(a_0))'_j & \\cdots & (a_{m-1}q_i\\tau_i(a_{m-1}))'_j & \\cdots& \\vert & a_{m+k} & \\cdots & a_{n-1}  \\\\\n\\cdots  & \\vert& \\cdots & \\tau'_i &\\cdots & \\tau'_i & \\cdots & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nOn the other hand, the table for $\\operatorname{exp}(\\iota(g))$ is:\n\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_ia_0 & \\cdots & a_{m-1}p_ia_{n-1} & \\cdots & a_{m-1}p_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\cdots & Id &\\cdots  & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_i\\tau'_i(a_0) & \\cdots & a_{m-1}q_i\\tau'_i(a_{n-1}) & \\cdots & a_{m-1}q_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_i  &\\cdots & \\tau'_i  &\\cdots & \\tau'_l & \\vert & \\cdots  \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\\color{white} =  \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots  & \\vert &\\cdots & (a_{m-1}p_1)'_j & \\cdots & (a_{m-1}p_i)'_j & \\cdots & (a_{m-1}p_l)'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &\\cdots &  Id &\\cdots &  Id &\\cdots & Id &\\cdots & \\vert & Id & \\cdots & Id \\\\\n\\cdots  & \\vert & \\cdots & (a_{m-1}q_1)'_j  & \\cdots & (a_{m-1}q_i)'_j & \\cdots & (a_{m-1}q_l)'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &\\cdots & \\tau'_1 &\\cdots & \\tau'_i &\\cdots & \\tau'_l & \\cdots & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\nIt is straightforward to check that for both tables the columns starting at $a_{m-1}p_ia_s$ for $0 \\leq s \\leq m-1$ are equal, as $\\tau'_i(s) = \\tau_i(s), \\forall s \\in  \\{0,\\dots, m-1\\}$. \n\n\nFor $ m \\leq s \\leq n-1$ by Lemma \\ref{lem:successor}, we know there is a correspondence between the first two rows of $\\iota(\\operatorname{exp}(g))$ and the first two rows of $\\operatorname{exp}(\\iota(g))$. On the other hand, from the description of the algebraic embedding, the order in which successors for every $q_j$ are selected depends only on the order of $\\{p_1, \\dots, p_l\\}$, so we have the following calculations:\n$$\\begin{array}{lllll}\n(a_{m-1}q_i\\tau_i(a_{m-1}))'_1 & = a_{m-1}q_ia_{m} & = a_{m-1}q_i\\tau'_i(a_{m}) \\\\\n& \\vdots & \\vdots &\\\\ \n(a_{m-1}q_i\\tau_i(a_{m-1}))'_k & = a_{m-1}q_ia_{m+k-1} & = a_{m-1}q_i\\tau'_i(a_{m+k-1}) \\\\\n(a_{m-1}q_i\\tau_i(a_{m-2}))'_1 & = a_{m-1}q_ia_{m+k} & = a_{m-1}q_i\\tau'_i(a_{m+k}) \\\\\n& \\vdots &\\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{m-2}))'_k  & = a_{m-1}q_ia_{m+2k-1} & = a_{m-1}q_i\\tau'_i(a_{m+2k-1})\\\\\n& \\vdots& \\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{1}))'_1  & = a_{m-1}q_ia_{m+(m-2)k} & = a_{m-1}q_i\\tau'_i(a_{m+(m-2)k})\\\\ \n& \\vdots &\\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{1}))'_k & = a_{m-1}q_ia_n & = a_{m-1}q_i\\tau'_i(a_{m+(m-1)k-1}) \\\\\n(a_{m-1}q_i\\tau_i(a_{0}))'_1 & = (a_{m-1}q_i)'_1\\\\ \n& \\vdots \\\\\n(a_{m-1}q_i\\tau_i(a_{0}))'_k & =  (a_{m-1}q_i)'_k.\\\\\n\\end{array}$$\nThat is, e.g., $(a_{m-1}q_i\\tau_i(a_{m-1}))'_1$ comes first in the choice of successor as $(a_{m-1}p_ia_{m-1})'_1$ appears first under the order of the $p_i$, independent of $\\tau_i$.  Thus, the latter two rows of these tables are also equivalent.\n\nFinally, it is easy to see that $\\iota(h \\circ g) = \\iota(h) \\circ \\iota(g)$, as $\\iota$ commutes with expansions and pushings. We only need to obtain row equality on the first part of the table as the remaining part depends entirely on $P$ (resp. on $Q$ for the element $h$): \n\\begin{align*}\n\\iota(g) & = \\begin{bmatrix}\n\\cdots  & a_{m-1}p_i & \\cdots & (a_{m-1}p_i)'_j & \\cdots \\\\\n\\cdots  & Id & \\cdots & Id &\\cdots  \\\\\n\\cdots  & a_{m-1}q_i& \\cdots & (a_{m-1}q_i)'_j & \\cdots\\\\\n\\cdots &\\tau'_i & \\cdots & \\tau'_i&\\cdots\\\\ \n\\end{bmatrix},\\\\ \n\\end{align*}\n\\begin{align*}\n\\iota(h) & = \\begin{bmatrix}\n \\cdots & a_{m-1}q_i& \\cdots & (a_{m-1}q_i)'_j& \\cdots\\\\\n\\cdots & \\tau'_i  & \\cdots& \\tau'_i & \\cdots\\\\\n \\cdots & a_{m-1}r_i & \\cdots & (a_{m-1}r_i)'_j& \\cdots\\\\\n \\cdots & \\tau''_i & \\cdots& \\tau''_i& \\cdots\\\\ \n\\end{bmatrix},\\\\ \n\\iota(h) \\circ \\iota(g) & = \\begin{bmatrix}\n \\cdots & a_{m-1}p_i& \\cdots & (a_{m-1}p_i)'_j& \\cdots\\\\\n\\cdots & Id  & \\cdots& Id & \\cdots\\\\\n \\cdots & a_{m-1}r_i & \\cdots & (a_{m-1}r_i)'_j& \\cdots\\\\\n \\cdots & \\tau''_i & \\cdots& \\tau''_i& \\cdots \\\\ \n\\end{bmatrix} = \\iota(h \\circ g).\n\\end{align*}  Thus the result follows. \n\\end{proof}\n\\label{Bibliography}\n\\bibliographystyle{abbrv} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results}\nWe consider the Cauchy problem of the generalized SQG (Surface Quasi-geostrophic) equation in the plane as follows\n\\begin{equation}\\label{SQG}\\tag{SQG}\n\\left\\{\\ba\n&\\omega_{t}+u\\cdot\\nabla \\omega =0, \\qquad \\qquad(x,t)\\in \\R^{2}\\times\\R^{+},\\\\\n&u=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega, \\\\\n&\\omega(x,0)=\\omega_0 \\ea\\ \\right.\n\\end{equation}\n with $0\\le \\alpha\\le\\frac 12$. Here, according to the second equation in $\\eqref{SQG}$, the unknown scalar function   $\\omega=\\omega(x,t)$ and vector field $u=(u_1(x,t),u_2(x,t))$  can be expressed as the singular integral\n\\begin{equation}\\label{Int}\nu(x)=\\int_{\\R^2} \\frac{(x-y)^\\perp}{|x-y|^{2+2\\alpha}}\\omega(y) \\,\\mathrm{d}y.\n\\end{equation}\nThroughout our paper,  we omit some constants before the singular integral \\eqref{Int}  for conciseness. Meanwhile, the expression \\eqref{Int} implies that $u$ is divergence-free, that is, $\\nabla\\cdot u=\\partial_{x_1}u_1+\\partial_{x_2}u_2=0.$\n\nWhen $\\alpha=0$, it is well-known that  \\eqref{SQG} corresponds to the two-dimensional incompressible Euler equations. In this case, the unknown functions  $\\omega=\\omega(x,t)$ and $u=u(x,t)$ are the vorticity and the velocity field, respectively. When $\\alpha=\\frac12$, \\eqref{SQG} corresponds to the surface quasi-geostrophic (SQG) equation which describes a famous approximation model of the nonhomogeneous fluid flow in a rapidly rotating 3D half-space (see \\cite{[P87]}). In this case, the unknown functions  $\\omega=\\omega(x,t)$ and $u=u(x,t)$  represent  potential temperature and velocity field, respectively. When $0<\\alpha<\\frac12$, \\eqref{SQG} is called the generalized (or modified) SQG equation.\n\n\nThe classical SQG and the generalized SQG equations  have been widely studied in the past years and much more progress has been  made. In \\cite{[KYZ],[KRYZ]}, it is proved that the generalized SQG in half space $\\R^{2+}=\\{x=(x_1,x_2)| x_2>0\\}$ has a unique local solution for vortex-patch initial data and will appear  singularity in finite time for some such kind of initial data when $0<\\alpha<\\frac{1}{24}$. This strongly implies that the SQG equation will appear finite-time singularity (even for smooth initial data) since the velocity has  less regularity when $\\alpha=\\frac12$. In fact, the singularity or formation of strong fronts has been suggested in \\cite{[CMT]} although the rigorous derivations have not been reached so far. We note that the global well-posedness or blow-up of  the SQG equation is an important issue. As pointed out in \\cite{[CMT]}, the singularity of the SQG equation will be similar to that of the three-dimensional Euler equations. Concerning the  dissipative SQG equation, which enjoys  a fractional dissipation term $-(-\\Delta)^{\\beta}\\omega$ on the right hand side of the second equation of \\eqref{SQG}, the global well-posedness in the critical case $\\beta=\\frac12$ was proved independently by Caffarelli and Vasseur \\cite{[CV]} and by Kiselev, Nazarov and Volberg \\cite{[KNV]} (see \\cite{[CVi],[KN]} for different approaches). The proof of global regularity for the subcritical case $\\beta>\\frac12$ is standard (see e.g. \\cite{[Con-Wu]}), while in the supercritical case $\\beta<\\frac12$  the global   regularity of small solutions is obtained (see e.g. \\cite{[CMZ],[HK],[Ju],[Wu]}) and the slightly supercritical case is studied recently in \\cite{[DKSV]}.\n\n\nIn this paper, our target is to show that, for any $T>0$, if $\\{\\omega^{\\alpha_0}, u^{\\alpha_0}\\}$ defined on $[0,T]$ is the unique smooth solution  of \\eqref{SQG}  for some $0<\\alpha_0\\le \\frac12$, then there exists $\\delta>0$ such that when $0<\\alpha<\\frac12$ and  $0<|\\alpha_0-\\alpha|\\le \\delta$, the problem \\eqref{SQG} with same initial data has also a unique smooth solution $\\{\\omega^\\alpha,u^\\alpha\\}$ defined on $[0,T]$, where we denote the solution of problem \\eqref{SQG} corresponding to $0\\le \\alpha\\le \\frac12$ by $\\{\\omega^\\alpha, u^\\alpha\\}$ satisfying $u^\\alpha=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^\\alpha$.  This is motivated by \\cite{[C86]} in which it is shown that if the Cauchy problem to the three-dimensional incompressible Euler equations have a unique smooth solution on $[0,T]$, then the corresponding three-dimensional incompressible Navier-Stokes equations with the same initial data will also have a unique smooth solution defined on $[0,T]$ when the viscosity is suitably small. Furthermore, our result  implies that the construction of the possible singularity of the smooth solution of the Cauchy problem to the generalized SQG with $\\alpha>0$  will be  subtle (see Corollary \\eqref{Cor1+}), in comparison with the singularity result presented in  \\cite{[KRYZ]}. To prove our main results, we consider the behavior of the difference between $u^\\alpha$ and $u^{\\alpha_0}$. Let us denote\n $$\\overline{\\omega}=\\omega^{\\alpha}-\\omega^{\\alpha_{0}}\\quad\\text{and}\\quad\\overline{u}=u^{\\alpha}-u^{\\alpha_{0}},$$\nwe easily find that the couple $(\\overline{\\omega},\\,\\overline{u})$ satisfies\n\\begin{equation*}\n\\overline{\\omega}_{t}+(u^{\\alpha_{0}}\\cdot\\nabla) \\overline{\\omega}+(\\overline{u}\\cdot\\nabla) \\overline{\\omega}+(\\overline{u}\\cdot\\nabla) \\omega^{\\alpha_{0}} =0.\n\\end{equation*}\nWith this equation, we can establish the following $H^s$-estimate of $\\overline{\\omega}(t)$:\n\\begin{align*}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}^2=&-\\int_{\\R^2} J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\\n&-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x.\n\\end{align*}\nThe difficulty for us is  how to use the information of $\\|\\overline{\\omega}(t)\\|_{H^s}$ to control $\\bar u$ in the nonlinear term. This requires us to consider the behavior in different scale in physical space or in different frequency regime in frequency space. Thus, we decompose $\\bar u$ in two parts\n\\begin{equation}\\label{diff-1-1}\n\\begin{split}\n\\overline{u}=&u^{\\alpha}-u^{\\alpha_{0}}\n\\\\=&\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}\n+(\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}-\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0})\\omega^{\\alpha_0}\n\\\\:=&\\overline{u}_I+\\overline{u}_{II},\n\\end{split}\\end{equation}\nwhere $\\bar\\omega=\\omega^\\alpha-\\omega^{\\alpha_0}$. In the above equalities \\eqref{diff-1-1}, we see that the part $u_I$ is related to the difference between the solutions, while the part $u_{II}$ corresponds to the difference between the singular integrals. This observation together with the scale analysis enables us to establish some technique propositions, see Proposition \\ref{uni-est}, Proposition \\ref{add-0} and Proposition \\ref{add1} which will play key roles in our proof of Theorem \\ref{th2} and Theorem \\ref{th3} respectively. More precisely,  in view of Riesz potential (see \\eqref{Riesz}), the  term $\\overline{u}_I$  can be estimated as\n\\begin{equation}\\label{Riesz1}\n\\|\\overline{u}_I\\|_{L^q(\\R^2)}\\le C(\\alpha)\\|\\bar\\omega\\|_{L^p(\\R^2)}, \\ \\frac1q=\\frac1p-\\frac{1-2\\alpha}{2},\n\\end{equation}\nfor $0<\\alpha<\\frac12$. However, when $\\alpha\\to\\frac12$, the constant $C(\\alpha)$ in \\eqref{Riesz1} will be unbounded. To overcome this difficulty, we establish  Propositions \\ref{uni-est}- \\ref{add1} to obtain some new uniform estimates as $\\alpha\\to \\frac12$.\n\nOur main results are stated as follows.\n\\begin{theorem}\\label{th1}\nLet $0< \\alpha_{0}<\\frac12.$  Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+1},$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ such that if $0<\\alpha<\\frac12$ and  $|\\alpha_0-\\alpha|\\le \\delta$,  the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\\begin{equation*}\n\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log|\\alpha_0-\\alpha||\\right),\n\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$.\n\\end{theorem}\n\n\\begin{theorem}\\label{th2}\nLet $0<\\alpha<\\alpha_{0}=\\frac12.$  Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+1}\\cap L^1,$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}\\cap L^1} dt$ such that if $0<\\alpha_0-\\alpha<\\delta$, the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\\begin{equation*}\n\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right),\n\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}\\cap L^1} dt$.\n\\end{theorem}\n\n\\begin{theorem}\\label{th3}\n\tLet $0<\\alpha<\\alpha_{0}=\\frac12.$  Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+2},$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+2}} dt$ such that if $0<\\alpha_0-\\alpha<\\delta$, the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n\t$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\t\\begin{equation*}\n\t\t\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right),\n\t\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+2}} dt$.\n\\end{theorem}\n\\begin{remark}\\label{Rm1}\nIn Theorem \\ref{th1}, we consider the case $0<\\alpha_0<\\frac12$.\nIn Theorem \\ref{th2} and Theorem \\ref{th3}, we deal with the case $\\alpha_0=\\frac12$.\n It is noted that in the proof of Theorem \\ref{th1},  the Hardy-Littlewood-Sobolev inequality (see Lemma \\ref{Hardy}) will be used. One point is that the expression $\\bar u_I$ in \\eqref{diff-1-1} can be reduced to $I_{1-2\\alpha}\\bar\\omega$, where $I_{1-2\\alpha}$ is a  Riesz operator  (see  \\eqref{Riesz}) which is bounded from $L^p$ to $L^q$ with $\\frac1q=\\frac1p-\\frac{1-2\\alpha}{n}$ satisfying $0<1-2\\alpha<n$ and $1<p<q<\\infty$. However,  it does not hold bounded uniformly with respect to $\\alpha$ when $\\alpha$ tends to $\\frac12$. Hence, in the proof of Theorem \\ref{th2} and Theorem \\ref{th3} we can not use Hardy-Littlewood-Sobolev inequality directly. To prove Theorem \\ref{th2}, we will establish some new and uniform estimates with respect to $\\alpha$ on the singular integral \\eqref{Int} (see Proposition \\ref{uni-est}). To prove Theorem \\ref{th3}, we will obtain some elegant estimates concerning $u_I$  with the help of Besov spaces of which definition is given in Appendix \\ref{sec4} (see Propositions \\ref{add-0} and \\ref{add1}).\n\\end{remark}\n\\begin{remark}\\label{Rm3}\nIn Theorem \\ref{th2}, we require that the initial data $\\omega_0\\in H^{s+1}\\cap L^1$ with $s>2$.  Thanks to the incompressible condition $\\nabla\\cdot u=0$,  the solution will stay in $L^1$. In Theorem \\ref{th3}, we drop the restriction on the initial data $\\omega_0\\in L^1$, but more regularity of the initial data $\\omega_0\\in H^{s+2}$ with $s>2$ will be needed.\n\\end{remark}\n\n\\begin{remark}\\label{Rm2+}\nAs mentioned above, \\eqref{SQG} becomes the two-dimensional incompressible Euler equations when $\\alpha_0=0$, of which the global existence of smooth solutions has been known (see \\cite{Majda1} and references therein). In comparison with the singularity for the patch solution with $0<\\alpha<\\frac{1}{24}$ in half space  obtained in \\cite{[KRYZ]}, whether the patch or smooth solution  of the Cauchy problem  to \\eqref{SQG} when $\\alpha>0$ appears singularity in finite time remains open. Theorem \\eqref{th1} implies that the possible blow-up time of the smooth solution to the Cauchy problem of  \\eqref{SQG} with $\\alpha>0$ can not be uniformly bounded when $\\alpha\\to 0+$.  More precisely, as a corollary of Theorem \\ref{th1}, we have\n\\end{remark}\n\n\\begin{corollary}\\label{Cor1+}\nLet $T^*_\\alpha>0$ the maximal existence time (may be $+\\infty$) of the  solution $\\omega^\\alpha\\in C([0,T]; H^{s+1}) (s>2)$  to \\eqref{SQG} with $\\alpha>0$. Then  $\\displaystyle\\limsup_{\\alpha\\to 0+}T^*_\\alpha=+\\infty$.\n\\end{corollary}\n\n\nThe rest of the paper is organized as follows. In Section \\ref{pre}, we will present some basic facts  which will be needed later. In Section \\ref{SI}, we will investigate a singular integral which can be viewed as an approximation of the Riesz transform. In Section \\ref{BesovE}, we will obtain  nonlinear terms and commutator estimates  related to $u_I=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}$ in \\eqref{diff-1-1}. The proof of the main results will given in Section \\ref{sec5}. In the end of the paper,  Appendix \\ref{sec4} on the Littlewood-Paley decomposition, Besov spaces will be given.\n\n\n\n\n\\section{Preliminaries}\\label{pre}\n\\setcounter{section}{2}\\setcounter{equation}{0}\n\n\nIn this section, we present some basic analysis facts. First of all, we introduce\n\\[\\Lambda^s=(-\\Delta)^\\frac{s}{2}\\quad\\text{and}\\quad\nJ^s=(I-\\Delta)^\\frac{s}{2},\\] where\n $$\\widehat{\\Lambda^sf}(\\xi)=|\\xi|^s\\widehat{f}(\\xi)\\quad\\text{and}\\quad\\widehat{J^sf}(\\xi)\n =\\big(1+|\\xi|^2\\big)^{\\frac{s}{2}}\\widehat{f}(\\xi),~~ s\\in \\R.$$\n\n\\begin{definition}[\\cite{[stein]}]\nLet  $s\\in \\R$ and $1\\leq p\\leq \\infty.$ We write\n$$\\|f\\|_{W^{s,p}(\\R^n)}:=\\norm{J^s f}_{L^p(\\R^n)}, ~~\\|f\\|_{\\dot{W}^{s,p}(\\R^n)}:=\\norm{\\Lambda^s f}_{L^p(\\R^n)}.$$\nThe nonhomogeneous Sobolev space $W^{s,p}(\\R^n)$ is defined as\n$$\nW^{s,p}(\\R^n)=\\{f\\in \\mathcal{S'}(\\R^n):\\|f\\|_{W^{s,p}(\\R^n)}<\\infty\\}.\n$$\nThe homogeneous Sobolev space $\\dot{W}^{s,p}(\\R^n)$ is defined as\n$$\n\\dot{W}^{s,p}(\\R^n)=\\{f\\in \\mathcal{S'}(\\R^n):\\|f\\|_{\\dot{W}^{s,p}(\\R^n)}<\\infty\\}.\n$$\nHere $\\mathcal{S'}$ is the Schwarz distributional function space.\n\\end{definition}\nWith this definition in hand, we give a commutator estimate and product estimate (see, e.g., Kenig, Ponce and Vega \\cite{[KPV]}).\n\\begin{lemma}\\label{commutator}\nLet $s>0$ and $1<p<\\infty.$ Then\n\\begin{equation}\\label{com-1}\n\\|J^s(fg)\\|_{L^p(\\R^n)}\\leq C(\\|f\\|_{W^{s,p_1}(\\R^n)}\\|g\\|_{L^{p_2}}+\\|g\\|_{W^{s,p_3}(\\R^n)}\\|f\\|_{L^{p_4}(\\R^n)}),\n\\end{equation}\n\\begin{equation}\\label{com-2}\n\\|J^s(fg)-fJ^sg\\|_{L^p(\\R^n)}\\leq C(\\|f\\|_{W^{s,p_1}(\\R^n)}\\|g\\|_{L^{p_2}(\\R^n)}+\\|g\\|_{W^{s-1,p_3}(\\R^n)})\\|\\nabla f\\|_{L^{p_4}(\\R^n)},\n\\end{equation}\nwhere $\\frac1p=\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{p_3}+\\frac{1}{p_4}$ with $p_2,$ $p_4\\in [1,\\infty]$ and $p_1,$ $p_3\\in (1,\\infty)$, and $C^{'}s$ are constants depending on $s,$\n$p_1,$ $p_2,$ $p_3$ and $p_4$. In addition, these inequalities remain valid when $J^s$ is replaced by $\\Lambda^s$.\n\\end{lemma}\nWe continue with the  Sobolev embedding theorem \\cite{[MWZ]}.\n\\begin{lemma}\\label{embedding1}\nFor $p_2\\neq\\infty,$ $W^{s_1,p_1}(\\R^n)\\hookrightarrow W^{s_2,p_2}(\\R^n)$ if and only if\n$$s_1-\\frac{n}{p_1}\\geq s_2-\\frac{n}{p_2},~~\\frac{1}{p_1}\\geq\\frac{1}{p_2}.$$\n\n\\end{lemma}\n\nThe following lemma is the so called Hardy-Littlewood-Sobolev inequality of fractional integration \\cite{[stein]}. We begin with definition of the Riesz potential $I_{\\alpha}$.\n\\begin{definition}\nLet $0<\\alpha<n.$ The Riesz potential $I_{\\alpha}=(-\\Delta)^{-\\frac{\\alpha}{2}}$ is defined by\n\\begin{equation}\\label{Riesz}\nI_{\\alpha}f(x)=\\frac{1}{\\gamma(\\alpha)}\\int_{\\R^n}\\frac{f(y)}{|x-y|^{n-\\alpha}}\\,\\mathrm{d}y,\n\\end{equation}\nwith\n$$\\frac{1}{\\gamma(\\alpha)}=\\pi^{\\frac n2}2^\\alpha \\frac{\\Gamma(\\frac{\\alpha}{2})}{\\Gamma(\\frac n2-\\frac \\alpha 2)}.$$\n\\end{definition}\n\n\n\\begin{lemma}\\label{Hardy}\nLet $0<\\alpha<n,$ $1<p<q<\\infty$, $\\frac1q=\\frac1p-\\frac{\\alpha}{n}.$\nThen, there exists a constant $A_{p,q}$ depending on $p,$ $q$ such that\n\\begin{equation}\\label{H}\n\\|I_{\\alpha}f\\|_{L^q(\\R^n)}\\leq A_{p,q}\\norm{f}_{L^p(\\R^n)}.\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{remark}\nIt is noted that the constant $A_{p,q}$ is unbounded as $\\alpha\\to 0$ here.\n\\end{remark}\nThe following is an elementary result from \\cite{[C86]} in which the case $m=1$ is proved.\n\\begin{proposition}\\label{o}\nLet $T>0,$ $G>0$ and $m>0$ be given constants and let $F(t)$ be a nonnegative continuous function on $[0,T).$ Let $\\nu_0$\nbe defined by $$\\nu_0=\\frac{1}{4m(2m TG)^\\frac1m\\int_0^T F(t)\\,\\mathrm{d}t}.$$ Then, for all $0<\\nu\\leq\\nu_0,$ all nonnegative solution $y(t)$ of\nthe system \\begin{equation}\\label{ODE}\n\\left\\{\\ba\n&\\frac{\\mathrm{d}\\,y(t)}{\\mathrm{d}t}\\leq \\nu F(t)+Gy(t)^{1+m}\\\\\n&y(0)=0\\ea\\ \\right.\n\\end{equation} is uniformly bounded on $[0,T)$ and\n \\begin{equation}\\label{ODE-0}\ny(t)\\leq\\min\\Big\\{\\frac{4^{\\frac1m}-1}{(2m TG)^\\frac1m},\\,\\, 4m\\big(4^\\frac1m-1\\big)\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\Big\\}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet us define $$\\sigma=\\min\\Big\\{\\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}}, \\Big(4m\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\Big)^{1+m}G\\Big\\}.$$\nDividing the first equation of  \\eqref{ODE} by $\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{1+m}$  yields\n\\begin{equation}\n\\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Big(\\frac{1}{\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{m}}\\Big)\\geq -\\nu F(t)-\\sigma.\n\\end{equation}\nBy integrating from $0$ to $t$, we obtain\n\\begin{equation}\\label{ODE-1}\n\\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}\\Big(\\frac{1}{(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y)^{m}}\\Big)\\geq\n \\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}-\\nu \\int_0^T F(t)\\,\\mathrm{d}t-\\sigma T.\n\\end{equation}\nThe choice $\\sigma\\leq \\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}}$ implies $\\sigma T\\leq \\frac{1}{2m}(\\frac{\\sigma}{G})^{\\frac{1}{m+1}}.$\n\nFor $\\nu\\leq \\nu_0,$ we have\n$$\\nu \\int_0^T F(t)\\,\\mathrm{d}t \\leq  \\frac{1}{4m}\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}.$$\nIndeed, if $\\sigma=\\big(4m\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\big)^{1+m}G,$ the last inequality is indeed an equality and if $\\sigma= \\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}},$\nit follows from $$\\nu \\int_0^T F(t)\\,\\mathrm{d}t \\leq\\nu_0 \\int_0^T F(t)\\,\\mathrm{d}t=\\frac{1}{4m(2m TG)^\\frac1m}=\\frac{1}{4m}\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}.$$\nThus, we get by \\eqref{ODE-1} that\n \\begin{equation*}\n\\frac{1}{\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{m}}\\geq\\frac14\n\\end{equation*}\nwhich implies \\eqref{ODE-0}.\n\\end{proof}\n\\section{Estimates on A Singular Integral}\\label{SI}\n\\setcounter{section}{3}\\setcounter{equation}{0}\n\nIn this section, we present some new results on a singular integral which will be needed in the proof of Theorem \\ref{th2}. We denote\n\\begin{equation}\\label{T-operator}\nTf(x)=K*f(x)=\\int_{\\R^n} K(x-y)f(y) dy \\quad\\text{with}\\quad K(x)=\\frac{x}{|x|^{n+1-\\beta}},\\quad 0<\\beta<n,\n\\end{equation}\nwhere $x\\in \\R^n$.\n\nLet \\begin{equation}\\label{Chi-L}\n\\chi_\\lambda(s)=\\chi(\\lambda s).\n\\end{equation}\nwhere $\\chi(s)\\in C_0^\\infty(\\R)$ is the usual smooth cutting-off function which is defined as\n$$\n\\chi(s)=\n\\left\\{\n\\begin{array}{ll}\n1, \\quad& |s|\\le 1,\\\\[3mm]\n0,\\quad & |s|\\ge 2,\n\\end{array}\n\\right.\n$$\nsatisfying $|\\chi'(s)|\\le 2$.\n\nSetting\n\\begin{eqnarray}\n&& T_1f(x):=K_1*f(x)\\quad\\text{with}\\quad K_1(x)=K(x)\\chi_\\beta(|x|),\\label{T1-operator}\\\\\n&& T_2f(x):=K_2*f(x)\\quad\\text{with}\\quad K_2(x)=K(x)(1-\\chi_\\beta(|x|)).\\label{T2-operator}\n\\end{eqnarray}\nThen we have the following proposition which  holds for general $n$-dimensional case.\n\\begin{proposition}\\label{uni-est}\n There exists a constant $C=C(n,s)$ independent of $\\beta$ such that\n\\begin{equation}\\label{T1-1}\n\\|T_1f\\|_{H^s(\\R^n)}\\leq C\\|f\\|_{H^s(\\R^n)},\\quad \\, s>0, \\,\\,0<\\beta<n;\n\\end{equation}\n\\begin{equation}\\label{T2-1}\n\\|T_2f\\|_{L^2(\\R^n)}\\leq C\\frac{\\beta^{\\frac n2}}{\\sqrt{n-2\\beta}}\\|f\\|_{L^1(\\R^n)}, \\quad s>0, \\,\\,0<\\beta<\\frac n2;\n\\end{equation}\n\\begin{equation}\\label{T2-2}\n\\|T_2f\\|_{\\dot{H}^s(\\R^n)}\\leq C(\\frac{\\beta}{1-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta})\\|f\\|_{\\dot{H}^{s-1}(\\R^n)}, \\quad s\\geq1,\\,\\, 0<\\beta<1.\n\\end{equation}\n\\end{proposition}\n\\begin{remark}\\label{Rm2-1}\nWhen $n=2$, the result of Proposition \\ref{uni-est} holds true if $K(x)$ in \\eqref{T-operator} is replaced by $ K(x)=\\frac{x^\\perp}{|x|^{3-\\beta}}$.\n\\end{remark}\n\n\\begin{remark}\nIt is emphasized that the constants $C$ is independent of $\\beta$, and  what is more,\n$\\frac{\\beta^{\\frac n2}}{\\sqrt{n-2\\beta}}$ in \\eqref{T2-1} is sufficiently small, $\\frac{\\beta}{1-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}$ is uniformly bounded\nin \\eqref{T2-2} when $\\beta$ tends to zero.\n\\end{remark}\n\\begin{remark}\n  When $\\beta=0$, it follows from \\eqref{T-operator} that $Tf=\\mathcal{R} f$ (in the sense that the integral takes principle values), where $\\mathcal{R}$ is a Riesz transformation which is a strong $(p,p)$ type operator with $1<p<\\infty$, that is,\n  \\begin{equation}\\label{Riesz0}\n  \\|\\mathcal{R} f\\|_{L^p}\\le C\\|f\\|_{L^p}, \\ 1<p<\\infty\n   \\end{equation}\n  for some constant $C>0$. By Proposition \\ref{uni-est}, it holds\n\\begin{equation}\\label{Riesz2}\n\\|Tf\\|_{L^2}\\le C(\\|f\\|_{L^2}+\\beta\\|f\\|_{L^1}),\n\\end{equation}\nwhere $C>0$ is an absolute constant. This means that the estimate \\eqref{Riesz2} recovers the corresponding one in \\eqref{Riesz0} with $p=2$.\n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{uni-est}]\nWe firstly prove \\eqref{T2-1} and \\eqref{T2-2}.\nNote that\n\\begin{equation*}\nT_2f(x)=\\int_{\\R^n}\\frac{x-y}{|x-y|^{n+1-\\beta}}\\big(1-\\chi_\\beta(|x-y|)\\big)f(y)\\,\\mathrm{d}y.\n\\end{equation*}\nFor $0<\\beta<\\frac n2$, we have\n\\begin{equation*}\n\\begin{split}\n\\big\\|T_2f\\big\\|_{L^2(\\R^n)}&\\leq\\Big\\|\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}|f(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq\\|f\\|_{L^1(\\R^n)}\\Big(\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{2(n-\\beta)}}\\,\\mathrm{d}y\\Big)^{\\frac12}\n\\\\&\\leq\\frac{C}{\\sqrt{n-2\\beta}}\\beta^{\\frac n2-\\beta}\\|f\\|_{L^1(\\R^n)}.\n\\end{split}\n\\end{equation*}\nSince\n$$\\lim_{\\beta\\to0+}\\beta^{-\\beta}=1,$$\nthere exists an absolutely constant $C(n)>0$ such that, for any $0\\leq\\beta<\\frac n2,$\n\\begin{equation*}\\begin{split}\n\\big\\|T_2f\\big\\|_{L^2(\\R^n)}&\\leq  C\\frac{\\beta^{\\frac n2}}{\\sqrt{n-2\\beta}}\\big\\|f\\big\\|_{L^1(\\R^n)}.\n\\end{split}\n\\end{equation*}\nThis means \\eqref{T2-1}.\n\nTo prove \\eqref{T2-2}, we note that for $s\\ge 1$ and $i=1, 2, \\cdots, n$,\n\\begin{equation}\\label{T2-E}\n \\begin{split}\n\\partial_i\\Lambda^{s-1}T_2f(x)=&\\int_{\\R^n}\\partial_i\\Big(\\frac{x-y}{|x-y|^{n+1-\\beta}}\\big(1-\\chi_\\beta(|x-y|)\\big)\\Big)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\n\\\\=&\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\partial_i\\Big(\\frac{x-y}{|x-y|^{n+1-\\beta}}\\Big)\\big(1-\\chi_\\beta(|x-y|)\\big)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\\\\&\n+\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{x-y}{|x-y|^{n+1-\\beta}}\\partial_i\\chi_\\beta(|x-y|)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\n\\\\:=&J_1+J_2,\n\\end{split}\n\\end{equation}\nwhere $\\partial_i=\\partial_{x_i}, i=1,2,\\ldots, n$.\n\nThen, for $0<\\beta<1$, we obtain\n\\begin{equation}\\label{J1}\n \\begin{split}\n\\big\\|J_1\\big\\|_{L^2(\\R^n)}&\\leq C\\Big\\|\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n+1-\\beta}}|\\Lambda^{s-1}_yf(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq C\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n+1-\\beta}}\\,\\mathrm{d}y\n\\\\&\\leq\\frac{C}{1-\\beta}\\beta^{1-\\beta}\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}.\n\\end{split}\n\\end{equation}\nThe term $J_2$ can be bounded as\n\\begin{equation}\\label{J2}\n \\begin{split}\n\\big\\|J_2\\big\\|_{L^2(\\R^n)}&\\leq C\\Big\\|\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}|\\Lambda^{s-1}_yf(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq C\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}\\,\\mathrm{d}y\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}.\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{J1} and \\eqref{J2} into \\eqref{T2-E} and using the fact that $$\\|\\Lambda^sT_2f\\|_{L^2}\\le \\|\\nabla\\Lambda^{s-1}T_2f\\|_{L^2},$$ we finish the proof of \\eqref{T2-2}.\n\nNow we turn to prove \\eqref{T1-1}. To do this, it suffice to show that there exists an absolute constant $C>0$ independent $\\beta$ such that\n\\begin{equation}\\label{K1-F}\n\\big\\|\\widehat{K_1}(y)\\big\\|_{L^\\infty(\\R^n)}\\leq C, \\, \\, 0<\\beta<n.\n\\end{equation}\nIn fact, since $\\int_{\\mathbb{S}^1} K_1(x) \\,\\mathrm{d}s(x)=0$ (here $\\mathbb{S}^1$ is the unit sphere surface in $\\mathbb{R}^n$) and $K_1(x)$ is supported on $|x|\\le \\frac 2\\beta$, we have\n\\begin{equation}\n\\widehat{K_1}(y)=\\int_{\\R^n} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x=\\int_{|x|\\leq\\frac{2}{\\beta}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x\n\\end{equation}\nIf $|y|<\\frac{\\beta}{2}$,  it is direct to estimate\n\\begin{equation}\\label{s-1}\n\\begin{split}\n\\big|\\widehat{K_1}(y)\\big|&\\leq C|y|\\int_{|x|\\leq \\frac 2\\beta}|x|\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n \\leq \\frac{2^\\beta}{\\beta+1}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nThen there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K1-F1}\n\\big|\\widehat{K_1}(y)\\big|\\leq C,\\,\\,  0<\\beta<n,\\ \\ |y|< \\frac{\\beta}{2}.\n\\end{equation}\nFor $\\frac{\\beta}{2}\\leq |y|\\leq\\beta,$ we rewrite $\\widehat{K_1}(y)$ as\n\\begin{equation*}\\begin{split}\n\\widehat{K_1}(y)&=\\int_{|x|<\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\\\&=\\int_{|x|<\\frac{1}{|y|}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\end{split}\n\\end{equation*}\nSimilar to \\eqref{s-1}, it deduces\n\\begin{equation*}\\begin{split}\n\\Big|\\int_{|x|<\\frac{1}{|y|}}\\big (e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x\\Big|&\\leq C|y|\\int_{|x|<\\frac{1}{|y|}}|x|\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq \\frac{1}{\\beta+1}\\frac{1}{|y|^\\beta} \\leq \\frac{2^\\beta}{\\beta+1}\\beta^{-\\beta}\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\Big|\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\\Big|&\\leq \\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n \\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation*}\nConsequently, there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K1-F2}\n\\big|\\widehat{K_1}(y)\\big|\\leq C\\left(\\frac{2^\\beta}{\\beta+1}\\beta^{-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}\\right)\\le C, \\,\\, 0<\\beta<n, \\,\\,\\frac{\\beta}{2}\\le |y|\\le\\beta.\n\\end{equation}\nAs for $ |y|>\\beta,$ $\\widehat{K_1}(y)$ can be divided into\n\\begin{equation}\\label{K1-F31}\n\\begin{split}\n\\widehat{K_1}(y)&=\\int_{|x|<\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\\\&=\\int_{|x|<\\frac{1}{|y|}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x.\n\\end{split}\n\\end{equation}\nFor the first term on the right hand of the above equality, we easily find that\n\\begin{equation}\\label{K1-F311}\n\\begin{split}\n\\Big|\\int_{|x|<\\frac{1}{|y|}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x\\Big|&\\leq C|y|\\int_{|x|<\\frac{1}{|y|}}|x|\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq \\frac{1}{\\beta+1}\\frac{1}{|y|^\\beta} \\leq \\frac{1}{\\beta+1}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nFor the second term,\nwe  choose $z=\\frac{y}{2|y|^2}$ with $|z|=\\frac{1}{2|y|}<\\frac{1}{2\\beta}$ such that $e^{2\\pi iy\\cdot z}=-1$\nand\n\\begin{equation*}\n\\int_{\\R^n} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x=\\frac12\\int_{\\R^n} e^{2\\pi ix\\cdot y}\\big(K_1(x)-K_1(x-z)\\big)\\,\\mathrm{d}x,\n\\end{equation*}\nmoreover, we have\n\\begin{equation}\\label{K1-F321}\n\\begin{split}\n\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x=&\\frac12\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}\\big(K_1(x)-K_1(x-z)\\big)\\,\\mathrm{d}x\n\\\\&-\\frac12\\int_{\\frac{1}{|y|}\\leq|x+z|,~ |x|\\leq\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\\\&+\\frac12\\int_{|x+z|\\leq\\frac{1}{|y|},~ |x|\\geq\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\\\\\n&+\\frac12\\int_{|x+z|\\ge\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x) \\,\\mathrm{d}x\n\\\\:= &I+J+K+L.\n\\end{split}\n\\end{equation}\nTo estimate the term $I$, we see that\n\\begin{equation}\\label{F321-I}\n\\begin{split}\nI&=\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\chi_\\beta(x)-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\geq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\chi_\\beta(x-z)\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta},~|x-z|\\geq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\chi_\\beta(x)-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\chi_\\beta(x-z)\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&:=I_1+I_2+I_3+I_4.\n\\end{split}\n\\end{equation}\nWe first estimate $I_2.$ Thanks to $|x-z|\\geq|x|-|z|\\geq\\frac{1}{\\beta}-\\frac{1}{2|y|}\\geq\\frac{1}{2\\beta},$ one has\n\\begin{equation}\\label{I-2}\n\\begin{split}\n|I_2|&\\leq\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{2\\beta}\\leq|x-z|\\leq\\frac{1}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}+C\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nThen, thanks to  $|x|=|x-z+z|\\geq|x-z|-|z|\\geq\\frac{1}{\\beta}-\\frac{1}{2|y|}\\geq\\frac{1}{2\\beta},$ $I_3$ can be estimated as follows:\n\\begin{equation}\\label{I-3}\n\\begin{split}\n|I_3|&\\leq\\int_{\\frac{1}{2\\beta}\\leq|x|\\leq\\frac{1}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{\\beta}\\leq|x-z|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{1 -2^{-\\beta}}{\\beta}\\beta^{-\\beta}+C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nThe term $I_4$ is directly estimated as\n\\begin{equation}\\label{I-4}\n\\begin{split}\n|I_4|&\\leq\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{\\beta}\\leq|x-z|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nNow we deal with $I_1.$  Note that\n\\begin{equation*}\n\\partial_i\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\Big)=\\frac{\\mathbf{e}_i}{|x|^{n+1-\\beta}}\n+(-n-1+\\beta)\\frac{xx_i}{|x|^{n+3-\\beta}},\\quad i=1,2,\\ldots,n.\n\\end{equation*}\nIn this case, since $|x-z|\\geq|x|-|z|\\geq2|z|-|z|\\geq|z|,$ by Taylor's expansion, one has\n\\begin{equation*}\\begin{split}\n&\\Big|\\frac{x-z}{|x-z|^{n+1-\\beta}}-\\frac{x}{|x|^{n+1-\\beta}}\\Big|\\\\ \\le&\n\\Big|\\sum_{i=1}^n\\Big(\\frac{z_i\\mathbf{e}_i}{|x-z|^{n+1-\\beta}}+\n(-n-1+\\beta)\\frac{(x-z)(x_i-z_i)z_i}{|x-z|^{n+3-\\beta}}\\Big)\\Big|+C\\sum_{k=2}^\\infty \\frac{|z|^k}{k!|x-z|^{n+k-\\beta}}.\n\\end{split}\n\\end{equation*}\nConsequently,\n\\begin{equation}\\label{I-1}\n\\begin{split}\n|I_1|\\leq &\\big(n+2-\\beta\\big)|z|\\int_{|z|\\leq|x-z|<\\frac{1}{\\beta}<\\infty}\\frac{1}{|x-z|^{n+1-\\beta}}\\,\\mathrm{d}x\\\\\n&+\nC\\sum_{k=2}^\\infty \\int_{|z|\\leq|x-z|<\\frac{1}{\\beta}<\\infty}\\frac{|z|^k}{k!|x-z|^{n+k-\\beta}}\\,\\mathrm{d}x\n\\\\ \\leq & C\\frac{|z|^\\beta}{1-\\beta}\\leq C\\frac{\\beta^{-\\beta}}{1-\\beta}.\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{I-2}-\\eqref{I-1} into \\eqref{F321-I} yields\n\\begin{equation}\\label{I-E}\n\\begin{split}\n|I|&=\\Big|\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\\Big|\\\\\n&\\le  C\\Big(\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}+\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}+\\frac{\\beta^{-\\beta}}{1-\\beta}\\Big)\n\\end{split}\n\\end{equation}\nfor some absolute constant $C>0$.\n\nConcerning the term $J$, thanks to $|x|\\geq|x+z|-|z|\\geq 2|z|-|z|\\geq |z|$, one has\n\\begin{equation}\\label{J-E}\n|J|\\leq\\int_{|z|\\leq|x|\\leq2|z|}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\leq\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}.\n\\end{equation}\nUtilizing $|x|\\leq|x+z|+|z|\\leq 2|z|+|z|\\leq 3|z|$, the term $K$ can be bounded by\n\\begin{equation}\\label{K-E}\n|K|\\leq\\int_{2|z|\\leq|x|\\leq3|z|}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\leq\\frac{(\\frac32)^{\\beta}-1}{\\beta}\\beta^{-\\beta}.\n\\end{equation}\nAs for the term $L$, the fact that $\\frac{2}{\\beta}\\ge |x|\\ge |x+z|-|z|\\ge \\frac{2}{\\beta}-\\frac{1}{2\\beta}=\\frac{3}{2\\beta}$ enables us to conclude\n\\begin{equation}\\label{L-E}\n\\begin{split}\n|L| \\leq\\int_{\\frac{3}{2\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n \\leq\\frac{1}{\\beta}\\left(\\Big(\\frac{3}{2\\beta}\\Big)^\\beta-\\Big(\\frac{2}{\\beta}\\Big)^\\beta\\right).\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{I-E}-\\eqref{L-E} into \\eqref{K1-F321}, we readily obtain that there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K12}\n\\Big|\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\\Big|\\le C.\n\\end{equation}\nIn view of \\eqref{K1-F311}, \\eqref{K12} and \\eqref{K1-F31}, there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K1-F3}\n\\big|\\widehat{K_1}(y)\\big|\\le C, \\,\\ \\ 0<\\beta<n,\\,\\, |y|>\\beta.\n\\end{equation}\nCombining  \\eqref{K1-F1}, \\eqref{K1-F2} with \\eqref{K1-F3}, we finish the proof of \\eqref{K1-F}. Applying \\eqref{K1-F}, one has\n\\begin{equation*}\\label{T1-1+}\\begin{split}\n&\\|T_1f\\|_{L^2(\\R^n)}=\\big\\|\\widehat{K_1}\\widehat{f}\\big\\|_{L^2(\\R^n)}\\le C\\big\\|\\widehat{f}\\big\\|_{L^2(\\R^n)}=C\\|f\\|_{L^2(\\R^n)},\\\\\n&\\|\\Lambda^sT_1f\\|_{L^2(\\R^n)}=\\big\\|\\widehat{K_1}\\widehat{\\Lambda^sf}\\big\\|_{L^2(\\R^n)}\\le C\\big\\|\\widehat{\\Lambda^sf}\\big\\|_{L^2(\\R^n)}=C\\|\\Lambda^sf\\|_{L^2(\\R^n)}\n\\end{split}\n\\end{equation*}\nfor any $0<s<n$. Hence \\eqref{T1-1} is proved and the proof of the lemma is complete.\n\\end{proof}\n\\section{Estimates via Besov Spaces}\\label{BesovE}\n\\setcounter{section}{4}\\setcounter{equation}{0}\nIn  this section, we will establish two key estimates concerning $$\\overline{u}_I=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega},\\qquad 0<\\alpha<\\frac12$$ in nonhomogeneous Besov spaces (see the Appendix in the end of the paper) which will be needed in the proof of Theorem \\ref{th3}.\nThe first proposition is to deal with the product of two functions. The second  proposition is about a commutator estimate.\n\\begin{proposition}\\label{add-0}\nFor any  $s>0,$  there exists a constant $C$ depending only on $s$ and $\\alpha$ such that\n\t\\begin{equation*}\n\t\\big\\|\\overline{u}_I\\cdot \\nabla \\omega^{\\alpha_0}\\big\\|_{H^s(\\R^2)}\\leq C\\left(\\|\\overline{\\omega}\\|_{L^2(\\R^2)}\\|\\omega^{\\alpha_0}\\|_{H^{s+2\\alpha+1}(\\R^2)}\n+\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\omega^{\\alpha_0} \\|_{B^{1+2\\alpha}_{2,1}(\\R^2)}\\right).\n\t\\end{equation*}\n\\end{proposition}\n\\begin{remark}\nLet us point out that the positive constant $C$ is uniformly bounded as parameter $\\alpha$ goes to $\\frac12.$\n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{add-0}]\nIn view of the Bony decomposition, one write\n\\begin{equation*}\n\t\\overline{u}_I\\cdot \\nabla \\omega^{\\alpha_0}=\\sum_{i=1}^2\\left(T_{\\partial_{i} \\omega^{\\alpha_0}}\\overline{u}^i_I +T_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}+R\\big(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0}\\big)\\right),\n\\end{equation*} where\n$$T_{\\partial_i \\omega^{\\alpha_0}}\\overline{u}^i_I=\\displaystyle{\\sum_{q>0}} \\Delta_q\\overline{u}^i_IS_{q-1}\\partial_i \\omega^{\\alpha_0}, \\quad\nT_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}=\\displaystyle{\\sum_{q>0}}S_{q-1} \\overline{u}^i_I\\Delta_q\\partial_i \\omega^{\\alpha_0},$$\n$$R\\big(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0}\\big)=\\displaystyle{\\sum_{q\\geq-1}}\\Delta_{q}\\overline{u}^i_I \\widetilde{\\Delta}_{q}\\partial_i \\omega^{\\alpha_0}.$$\nAccording to the H\\\"{o}lder inequality and  Lemma \\ref{B}, we obtain that for $q>0,$\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_q\\overline{u}^i_I S_{q-1}\\partial_i \\omega^{\\alpha_0}\\big\\|_{L^2}&\\leq 2^{qs}\\big\\|S_{q-1}\\nabla \\omega^{\\alpha_0} \\big\\|_{L^\\infty}\\big\\|\\Delta_q\\overline{u}_I\\big\\|_{L^2}\n\\\\&\\leq 2^{qs}\\sum_{-1\\leq k\\leq q-2}\\|\\Delta_{k}\\nabla \\omega^{\\alpha_0} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq2^{qs}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\sum_{-1\\leq k\\leq q-2}2^{(q-k)(2\\alpha-1)}2^{k(1+2\\alpha)}\\|\\Delta_{k} \\omega^{\\alpha_0} \\|_{L^2}.\n\\end{split}\n\\end{equation*}\nTherefore, Lemma \\ref{annulus} and the Young inequality for series yields\n\\begin{equation}\\label{p-1}\n\\begin{split}\n\\norm{T_{\\nabla \\omega^{\\alpha_0}}\\overline{u}_I}_{H^s}&\\leq C_s\\norm{\\big\\{2^{qs}\\big\\|S_{q-1}\\partial_i \\omega^{\\alpha_0} \\Delta_q\\overline{u}^i_I\\big\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C_s2^{2(2\\alpha-1)}\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0} \\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation}\nSimilarly, for $0<\\epsilon<2\\alpha,$\n\\begin{align*}\n2^{qs}\\big\\|S_{q-1} \\overline{u}^i_I\\Delta_q\\partial_i \\omega^{\\alpha_0}\\big\\|_{L^2}&\\leq2^{qs}\\|S_{q-1}\\overline{u}_I \\|_{L^\\infty}\\|\\Delta_q\\nabla \\omega^{\\alpha_0}\\|_{L^2}\n\\\\&\\leq\\sum_{-1\\leq k\\leq q-2}\\|\\Delta_{k} \\overline{u}_I \\|_{L^\\infty}\\|\\Delta_q \\omega^{\\alpha_0}\\|_{L^2}2^{q(s+1)}\n\\\\&\\leq C\\|\\Lambda^{-(1-2\\alpha+\\epsilon)}\\overline{\\omega}\\|_{L^{\\frac{2}{2\\alpha-\\epsilon}}}\\sum_{k\\leq q-2}2^{2k\\alpha}2^{q(s+1)}\\|\\Delta_{q} \\omega^{\\alpha_0} \\|_{L^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2}2^{q(s+1+2\\alpha)}\\|\\Delta_{q} \\omega^{\\alpha_0} \\|_{L^2}\\sum_{k\\leq q-2} 2^{2\\alpha(k-q)},\n\\end{align*}\nwhere Lemma \\ref{Hardy} has been used in the last inequality. In addition, when $\\alpha\\rightarrow\\frac12,$ the constant $C$ is uniformly bounded.\nHence, by Lemma \\ref{annulus}, we get\n\\begin{equation}\\label{p-2}\n\\begin{split}\n\\big\\|T_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}\\big\\|_{H^s}&\\leq C_s\\norm{\\big\\{2^{qs}\\|S_{q-1}\\overline{u}_I \\Delta_q\\nabla \\omega^{\\alpha_0}\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2} \\|\\omega^{\\alpha_0} \\|_{H^{s+1+2\\alpha}}.\n\\end{split}\n\\end{equation}\n\nFor the reminder term, we see that\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_{q}R(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0})\\big\\|_{L^2}&\\leq\n\\sum_{q\\leq q'+N_0}2^{qs}\\big\\|\\Delta_{q}(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\omega^{\\alpha_0})\\big\\|_{L^2}\n\\\\&\\leq\\sum_{q\\leq q'+N_0}2^{qs}\\big\\|\\Delta_{q'}\\overline{u}_I\\big\\|_{L^\\infty} \\big\\|\\widetilde{\\Delta}_{q'}\\nabla \\omega^{\\alpha_0}\\big\\|_{L^2}\n\\\\&\\leq C\\big\\|\\Lambda^{-(1-2\\alpha+\\epsilon)}\\overline{\\omega}\\big\\|_{L^{\\frac{2}{2\\alpha-\\epsilon}}}\\sum_{q\\leq q'+N_0}2^{qs}\n2^{(1+2\\alpha)q'}\\big\\|\\widetilde{\\Delta}_{q'}\\omega^{\\alpha_0}\\big\\|_{L^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^{2}}\\sum_{q\\leq q'+N_0}2^{(q-q')s}\n2^{(s+1+2\\alpha)q'}\\|\\widetilde{\\Delta}_{q'}\\omega^{\\alpha_0}\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nThis ensures that by Lemma \\ref{B}, for $s>0,$\n\\begin{equation}\\label{p-3}\n\\begin{split}\n\\big\\|R(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0})\\big\\|_{H^s}&\\leq\\big\\|\\big\\{2^{qs}\\|\\Delta_{q}R(\\overline{u}^i_I, \\nabla \\omega^{\\alpha_0})\\|_{L^2}\\big\\}_{q\\geq-1}\\big\\|_{\\ell^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2} \\big\\|\\omega^{\\alpha_0}\\big \\|_{H^{s+1+2\\alpha}}.\n\\end{split}\n\\end{equation}\nCollecting   \\eqref{p-1}, \\eqref{p-2} and \\eqref{p-3} above gives the proof of this proposition.\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{add1}\nFor any $s>0,$  there exists a  constant $C$ depending only on $s$ such that,\n\t\\begin{equation}\\begin{split}\\label{add-2}\n\t&\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2(\\R^2)}\\\\ \\leq& C(\\|\\overline{\\omega}\\|_{H^{2\\alpha}(\\R^2)}^2\n+\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\overline{\\omega}\\|_{H^{2\\alpha+1}(\\R^2)}+\n\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\overline{\\omega} \\|_{B^{1+2\\alpha}_{2,1}(\\R^2)}).\n\t\\end{split}\\end{equation}\nIn particular, if $s>2,$ then we have\n\\begin{equation}\\label{c-add}\n\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2(\\R^2)}\\leq C\\|\\overline{\\omega}\\|_{H^{s}(\\R^2)}^2.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWith the help of Bony's decomposition, one writes\n\\begin{equation*}\\begin{split}\n&J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\\\=&\n\\sum_{i=1}^2\\Big([J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}+J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I)-T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I+J^s\\big(R(\\overline{u}^i_I,\\partial_i \\overline{\\omega})\\big)\n-R(\\overline{u}^i_I,J^s\\partial_i \\overline{\\omega})\\Big).\n\\end{split}\n\\end{equation*}\nThe last two terms can be further decomposed into three parts\n\\begin{equation*}\\begin{split}\n&J^s\\big(R(\\overline{u}^i_I,\\partial_i \\overline{\\omega})\\big)-R(\\overline{u}^i_I,J^s\\partial_i \\overline{\\omega})\n\\\\=&\\sum_{q'\\geq0}J^s(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega})-\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}+[J^s, {\\Delta}_{-1}\\overline{u}^i_I \\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}.\n\\end{split}\n\\end{equation*}\nNext, we are going to establish the standard inner $L^2$-norm of the six terms above one by one.\n\n\\textbf{Bounds for the term $[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}$}.\nBy virtue of Proposition \\ref{Dy}, we can rewrite  $[J^s, T_{\\overline{u}^i_I}\\cdot \\partial_i]$ as a convolution operator. Indeed,\n\\begin{equation*}\\begin{split}\n [J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\n =&\\sum_{q>0}[J^s\\widetilde{\\Delta}_q, S_{q-1}{u}^i_I \\partial_i]\\Delta_q\\overline{\\omega}\n\\\\=&\\sum_{q>0}\\int_{\\R^2} 2^{2q}G_s(2^qy)\\big(S_{q-1}\\overline{u}^i_I(x-y)-S_{q-1}\\overline{u}^i_I(x)\\big)\\Delta_q\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}y\n,\\end{split}\n\\end{equation*}\nwhere $G_s$ is the inverse Fourier transform of $\\xi\\mapsto \\langle2^{q}\\xi\\rangle^{s}\\varphi(\\xi)$.\n\nFrom the first order Taylor formula, we deduce that\n\\begin{equation*}\\begin{split}\n\\big|[J^s, T_{\\overline{u}^i_I} \\partial_i]\\overline{\\omega}\\big|&\\leq\n\\sum_{q>0}\\int_{\\R^2}\\int_{0}^{1} 2^{2q}|G_s(2^qy)y|\n|\\nabla S_{q-1}\\overline{u}^i_I(x-\\tau y)|\\big|\\Delta_q\\partial_i\\overline{\\omega}(x-y)\\big|\\,\\mathrm{d}\\tau\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nNow, taking the $L^2$ norm of the above inequality, using the fact that $L^2\\sim B^0_{2,2}$, and using Lemma \\ref{annulus}, we get\n\\begin{equation*}\\begin{split}\n&\\big\\|[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\\big\\|_{L^2}\n\\\\ \\leq&\\Big(\\sum_{q>0}\\Big\\|\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n|\\nabla S_{q-1}\\overline{u}_I(\\cdot-\\tau y)|\n|\\Delta_q\\nabla\\overline{\\omega}(\\cdot-y)|\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}^2\\Big)^{\\frac12}.\n\\end{split}\n\\end{equation*}\nAdopting to  the fact that the\nnorm of an integral is less than the integral of the norm and using H\\\"{o}lder's\ninequality yield\n\\begin{equation*}\\begin{split}\n&\\Big\\|\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n\\big|\\nabla S_{q-1}\\overline{u}_I(x-\\tau y)\\big|\n\\big|\\Delta_q\\nabla\\overline{\\omega}(x-y)\\big|\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}\\\\ \\leq&\n\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n\\|\\nabla S_{q-1}\\overline{u}_I(\\cdot-\\tau y)\\|_{L^\\infty}\n\\|\\Delta_q\\nabla\\overline{\\omega}(\\cdot-y)\\|_{L^2}\\,\\mathrm{d}\\tau\\mathrm{d}y\\\\ \\leq&\n2^{qs}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}\\|\\Delta_q\\overline{\\omega}\\|_{L^2},\n\\end{split}\n\\end{equation*}where the translation invariance of the Lebesgue measure is used in the last inequality.\n\nHence, the H\\\"{o}lder inequality and Bernstein's inequality enable us to conclude that\n\\begin{equation*}\\begin{split}\n\\big\\|[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\\big\\|_{L^2}&\\leq\n\\Big(\\sum_{q>0}2^{2qs}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}^2\\|\\Delta_q\\overline{\\omega}\\|_{L^2}^2\\Big)^{\\frac12}\n\\\\&\\leq \\sup_{q>0}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}\\|\\overline{\\omega}\\|_{H^s}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{Bounds for $J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I).$} By virtue of the H\\\"{o}lder inequality and  Bernstein's inequality, we get\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_q\\overline{u}_I\\cdot S_{q-1}\\partial_i \\overline{\\omega}\\big\\|_{L^2}&\\leq 2^{qs}\\|S_{q-1}\\partial_i \\overline{\\omega} \\|_{L^\\infty}\\|\\Delta_q\\overline{u}^i_I\\|_{L^2}\n\\\\&\\leq 2^{qs}\\sum_{k\\leq q-2}\\|\\Delta_{k}\\nabla \\overline{\\omega} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq2^{qs}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\sum_{k\\leq q-2}2^{(q-k)(2\\alpha-1)}2^{k(1+2\\alpha)}\\|\\Delta_{k} \\overline{\\omega}\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nHence, we have by Lemma \\ref{annulus} that\n\\begin{equation*}\\begin{split}\n\\big\\|J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I)\\big\\|_{L^2}&\\leq\nC_s\\norm{\\big\\{2^{qs}\\|\\Delta_q\\overline{u}^i_I S_{q-1}\\partial_i \\overline{\\omega}\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C_s2^{2(2\\alpha-1)}\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{A similar bound holds for both terms $T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I$}, $\\displaystyle\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\cdot\\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}$. By the H\\\"older inequality, one has\n\\begin{equation*}\n\\begin{split}\n\\big\\|\\Delta_q\\overline{u}^i_I  S_{q-1}\\partial_i J^s\\overline{\\omega}\\big\\|_{L^2}&\\leq \\|S_{q-1}\\nabla J^s\\overline{\\omega} \\|_{L^\\infty}\\|\\Delta_q\\overline{u}_I\\|_{L^2}\n\\\\&\\leq \\sum_{k\\leq q-2}\\|\\Delta_{k}\\nabla J^s\\overline{\\omega} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq 2^{q(1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\|J^s\\overline{\\omega}\\|_{L^2}\\sum_{k\\leq q-2}2^{2(k-q)},\n\\end{split}\n\\end{equation*}\nfrom which it follows that\n\\begin{equation*}\n\\big\\|T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I\\big\\|_{L^2} \\leq C\\|\\overline{\\omega}\\|_{H^s} \\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{equation*}\nFor the term $\\displaystyle\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}$, by the H\\\"older inequality, we immediately obtain\n\\begin{equation*}\n\\begin{split}\n\\Big\\|\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}\\Big\\|_{L^2}&\\leq\n\\sum_{q'\\geq 0}\\big\\|\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i  J^s\\overline{\\omega}\\big\\|_{L^2}\n\\\\&\\leq\\sum_{q'\\geq 0}\\|\\Delta_{q'}\\overline{u}_I\\|_{L^2}\\big\\|\\widetilde{\\Delta}_{q'}\\nabla  J^s\\overline{\\omega}\\big\\|_{L^\\infty}\n\\\\&\\leq\\|J^s\\overline{\\omega}\\|_{L^2}\\sum_{q'\\geq 0}\\|\\Delta_{q'}\\overline{\\omega}\\|_{L^2}\n2^{q'(1+2\\alpha)}\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{Bounds for the term $\\displaystyle\\sum_{q'\\geq0}J^s\\big(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega}\\big)$}. Utilizing again the H\\\"{o}lder inequality and  Bernstein's inequality gives\n\\begin{align*}\n\\Big\\|\\sum_{q'\\geq0}J^s\\big(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega}\\big)\\Big\\|_{L^2}&\\leq\n\\sum_{q'\\geq0}\\Big(\\sum_{q\\leq q'+N_0}2^{2qs}\\big\\|\\Delta_{q}(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega})\\big\\|_{L^2}^2\\Big)^\\frac12\n\\\\&\\leq\\sum_{q'\\geq0}\\Big(\\sum_{q\\leq q'+N_0}2^{2qs}\\Big)^\\frac12\\|\\Delta_{q'}\\overline{u}_I\\|_{L^2} \\big\\|\\widetilde{\\Delta}_{q'}\\nabla \\overline{\\omega}\\big\\|_{L^\\infty}\n\\\\&\\leq\\sum_{q'\\geq0}2^{q's}\\|\\Delta_{q'}\\overline{\\omega}\\|_{L^2}\n2^{q'(1+2\\alpha)}\\|\\widetilde{\\Delta}_{q'}\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{H^{1+2\\alpha}}.\n\\end{align*}\n\n\n\n\n\\textbf{Bounds  for the last term $[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}$}.  Adopting to the  similar method to estimate $[J^s, T_{\\overline{u}^i_I} \\partial_i]\\overline{\\omega}$, we get\n\\begin{align*}\n&[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\n\\\\=&\\sum_{|q+1|\\leq2}[J^s\\Delta_q, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\n\\\\=&\\sum_{|q+1|\\leq2}\\int_{\\R^2} 2^{2q}G_s(2^qy)\\left({\\Delta}_{-1}\\overline{u}^i_I(x-y)-{\\Delta}_{-1}\\overline{u}^i_I(x)\\right)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}y\n\\\\=&\\sum_{|q+1|\\leq2}\\int_{\\R^2}\\int_{0}^{1} 2^{2q}G_s(2^qy)\\big(y\\cdot\\nabla  {\\Delta}_{-1}\\overline{u}^i_I(x-\\tau y)\\big)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}\\tau\\mathrm{d}y.\n\\end{align*}\nBased on this, the Minkowski inequality and the H\\\"older inequality allow us to infer that\n\\begin{equation*}\\begin{split}\n&\\big\\|[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\\big\\|_{L^2}\n\\\\ \\leq&\\Big(\\sum_{|q+1|\\leq2}\\Big\\|\\int_{\\R^2}\\int_{0}^{1} 2^{2q}G_s(2^qy)\\big(y\\cdot\\nabla  {\\Delta}_{-1}\\overline{u}^i_I(x-\\tau y)\\big)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}^2\\Big)^{\\frac12}\n\\\\ \\leq&C_s\\|{\\Delta}_{-1}\\nabla\\overline{u}_I\\|_{L^\\infty}\n\\big\\|\\widetilde{\\Delta}_{-1}\\nabla\\overline{\\omega}\\big\\|_{L^2}\n  \\leq C_s\\big\\|\\Lambda^{2\\alpha}\\overline{\\omega}\\big\\|_{L^2}\\|\\overline{\\omega}\\|_{L^2}\\leq\nC_s\\|\\overline{\\omega}\\|_{H^{2\\alpha}}^2.\n\\end{split}\n\\end{equation*}\nCombining these estimates above yields \\eqref{add-2}. This ends the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Proof of main Theorems }\\label{sec5}\n\\setcounter{section}{5}\\setcounter{equation}{0}\n\n This section is devoted to showing the main theorems. Let us begin by  proving Theorem~\\ref{th1}.\n \\subsection{Proof of Theorem \\ref{th1}}\n\nFirst of all, let us denote\n$$\\overline{\\omega}=\\omega^{\\alpha}-\\omega^{\\alpha_{0}}\\quad\\text{and}\\quad\\overline{u}=u^{\\alpha}-u^{\\alpha_{0}}.$$\nThen,  the couple $(\\overline{\\omega},\\,\\overline{u})$ satisfies\n\\begin{equation}\\label{diffrence}\n\\overline{\\omega}_{t}+u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega}+\\overline{u}\\cdot\\nabla \\overline{\\omega}+\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}} =0.\n\\end{equation}\nOperating $J^s$ on \\eqref{diffrence} and taking the scalar product of the resulting\nequation with $J^s\\overline{\\omega}$ in $L^2,$ we get\n\\begin{equation}\\label{11-1}\\begin{split}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}^2=&-\\int_{\\R^2} J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\\n&-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\:=&\nI_1+I_2+I_3.\n\\end{split}\\end{equation}\n\nWe are going to estimate the three terms on the right hand side of \\eqref{11-1} one by one. By the divergence-free condition and \\eqref{com-2}, we have\n\\begin{equation*}\\begin{split}\nI_1=&-\\int_{\\R^2} (J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega})\nJ^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\n\\big\\|J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\big\\|_{L^2}\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\Big(\\|J^su^{\\alpha_{0}}\\|_{L^\\frac{1}{\\alpha_0}}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha_0}}+\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\Big)\n\\|J^s\\overline{\\omega}\\|_{L^2}.\n\\end{split}\\end{equation*}\nNote that $0<\\alpha_0<\\frac12$ and\n$$\nu^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}.\n$$\nUsing Lemma \\ref{Hardy} yields\n$$\\|J^su^{\\alpha_{0}}\\|_{L^\\frac{1}{\\alpha_0}}\\leq C\\norm{\\omega^{\\alpha_0}}_{H^s}.$$\nApplying Lemma \\ref{embedding1} and Lemma \\ref{Hardy} gives\n$$\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha_0}}\\leq C\\|\\overline{\\omega}\\|_{H^s},$$\n\\begin{equation*}\\begin{split}\n\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\leq \\norm{u^{\\alpha_{0}}}_{W^{s,\\frac{1}{\\alpha_0}}}\\leq C\\norm{\\omega^{\\alpha_0}}_{H^{s}}.\n\\end{split}\n\\end{equation*}\nThus, \\begin{equation}\\label{wsp-1}\\begin{split}\n|I_1|\\leq C\\|\\overline{\\omega}\\|_{H^s}^2\\norm{\\omega^{\\alpha_0}}_{H^{s}}.\n\\end{split}\\end{equation}\nBy the decomposition \\eqref{diff-1-1}, the second term can be written as\n\\begin{equation*}\\begin{split}\n I_2=&-\\int_{\\R^2}(J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})-\\overline{u}\\cdot\\nabla J^s\\overline{\\omega})\n J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\=&\n-\\int_{\\R^2}(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega})\nJ^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2}\\big( J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)\nJ^s\\overline{\\omega}\\,\\mathrm{d}x.\n\\end{split}\n\\end{equation*}\nUsing  \\eqref{com-2} and the Sobolev embedding inequalities, we obtain\n\\begin{equation*}\\begin{split}\n&-\\int_{\\R^2}\\big(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\\big(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha}}+\\|\\nabla \\overline{u}_I\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}.\n\\end{split}\n\\end{equation*}\nSimilarly,  for $p>\\frac1\\alpha>2,$\n\\begin{equation*}\\begin{split}\n&-\\int_{\\R^2}\\big(J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\\big(\\|J^s\\overline{u}_{II}\\|_{L^p}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2p}{p-2}}+\n\\|\\nabla \\overline{u}_{II}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\n\\end{equation*}\nTherefore,\n\\begin{equation}\\label{wsp-3}\n|I_2|\\leq\n\\|\\overline{\\omega}\\|_{H^s}^2(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p}).\n\\end{equation}\nConcerning the third term, we use  \\eqref{com-1} and the Sobolev embedding inequalities to get\n\\begin{equation}\\label{wsp-4}\\begin{split}\n|I_3|=&\\Big|\\int J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\\n=&\n\\Big|\\int J^s(\\overline{u}_{I}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x+\\int J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\n\\\\ \\leq&\n\\|\\overline{\\omega}\\|_{H^s}(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p})\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}.\n\\end{split}\\end{equation}\nPlugging  these estimates \\eqref{wsp-1},    \\eqref{wsp-3},  \\eqref{wsp-4} into \\eqref{11-1} yields\n\\begin{equation}\\label{Hs-est}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}  \\leq&\n\\|\\overline{\\omega}\\|_{H^s}\\big(\\|\\omega^{\\alpha_{0}}\\|_{H^{s}}\n+\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p}\\big)\\\\\n&+\n\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\big(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\n\\|J^s\\overline{u}_{II}\\|_{L^p}\\big)\\\\\n:=&\\tilde {I_1}+\\tilde {I_2}.\n\\end{split}\n\\end{equation}\nThe integral form of $\\overline{u}_I$ can be written as\n\\begin{equation*}\nJ^s\\overline{u}_I(x)=\\int_{\\R^2}\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\nJ^s\\overline{\\omega}(y)\\,\\mathrm{d}y.\n\\end{equation*}\nThen, using  Lemma \\ref{Hardy} enables us to get\n\\begin{equation}\\label{4-16-1}\n\\begin{split}\n\\|J^s\\overline{u}_I\\|_{L^{\\frac1\\alpha}}&\\leq\\Big\\|\\int_{\\R^2}\\frac{1}{|x-y|^{2-(1-2\\alpha)}}\n|J^s\\overline{\\omega}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\&\\leq\nC(\\alpha)\\|J^s\\overline{\\omega}\\|_{L^{2}},\n\\end{split}\\end{equation}\nwhere $C(\\alpha)$ depends on $\\alpha$ and will be bounded if $0\\le \\alpha<\\alpha_0<\\frac12$ (but will be unbounded if $\\alpha$ tend to $\\frac12$).\nWhen $p>\\frac1\\alpha>2,$ adopting to the similar way to \\eqref{4-16-1} gives\n\\begin{equation}\\label{4-16-2}\\begin{split}\n\\|J^s\\overline{u}_{II}\\|_{L^p}&\\leq C(\\alpha)\\big(\\|J^s\\omega^{\\alpha_0}\\|_{L^{\\frac{2p}{2+p(1-2\\alpha)}}}+\n \\|J^s\\omega^{\\alpha_0}\\|_{L^{\\frac{2p}{2+p(1-2\\alpha_0)}}}\\big)\n \\\\&\\leq C(\\alpha)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\\end{equation}\nHence,\n\\begin{equation}\\label{Hs-1}\n\\tilde {I_1} \\leq  C\\|\\overline{\\omega}\\|_{H^s}^{2}+C\\|\\overline{\\omega}\\|_{H^{s}}\n\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{equation}\nOn the other hand, the estimate \\eqref{4-16-2} is not adaptable to $\\tilde {I_2}$ in \\eqref{Hs-est}. We will use a different way to estimate\n$\\|J^s\\overline{u}_{II}\\|_{L^p}$ in \\eqref{Hs-est}.\nFor $0<\\epsilon<1$ to be determined later, we write  $\\overline{u}_{II}$ as\n\\begin{equation*}\\begin{split}\n \\overline{u}_{II}=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y.\n\\end{split}\\end{equation*}\nTherefore,\n\\begin{equation}\\label{11-6-0}\\begin{split}\n J^s\\overline{u}_{II}&=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=\\Big(\\int_{|x-y|\\leq\\epsilon}+\\int_{1>|x-y|\\geq\\epsilon}+\\int_{|x-y|\\geq1}\\Big)\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&:=H_{1}+H_{2}+H_{3}.\n\\end{split}\\end{equation}\nFor the first term $H_1,$ using the Young inequality and the Sobolev embedding, we get\n\\begin{equation*}\\begin{split}\n\\|H_1\\|_{L^{p}}&= \\Big\\|\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^{p}}\n\\\\&\\leq C\\Big(\\frac{\\epsilon^{1-2\\alpha}}{1-2\\alpha}+\\frac{\\epsilon^{1-2\\alpha_0}}{1-2\\alpha_0}\\Big)\n\\|J^s\\omega^{\\alpha_0}\\|_{L^{p}}\n\\\\&\\leq C\\Big(\\frac{\\epsilon^{1-2\\alpha}}{1-2\\alpha}\n+\\frac{\\epsilon^{1-2\\alpha_0}}{1-2\\alpha_0}\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nAs for $H_2$ and $H_3$, it is divided into two cases.\n\n{\\it Case 1: $\\alpha_0>\\alpha.$} By the mean value theorem, we can obtain\n\\begin{equation*}\\begin{split}\nH_2\\leq|\\alpha_0-\\alpha|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nUtilizing the Young inequality yields\n\\begin{equation*}\\begin{split}\n\\|H_2\\|_{L^p}&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\\,\\mathrm{d}y,\n\\\\&\\leq\\frac{C}{1-2\\alpha_0}|\\alpha_0-\\alpha||\\log\\epsilon|\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\n  To deal with $H_3$, we fix a small number $\\sigma>0$ such that $p>\\frac{2}{2\\alpha-\\sigma}>2.$ Thanks to the fact that $\\log |x-y|\\le C|x-y|^\\sigma$ for any $\\sigma>0$ and $|x-y|\\ge 1$, we have\n\\begin{equation*}\\begin{split}\n\\|H_3\\|_{L^p}&\\leq |\\alpha_0-\\alpha|\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{1}{|x-y|^{1+2\\alpha-\\sigma}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}where $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, then we can choose some $r>2$ such that\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}holds (see Lemma \\ref{embedding1}).\n\n{\\it Case 2: $\\alpha_0<\\alpha<\\frac12.$} It is similar to {\\it Case 1} by exchanging the position of $\\alpha$ and $\\alpha_0$. For instance, by the mean value theorem, we can obtain\n\\begin{equation*}\\begin{split}\nH_2\\leq|\\alpha_0-\\alpha|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nThen\n\\begin{equation*}\\begin{split}\n\\|H_2\\|_{L^p}&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\\,\\mathrm{d}y,\n\\\\&\\leq\\frac{C}{1-2\\alpha}|\\alpha_0-\\alpha||\\log\\epsilon|\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nNow  we fix a small number $\\sigma>0$ such that $p>\\frac{2}{2\\alpha_0-\\sigma}>2.$ Similarly, we have\n\\begin{equation*}\\begin{split}\n\\|H_3\\|_{L^p}\n&\\leq |\\alpha_0-\\alpha|\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\n\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha_0-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}where $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, then we can choose some $r>2$ such that\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}holds (see Lemma \\ref{embedding1}).\n\nAs a consequence, we get\n\\begin{equation*}\n\\|J^s\\overline{u}_{II}(t)\\|_{L^p}\\leq C\\left(\\epsilon^{1-2\\alpha}\n+\\epsilon^{1-2\\alpha_0}+|\\alpha_0-\\alpha|+|\\alpha_0-\\alpha||\\log\\epsilon|\\right)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{equation*}\nHence,\n\\begin{equation}\\label{Hs-2}\n\\begin{split}\n\\tilde {I_2}=&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\n\\|J^s\\overline{u}_{II}\\|_{L^p})\\\\ \\leq & C\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}\n+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\left(\\epsilon^{1-2\\alpha}+\\epsilon^{1-2\\alpha_0}\n+|\\alpha_0-\\alpha|+|\\alpha_0-\\alpha||\\log\\epsilon|\\right).\n\\end{split}\\end{equation}\nSet  $\\epsilon=\\alpha_0-\\alpha$.\nBy plugging \\eqref{Hs-1} and \\eqref{Hs-2} into \\eqref{Hs-est}, we get\n\\begin{equation}\\label{Hs-3}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}  \\leq&\nC\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}\n +C\\|\\overline{\\omega}\\|_{H^s}^{2}\\\\\n&+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log |\\alpha_0-\\alpha||\\right).\n\\end{split}\\end{equation}\nMultiply \\eqref{Hs-3} by $\\exp(-C\\int_0^t\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\,\\mathrm{d}s)$ and consider the quantity\n$$y(t)=\\|\\overline{\\omega}\\|_{H^s}\\exp\\Big(-C\\int_0^t\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\,\\mathrm{d}s\\Big).$$\nWe then get the inequality\n\\begin{equation*}\n\\frac{\\mathrm{d}y(t)}{\\mathrm{d}t}\\leq \\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}\\right)F(t)+Gy^2(t),\n\\end{equation*}\nwhere $$F(t)=\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\exp\\Big(-C\\int_0^t\\|\\omega^{\\alpha_0}(s)\\|_{H^{s+1}}\\,\\mathrm{d}s\\Big),$$\nand $$G=C\\exp\\Big(C\\int_0^T\\|\\omega^{\\alpha_0}(t)\\|_{H^{s+1}}\\,\\mathrm{d}t\\Big).$$\nBy Proposition  \\ref{o}, there exists a $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ such that when $0<\\alpha<\\frac12$ and $|\\alpha-\\alpha_0|<\\delta$,\n\\begin{equation*}\ny(t)\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}\\right)\\int_0^T F(t)\\,\\mathrm{d}t,\n\\end{equation*}\nwhich implies that\n\\begin{equation*}\n\\|\\overline{\\omega}\\|_{H^s}\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log|\\alpha_0-\\alpha||\\right).\n\\end{equation*}\nHere $C>0$ is a constant depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ as well.\n\nAssume that  $\\omega^{\\alpha_{0}}\\in C([0,T_0];H^{s+1})$, $s>2.$ According to the local well-posedness theory,  for $\\alpha<\\alpha_0,$ $\\omega^{\\alpha}\\in C([0,T];H^{s+1})$ for some $T>0$.\nIf $T\\geq T_0$, the proof is finished. If $T<T_0$, we are going  to prove that $T$ can be extended to $T_0$. Denote $T_{\\max}>0$ the maximal existence time satisfying $\\omega^{\\alpha}\\in C([0,T_{\\max});H^{s+1})$. By performing  the $(s+1)$-order energy estimate, we get\n\\begin{equation*}\\begin{split}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\omega^\\alpha(t)\\|_{H^{s+1}}^2\n&=-\\int_{\\R^2}\\Lambda^{s+1}\\big(u^{\\alpha}\\cdot\\nabla \\omega^\\alpha\\big)\\Lambda^{s+1}\\omega^\\alpha\\,\\mathrm{d}x\n\\\\&=-\\int_{\\R^2}\\Lambda^{s+1}\\big(u^{\\alpha}\\cdot\\nabla \\omega^\\alpha-u^{\\alpha}\\cdot\\Lambda^{s+1}\\nabla \\omega^\\alpha\\big)\\Lambda^{s+1}\\omega^\\alpha\\,\\mathrm{d}x\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}\\left(\\|\\Lambda^{s+1}u^\\alpha\\|_{L^{\\frac1\\alpha}}\\|\\nabla \\omega^\\alpha\\|_{L^{\\frac{2}{1-2\\alpha}}}\n+\\|\\nabla u^\\alpha\\|_{L^\\infty}\\|\\omega^\\alpha\\|_{H^{s+1}}\\right)\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}^2\\left(\\|\\nabla \\omega^\\alpha\\|_{L^{\\frac{2}{1-2\\alpha}}}+\\|\\nabla u^\\alpha\\|_{L^\\infty}\\right)\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}^2\\|\\omega^\\alpha\\|_{H^{s}}.\n\\end{split}\\end{equation*}\nBy the Gronwall inequality, we have\n\\begin{equation*}\\begin{split}\n\\|\\omega^\\alpha(t)\\|_{H^{s+1}}&\\leq e^{\\int_0^{T_{\\max}}\\|\\omega^\\alpha(t)\\|_{H^{s}}\\,\\mathrm{d}t}\\|\\omega^\\alpha_0\\|_{H^{s+1}}\n\\\\&\\leq e^{\\int_0^{T_{\\max}}(\\|\\overline{\\omega}(t)\\|_{H^{s}}+\\|\\omega^{\\alpha_0}(t)\\|_{H^{s}})\\,\\mathrm{d}t}\\|\\omega^\\alpha_0\\|_{H^{s+1}}\n\\leq C\n\\end{split}\\end{equation*}\nfor $t\\in [0,{T_{\\max}}]$ and hence $\\omega^\\alpha(T_{\\max})$ is finite. This deduces a contradiction with $T_{\\max}$ is the maximal existence time by using the local well-posedness theory. In consequence, $T=T_0$ as required and the proof of the theorem is finished.\n\n\n\\subsection{Proof of Theorem \\ref{th2}}\n\n\nTaking the scalar product of \\eqref{diffrence} with $\\overline{\\omega}$ in $H^s$ and using Lemma \\ref{commutator} and the Sobolev embedding inequalities enable us to get\n\\begin{equation}\\label{Hs-est-1-1}\\begin{split}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}&\\leq\n\\|\\overline{\\omega}\\|_{H^s}\\|u^{\\alpha_{0}}\\|_{H^{s}}+\n(\\|\\overline{\\omega}\\|_{H^s}+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}})\\|\\overline{u}\\|_{H^{s}}\\\\\n&\\leq\n\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_{0}}\\|_{H^{s}}+\n(\\|\\overline{\\omega}\\|_{H^s}+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}})(\\|\\overline{u}_{I}\\|_{H^{s}}+\\|\\overline{u}_{II}\\|_{H^{s}}).\n\\end{split}\\end{equation}\nHere we have used the decomposition \\eqref{diff-1-1} with\n\\begin{equation*}\n\\overline{u}_{I}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}, \\quad \\overline{u}_{II}=\\left(\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}-\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\right)\\omega^{\\alpha_0}.\n\\end{equation*}\nBy using Proposition \\ref{uni-est} (Remark \\ref{Rm2-1})  with $\\beta=1-2\\alpha$, we obtain\n\\begin{equation}\\label{u1-est}\n\\|\\overline{u}_{I}(t)\\|_{H^{s}}\\leq\nC\\big(\\|\\overline{\\omega}\\|_{H^s}+(1-2\\alpha)\\|\\overline{\\omega}\\|_{L^1}\\big),\n\\end{equation}where $C=C(\\alpha,s)$ is an absolutely constant when $\\alpha\\rightarrow\\frac12.$\nBy using Proposition \\ref{uni-est} again ($\\beta=1-2\\alpha$ and $\\beta=0$ respectively), there also exists a uniformly bounded constant $C=C(\\alpha,s)$ when $\\alpha\\rightarrow\\frac12$ such that\n \\begin{equation*}\\| \\overline{u}_{II}(t)\\|_{H^s}\\leq C\\big(\\|\\omega^{\\alpha_0}\\|_{H^s}+\\|\\omega^{\\alpha_0}\\|_{L^1}\\big).\n\\end{equation*}\nIt follows that \\begin{equation}\\label{equ-12}\n\\begin{split}\n\\|\\overline{\\omega}\\|_{H^s}(\\|\\overline{u}_{I}\\|_{H^{s}}+\\|\\overline{u}_{II}\\|_{H^{s}})\\leq C\\|\\overline{\\omega}\\|_{H^s}^2+\nC\\|\\overline{\\omega}\\|_{H^s}(\\|\\omega^{\\alpha_0}\\|_{H^s}+\n\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1}).\n\\end{split}\\end{equation}\n\nNow we adopt to anther way to estimate $\\|\\overline{u}_{II}\\|_{H^s}$ in order to deal with $\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|\\overline{u}_{II}\\|_{H^s}$ on the right hand side of \\eqref{Hs-est-1-1}. The decomposition \\eqref{11-6-0} will be applied, which is\n\\begin{equation}\\label{11-6}\\begin{split}\n J^s\\overline{u}_{II}&=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=\\Big(\\int_{|x-y|\\leq\\epsilon}+\\int_{1>|x-y|\\geq\\epsilon}+\\int_{|x-y|\\geq1}\\Big)\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=H_{1}+H_{2}+H_{3},\n\\end{split}\\end{equation}\nwhere  $0<\\epsilon<1$ is to be determined later.\n\nPerforming the fact that $$\\int_{|x|=1}\\frac{x^\\perp}{|x|^{2+2\\alpha}}\\,\\mathrm{d}s=\\int_{|x|=1}\\frac{x^\\perp}{|x|^{3}}\\,\\mathrm{d}s=0,$$\nwe get\n\\begin{equation*}\\begin{split}\nH_{1}&=\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}-\\frac{(x-y)^{\\perp}}{|x-y|^{3}}\\Big)\n\\Big(J^s\\omega^{\\alpha_0}(y)-J^s\\omega^{\\alpha_0}(x)\\Big)\\,\\mathrm{d}y\n\\\\&=\\int_{|z|\\leq\\epsilon}\\Big(\\frac{z^{\\perp}}{|z|^{2+2\\alpha}}-\\frac{z^{\\perp}}{|z|^{3}}\\Big)\\big(J^s\\omega^{\\alpha_0}(x-z)-J^s\\omega^{\\alpha_0}(x)\\big)\\,\\mathrm{d}z.\n\\end{split}\n\\end{equation*}\nFrom the mean value theorem, we deduce that\n\\begin{equation*}\\begin{split}\n|H_{1}|&\\leq\\int_0^1\\int_{|z|\\leq\\epsilon}\n\\Big(\\frac{1}{|z|^{2\\alpha}}+\\frac{1}{|z|}\\Big)\\big|\\nabla J^s\\omega^{\\alpha_0}(x-\\tau z)\\big|\\,\\mathrm{d}z\\mathrm{d}\\tau.\n\\end{split}\n\\end{equation*}\nNow, taking the $L^2$ norm of the above inequality, and using the fact that the norm of an integral is less that the integral of the norm, we get\n\\begin{equation*}\\begin{split}\n\\|H_{1}\\|_{L^2}&\\leq \\int_0^1\\int_{|z|\\leq\\epsilon}\n\\Big(\\frac{1}{|z|^{2\\alpha}}+\\frac{1}{|z|}\\Big)\\big\\|\\nabla J^s\\omega^{\\alpha_0}(\\cdot-\\tau z)\\big\\|_{L^2}\\,\\mathrm{d}z\\mathrm{d}\\tau.\n\\end{split}\n\\end{equation*}\nThe translation invariance of the Lebesgue measure then ensures that\n\\begin{equation}\\label{11-7}\\begin{split}\n\\|H_{1}\\|_{L^2}&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\big\\|J^{s+1}\\omega^{\\alpha_0}\\big\\|_{L^2}.\n\\end{split}\n\\end{equation}\nFor $2+2\\alpha\\leq\\xi\\leq3,$ we estimate $H_{2}$ as follows,\n\\begin{equation}\\label{11-8}\\begin{split}\n\\|H_{2}\\|_{L^2}&= \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{(x-y)^{\\perp}\\big(|x-y|^{\\xi}\\log|x-y|\\big)}{|x-y|^{2+2\\alpha}|x-y|^{2+2\\alpha_0}}\nJ^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^2}\n\\\\&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{2}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\\n&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|J^s\\omega^{\\alpha_0}\\|_{L^2},\n\\end{split}\n\\end{equation}\nwhere the Young inequality and the mean value theorem have been used.\nAdopting to the similar method to estimate $H_{2}$, we get\n\\begin{equation}\\label{11-9}\\begin{split}\n\\|H_{3}\\|_{L^2}&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\&\\leq(\\frac12-\\alpha)\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation} where $\\frac32=\\frac1p+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$ whence $p<\\frac{2}{2-2\\alpha+\\sigma}<2$.\nBy the Gagliardo-Nirenberg inequality, we have\n\\begin{equation*}\\begin{split}\n\\|\\Lambda^s\\omega^{\\alpha_0}\\|_{L^p}\\leq\\|\\omega^{\\alpha_0}\\|_{L^1}^{1-\\theta} \\|\\Lambda^{s+1}\\omega^{\\alpha_0}\\|_{L^2}^{\\theta},\n\\end{split}\n\\end{equation*}\nwhere $\\theta=1-\\frac{2}{p(s+2)},$ $\\frac1p\\leq\\frac{1-\\theta}{1}+\\frac{\\theta}{2}$. Then, we conclude that $p\\geq \\frac{2(s+1)}{s+2}.$\nThis enables us to choose some $\\frac{2(s+1)}{s+2}\\leq p<\\frac{2}{2-2\\alpha+\\sigma}.$\nCombining the estimates \\eqref{11-7}-\\eqref{11-9} with \\eqref{11-6}\nand choosing $\\epsilon=(\\frac12-\\alpha)$, we get\n\\begin{equation}\\label{u2-est}\\begin{split}\n\\|\\overline{u}_{II}(t)\\|_{H^{s}}\\leq CL(\\alpha)\\big(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}+\\|\\omega^{\\alpha_0}\\|_{L^1}\\big).\n\\end{split}\n\\end{equation}\nHere and what in follow,\n\\[L(\\alpha):=\\Big(\\frac12-\\alpha\\Big)^{2-2\\alpha}\n+\\Big(\\frac12-\\alpha\\Big)\\Big|\\log\\Big(\\frac12-\\alpha\\Big)\\Big|^2\n+\\Big(\\frac12-\\alpha\\Big). \\]\nHence, combining \\eqref{u1-est} and \\eqref{u2-est} yields\n\\begin{equation}\\label{equ-12-1}\\begin{split}\n&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{u}_{I}\\|_{H^{s}}\n+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{u}_{II}\\|_{H^{s}}\n\\\\ \\leq&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}+\n\\Big(\\frac12-\\alpha\\Big)\\|\\overline{\\omega}\\|_{L^1}\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n +CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\\|\\omega^{\\alpha_0}\\|_{L^1}^2)\n\\\\ \\leq&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n\\|\\overline{\\omega}\\|_{H^s}+CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\n\\|\\omega^{\\alpha_0}\\|_{L^1}^2+\\|\\omega^{\\alpha}\\|_{L^1}^2).\n\\end{split}\n\\end{equation}\nPlugging \\eqref{equ-12} and  \\eqref{equ-12-1} into \\eqref{Hs-est-1-1} gives\n\\begin{equation}\\label{Hs-est-s}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}\\leq&\n\\|\\overline{\\omega}\\|_{H^s}(\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n+\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1})+\\|\\overline{\\omega}\\|_{H^s}^2\n\\\\&+CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\n\\|\\omega^{\\alpha_0}\\|_{L^1}^2+\\|\\omega^{\\alpha}\\|_{L^1}^2).\n\\end{split}\n\\end{equation}\nThanks to the incompressible condition that $\\nabla\\cdot u=0$, it follows that  $\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1}$ is bounded if  the initial data $\\omega_0\\in L^1$. Arguing similarly as the last part in the proof of Theorem \\ref{th1}, we obtain\n\\begin{equation*}\n\\|\\overline{\\omega}(t)\\|_{H^s}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{equation*}\nMoreover, we can prove that $\\omega^{\\alpha}\\in C([0,T];H^{s+1})$. The proof of the theorem is finished.\n\n\n\n\n\\subsection{Proof of Theorem \\ref{th3}}\n\nSimilar to the proof Theorem \\ref{th1}, it follows from the difference equation \\eqref{diffrence} that \\eqref{11-1} holds true.\nThe three terms $I_1,\\, I_2,\\, I_3$ on the right side of \\eqref{11-1} will be estimated as follows. Applying the commutator estimates in Lemma \\ref{commutator} and the Sobolev embedding inequalities, we immediately have\n\\begin{equation}\\label{11-2}\\begin{split}\n|I_1|=&\\Big|\\int_{\\R^2} \\big(J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\big)\nJ^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\n\\|J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\|_{L^2}\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&(\\|J^su^{\\alpha_{0}}\\|_{L^2}\\|\\nabla\\overline{\\omega}\\|_{L^\\infty}+\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2})\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|u^{\\alpha_{0}}\\|_{H^{s}}.\n\\end{split}\\end{equation}\nSubstituting the decomposition \\eqref{diff-1-1} into $I_2$ and $I_3$, respectively, we have\n\\begin{equation}\\label{I_2}\\begin{split}\n&I_2=-\\int_{\\R^2} J^s(\\overline{u}_{I}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x:= I_{21}+I_{22};\\\\\n&I_3=-\\int_{\\R^2} J^s(\\overline{u}_{I}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x:= I_{31}+I_{32}.\n\\end{split}\\end{equation}\nChoose some $\\sigma\\in (0,2\\alpha).$  For any $p>\\frac{2}{2\\alpha-\\sigma}>2,$ applying Lemma \\ref{commutator} and the Sobolev embedding inequalities again, we obtain\n\\begin{equation*}\\begin{split}\n|I_{22}|=&\\Big|\\int_{\\R^2}\\big(J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\\|J^s(\\overline{u}_{II}\\cdot \\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\ \\leq &\\big(\\|J^s\\overline{u}_{II}\\|_{L^p}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2p}{p-2}}+\n\\|\\nabla \\overline{u}_{II}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\\begin{split}\n|I_{32}|&\\leq\n\\|J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\\end{equation*}\nApplying Proposition \\ref{add1} gives\n\\begin{equation*}\\begin{split}\n|I_{21}|=&\\Big|\\int_{\\R^2}\\big(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n \\leq \\|\\overline{\\omega}\\|^3_{H^s}.\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{add-0}, one has\n\\begin{equation*}\\begin{split}\n|I_{31}|&\\leq\n\\|J^s(\\overline{u}_I\\cdot\\nabla \\omega^{\\alpha_{0}})\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}^2\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}.\n\\end{split}\\end{equation*}\nInserting the estimates of $I_{21},\\,I_{22}, \\,I_{31}$ and $I_{32}$ into \\eqref{I_2}, we arrive at\n\\begin{equation}\\label{11-3}\\begin{split}\n&|I_2|+|I_3|\\\\\n\\leq& C\\left(\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p}+\n\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}+\n\\|\\overline{\\omega}\\|^3_{H^s}+\\|\\overline{\\omega}\\|_{H^s}^2\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\\right).\n\\end{split}\\end{equation}\nIn view of \\eqref{11-1}, \\eqref{11-2} and \\eqref{11-3}, it deduces\n\\begin{equation}\\label{th3-proof}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}  \\leq&\\|\\overline{\\omega}\\|_{H^s}(\\|u^{\\alpha_{0}}\\|_{H^{s}}+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}})\n+\\|\\overline{\\omega}\\|^2_{H^s}\\\\\n&+\\|\\overline{\\omega}\\|_{H^s}\\|J^s\\overline{u}_{II}\\|_{L^p}\n+\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\\end{equation}\n\nNow we estimate $\\|J^s\\overline{u}_{II}\\|_{L^p}$. Similar to the proof of Theorem \\ref{th2}, we use the integral form \\eqref{11-6}.\nNote that\n\\begin{equation*}\\begin{split}\nH_{1}&=\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}-\\frac{(x-y)^{\\perp}}{|x-y|^{3}}\\Big)\\Big(J^s\\omega^{\\alpha_0}(y)-J^s\\omega^{\\alpha_0}(x)\\Big)\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nBy the mean value formula and the H\\\"{o}lder inequality, we obtain\n\\begin{equation*}\\begin{split}\n\\|H_{1}\\|_{L^p}&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|J^{s+1}\\omega^{\\alpha_0}\\|_{L^p}\n\\\\&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+2}},\n\\end{split}\n\\end{equation*}\nwhere Lemma \\ref{embedding1} is used in the last inequality.\nSimilarly, for $2+2\\alpha\\leq\\gamma\\leq3,$\n\\begin{equation*}\\begin{split}\n\\|H_{2}\\|_{L^p}&= \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{(x-y)^{\\perp}(|x-y|^{\\gamma}\\log|x-y|)}{|x-y|^{2+2\\alpha}|x-y|^{2+2\\alpha_0}}\nJ^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^p}\n\\\\&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{2}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nNote that $\\ln |x-y|\\le C|x-y|^\\sigma$ for $|x-y|\\ge 1$ and any $\\sigma>0$, where $C$ may depend on $\\sigma$. We can apply the mean value formula and the Young inequality to obtain\n\\begin{equation*}\\begin{split}\n\\|H_{3}\\|_{L^p}&\\leq\\Big (\\frac12-\\alpha\\Big)\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}\nwhere $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, and we can choose some $r>2$ such that the following embedding\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}\nholds (see Lemma \\ref{embedding1}).\nConsequently,\n\\begin{equation}\\label{11-5}\\begin{split}\n \\|J^s\\overline{u}_{II}(t)\\|_{L^p}\\leq &\\|H_{1}\\|_{L^p}+\\|H_{2}\\|_{L^p}+\\|H_{3}\\|_{L^p}\n \\\\\\leq & C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\n\\\\\n&+\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}+\\Big(\\frac12-\\alpha\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation}\nLet  $\\epsilon=\\frac12-\\alpha$. The estimate \\eqref{11-5} combined with \\eqref{th3-proof} yields\n\\begin{equation*}\\begin{split}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}  \\leq &\\|\\overline{\\omega}\\|_{H^s}\\big(\\|u^{\\alpha_{0}}\\|_{H^{s}}\n+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\\big)\\\\\n&+\\|\\overline{\\omega}\\|^2_{H^s}+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}^2\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{split}\\end{equation*}\nArguing similarly as the last part in the proof of Theorem \\ref{th1}, we obtain\n\\begin{equation*}\n\\|\\overline{\\omega}(t)\\|_{H^s}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{equation*}\nMoreover, we can prove that $\\omega^{\\alpha}\\in C([0,T]; H^{s+2})$ for any $t\\in [0,T]$. The proof of the theorem is finished.\n\n\\subsection{Proof of Corollary \\ref{Cor1+}}\n\nSuppose that the result is not true. Then there exists a $M>0$ and a $\\delta_0>$ such that $T^*_\\alpha\\le M$ for all $\\alpha\\in (0,\\delta_0)$. But it is known that for any $T>0$, the smooth solution of the Euler equations exists on $[0,T]$. Take $T=M+1$. According to Theorem \\ref{th1}, there exists a $0<\\delta\\le \\delta_0$ depending on $T$ such that the smooth solution of the generalized SQG exists on $[0,T]$ as well. This contradicts with the assumption that  the maximal existence time $T^*_\\alpha\\le M$. The proof of the corollary is complete.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Galactic Center}\nThe DSO source approaching the super-massive black hole SgrA* at the center of the Milky Way \nhas spawned great activities in observing that region covering the \nentire electromagnetic spectrum from radio, via infrared to X-ray wavelengths using \ntelescopes across the world.\nThe upcoming events underline the importance of the Galactic Center as a laboratory to \ninvestigate and understand phenomena in the immediate environment of super massive black holes\n(Eisenhauer 2010, Ghez 2009).\nGas and stars within the Galactic Center stellar cluster provide the fuel\nfor the central super-massive black hole and hence the reason for most of the\nflux density variability observed from it.\n\n\n\\subsection{The variability of Sagittarius A*}\nProgress has been made in the understanding of the emission process \nassociated with the immediate surroundings of the super-massive black hole counterpart SgrA*  \nas well as the three-dimensional dynamics and the population of the central stellar cluster.\nThere is also ample evidence of interactions between the cluster and SgrA*.\n\nSeries of monitoring observations in the near-infrared (NIR), X-ray, and sub-millimeter (sub-mm) regimes\naccumulated over the years allowed us to perform for the first time detailed\nstatistical studies of the variability of SgrA*\n(Witzel et al. 2012; Eckart et al. 2012).\nThe analyses show that the histogram of the near-infrared flux density is a\npure power-law and the emission process is most likely dominated by \nsynchrotron radiation. \nIn Witzel et al. (2012), we present a comprehensive data description for \nNIR measurements of SgrA*. \nWe characterized the statistical properties of the variability of SgrA* in the near-infrared, which we found to be \nconsistent with a single-state process forming a flux density power-law distribution. \nWe discovered a linear rms-flux relation for the flux density range up to 12 mJy on a time scale of 24 minutes. \nThis and the structure function of the flux density variations imply a phenomenological, formally nonlinear statistical \nbehavior that can be modeled. In this way, we can simulate the observed variability and extrapolate its behavior to higher \nflux levels and longer time scales. \nSgrA* is also strongly variable in the X-ray domain\n(Baganoff et al. 2003, Baganoff et al. 2001, Porquet et al. 2008, Porquet et al. 2003, Eckart et al. 2012 \nand references there in, as well as, Nowak et al. 2012, Barriere et al. 2014 for a recent strong flares observed with\nChandra and NuSTAR).\nThe detailed statistical investigation by Witzel et al. (2012) also suggests that the past strong X-ray variations\nthat give rise to the observed X-ray echos can in fact be explained by the NIR variability histogram under the \nassumption of a Synchrotron Self Compton (SSC) process. \nSgrA* is extremely faint in the X-ray bands, \nthough strong activity has been revealed through the detection of flares. \nTherefore, SgrA* is the ideal target to investigate the \nmass accretion and the ejection physics in the case of an extremely low \naccretion rate onto a super-massive black hole. This is actually the phase\nin which super-massive black holes are thought to spend most of their lifetime.\nThe activity phase onset of a magnetar (see section below) at a separation of only \nabout 3 arcseconds from the Galactic Center presented a problem \nfor the SgrA* monitoring program in 2013\n(Mori et al. 2013, Shannon et al. 2013, Rea et al. 2013).\n\n\nEckart, et al. (2012) present simultaneous observations and modeling \nof the millimeter (mm), NIR, and X-ray flare emission of SgrA*.\nThese data allowed us to investigate physical processes giving rise to the variable emission \nof SgrA* from the radio to the X-ray domain.\nIn the radio cm-regime SgrA* is hardy linearly polarized but shows a fractional circular polarization\nof around 0.4\\% (Bower, Falcke \\& Backer 1999, Bower et al. 1999).\nThe circular polarization decreases towards the mm-domain (Bower 2003), where as\nMacquart et al. (2006) report variable linear polarization from SgrA* of a few percent in the mm-wavelength domain.\nThe observations reveal flaring activity in all wavelength bands.\nThe polarization degree and angle in the sub-mm are likely linked to the magnetic field structure \nor the general orientation of the source.\nIn general - the NIR emission is leading the sub-mm with a delay of about one to two hours (see below)\nand the excursions in the NIR and X-ray emission are rather simultaneous.\nAs a result we found that the observations can be modeled as the signal from an \nadiabatically expanding source component\n(Eckart et al. 2008b, Yusef-Zadeh et al. 2006, Eckart et al. 2006b).\nof relativistic electrons emitting via the synchrotron\/SSC process.\nA large fraction of the lower energy mm\/cm- flux density excursions is not necessarily correlated \nwith the NIR\/X-ray variability\n(e.g. Dexter \\&  Fragile 2013, Dexter et al. 2013, and details and further references Eckart et al. 2012).\nOne may compute the SSC spectrum produced by up-scattering\nof a power-law distribution of\nsub-mm-wavelength photons into the NIR and X-ray domain by using the\nformalism given by Marscher (1983) and Gould (1979).\nSuch a single SSC component model may be too simplistic,\nalthough it is considered\nas a possibility  in most of the recent modeling approaches.\nIt does not take into account possible deviations from the\noverall spectral index of $\\alpha$=1.3\nat any specific wavelength domain like the NIR or X-ray regime.\n\n\nThe number density distribution of the relativistic electrons responsible for the \nsynchrotron spectrum can be described by \n\\begin{equation}\\label{Equation:Gamma}\nN(E) = N_0 E^{-2 \\alpha +1}~~~,\n\\end{equation}\nwith  $\\gamma_e$ between  $\\gamma_1$ and  $\\gamma_2$\nwhich limit the lower and upper bound of the relativistic electron spectrum\n\\begin{equation}\\label{Equation:spectrum}\n\\gamma_1 mc^2 < E= \\gamma_e m c^2 <  \\gamma_2 m c^2~~~.\n\\end{equation}\nLorentz factors $\\gamma_e$ for the emitting electrons of the order of\na few thousand are required to produce a sufficient SSC flux in the\nobserved X-ray domain. In addition the relativistic bulk motion of \nthe orbiting or outward traveling component\nis described by the bulk  Lorentz factor $\\Gamma$ (see Fig.~\\ref{Fig:models}).\nOn the left of this figure we show a sketch of a typical relativistic electron \ndensity distribution resulting from MHD calculations\n(e.g. Dexter et al. 2010,  Dexter \\&  Fragile 2013, Moscibrodzka \\& Falcke 2013). We show a cut through only \none side of the three dimensional structure. The central plane and outflow region \nresulting from these calculations can be modeled in different, dedicated approaches (top and bottom right).\n\nIn the central plane modeling the flux variations are assumed to be the result of the motion of an orbiting blob\nor a hot spot (e.g. Eckart et al. 2006a).\nIn the outflow model the flux variations are assumed to be due\nto the ejection of a blob and its motion along the jet \n(with bulk motions close to the speed of light) or \na much slower overall outflow component. \nIn this case - for VLBI observations - the larger outflow extent at increasingly lower radio frequencies would be\nhidden by the decreasingly lower angular resolutions due to interstellar scattering.\nIn Fig.\\ref{Fig:diskmodel} the accretion disk (here assumed to be edge-on) is shown as a  vertical thick line \nto the right, the dashed part indicates the disk sections in the back- and foreground.\nExtending to the left we show one side above the disk.\nHere higher energy flare emission (lower part) is assumed to be responsible for the observed\nNIR\/X-ray flare emission.\nLower energy flare emission (upper part) may substantially contribute to long wavelength\ninfrared emission.\nIn addition to the expansion towards and beyond the the mm-source size, radial and azimuthal\nexpansion within the disk may occur.\nHence, long mm\/cm-wavelength variability may originate from different source components of SgrA*\nand may be difficult to be disentangled based on radio data alone.\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{diskmodel.eps}\n  \\caption{\\small\nSketch of a possible source structure for the accretion disk around the SMBH\nassociated with SgrA* following Fig.12 in Eckart et al. (2008a).\n}\n  \\label{Fig:diskmodel}    \n\\end{figure}\n\nIn Fig.\\ref{Fig:models}\nthe relativistic boosting vectors for the electrons $\\gamma_e$ and the bulk motion $\\Gamma_{bulk}$ are not \ndrawn to a proper relative scale. One can assume $\\gamma_e$ $\\ge$ $\\Gamma_{bulk}$.\nIn the case of relativistically orbiting gas as well as relativistic outflows\none may use modest values for $\\Gamma$.\nBoth dynamical phenomena are likely to be relevant in the case of SgrA*.\nThe size of the central plane synchrotron component is assumed to be of the\norder of or at most a few times the Schwarzschild radius.\nFrom the overall variable radio\/sub-mm spectrum spectrum we can assume a turnover frequency $\\nu_m$ of a few \n100~GHz (see details in Eckart et al. 2012).\nThe motion of the synchrotron emitting cloud can be described via\n\n\\begin{equation}\\label{Equation:delta}\n\\delta=\\Gamma_{bulk}^{-1}(1-\\beta cos \\phi )^{-1}~~,\n\\end{equation}\n\\begin{equation}\\label{Equation:Gamma}\n\\Gamma_{bulk}= (1-\\beta^2)^{-1\/2}~~.\n\\end{equation}\nHere  $\\beta= v\/c $ and $v$ is the speed of the bulk motion of the synchrotron cloud,\n$\\delta$ is the Doppler factor and $\\phi$ the angle to the line of sight.\nFor a relativistic bulk motion with $\\Gamma_{bulk}$ around \n 1.7$\\pm$0.3 (i.e. angles $\\phi$ of about $30^{\\circ} \\pm 15^{\\circ}$) the corresponding magnetic field\nstrengths are often assumed to be of the order of a few ten Gauss,\nwhich is also within the range of magnetic fields expected for RIAF models\n(e.g.  Yuan et al. 2006, Narayan et al. 1998).\n\n\nModeling of the light curves shows that (at least for the brighter events) \ntypically the sub-mm flux density excursions follow the NIR emission with a delay of \nabout one to two hours with an expansion velocity of about 0.01c-0.001c\n(Eckart et al. 2008b, Yusef-Zadeh et al. 2006, Eckart et al. 2006b).\nWe find source component sizes of around one Schwarzschild radius, flux densities of a few Janskys, \nand steep spectral indices. \nTypical model parameters suggest that either the adiabatically expanding source components have \na bulk motion larger than its expansion velocity \nor the expanding material contributes to a corona or disk, \nconfined to the immediate surroundings of SgrA*. \nFor the bulk of the synchrotron and SSC models, we find synchrotron turnover frequencies in the \nrange of 300-400 GHz. For the pure synchrotron models, this results in densities of relativistic \nparticles in the mid-plane of the assumed accretion flow of the order of 10$^{6.5}$ cm$^{-3}$, \nand for the SSC models the median densities are about \none order of magnitude higher. However, to obtain a realistic description of the frequency-dependent \nvariability amplitude of SgrA*, models with higher turnover frequencies and even higher \ndensities are required. \nThis modeling approach also successfully reproduces the degree of flux density variability across \nthe radio to far-infrared spectrum of SgrA*. \nIn Fig.~\\ref{Fig:spectrum} we show observed flux densities of SgrA* taken from the \nliterature (blue) compared to a combined\nmodel that consists of the fit given by Falcke et al. (2000), Marrone et al. (2008) (black line), \nand Dexter et al. (2010) (black dashed line). We plotted in red the spectra of synchrotron self-absorption frequencies for\nthe range of models. Here we show results for the preferred synchrotron plus SSC (SYN-SSC) model\nthat most closely represents the observed variability of SgrA*.\n\nValencia-S. et al. (2012) present theoretical polarimetric light curves expected in the \ncase of optically thin NIR emission from over-dense regions close to the marginal stable orbit\n(see also Broderick et al. 2005, Eckart et al. 2006a, Zamaninasab et al. 2010, 2011).\nUsing a numerical code the authors track the time evolution of detectable polarization \nproperties produced by synchrotron emission of compact sources in the vicinity of the black hole. \nThey show that the different setups lead to very special patterns in the time-profiles \nof polarized flux and the orientation of the polarization vector. As such, they may be used \nfor determining the geometry of the accretion flow around SgrA* \n(see also Karas et al. 2011, Zamaninasab, et al. 2011).\n\nDuring the 2013 Bad Honnef and the Granada conference (see Acknowledgments) \nefforts to monitor SgrA* during the DSO fly-by and first observational results from 2013 were reported by\nAkiyama, et al. (2013ab), Eckart et al. (2013abc), Jalali et al. (2013), Meyer et al. (2013), Phifer et al. (2013).\nThe NRAO Karl G. Jansky Very Large Array (VLA) is undertaking an ongoing community \nservice observing program to follow the expected encounter of the DSO \ncloud with the black hole SgrA* in 2013\/14 (Chandler \\& Sjouwerman 2013).\nThe NRAO VLA has been observing the Sgr~A region since \nOctober 2012 on roughly a bi-monthly interval, \ncycling through eight observing bands.\nFor monitoring the flux densities and in particular the radio spectral indices \nthe short wavelength observations ($\\lambda$$<$6cm) are most useful. \nFor 2012\/13 no particular flux density variation \nwas detected that could be attributed to the interaction between SgrA* and the DSO. \nThis may be linked with the fact that the newly determined periapse passage \nis now expected to happen in April\/May of 2014 (Phifer et al. 2013), i.e. later than originally anticipated. \nHowever, in the radio-shock\nframe in which variations of up to several Janskys were expected even during the pre-periapse time. \nHence, the lack of strong radio flares \nindicates that the medium is less dense than expected and\/or that the bow-shock size i.e. the\ncross-section of the dust source is much smaller than assumed\n(Narayan, et al. 2013, Crumley, et al. 2013, Sadowski, et al. 2013, Yusef-Zadeh, et al. 2013, Shcherbakov, et al. 2014).\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{specgraph1.eps}\n  \\caption{\\small\nThe variable radio spectrum of SgrA*: Measurements and model results\n(see text and Eckart et al. 2012 for details).\n}\n  \\label{Fig:spectrum}    \n\\end{figure}\n\n\n\n\\subsection{VLBI imaging of SgrA*}\nThere is also profound progress in imaging and modeling of the central putative accretion disk\nof SgrA* as well as the jet that may be associated with the source\n(Falcke \\& Markoff, 2013, Moscibrodzka \\& Falcke, 2013, Valencia-S, M. et al. 2012).\nIn fact imaging of SgrA* may turn out to be a Rosetta Stone in the attempts of distinguishing between \ndifferent relativity theories of black holes\n(e.g. Boller \\&  M\\\"uller, 2013, on astronomical tests of general relativity and the\n pseudo-complex theory).\n\nVLBI (Very Long Baseline Interferometry) observations at very short millimeter \nradio wavelengths can overcome the effects of interstellar scattering \nand allow us to study the source intrinsic structure of SgrA*.\nLarge mm\/sub-mm facilities like the \nVLBA (Very Long Baseline Array), \nVERA (VLBI Exploration of Radio Astrometry), \nALMA (Atacama Large Millimeter Array), \nPdBI (Plateau de Bure Interferometer) and the sensitive mm-telescopes in the\nEVN (European VLBI Network) - such as the IRAM 30m and the 100m Effelsberg telescopes - are participating in this effort,\nwhich will eventually culminate in the project EHT (Event Horizon Telescope), \na VLBI array especially designed to image the structures close to the\nevent horizons of the larges SMBHs in the sky - namely SgrA* and M87,\nwith 1 Schwarzschild radius extending to an angular size of about 10$\\mu$as and 3.7$\\mu$as, respectively.\nMulti-epoch imaging observations will allow to constrain the locations and sizes of the flaring region of SgrA* \nwithin the putative temporal accretion disk of the accretion stream\/flow towards or an accretion wind from SgrA*.\nThese measurements will also constrain the acceleration processes (e.g. magnetic reconnection events or \nnon-axisymmetric standing shocks) that give rise to the population of relativistic electrons and the variable \nemission we see from SgrA*.\nUltimatly, alternative black hole models will be probed and attempts to test the black hole no-hair theorem \nwill be possible with the new VLB mm\/sub-mm facilities\n(Broderick, et al. 2014, Fish et al. 2014, Akiyama, et al. 2013ab, Huang, et al. 2012, Broderick, et al. 2011,\nBroderick, et al.  2011, Fish et al. 2011, Lu, R.-S., et al.  2011).\n\nThese VLBI experiments will eventually enable spatially resolved studies on sub-horizon scales, leading \nto an unprecedented exploration of a putative predicted black-hole shadow \n(e.g. Huang et al. 2007)\nas an evidence for light trapping by the black hole as well as its interaction with \nthe surrounding material. \nIt will be possible to monitor the possible expansion of source components \nduring flare activity.\nFurthermore, when a rotating black hole is\n immersed in a magnetic field of external origin, the gravito-magnetic \ninteraction is capable of triggering the magnetic reconnection, \naccelerating the particles to very high energy \n(Karas et al. 2012, 2013; Morozova et al. 2014). \nThis frame-dragging phenomenon is particularly interesting in the \ncontext of exploring the strong-gravity effects in astrophysical black holes \nbecause the effect does not have a Newtonian counterpart and it operates \non the border of the ergospheric region (Koide \\& Arai 2008), \ni.e. very close to the black hole horizon, and it can be probed with \nthe future EHT. Also, one can investigate if the black hole proximity generates \nconditions favourable to incite the magnetic reconnection that eventually \nleads to plasma heating and particle acceleration. \nThis effect could contribute to the flaring activity.\n\n\n\\subsection{The importance of dusty sources close to the center}\nA major discussion point is if and how the DSO source will be disrupted during\nits peri-bothron\\footnote{Peri- or apo-bothron is the term used for \nperi- or apoapsis - i.e. closest or furthest separation - for an elliptical orbit with a black hole present \nat one of the foci.  \nAs already mentioned by Frank and Rees (1976) word 'bothros' was apparently first suggested in the context of \nblack holes by W.R. Stoeger.\nIt originates from the greek word \n\\`o $\\beta$\\'o$\\theta \\rho o \\varsigma$ \nwith the equivalent meaning of 'the sink' or 'the deep dark pit'.\n} passage.\nIt may be only its dusty envelope that will be disrupted\nsince the K$_s$-band identifications\nof the source suggest that it can also be associated with a star (Eckart et al. 2013a).\nIn addition to the VLT NACO and the Keck NIRC detections of the \nDSO NIR continuum emission (Eckart et al. 2013bc),\nhere we show the detection of the DSO continuum at about K$\\sim$19 using SINFONI data (Fig.~\\ref{Fig:DSO}).\nThe detection of the continuum emission in data sets taken with three different \ninstrumental setups over many years strengthens the case for a substantial\ncontinuum emission from that dusty source.\nAs posted in the astronomer's telegram No.6110 on 2 May 2014 (Ghez et al. 2014), the DSO was detected \n 3.8$\\mu$m during its peri-bothron passage around the central black hole SgrA*. Hence, it appears to be intact\nand up to this point not yet heavily affected by tidal effects. \nThis clearly supports our finding (Eckart et al. 2013bc) that it may very well be a dusty star\nrather than a pure gas and dust cloud.\n\nIn contrast to a pure dust and gas nature of the DSO its possible \nstellar (i.e. a dust enshrouded star) nature is discussed and partially favoured in\nMeyer et al. (2013, 2014), Eckart et al. (2013abc), Scoville \\& Burkert (2013)\nBallone et al. (2013), Phifer et al. (2013).\nEckart et al. (2013a) investigate the possible mass transfer across \nLagrange point L1 in a simple Roche model.\nIf the star has a mass of about 1M$_{\\odot}$,  the separation of $L1$ from it will be\nabout 0.1~AU. For a Herbig Ae\/Be stars with 2-8~M$_{\\odot}$ ~that distance will be between\n0.2 and 0.5~AU. For a typical S-cluster stellar mass of $\\sim$20-30~M$_{\\odot}$ ~the separation \nwill be closer to one AU. \nThe interferometrically determined  inner ring sizes that one typically finds for young Herbig Ae\/Be and\nT~Tauri stars can indeed be as small as 0.1-1~AU (Monnier \\& Millan-Gabet 2002).\nAny stellar disk or shell may already have been stripped \nsubstantially if the DSO has performed more than a single orbit.\nIf the source has a size of about 1~AU \n(as determined from its MIR-luminosity; Gillessen et al. 2012a) \nthen a significant amount of the dusty circumstellar material\nmay pass beyond $L1$ during peri-bothron passage. \nThis material will then start to move into the Roche lobe associated with SgrA*. \n\n\n\nHowever, it is not at all clear what will happen to the transferred material  after \nthe peri-bothron passage around May 2014 or beyond. \nThe fact that this dusty object may be a dust enshrouded star rather than a dust \ncloud will have an influence on the expected flux density variations resulting \nfrom the close approach. They may be much weaker than expected.\nSimulations (e.g.  Burkert et al. 2012, Schartmann et al. 2012,\nsee also Zajacek, Karas \\& Eckart 2014) that\nhave discussed the feeding rate of SgrA* as a function of radius\nindicate that a portion of the material may fall towards SgrA*.\nIf SgrA* is associated with a significant wind on scale of the peri-bothron separation, \nthen a large part of the material may be blown away again by an out-bound accretion wind.\nShcherbakov \\& Baganoff (2010) have discussed the feeding rate of SgrA* as a function of radius.\nBased on their modeling one may suggest that the bow-shock sources X3 and X7 (Muzic et al. 2010) are  \nstill in the regime in which most of the in-flowing mass is blown away again.\nAnother case for comparison is the star S2. During its peri-bothron passage the star has been well \nwithin the zone in which matter of its (weak) stellar wind could have been accreted by SgrA*. \nThe DSO peri-bothron will be at a larger radius than that of S2 (Phifer et al. 2013). \nThis may imply that \nno enhanced accretion effect will result from it during the peri-bothron passage. \nUntil May 2014 no increase in variability and no significant flux density increase well above normal levels \nhas been reported in the radio to X-ray domains.\n\n\nThe fate of the DSO and the cometary sources X3 and X7 underline the importance\nof investigating the wind properties in the vicinity of SgrA* in more detail.\nIRS~8 is a unique possibility to study the bow shock properties and polarization \nfeatures in the dusty environment at the Galactic Center.\nBased on a detailed study of near-infrared emission \nRauch et al. (2013) present interstellar dust properties for \nthe northern arm in the vicinity of the IRS~8 bow shock. \nThis study allowed us for the first time to determine the relative positioning of \nIRS~8 with respect to the northern arm and the \nsuper-massive black hole SgrA*. The result indicates that the\ncentral star of IRS~8 is in fact located closer towards the observer than the northern arm.\nIn Eckart et al. (2013a) we investigated the near-infrared \nproper motions and spectra of infrared excess sources at the Galactic Center.  \nThe work concentrated on a small but dense cluster of comoving \nsources (IRS13N) located ~3'' west of SgrA*.  Our analysis shows that \nthese stars are spectroscopically and dynamically young and can indeed be \nidentified with continuum emission at 2 microns and shortward, indicating that these mid-infrared \nsources are not only dust sources but young stars.\nThe possibility of ongoing star formation at the Galactic Center is \nsupported through simulations by Jalali et al. 2014 (submitted) and  Jalali et al. (2013).\nIn fact the DSO may be a representative of dusty sources similar to those\ndiscussed in Eckart et al. (2006b; see their Fig.14;\ncompare also to the discussion of sources X3 and X7 given by Muzic et al. 2010).\nMeyer et al. (2014) present NIR spectroscopic data of several of these sources.\nThey also show that the DOS does not seem to be unique, since several red emission-line objects \ncan be found in the central arcsecond. \nIn summary, Meyer et al. (2014) conclude that it seems more likely that G2 is ultimately a\nstellar source that is clearly associated with gas and dust (see also Eckart et al. 2013abc).\n\n\n  \\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{DSOsumme.eps}\n  \\caption{\\small\nThe DSO detected in its K-band continuum emission in 2010 SINFONI data.\nLeft: The original image (positive greyscale); \nRight: A LUCY deconvolved image (negative greyscale) shown at an \nangular resolution close to the diffraction limit of the VLT UT4.\n}\n  \\label{Fig:DSO}    \n\\end{figure}\n\n\n\n\\subsection{Stellar dynamics and tests of relativity}\nSgrA*, the super-massive black hole at the center of the Milky Way, \nis surrounded by a small cluster of high velocity stars, known as the S-stars\n(Eckart \\& Genzel 1997).\nSabha et al. (2012) aimed at constraining  the amount and nature of the stellar \nand dark mass that is associated with the cluster in the immediate vicinity of SgrA*. \nThe authors use near-infrared imaging to determine the Ks-band luminosity function \nof the S-star cluster members, the distribution of the diffuse background \nemission and the stellar number density counts around the central black hole. \nThis allows us to determine the stellar light and mass contribution expected \nfrom the faint members of the cluster. \nSabha et al. (2012) then use post-Newtonian N-body simulations to investigate the effect of stellar \nperturbations on the motion of S2, as a means of detecting the number and masses \nof the perturbers. The authors find that the stellar mass derived from the Ks-band \nluminosity extrapolation is much smaller than the amount of mass that might \nbe present considering the uncertainties in the orbital motion of the star S2. \nAlso the amount of light from the fainter S-cluster members is below the \namount of the residual light at the position of the S-star cluster after one removes \nthe bright cluster members. If the distribution of stars and stellar remnants \nis peaked near SgrA* strongly enough, observed changes in the orbital \nelements of S2 can be used to constrain both the masses and the number of objects inside its orbit. \nBased on simulations of the cluster of high velocity stars we find that in\nthe NIR K-band - close to the confusion level for 8 m class telescopes -\nblended stars will occur preferentially near the position of SgrA*\nwhich is the direction towards which we find the highest stellar density.\nThese blended stars consist of several faint, (with the current facilities) \nindividually undetectable stars that get aligned along the line-of-sight, \nproducing the visual effect of a new point source. \nThe proper motion of stars and the corresponding velocity dispersion\nleads to the fact that such a blended star configuration dissolves typically after 3 years.\n\n\nStars that get very close to the super massive black hole are ideal probes to\nanalyse the gravitational field and to search for effects of relativity due \nto the presence of the high mass concentration and its effect on space time.\nThis can be done by tracing the orbit of stars through proper motions and radial velocities.\nAs discussed in Zucker et al. (2006) relativistic effects should express themselves \nspectroscopically. The redshift $z$ of a black hole orbiting star can be written as:\n\\begin{equation}\\label{eq:aa1}\nz = \\Delta\\lambda\/\\lambda = B_0 + B_1\\beta + B_2\\beta^2 + O(\\beta^3)\n\\end{equation}\nwith $B_0$ being an offset,\n$B_1\\beta$ describing the Doppler velocity  \nand $B_2\\beta^2$ expressing the relativistic effects.\nHere the value $B_2$ contains equal contributions from the gravitational redshift and the \nspecial relativistic transverse Doppler effect.\nThe combined effect gives a redshift that is about an order of magnitude larger than\nthe currently achieved spectral resolution of $\\delta\\lambda\/\\lambda$$\\sim$10$^{-4}$.\nFor S2 one expects about a 150-200 km\/s signal measurable over a few months \non top of an orbit-depending radial velocity of more than 4000 km\/s.\nExpectations are high that this will be observable during the next peri-bothron\nfor S2 around 2017.9$\\pm$0.35 (Gillessen et al. 2009b, Eisenhauer et al. 2003) \nor S2-102 around 2021.0$\\pm$0.3 (Meyer et al. 2013).\nRealistically, however, one needs several stars  on different orbits \nto detect the relativistic effect with certainty (Zucker et al. 2006; see also  \nRubilar \\&  Eckart 2001 for peri-bothron shift).\nAlternatively, one has to find stars that are (or get) closer than S2 \nand S2-102 (Meyer et al. 2013; see below) to SgrA*.\n\nDetailed imaging and the analysis of proper motions may be another way to trace relativistic effects.\nAn important deviation from Keplerian motion occurs as a result of\nrelativistic corrections to the equations of motion, which to the lowest\norder predict a certain advance of the argument of \nperi-bothron each orbital period.\nChoosing $a = 5.0$~mpc and $e= 0.88$ for the semi-major axis and eccentricity \nof S2, respectively, and assuming a black hole mass of $M_\\bullet=4.0\\times 10^6 $~M$_\\odot$ this advance will be\n\\begin{equation}\\label{Equation:DomegaGR}\n\\left(\\Delta\\omega\\right)_\\mathrm{GR} \n= \\frac{6\\pi GM_\\bullet}{c^2a(1-e^2)} \\approx 10.8^\\prime.\n\\end{equation}\nThe relativistic precession is prograde, and leaves the orientation of the orbital\nplane unchanged.\n\nThe location of the peri-bothron advances for each orbital period\ndue to the spherically-symmetric component of the distributed mass that \nis resolved by the elliptical orbit of the star.\nThe amplitude of this Newtonian ``mass precession'' is\n\\begin{equation}\\label{eq:nuM}\n\\left(\\Delta\\omega\\right)_\\mathrm{M} = -2\\pi G_\\mathrm{M}(e,\\gamma)\\sqrt{1-e^2}\\left[\\frac{M_\\star}{M_\\bullet}\\right].\n\\end{equation}\nHere, \n$M_\\star =  M_\\star(r<a)$\nis the distributed mass within a radius $r=a$, and $G_\\mathrm{M}$ is\na dimensionless factor of order unity that depends on $e$ and on the power-law index\nof the density, $\\rho\\propto r^{-\\gamma}$ (Merritt 2012).\nIn the special case $\\gamma=2$,\n\\begin{equation}\nG_\\mathrm{M} = \\left(1+\\sqrt{1-e^2}\\right)^{-1}\n\\approx 0.68\\ \\mathrm{for\\ S2}~~,\n\\end{equation}\nso that\n\\begin{equation}\\label{Equation:DomegaM}\n\\left(\\Delta\\omega\\right)_\\mathrm{M} \\approx \n-1.0^\\prime\\; \\left[\\frac{M_\\star}{10^3 \\mathrm{~M}_\\odot}\\right].\n\\end{equation}\nMass precession is retrograde, i.e., opposite in sense to the relativistic precession.\n\nThe contribution of relativity to the peri-bothron advance is determined \nuniquely by the known values of $a$ and $e$.\nA measured $\\Delta\\omega$ can then be used to constrain the \nmass enclosed within S2's orbit, by subtracting\n$(\\Delta\\omega)_\\mathrm{GR}$ and comparing the result with Equation~(\\ref{Equation:DomegaM}).\nSo far, this technique has yielded only upper limits on $M_\\star$ of \n$\\sim 10^{-2}M_\\bullet$ (Gillessen et al. 2009a).\nFirst robust upper limits of that order have been derived by Mouawad et al. (2003, 2005).\nThe role of relativistic effects versus the gravitational effects of the nuclear star cluster \nand a ring of stars are summarized by Subr et al. (2004).\n\n\n\n  \\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{exploreorbits.eps}\n  \\caption{\\small\nAn example for the location of the Galactic Center S-stars on the $(a,e)$-plane.\nSee details in the text and Antonini \\& Merritt (2013), in particular the left panel\nof their Fig.1.\n}\n  \\label{Fig:stellarorbits}    \n\\end{figure}\n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{Figure_S2histogram.eps}\n  \\caption{\\small\nHistograms of the predicted change in S2's argument of peri-bothron, $\\omega$,\nover the course of one orbital period ($\\sim15.6$~yr; see also Sabha et al. 2012).\nSee text for a detailed description of the figure.\n}\n  \\label{Fig:S2histogram}    \n\\end{figure}\n\n\n\n\n\nIn addition to the relativistic effects on the orbit of a test star, and the precession due to the smooth matter distribution, the granularity of the cluster stars and stellar remnants needs to be considered. The granularity of the distributed mass makes itself felt via the phenomenon of \"resonant relaxation\" (RR; Rauch \\& Tremaine 1996, Hopman \\& Alexander 2006). On current observational time scales the stellar orbits near SgrA* remain nearly fixed in their orientations. The perturbing effect of each field star on the motion of a test star (e.g. S2) can be approximated as a torque that is fixed in time and proportional to the mass $m$ of the field star. \nSabha et al. (2012) show that the effects of RR are competing with the relativistic and Newtonian periastron effects on the orbits of the known high velocity S-cluster stars. \nFollowing Sabha et al. (2012), we show in Fig.~\\ref{Fig:S2histogram}    \nthe predicted change in S2's orbital elements over the course of one orbital period ($\\sim 15.6$ yr). The shift due to relativity, $\\Delta\\omega_\\mathrm{GR} \\approx 11^\\prime$,\nhas been subtracted from the total; what remains is due to Newtonian perturbations from the field stars. Each histogram was constructed from integrations of 100 random realizations of the same initial model, with field-star mass $m = 10M_\\odot$, \nand four different values of the total number: $N = 200$ (solid\/black); \n$N = 100$ (dotted\/red); $N = 50$ (dashed\/blue); and $N = 25$ (dot-dashed\/green). \nThe average value of the peri-bothron shift increases with increasing $Nm$, as predicted by Equation (10). There is also a separate contribution that scales approximately as $\\sim 1\/\\sqrt{N}$ and that results in a variance about the mean value. \n\nSabha et al. (2012) show that the net effect of the torques from $N$ field stars is to change the angular momentum, $\\mathbf{L}$, of S2's orbit according to\n\\begin{equation}\\label{Equation:DeltaLRR}\n\\frac{|\\Delta\\mathbf L|}{L_c} \\approx K\\sqrt{N}\\frac{m}{M_\\bullet}\\frac{\\Delta t}{P}\n\\end{equation}\nwhere $L_c$ is the angular momentum of a circular orbit having the same semi-major\naxis as that of the test star.\nHere Equation \\ref{Equation:DeltaLRR} describes ``coherent resonant relaxation''.\nSabha et al. (2012) state that the normalizing factor $K$ should be of order unity (Eilon et al. 2009).\n\nChanges in $\\mathbf{L}$ imply changes in both the eccentricity of the test star orbit, \nas well as changes in its orbital plane.\nChanges in the orbital plane can be described in a coordinate-independent way via the\nangle $\\Delta\\theta$, where\n\\begin{equation}\n\\cos\\left(\\Delta\\theta\\right) = \\frac{\\mathbf{L}_1\\cdot\\mathbf{L}_2}{L_1L_2}\n\\end{equation}\nand $\\{\\mathbf{L}_1, \\mathbf{L}_2\\}$ are the values of $\\mathbf{L}$ at two times\nseparated by $\\Delta t$ (Sabha et al. 2012).\nFor small values of $\\Delta \\theta$ this can be written as\n\\begin{equation}\n\\Delta\\theta \\approx \\sqrt{2 - 2\\frac{\\mathbf{L}_1\\cdot\\mathbf{L}_2}{L_1L_2}} \\approx\n1 - \\frac{\\mathbf{L}_1\\cdot\\mathbf{L}_2}{L_1L_2} .\n\\end{equation}\nIf we set $\\Delta t$ equal to the orbital period of the test star, the changes\nin its orbital elements due to resonant relaxation are expected to be\n\\begin{eqnarray}\n\\left|\\Delta e\\right|_\\mathrm{RR} &\\approx& K_e \\sqrt{N} \\frac{m}{M_\\bullet} , \\\\\n\\left(\\Delta \\theta\\right)_\\mathrm{RR} &\\approx& 2\\pi K_t\\sqrt{N} \\frac{m}{M_\\bullet} ,\n\\end{eqnarray}\nwhere $N$ is the number of stars having $a$-values similar to, or less than, that of\nthe test star (Sabha et al. 2012).\nThe constants $K_e$ and $K_t$ may depend on the properties of the orbits of the\nfield star distribution in the Galactic Center stellar cluster.\nMore details are given in Sabha et al. (2012).\nThe Kozai mechanism is another kind of resonant process that is thought to operate in the Galactic Center environment (Subr \\& Karas 2005, Chang 2009, Chen \\& Amaro- Seoane 2014).\n\nFigure Fig.~\\ref{Fig:stellarorbits}    (following Antonini \\& Merritt 2013)\nplots the location of the S stars in the $(a,e)$-plane (semimajor axis - ellipticity).\nThe data for the two currently closest known stars S2 and S2-102 \nare indicated by pattern filled circles. \nFor the two currently closest sources one finds: \n\\begin{eqnarray}\n\\mathrm{S0-102}:  T_\\mathrm{orbit} &=& 11.5\\pm0.3\\; \\mathrm{yr} \\ \\ e=0.680\\pm0.020 \\ \\  a=0.100''\\pm0.010''\\; \\nonumber \\\\\n\\mathrm{S2}: T_\\mathrm{orbit} &=& 15.6\\pm 0.4\\; \\mathrm{yr} \\ \\  e = 0.883\\pm0.003 \\ \\  a = 0.125''\\pm 0.002''\\;  \\nonumber\n\\end{eqnarray}\n(Meyer et al. 2013, Gillessen et al. 2009b, Eisenhauer et al. 2003; \n1'' at the distance of the Galactic Center corresponds to  alinear scale of about $39~mpc = 0.127~lyr = 8044~AU$).\n\n\nAlso plotted in Fig.~\\ref{Fig:stellarorbits} are curves indicating where the effects of RR begin to be\nmediated by relativistic precession of the ``test'' star.\nThe upper curve is the ``Schwarzschild barrier'' (Merritt et al. 2011); \nstars below this curve precess due to GR so rapidly that their precession,\nrather than the mean precession rate of the field stars, determines the coherence\ntime over which the RR torques can act.\nThis is reflected in the horizontal tick marks which give the expected amplitude of eccentricity changes as an orbit precesses in the essentially fixed torquing field due to the field stars.\nThe location of the Schwarzschild barrier in the $(a,e)$-plane\ndepends somewhat on the assumed\nspatial distribution of the stars in the nuclear cluster, but for most reasonable\ndistributions, S2 lies below the barrier (Antonini \\& Merritt 2013).\nCloser to SgrA*, frame-dragging torques due to the spin of SgrA* begin to make\nthemselves felt. \nThe two lower curves on Fig. 6 mark where these torques begin to compete with\nRR torques.\nIn this regime, orbits precess so rapidly that their eccentricities are essentially\nunaffected by the $\\sqrt{N}$ torques, but their orbital planes can still change\n(``vector RR'').\nBelow the curves marked ``Kerr'' in \nFig.~\\ref{Fig:stellarorbits} are curves indicating where the effects of RR begin to be\nchanges in orbital planes due\nto frame dragging dominate the changes due to vector RR (Merritt et al. 2010).\nStars in this regime can be used to test theories of gravity, e.g., ``no-hair''\ntheorems.\n\nFig.~\\ref{Fig:stellarorbits} \nalso shows the $(a,e)$-regions that will be experimentally accessible with the new upcoming interferometric instrumentation in the NIR like GRAVITY at the VLTI and LINC-NIRVANA at the LBT (see below). \nOnly for stars at much smaller orbital separations from SgrA* than S2 or S2-102, \nas they may be found with infrared interferometers like GRAVITY at the VLTI or LINC-NIRVANA at the LBT (see below), are Newtonian perturbations (mass precession,\nRR) negligible compared with the effects of GR.\nHowever, even then one will need more than one star with different orbital elements in order to clearly \ndemonstrate that the eventually observed orbital changes are truly due to relativistic effects \n(see e.g.  Merritt et al. 2010, Will 2008 , Zucker et al. 2006, Rubilar \\& Eckart 2001).\n\n\n\n\n\n\\subsection{Effects due to stellar collisions}\nThe stellar density close to center is in excess of 10$^5$ M$_{\\odot}$ ~~per cubic-parsec.\nThis implies that in this region the evolution of stars is influenced by stellar collisions\n(e.g.  Davies et al. 2011, Church et al. 2009).\nObservations of the stars \nat the Galactic Center show that there is a lack of red giants \nwithin about 0.5~pc of the super-massive black hole. \nThe very high stellar number densities this close to the SMBH \nimply that the giants -- or their progenitors -- may have been \ndestroyed by stellar collisions. \nThe derivation of collisional rates between different \ntypes of stars at the Galactic Center and their likely effects\nare currently a field of intense investigation\nin order to quantify how large a contribution stellar \ncollisions could make to the puzzle of the missing red giants\n(Dale et al. 2009).\nAs an example see recent contributions on star formation (see also references below), \nand on the structure of the \ncentral stellar cluster and stellar interactions in the Galactic Center area by\nSabha et al. (2012), Perets et al. (2014), Mastrobuono-Battisti \\&  Perets (2013),\nHaas \\& Subr (2012), Subr \\& Haas (2012), Haas, Subr \\& Vokrouhlicky (2011), Feldmeier et al. (2013).\n\n\\subsection{Pulsars at the Galactic Center}\nAnother very promising way of investigating the super-massive black hole properties in \ngreat detail is to find pulsars orbiting SgrA*. \nIf close enough, they\nwould allow us to measure the spin and quadrupole moment of the \ncentral black hole with superb precision\nenabling us to test different theories of gravity\n(Psaltis et al. 2012).\nThe number of pulsars in the central cluster will depend on the \nstar formation history in the overall region.\nThe discovery of radio pulsations from a magnetar PSR~J1745-2900 with the \nEffelsberg telescope has highlighted the great value and the efforts \nthat are currently being undertaken to find pulsars in the Galactic Center region\n(Spitler et al. 2014, Lee et al. 2013, Mori et al. 2013, Eatough et al. 2013a, Eatough et al. 2013b).\n\n\n\\subsection{Star formation and the Galactic Center}\nStar formation activity in the Galactic Center is an ongoing research topic\n(e.g. Yusef-Zadeh et al. 2009, 2010, 2013).\nGiven the deep gravitational potential towards the very center,\nstars cannot form in the same way they do throughout the Milky Way.\nHowever, infrared observations have shown evidence that massive \nstars were formed in the hostile environment of SgrA* a \nfew million years ago. \nVLA, ALMA and CARMA measurements suggest that \nstar formation is still taking place in this region and has been going on during the\npast few 10$^5$ years. A broad variety of possible star formation scenarios\ncan be discussed. Massive stars may have formed within a disk of \nmolecular gas resulting from a passage of a giant molecular\ncloud interacting with SgrA* and then dispersed after having formed these stars\n(Nayakshin, Cuadra \\& Springel 2007).\nDense clumps originating from the CND\nloosing angular momentum and falling towards the very center may also\nbe a source of constant or episodic star formation in the Galactic Center \n(Jalali et al. 2013).\nA fraction of the material associated with these phenomena  \nmay have accreted onto SgrA* probably in an episodic or transient manner (Czerny et al. 2013). \nThis process may in part be responsible for the origin of the \n$\\gamma$-ray emitting Fermi bubbles (Su,  Slatyer \\&  Finkbeiner 2010).\n\n\n\n\\subsection{The Galactic Center on larger scales}\n\nRadio polarization observations by the Parkes radio telescope in Australia have recently \nled to the discovery of giant radio lobes emanating from the Galactic \nnucleus (Zubovas \\&  Nayakshin 2012, Bland-Hawthorn et al. 2013).\nThese lobes are largely coincident with the Fermi Bubbles discovered in the\n$\\gamma$-ray domain (Su,  Slatyer \\&  Finkbeiner 2010).\nHowever, the radio outflows extend to even larger angular scales, \ncovering about 55 to 60 degrees north and south of the Galactic plane. \nIt is likely that the radio lobes -- and the Fermi Bubbles -- are \nthe result of the concentrated star formation occurring in the \nCentral Molecular Zone of the Milky Way rather than signatures \nof putative activity of the super-massive black hole. \nThese Fermi bubbles may be related to \nthe giant magnetized outflows from the Galactic Center \n(Crocker 2012, Crocker et al. 2011, Crocker \\& Aharonian 2011).\nThese phenomena link the Galactic Center nuclei of nearby active galaxies and \nlarger scale events (e.g. amounts of gas escaping from the potential well of the galaxy).\nMassive inflow of gas towards the centers, as well as\nstar formation or jet driven outflows from the nuclear regions, are frequently observed \nand are directly or indirectly linked with the accretion processes onto super-massive black holes.\n\n\\section{Extragalactic Nuclei}\nThe observational results obtained on the Galactic Center during the past \nfew years need to be put into perspective. \nThe nucleus of our Galaxy has an extremely low luminosity.  \nHowever, it also demonstrates that violent activity may take place at times. \nHence, a comparison between the center of the Milky Way and the nuclei of galaxies hosting \nLow Luminosity Active Galactic Nuclei (LLAGN) appears to be imperative.\n\nFootprints of AGN feeding and feedback in LLAGN can be observed in many cases\n(Garcia-Burillo \\& Combes 2012, Garcia-Burillo et al. 2012, Marquez \\& Masegosa 2008).\nThe study of the content, distribution and kinematics of interstellar \ngas is a key to understanding the fueling of AGN and star formation activity \nin galaxy disks. Current mm-interferometers provide a sharp view of the \ndistribution and kinematics of molecular gas in the circumnuclear disks \nof galaxies through extensive mapping of molecular line (mainly CO and to some extent\nhigh density tracers like HCN, HCO$^+$, CS etc). \nThe use of molecular tracers specific to the dense gas phase \ncan probe the influence of feedback on the chemistry and energy \nbalance in the interstellar medium of galaxies. \nRadiative and mechanical feedback are often used as a mechanism \nof self-regulation in galaxy evolution \nas well as thermal and non-thermal AGN activity.\nIn the disks of galaxies the evolution is predominantly expressed in star formation and \nthe evolution of stellar populations and the ISM properties.\nIn galactic nuclei the thermal part is represented mainly by the properties of the nuclear ISM\nand by the NLR and BLR regions while the non-thermal part is dominated by \nsynchrotron\/SSC emission due to SMBH accretion and the presence of jets.\n\n\n\n\nHigh angular resolution ($<$1'') observations with sub(mm)-interferometers \n(IRAM PdBI, ALMA) in the context of the NUclei of GAlaxies (NUGA) survey \n(e.g. Garcia-Burillo \\& Combes 2012, Garcia-Burillo et al. 2012, \nGarcia-Burillo et al. 2003, Krips et al. 2007, Moser et al. 2013,\nand upcoming more ALMA NUGA papers)\nallow us to study the mechanisms responsible for fueling AGN and star formation \nactivity in the central R$<$1 kpc disks of a \nsample of 25 active galaxies at the 10-100 pc scale. \nThis study has revealed streaming motions towards and away from the nuclear \nregion that do not necessarily have to be co-phased with current AGN activity.\nIn several cases these observations also reveal the presence of \nmolecular circumnuclear disks and massive molecular outflows.\n\nThese phenomena are directly coupled to black-hole fueling, feedback and \nduty cycles that are essential for understanding the activity in galactic nuclei\n(Mundell et al. 2003, 2004, 2007, 2009;\nsee also Combes et al. 2014, Heckman \\& Best 2014).\nThe distribution of \ngas and stars in nearby galaxies traced by 3-D studies of molecular, \nneutral and ionized gas provides a unique view of the role of the \nmulti-phase medium in triggering and fueling nuclear activity in \ngalactic nuclei on scales ever closer to the central black hole. \nRadio- and mm-interferometers and in particular 3-dimensional imaging \ndevices in the infrared and optical domain (like integral field units) \nallow us to obtain spectroscopic and photometric information at every position\nof the field-of-view. Such observations have been used to elaborate\ncomparative studies of gaseous \nand stellar dynamics in active and quiescent galaxies. \nStudying the inner kiloparsec of AGN is particularly important, since \nthe activity and dynamical time scales become comparable in this region.\nThese investigations also show that the variability observed \nin nearby radio-quiet AGN may provide \nparallels to our own Galactic Center. \nThey also provide evidence that AGN duty cycles \nmay be shorter than previously thought.\n\n\n\n\n\n\\begin{figure}[!ht]\n  \\centering\n   \\includegraphics[width=0.99\\textwidth]{Spectrometer.eps}\n  \\caption{\\small\nThe fringe tracking spectrometer of GRAVITY being under \nconstruction at the University of Cologne (Straubmeier et al. 2012).\n}\n  \\label{Fig:Gravity}    \n\\end{figure}\n\n\\section{Black Hole Laboratory: New instrumentation and perspective}\nAn improved link between theory and observations will be provided by new and upcoming \ninstrumentation such as the future capability of measuring polarization in the X-ray \ndomain (Soffitta et al. 2012, 2013),\nand direct VLBI imaging of the event horizon region for e.g. M87 as a representative of the largest\nsuper massive black holes, and SgrA* as a representative for low-mass SMBHs \n(Dexter et al. 2012, Fish et al. 2014, Broderick et al. 2014).\nThe SKA (Square Kilometer Array) will be key instrument to stablish\nthe census of pulsars at the Galactic Center, to determine the star\nformation history, and to provide the required precision to determine\nthe spin and quadrupole of the central BH, testing the theories\nof gravity (e.g. Aharonian et al. 2013).\n\nThe beam combining instrument GRAVITY\\footnote{http:\/\/www.mpe.mpg.de\/2441478\/Team}\n at the VLTI (P.I. Frank Eisenhauer, MPE, Garching)\nwill be able to measure the NIR image centroid paths during flux density excursions of SgrA*\n(Eisenhauer et al. 2011, Straubmeier et al. 2012, Eckart et al. 2010, 2012).\nThese paths will depend on the geometrical structure and time evolution of the \nemitting region i.e. spot shape, e.g. presence of a torus or spiral-arm patterns, an emerging jet component.\nFig.~\\ref{Fig:Gravity} shows the fringe tracking spectrometer of GRAVITY being under \nconstruction at the University of Cologne. Operating on six interferometric baselines, \ni.e. using all four UTs, the 2nd generation VLTI instrument GRAVITY will deliver narrow \nangle astrometry with 10$\\mu$as accuracy at the infrared K-band.\n\nWhile most of the mentioned geometries are currently able to fit the observed variable emission from SgrA*, \nfuture NIR interferometry with GRAVITY at the VLTI will break some of the degeneracies \nbetween different emission models. \nGRAVITY will be able to \ndetect the positional shift of the photo-center of a flare at the Galactic Center \nwithin the $\\sim$20~min orbital time scale of a source component close to the last stable orbit, \nusing the flares as dynamical probes of the gravitational field around SgrA*. \nIn particular GRAVITY observations of polarized NIR light could reveal a clear centroid \ntrack of bright spot(s) orbiting in the mid-plane of the accretion disk (e.g. Zamaninasab et al. 2010, 2011).\nA non-detection of centroid shifts may point at a multi-component \nmodel or spiral arms scenarios. However, a clear wander between alternating centroid positions \nduring the flares will strongly support the idea of bright long-lived spots occasionally orbiting \nthe central black hole. \n \n\n\n\n\\vspace*{1cm}\n\\noindent\n{\\small\n{\\bf Acknowledgments:}\nIn this article we summarize contributions on the Galactic Center, held at \nthe conference 'Equations of Motion in Relativistic Gravity' at the \nPhysikzentrum Bad Honnef, Bad Honnef, Germany, (February 17-23, 2013)\n\\footnote{http:\/\/www.puetzfeld.org\/eom2013.html}\nand the COST action MP0905 - \"Black Holes in a Violent Universe\" - Galactic Center  \nGC2013 conference\\footnote{http:\/\/www.astro.uni-koeln.de\/gc2013  and http:\/\/galacticcenter.iaa.es\/}\ntitled 'The Galactic Center Black Hole Laboratory' held\nat the Instituto de Astrofisica de Andalucia (IAA-CSIC) in Granada, Spain \n(November 19 - 22, 2013).\nThe main goal of the COST conference and the contribution in Bad Honnef \nwas to focus on the new observational and\ntheoretical results that are linked to the Galactic Center and to point out\npossible consequences for our understanding of galactic nuclei in general.\nThe COST conference brought together scientists working on all mass-scales of Black Holes \n(Quantum to super-massive), observers and theoreticians as well as particle physicists.\n\n\nWe are grateful to D. Merritt for constructive comments on the draft.\nthank especially the Local Organizing Committees of the Bad Honnef and the Granada conferences.\nWe thank especially the Local Organizing Committees of the Bad Honnef and the Granada conferences.\nFor Bad Honnef we thank the organizers D. Puetzfeld, \nC. Laemmerzahl, B.F. Schutz as well as the \nstaff of the Physikzentrum Bad Honnef, Bad Honnef, Germany.\nThe Granada conference was hosted by and held at the \nInstituto de Astrofisica de Andalucia (IAA-CSIC), Spain.\nWe thank especially the Local Organizing Committee at the IAA-CSIC that was lead by \nDr. Antxon Alberdi (antxon@iaa.es).\nThe Granada conference was also closely linked to and substantially supported by the COST Action \nMP1104 - \"Polarization as a tool to study the Solar System and beyond\", the \nFP7 action \"Probing Strong Gravity by Black Holes Across the Range of Masses\"\n[FP7-SPACE-2012-1 (FP7-312789)], and the \ncollaborative research center SFB956 \"Conditions and Impact of Star \nFormation - Astrophysics, Instrumentation and Laboratory Research\".\nWe also thank RadioNet for support.\nWe are also grateful to all SOC members of the Granada conference:\nVladimir Karas, Michal Dovciak, Devaky Kunneriath, Delphine Porquet, Andreas Eckart, Silke Britzen,\nAnton Zensus, Michael Kramer, Wolfgang Duschl, Antxon Alberdi , Rainer Sch\\\"odel, Murat H\\\"udaver,\nCarole Mundell, Farhad Yusef-Zadeh, Andrea Ghez, Mark Morris.\nThis work was supported\nin part by the Deutsche Forschungsgemeinschaft (DFG) via the Cologne Bonn\nGraduate School (BCGS), the Max Planck Society through the International Max\nPlanck Research School (IMPRS) for Astronomy and Astrophysics, as well as\nspecial funds through the University of Cologne. \n}\n\n\n\\vspace*{1cm}\n\\noindent\n{\\bf References:}\n\\par\\noindent\\hangindent 15pt {}{ Aharonian, F., et al.,\"Pathway to the Square Kilometer Array\", The German White Paper, 2013, arXiv:1301.4124}\n\\par\\noindent\\hangindent 15pt {}{ Akiyama, K.; Takahashi, R.; Honma, M.; Oyama, T.; Kobayashi, H.; 2013a, PASJ 65, 91}\n\\par\\noindent\\hangindent 15pt {}{ Akiyama, K.; Kino, M.; et al., 2013b, arXiv1311.5852A}\n\\par\\noindent\\hangindent 15pt {}{ Antonini \\& Merritt 2013, ApJ 763 ,L10}\n\\par\\noindent\\hangindent 15pt {}{ Baganoff, F. 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+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\newtheorem{teo}{Theorem}[section]\n\\newtheorem{lema}{Lemma}[section]\n\\newtheorem{prop}{Proposition}[section]\n\\newtheorem{defi}{Definition}[section]\n\\newtheorem{obs}{Remark}[section]\n\\newtheorem{coro}{Corollary}[section]\n\n\n\n\n\\begin{document}\n\n\n\n\t\n\t\\title[ Spectral stability of periodic waves.]\n\t{On the spectral stability of periodic traveling waves for the\n\t\tcritical Korteweg-de Vries and Gardner equations}\n\t\n\t\n\t\n\t\\subjclass[2000]{76B25, 35Q51, 35Q53.}\n\t\n\t\\keywords{spectral stability, spectral instability, periodic waves, KdV\n\t\ttype equations.}\n\t\\thanks{{\\it Date}: April, 2021.}\n\t\n\t\\maketitle\n\t\n\t\n\t\\begin{center}\n\t\t{\\bf F\\'abio Natali}\n\t\t\n\t\t{\\scriptsize    Department of Mathematics, State University of\n\t\t\tMaring\\'a, Maring\\'a, PR, Brazil. }\\\\\n\t\t{\\scriptsize fmanatali@uem.br }\n\t\t\n\t\t\n\t\t\n\t\t\\vspace{3mm}\n\t\t\n\t\t{\\bf Eleomar Cardoso Jr.}\n\t\t\n\t\t{\\scriptsize   Federal University of Santa Catarina, Blumenau, SC, Brazil.}\\\\\n\t\t{\\scriptsize eleomar.junior@ufsc.br}\n\t\t\n\t\t\n\t\t\\vspace{3mm}\n\t\t\n\t\t{\\bf Sabrina Amaral}\n\t\t\n\t\t{\\scriptsize    Department of Mathematics, State University of\n\t\t\tMaring\\'a, Maring\\'a, PR, Brazil. }\\\\\n\t\t{\\scriptsize sabrinasuelen20@gmail.com}\n\t\t\n\t\\end{center}\n\t\n\t\n\t\\begin{abstract}\n\t\tIn this paper, we determine spectral\n\t\tstability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our paper concerns in solving some auxiliary initial value problems to obtain the spectral stability.\n\t\\end{abstract}\n\t\n\t\\section{Introduction}\n\t\n\tSpectral stability of periodic waves is a subject of research\n\tin which several mathematicians have been interested in the last years. In\n\tthis paper, we establish this property for the critical Korteweg-de Vries (critical KdV henceforth) equation\n\t\\begin{equation}\\label{4kdv}\n\t\tu_t+(u^5)_x+u_{xxx}=0,\n\t\\end{equation}\n\tand the Gardner equation\n\t\\begin{equation}\\label{gardner1}\n\t\tu_t+(u^2)_x+(u^3)_x+u_{xxx}=0.\n\t\\end{equation}\n\t\\indent Both equations are examples of the generalized Korteweg-de Vries equation (gKdV henceforth), that is,\n\t\\begin{equation}\\label{gkdv} u_t+(g(u)+u_{xx})_x=0,\\ \\ \\ \\ \\ \\ \\ \\\n\t(x,t)\\in\\mathbb{R}\\times\\mathbb{R}, \\end{equation}\n\twhere $g:\\mathbb{R}\\rightarrow\\mathbb{R}$ is a smooth real function. This model is very important in several physical settings. For instance, it appears in shallow water surfaces, in internal water waves, in nonlinear optics, in the\n\tevolution of ion-acoustic waves, in unmagnetized plasma, and in\n\tnonlinear hydromagnetic waves in a cold collisionless plasma\n\t(\\cite{be1}, \\cite{be2}, \\cite{Dodd}, \\cite{kdv}).\\\\\n\t\\indent The general equation above admits traveling waves of\n\tthe form $u(x,t)=\\phi(x-\\omega t)$, where $\\omega$ indicates the\n\twave-speed. By substituting this type of solution into $(\\ref{gkdv})$\n\tand integrating it once, we obtain\n\t\\begin{equation}\\label{travkdv}\n\t\t-\\omega\\phi+\\phi''+g(\\phi)+A=0,\n\t\\end{equation} where $A$ is an arbitrary integration constant and\n\t$\\phi=\\phi_{(\\omega,A)}$ has $L-$periodic boundary conditions.\\\\\n\t\\indent Now, we present the basic framework about spectral stability, as well as some important contributions concerning this subject. The problem of spectral instability for the gKdV equation\n\tconsists in proving the existence of $\\lambda\\in\\mathbb{C}$ with\n\t$\\mbox{Re}(\\lambda)>0$ such that its corresponding eigenfunction\n\t$u\\neq0$ satisfies the linearized equation \\begin{equation}\\label{linprob}\n\t\\partial_x\\mathcal{L}u=\\lambda u.\n\t\\end{equation} Here,\n\t$\\mathcal{L}$ denotes the linearized operator around the traveling\n\twave $\\phi$ defined in $L^2$ given by \\begin{equation}\\label{linop}\n\t\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\phi),\\end{equation}\n\twhere $g'$ indicates the derivative of $g$ in terms of $\\phi$. In the affirmative case, $\\phi$ is said to be spectrally unstable. Otherwise, the periodic wave $\\phi$ is said to be spectrally stable\n\tif the spectrum of $\\partial_x\\mathcal{L}$ is entirely  contained in the imaginary axis of the complex plane $\\mathbb{C}$.  \\\\\n\t\\indent If we restrict to the case of solitary\n\twaves, $J=\\partial_x$ is a one-to-one operator with no  bounded inverse. This fact prevents the use of classical methods of spectral instability as in \\cite{grillakis1}. However,\n\tsufficient conditions have been established by some contributors to overcome this difficulty. For instance,\n\tin \\cite{kap} the authors determined results of spectral\n\tstability related to the problem $(\\ref{linprob})$ by using the\n\tKrein-Hamiltonian instability index. Moreover, it is possible to adapt the method to\n\tconclude similar facts for the BBM-type problems \\begin{equation}\\label{BBM}\n\tu_t+u_x-u_{txx}+(g(u))_x=0,\\end{equation} and the fractional models\n\trelated to those equations. In \\cite{lopes} the author presented\n\tsufficient conditions for the linear instability by\n\tusing the semigroup theory. Interesting results were also given by \\cite{A} and \\cite{lin}.\\\\\n\t\\indent In periodic setting we have the work \\cite{DK}, where sufficient conditions for the spectral stability\/instability have been determined.\n\tHowever in such case, since $J$ is not a one-to-one operator, the authors have\n\tconsidered the modified problem \\begin{equation}\\label{modspecp1}\n\tJ\\mathcal{L}\\big|_{H_0}u=\\lambda u, \\end{equation} where $H_0\\subset\n\tH:=L_{per}^2([0,L])$ is the closed subspace given by\n\t$$H_0=\\left\\{f\\in L^2([0,L]);\\ \\int_0^Lf(x)dx=0\\right\\}.$$ The\n\tKrein-Hamiltonian index formula was applied to deduce the spectral\n\tstability of periodic waves for the equation\n\t$(\\ref{gkdv})$ with $g(s)=\\pm s^3$. However, it was necessary to know the behavior of the first\n\tfive eigenvalues of the linear operator $\\mathcal{L}$ in\n\t$(\\ref{linop})$. \\\\\n\t\\indent In our analysis, the previous knowledge of the periodic wave\n\tis not necessary but it can be useful in order to obtain the spectral stability\/instability. We use a different way to compute the Krein-Hamiltonian index formula for some specific examples but the method can be adapted to other models contained in the regime of $(\\ref{gkdv})$ and related equations. Presented here are the critical KdV and Gardner equations.\\\\\n\t\\indent We are going to illustrate our two basic examples. First, if the critical KdV is considered, we prove the spectral stability\/instability results associated with the positive and zero mean periodic waves. It is well known that both periodic waves appear when, in the equation $(\\ref{travkdv})$, $g(\\phi)=\\phi^5$ and $A=0$. Concerning positive and periodic waves, we determine our results  using the explicit solution determined in \\cite{AN2}. After that, we solve numerically some auxiliary initial value problems which give us the precise information about the Krein-Hamiltonian index formula in order to obtain the spectral stability. Explicit zero mean periodic waves were unknown in the current literature until now. To fill this gap, we present a cnoidal wave profile.\\\\\n\t\\indent For the case of positive periodic waves (see ($\\ref{dn4kdv}$)), it is expected that $\\ker(\\mathcal{L})=[\\phi']$ and $n(\\mathcal{L})=1$, where $n(\\mathcal{L})$ indicates the number of negative eigenvalues of $\\mathcal{L}$. When periodic waves with the zero mean property are considered (see $(\\ref{cnoid4kdv})$), we have $n(\\mathcal{L})=2$ and $\\ker(\\mathcal{L})=[\\phi']$. In the second case, and using an explicit solution, it has been determined the same spectral property for the case $g(s)= s^3$ and $A=0$ as determined \\cite{AN1} and \\cite{DK}. It is worth mentioning that the authors had in hands the behavior of the first five eigenvalues of the linearized operator $\\mathcal{L}$ to calculate the sign of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ using Fourier series. This quantity plays an important role to deduce spectral stability results for the gKdV equation in the sense that it is possible to identify an eventual existence of points $(\\omega,A)$ contained in the parameter regime satisfying $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$. The change of sign establishes a rupture on the spectral stability scenario when cnoidal waves for the modified KdV equation are considered (\\cite{AN1} and \\cite{DK}). Concerning our zero mean periodic waves for the critical KdV we obtain, in the line $(\\omega,0)$, a threshold value $\\omega_1>0$ such that $\\langle\\mathcal{L}^{-1}1,1\\rangle=0$. In addition, $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$, if $\\omega<\\omega_1$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$, if $\\omega>\\omega_1$. In the first case, the wave is spectrally stable and in the second one, spectrally unstable. For dnoidal waves, there is no threshold value $\\omega_1$ for the quantity $\\langle\\mathcal{L}^{-1}1,1\\rangle$ and the periodic waves are spectrally stable. Summarizing our results, we have the following theorem:\n\t\\begin{teo}\\label{teoest} Let $L>0$ be fixed.\\\\\n\t\ta) For all $\\omega>\\frac{\\pi^2}{L^2}$, positive and periodic waves of dnoidal type for the critical KdV equation are spectrally stable.\\\\\n\t\tb) There exists a unique $\\omega_1>\\frac{4\\pi^2}{L^2}$ such that the zero mean periodic waves of cnoidal type for the critical KdV equation are spectrally stable for $\\omega\\in \\left(\\frac{4\\pi^2}{L^2},\\omega_1\\right)$ and spectrally unstable for $\\omega>\\omega_1$.\n\t\\end{teo}\n\\begin{obs}\n\t  Theorem $\\ref{teoest}$-a) establishes the spectral stability of the periodic dnoidal waves. This solution first appeared in \\cite{AN2} and the authors established the existence of a unique $\\omega_0>\\frac{\\pi^2}{L^2}$ such that the dnoidal wave is orbitally stable for $\\omega\\in \\left(\\frac{\\pi^2}{L^2},\\omega_0\\right)$ (using the classical argument in \\cite{grillakis1}) and orbitally unstable for $\\omega>\\omega_0$ (employing an adaptation of the arguments in \\cite{bona}). Since the Cauchy problem for the equation $(\\ref{4kdv})$ is not globally well posed in the energy space $H_{per}^1([0,L])$, we are in conformity with the arguments in \\cite{AN2}. In fact, we are attesting for KdV type equations that spectral stability implies the orbital stability provided that the global well posed in the energy space $H_{per}^1([0,L])$ of the associated Cauchy problem is verified.\n\\end{obs}\n\n\t\\indent Next, we shall give few words about the Gardner equation. We construct explicit periodic waves with cnoidal profile by using the modified KdV equation and its corresponding cnoidal solution. In fact, if $\\phi$ is a solution of the equation $(\\ref{travkdv})$ with $g(s)=s^2+s^3$, thus $\\varphi=\\phi+\\frac{1}{3}$ is a periodic solution with cnoidal profile as $\\varphi(x)=d{\\rm cn}(ex,k)$ and for the corresponding modified KdV equation\n\t\\begin{equation}\\label{mkdv12}\n\t-\\varphi''+\\left(\\omega+\\frac{1}{3}\\right)\\varphi-\\varphi^3=0,\n\t\\end{equation}\n\twhere $d$ and $e$ are smooth functions depending on the wave speed $\\omega+\\frac{1}{3}$. In equation $(\\ref{mkdv12})$, $k\\in(0,1)$ is called modulus of the elliptic function.\\\\\n\t\\indent It is well known that equation $(\\ref{mkdv12})$ admits periodic waves with dnoidal and cnoidal profiles. The corresponding solution with dnoidal profile for the Gardner equation and its respective orbital stability have been determined in \\cite{AP2}. Our intention is to determine spectral stability results of the associated cnoidal profile and we also present a threshold value $\\omega_1$ such that $\\langle\\mathcal{L}^{-1}1,1\\rangle=0$ at $\\omega=\\omega_1$. More specifically, we obtain the same threshold value $\\omega_1$ as obtained for the cnoidal waves for the modified KdV equation and the reason for that concerns a connection between modified KdV and Gardner equations using the Galilean invariance $\\varphi=\\phi+\\frac{1}{3}$. This fact produces that the linearized operator associated to both periodic waves $\\varphi$ and $\\phi$ are the same. As a consequence, if $\\mathcal{L}_{\\varphi}$ is the corresponding linearized operator around $\\varphi$ for the modified KdV equation and $\\mathcal{L}$ the linearized operator around $\\phi$, we have $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$, that is, the sign of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ is determined just by analysing $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle$. Thus, $\\omega<\\omega_1$  implies that $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ (spectral stability) while $\\omega>\\omega_1$ gives us $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$ (spectral instability). Summarizing our results, we have:\n\t\\begin{teo}\\label{teoestG} Let $L>0$ be fixed. There exists a unique $\\omega_2>\\frac{4\\pi^2}{L^2}$ such that the zero mean periodic waves of cnoidal type for the Gardner equation are spectrally stable for $\\omega\\in \\left(\\frac{4\\pi^2}{L^2},\\omega_2\\right)$ and spectrally unstable for $\\omega>\\omega_2$.\n\t\\end{teo}\n\t\n\t\\indent This paper is organized as follows. In Section 2 we give the\n\tbasic framework of the spectral stability following the ideas in \\cite{DK}. In Section 3, we study the existence of periodic waves and\n\ttheir dependence with respect to the parameters, as well as the\n\tspectral analysis of the linearized operator. Finally, Section 4 is devoted to our applications.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\section{Basic Framework of Spectral Stability of Periodic Waves.}\n\t\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\tIn this section we present the basic framework established in\n\t\\cite{DK} which provides us a criterion for determining the spectral\n\tstability of periodic waves related to the abstract Hamiltonian\n\tequations of the form\n\t\n\t\\begin{equation}\\label{Hamilt} u_t=J\\mathcal{E}'(u)\\end{equation} defined on a Hilbert space\n\t$H$, where $J:H\\to {\\rm{range}}(H)\\subset H$ is a skew symmetric, and\n\t$\\mathcal{E}:H\\to\\mathbb{R}$ is a $C^2-$functional. We restrict\n\tourselves to the specific case when $J=\\partial_x$ and\n\t$\\mathcal{E}(u)=\\frac{1}{2}\\int_0^Lu_x^2-2G(u)dx$,\n\twhere $G'=g$. In that case, the equation $(\\ref{Hamilt})$ becomes the well known\n\tgeneralized Korteweg-de Vries equation as in $(\\ref{gkdv})$.\\\\\n\t\\indent Let us consider again the spectral problem related to the\n\tgeneralized KdV equation\n\t\n\t\\begin{equation}\\label{specp}\n\t\\partial_x\\mathcal{L}u=\\lambda u,\n\t\\end{equation} where\n\t$\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\omega,A,\\phi)$ is the\n\tlinearized operator around the periodic wave $\\phi$ which is\n\ta periodic traveling wave solution of the equation $(\\ref{travkdv})$.  As we have mentioned before, the standard theories\n\tof spectral instability of traveling waves for the abstract\n\tHamiltonian system as in \\cite{grillakis1} and\n\t\\cite{lopes} can not be applied in this context. To overcome this\n\tdifficulty, we are going to give a brief explanation of the results in \\cite{DK}.\n\tIndeed, let us consider the modified spectral problem obtained from\n\t$(\\ref{specp})$\n\t\n\t\\begin{equation}\\label{modspecp} J\\mathcal{L}\\big|_{H_0}u=\\lambda u,\n\t\\end{equation}\n\twhere $H_0\\subset H=L_{per}^2([0,L])$ is the closed subspace given by\n\t\n\t$$H_0=\\left\\{f\\in L^2([0,L]);\\ \\int_0^Lf(x)dx=0\\right\\}.$$\n\tFor a fixed period $L>0$, we need to assume in this whole section that:\\\\\n\t\n\t{\\rm (a1)} There exists a fixed pair $(\\omega_0,A_0)$ and $\\phi:=\\phi_{(\\omega_0,A_0)}$ smooth even periodic solution for the equation $(\\ref{travkdv})$. Moreover, we assume that\n\t$\\phi'$ has only two zeros in the interval $[0,L)$. \\\\\n\t\n\t{\\rm (a2)} $\\ker(\\mathcal{L})=[\\phi']$.\\\\\n\t\n\t\n\t\\indent Assumption {\\rm (a1)} implies, from the classical Floquet theory in \\cite{Magnus} that $n(\\mathcal{L})=1$ or $n(\\mathcal{L})=2$ where\n\t$n(\\mathcal{L})$ indicates the number of negative eigenvalues of the linearized operator $\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\phi)$. In addition, assumption {\\rm (a2)} allows us to deduce the existence of a non-periodic even solution $\\bar{y}$ which satisfies the Hill equation\n\t\\begin{equation}\\label{Hill1}\n\t-\\bar{y}''+\\omega\\bar{y} -g'(\\phi) \\bar{y}=0,\n\t\\end{equation}\n\twhere $\\{\\bar{y},\\phi'\\}$ is a fundamental set of solutions for the linear equation $(\\ref{Hill1})$.\\\\\n\t\\indent According with Theorem \\ref{teo2} determined in Section 3, one can see that assumption {\\rm (a2)} will provide us the existence of a smooth surface of even periodic waves which solves $(\\ref{travkdv})$ and defined in an open subset $\\mathcal{O}\\subset\\mathbb{R}^2$,\n\t$$(\\omega,A)\\in \\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in H_{per}^s([0,L]),\\ \\ s\\gg1,$$ all of them with the same period $L>0$. In what follows and in the whole paper, we shall not distinguish the periodic wave $\\phi$ for a fixed pair $(\\omega_0,A_0)$ and $\\phi$ for a pair $(\\omega,A)\\in\\mathcal{O}$ since both have the \\textit{same fixed period $L>0$}. The intention is to simplify our presentation with easier notations.\\\\\n\t\\indent Next, we describe the arguments in \\cite{DK}. For the spectral problem in $(\\ref{modspecp})$ let $k_r$ be the number of real-valued and positive eigenvalues (counting multiplicities). The quantity $k_c$ denotes the number of complex-valued eigenvalues with a positive real part. Since ${\\rm Im}(\\mathcal{L})=0$, where ${\\rm Im}(z)$ indicates the imaginary part of the complex number $z$, we see that $k_c$ is an even integer. For a self-adjoint operator $\\mathcal{A}$, let $n(\\langle w,\\mathcal{A}w\\rangle)$ be the dimension of the maximal subspace for which $\\langle w,\\mathcal Aw\\rangle<0$. Also, let $\\lambda$ be an eigenvalue and $E_{\\lambda}$ its corresponding eigenspace. The eigenvalue is said to have negative Krein signature if\n\t$$k_i^{-}(\\lambda):=n(\\langle w,(\\mathcal{L}\\big|_{H_0})\\big|_{E_{\\lambda}}w\\rangle)\\geq1,$$\n\totherwise, if $k_i^{-}(\\lambda)=0$, then the eigenvalue is said to have a positive Krein signature. If $\\lambda$ is geometrically and algebraically simple with the eigenfunction $\\psi_{\\lambda}$, then\n\t$$k_i^{-}(\\lambda)=\\left\\{\\begin{array}{llll}\n\t\t0,\\ \\langle \\psi_{\\lambda},(\\mathcal{L}\\big|_{H_0})\\psi_{\\lambda}\\rangle>0\\\\\n\t\t1,\\ \\langle \\psi_{\\lambda},(\\mathcal{L}\\big|_{H_0})\\psi_{\\lambda}\\rangle<0.\\end{array}\n\t\\right.$$\n\t\n\tWe define the total Krein signature as\n\t$$k_i^{-}:=\\sum _{\\lambda\\in i\\mathbb{R}\\backslash\\{0\\}}k_{i}^{-}(\\lambda).$$\n\tThe fact ${\\rm Im}(\\mathcal{L})=0$ implies that $k_i^-(\\lambda)=k_i^{-}(\\overline{\\lambda})$ and $k_i^{-}$ is an even integer.\\\\\n\t\n\tLet us consider\n\t\\begin{equation}\\label{I}\n\t\\mathcal{I}=\\langle \\mathcal{L}^{-1}1,1\\rangle.\n\t\\end{equation}\n\tIf $\\mathcal{I}\\neq0$, denote $\\mathcal{D}$ as the $2\\times 2-$matrix given by\n\t\\begin{equation}\\label{Dmatrix}\n\t\\mathcal{D}=\\frac{1}{\\langle\\mathcal{L}^{-1}1,1\\rangle}\\left[\\begin{array}{llll}\n\t\t\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle & & \\langle\\mathcal{L}^{-1}\\phi,1\\rangle\\\\\\\\\n\t\t\\langle\\mathcal{L}^{-1}\\phi,1\\rangle & & \\langle\\mathcal{L}^{-1}1,1\\rangle\\end{array}\\right]\n\t\\end{equation}\n\tWe obtain, then, the following results:\n\t\\begin{teo}\\label{krein}\n\t\tSuppose that assumptions {\\rm (a1)-(a2)} hold. If $\\mathcal{I}\\neq0$ and $\\mathcal{D}$ is non-singular (i.e. $\\det(\\mathcal{D})\\neq 0$) we have for the eigenvalue problem $(\\ref{modspecp})$\n\t\t$$k_r+k_c+k_{i}^{-}=n(\\mathcal{L})-n(\\mathcal{I})-n(\\mathcal{D}).$$\n\t\tThe nonpositive integer $K_{{\\rm Ham}}=k_r+k_c+k_{i}^{-}$ is called \\textit{Hamiltonian-Krein index}.\n\t\\end{teo}\n\t\\begin{proof}\n\t\tSee Theorem 1 in \\cite{DK}.\n\t\\end{proof}\n\t\n\t\\begin{coro}\\label{coroest}\n\t\tUnder the assumptions of Theorem $\\ref{krein}$, if $k_c=k_r=k_{i}^{-}=0$ the periodic wave $\\phi$ is spectrally stable. In addition, if $K_{{\\rm Ham}}=1$ the refereed periodic wave is spectrally unstable.\n\t\\end{coro}\n\t\\begin{proof}\n\t\tSince $k_c=k_r=0$, there is no eigenvalues with positive real part for the problem $(\\ref{modspecp})$ and $\\phi$ is spectrally stable since the total Krein signature is zero. Now, if $K_{{\\rm Ham}}=1$ we deduce that $k_r=1$ since $k_c$ and $k_{i}^{-}$ are even nonnegative integers. So, operator $J\\mathcal{L}$ presented in the spectral problem $(\\ref{modspecp})$ has a positive eigenvalue which enable us to deduce the spectral instability of the periodic wave.\n\t\\end{proof}\n\t\n\tWe shall present some considerations concerning the result determined in Corollary $\\ref{coroest}$ applied to the case of the generalized KdV equation in $(\\ref{gkdv})$. In fact, by assuming that assumption {\\rm (a2)} is verified one has\n\t$$\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2dx,\\ \\ \\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx,$$\n\tand\n\t$$\\mathcal{I}=\\langle\\mathcal{L}^{-1}1,1\\rangle=\\frac{d}{dA}\\int_0^L\\phi dx.$$\n\tSo, we have\n\t$$n(\\mathcal{I})=\\left\\{\\begin{array}{llll}\n\t\t0,\\ \\frac{d}{dA}\\int_0^L\\phi dx\\geq0\\\\\\\\\n\t\t1,\\ \\frac{d}{dA}\\int_0^L\\phi dx<0.\\end{array} \\right.$$\n\t\n\tOn the other hand, in order to determine $n(\\mathcal{D})$ it is necessary\n\tto analyze the quantity $\\mathcal{D}$.\n\tIn fact, if $\\mathcal{D}<0$ we have that the associated matrix has a\n\tpositive and a negative eigenvalue and therefore $n(\\mathcal{D})=1$.\n\tHowever, if $\\mathcal{D}>0$, it is not possible to directly decide\n\tabout the quantity $n(\\mathcal{D})$ since we could have\n\t$n(\\mathcal{D})=0$ (two positive eigenvalues for the associated matrix) or $n(\\mathcal{D})=2$\n\t(two negative eigenvalues). In the next section,\n\twe determine sufficient conditions to obtain assumptions {\\rm\n\t\t(a1)-(a2)} for a general class of second order differential equations. In addition, we present two useful initial value problems used to determine a precise way to calculate $\\mathcal{I}$ and $\\mathcal{D}$.\n\t\n\t\n\t\n\t\\section{Basic Framework on Spectral Analysis.}\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\t\\indent In a general setting (without considering the arguments in the last section for a while), let us suppose that $\\phi$ is an even $L-$periodic solution of the general equation\n\t\\begin{equation}\\label{ode}\n\t\t-\\phi''+f(\\omega,A,\\phi)=0,\n\t\\end{equation}\n\twhere $f$ is a smooth function depending on $(\\omega,A,\\phi)$ and $(\\omega,A)$ is an element of an admissible set $\\mathcal{P}\\subset\\mathbb{R}^2$. This means that $\\mathcal{P}$ contains all the pairs $(\\omega,A)$ where $\\phi$ is a periodic solution of $(\\ref{ode})$.\\\\\n\t\\indent Let $\\mathcal{L}$ be the linearized equation around $\\phi$, where $\\phi$ is a periodic\n\tsolution of (\\ref{ode}) of period $L$. The linearized operator \\begin{equation}\n\t\\mathcal{L}(y) = - y'' + f'(\\omega, A, \\phi)\\, y,\n\t\\;\\;\\; (\\omega, A) \\in \\mathcal{P} \\label{hill} \\end{equation} is a Hill\n\toperator and $f'$ is the derivative in terms of $\\phi$. According to \\cite{Haupt} and \\cite{Magnus},\n\tthe spectrum of $\\mathcal{L}$ is formed by an unbounded\n\tsequence of real numbers\n\t\\[\n\t\\lambda_0 < \\lambda_1 \\leq \\lambda_2 < \\lambda_3 \\leq \\lambda_4\\;\\;\n\t...\\; < \\lambda_{2n-1} \\leq \\lambda_{2n}\\; \\cdots,\n\t\\]\n\twhere equality means that $\\lambda_{2n-1} = \\lambda_{2n}$  is a\n\tdouble eigenvalue. The spectrum of $\\mathcal{L}$ is\n\tcharacterized by the number of zeros of the eigenfunctions, if $\\Psi$\n\tis an eigenfunction for the eigenvalue $\\lambda_{2n-1}$ or\n\t$\\lambda_{2n}$, then $\\Psi$  has exactly $2n$ zeros in the half-open\n\tinterval $[0,L)$.\n\t\n\tIn order to apply the general theory of orbital stability,\n\t\\cite{bona}, \\cite{grillakis1} and \\cite{W1}, the spectrum of\n\t$\\mathcal{L}$ is of main importance and also of the major\n\tdifficulty in the applications. It is necessary to know exactly the\n\tnon-positive spectrum; more precisely, it is necessary to know the\n\tinertial index $in(\\mathcal{L})$ of\n\t$\\mathcal{L}$, where $in(\\mathcal{L})$ is\n\ta pair of integers $(n,z)$, where $n$ is the dimension of the\n\tnegative subspace of $\\mathcal{L}$ and $z$ is the\n\tdimension of the null subspace of $\\mathcal{L}$.\n\t\n\tThe results of this section are based on \\cite{natali2}, \\cite{neves} and \\cite{neves1} and the first\n\tone concerns the invariance of the index with respect to the parameters. Since the derivative  $\\phi'$ is an eigenfunction related to $\\lambda =0$ for every  $(\\omega, A)\n\t\\in \\mathcal{P}$, we can state the following result.\n\t\n\t\\begin{teo}\n\t\tLet $\\phi$ a smooth $L-$periodic solution of the equation $(\\ref{ode})$.\n\t\tThen the\n\t\tfamily of operators $\\mathcal{L}(y) = - y'' +\n\t\tf'(\\omega, A, \\phi)\\, y $ is isoinertial with respect to $(\\omega,A)$ in the parameter regime. \\label{teo0}\n\t\\end{teo}\n\t\\begin{proof}\n\t\tSee \\cite{natali2} and \\cite{neves1}.\n\t\\end{proof}\n\t\n\tIn order to calculate the inertial index of $\\mathcal{L}$ for a fixed value of $(\\omega_0,A_0)$, we shall consider the\n\tauxiliary function $\\bar{y}$ the unique solution of the problem \\begin{equation}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\t- \\bar{y}'' + f'(\\omega_0, A_0, \\phi) \\bar{y} = 0 \\\\\n\t\t\\bar{y}(0) = - \\frac{1}{\\phi''(0)} \\\\\n\t\t\\bar{y}'(0)=0,\n\t\\end{array} \\right.\n\t\\label{y} \\end{equation} and also the constant  $\\theta$ given by \\begin{equation} \\theta=\n\t\\frac{ \\bar{y}'(L)}{\\phi''(0)}, \\label{theta} \\end{equation} where $L$ is\n\tthe period of $\\phi=\\phi_{(\\omega_0,A_0)}$.\n\t\n\tWe know that the derivative $\\phi'$ is an eigenfunction\n\tfor  the eigenvalue $\\lambda = 0$, and also that $\\phi'(x)$\n\thas exactly two zeros in the half-open interval $[0, L)$.\n\tTherefore we have three possibilities:   \\begin{itemize}\n\t\t\\item[i)] $\\lambda_1 = \\lambda_2 = 0 \\Rightarrow in(\\mathcal{L}) = (1,2)$,\\\\\n\t\t\\item[ii)] $\\lambda_1 = 0 < \\lambda_2 \\Rightarrow in(\\mathcal{L}) = (1,1)$,\\\\\n\t\t\\item[iii)] $\\lambda_1 < \\lambda_2 = 0 \\Rightarrow  in(\\mathcal{L}) = (2,1)$,\n\t\\end{itemize}\n\t\n\tThe method we use to decide and calculate the inertial index is\n\tbased on Lemma 2.1 and Theorems 2.2 and 3.1 of \\cite{neves}. This\n\tresult can be stated as follows.\n\t\n\t\n\t\\begin{teo}\n\t\tLet $\\theta$ be the constant given by (\\ref{theta}), then the\n\t\teigenvalue $\\lambda=0$ is simple if and only if $ \\theta \\neq 0$.\n\t\tMoreover, if $\\theta \\neq 0$, then  $ \\lambda_{1}=0$ if $\\theta <\n\t\t0$, and $ \\lambda_{2}=0$ if $\\theta > 0$. \\label{teo1}\n\t\\end{teo}\n\t\\begin{flushright}\n\t\t$\\square$\n\t\\end{flushright}\n\t\n\t\n\t\n\t\n\tLet $L>0$ be fixed. In order to show our spectral stability results, it is convenient to show the existence\n\tof a family $\\phi$ of $L$-periodic solutions for the\n\tequation (\\ref{ode}) that smoothly depends on the parameters\n\t$(\\omega,A)$, for $(\\omega,A)$  in an open set $\\mathcal{O} \\subset\n\t\\mathcal{P}$.\n\t\n\t\n\t\\begin{teo}\n\t\tLet $\\phi_{(\\omega_0,A_0)}$ be an\n\t\teven periodic solution of $(\\ref{ode})$ defined in a fixed pair $(\\omega_0,A_0)$ in the parameter regime. If $\\theta \\neq 0$, where\n\t\t$\\theta$ is  the constant given in Theorem \\ref{teo1}, and $L$ is\n\t\tthe period of $\\phi_{(\\omega_0,A_0)}$, then there is an open\n\t\tneighborhood $\\mathcal{O}$ of $(\\omega_0,A_0)$,\n\t\tand a family $\\phi_{(\\omega,A)} \\in H_{per,e}^2([0,L])$ of\n\t\t$L$-periodic solutions of $(\\ref{ode})$, which smoothly depends on\n\t\t$(\\omega,A) \\in \\mathcal{O}$ in a $C^1$ manner. \\label{teo2}\n\t\\end{teo}\n\t\\begin{proof}\n\t\tLet $\\mathcal{P}$ the set of parameters and $\\mathcal{F}$ be the operator given by the equation (\\ref{ode}) restrict\n\t\tto the even functions, precisely, $\\mathcal{F}:\\mathcal{P}\\times\n\t\tH_{per,e}^2([0,L])  \\rightarrow  L_{per,e}^2([0,L]) $, \\begin{equation}\n\t\t\\mathcal{F}(\\omega,A,\\phi) = -\\phi''+ f(\\omega,A,\\phi). \\label{eq31}\n\t\t\\end{equation} Then $\\mathcal{F}(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)}) = 0$,\n\t\tsince $\\phi_{(\\omega_0,A_0)}$ is an even periodic solution of the\n\t\tequation (\\ref{ode}). If $\\theta \\neq 0 $, Theorem \\ref{teo1}\n\t\timplies that $ \\mathcal{L}_{(\\omega_0, A_0)}(y) = - y'' +\n\t\tf'(\\omega_0, A_0, \\phi_{(\\omega_0,A_0)})\\, y$, has an one-dimensional nullspace;\n\t\tand from the invariance, this nullspace is spanned by\n\t\t$\\phi'_{(\\omega_0, A_0)}$. Since $\\phi'_{(\\omega_0, A_0)}$ is odd,\n\t\tit is not an element of $H_{per,e}^2([0,L])$, it follows that $\n\t\t\\mathcal{F}_{\\phi}(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)}) =\n\t\t\\mathcal{L}_{(\\omega_0, A_0)}:H_{per,e}^2([0,L]) \\subset L_{per,e}^2([0,L])\\rightarrow\n\t\tL_{per,e}^2([0,L]) $ is invertible and its inverse is bounded.\n\t\tTherefore, the results of the Theorem \\ref{teo2} follows from the\n\t\timplicit function theorem. See Theorem 15.1 and Corollary 15.1 of\n\t\t\\cite{Deimling}.\n\t\\end{proof}\n\t\n\tNext, we turn back to the setting contained in Section 2 by considering $(\\ref{ode})$ as\n\t\n\t\\begin{equation}\\label{odekdv}\n\t\t-\\phi''+\\omega\\phi-g(\\phi)-A=0.\n\t\\end{equation}\n\tWe assume that $\\theta\\neq0$ in a single point $(\\omega_0,A_0)$  in the parameter regime. By Theorem $\\ref{teo2}$ we can define\n\t\\[\n\t\\psi = \\frac{\\partial \\phi}{\\partial \\omega} \\qquad\n\t\\mbox{and} \\qquad \\eta= \\frac{\\partial \\phi}{\\partial\n\t\tA}.\n\t\\]\n\t\n\t\n\tAgain by Theorem \\ref{teo2}, it is easy to see that $\\psi$\n\tabove is an even periodic smooth function which satisfies, for the case of the equation $(\\ref{gkdv})$\n\t\\begin{equation} -\n\t\\psi'' + \\omega \\psi -g'(\\phi)\\psi= -\n\t\\phi. \\label{psi1} \\end{equation} In addition, $\\eta$ is also an\n\teven periodic function satisfying\n\t\n\t\\begin{equation} - \\eta'' + \\omega \\eta\n\t-g'(\\phi)\\eta= 1. \\label{eta1} \\end{equation}\n\t\n\t\\begin{obs}\\label{obsiso}\n\t\tTheorem $\\ref{teo0}$ gives us an important property concerning the quantity and multiplicity of the first two eigenvalues associated to the linearized operator $\\mathcal{L}$ defined in $(\\ref{linop})$. Indeed, if $\\theta\\neq0$ in a certain point $(\\omega_0,A_0)$ in the parameter regime $\\mathcal{P}$, we can conclude that the kernel of $\\mathcal{L}$ is simple and $n(\\mathcal{L})$ is constant for all $(\\omega,A)$ in an open subset contained in $\\mathcal{P}$, that is, the value $in(\\mathcal{L})$ is constant in this subset.\n\t\t\n\t\\end{obs}\n\t\n\tNext result gives us an immediate converse of Theorem $\\ref{teo2}$ for the case $g(s)=s^{p+1}$.\n\t\\begin{prop}\\label{propsimp}\n\t\tLet $\\widetilde{\\mathcal{O}}\\subset\\mathbb{R}^2$ be an open subset. Suppose that $(\\omega,A)\\in\\widetilde{\\mathcal{O}}\\mapsto\\phi_{(\\omega,A)}$ is a smooth surface of even (odd) periodic traveling wave solutions which solves $(\\ref{odekdv})$ with $g(s)=s^{p+1}$  all of them with the same fixed period $L>0$. Then, $\\ker(\\mathcal{L})=[\\phi']$ and the value $n(\\mathcal{L})$ is constant for all $(\\omega,A)\\in\\widetilde{\\mathcal{O}}$. The same result remains valid for the case $A\\equiv0$, by considering $\\widetilde{I}\\subset\\mathbb{R}$ an open subset and $\\omega\\in \\widetilde{I}\\mapsto\\phi_{\\omega}$ a smooth curve of even periodic waves.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tTo simplify the notation, let us denote $\\phi=\\phi_{(\\omega,A)}$ and consider $\\{\\phi',\\bar{y}\\}$ the fundamental set of solutions related to the equation $-y''+\\omega y - (p+1)y^p=0$. By contradiction, assume that $\\bar{y}$ is $L-$periodic. Since $\\phi'$ is odd, the arguments in \\cite{Magnus} give us that $\\bar{y}$ can be considered even. The Wronskian of the set $\\{\\phi',\\bar{y}\\}$ and denoted by $\\mathcal{W}(\\phi',\\bar{y})$ satisfies $\\mathcal{W}(\\phi',\\bar{y})=1$ over $[0,L]$ (see \\cite{Magnus}). Moreover, since $\\bar{y}$ and $\\phi'$ are both periodic functions, we obtain from $(\\ref{travkdv})$ that\n\t\t\\begin{equation}\\label{wronsk}\\begin{array}{lllll}\n\t\t\t\tL&=&\\displaystyle\\int_0^{L}\\mathcal{W}(\\phi',\\bar{y})dx=-2\\int_0^{L} \\bar{y}\\phi''dx=-2\\int_0^{L}\\bar{y}\\left(\\omega\\phi-\\phi^{p+1}-A\\right)dx\\\\\\\\\n\t\t\t\t&=&\\displaystyle-2\\omega\\int_0^{L} \\bar{y}\\phi dx+2\\int_0^{L} \\bar{y}\\phi^{p+1}dx+2A\\int_0^{L}\\bar{y}dx.\\end{array}\n\t\t\\end{equation}\n\t\tSince $\\mathcal{L}\\phi=A-p\\phi^{p+1}$, by $(\\ref{psi1})$ and $(\\ref{eta1})$ we obtain from $(\\ref{wronsk})$\n\t\t\t\\begin{equation}\\label{wronsk1}\n\t\t\t\tL=2\\omega \\langle \\mathcal{L}\\psi,\\bar{y}\\rangle-\\frac{2}{p}\\langle\\mathcal{L}\\phi,\\bar{y}\\rangle+2A\\left(\\frac{1}{p}+1\\right)\\langle\\mathcal{L}\\eta,\\bar{y}\\rangle.\n\t\t\t\\end{equation}\n\\indent The fact that $\\mathcal{L}\\bar{y}=0$ allows us to deduce from the self-adjointness of $\\mathcal{L}$ and $(\\ref{wronsk1})$ that $L=0$. This contradiction shows that $\\ker(\\mathcal{L})=[\\phi']$.\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\tLet us suppose that $\\theta\\neq0$ in a single point $(\\omega_0,A_0)$ in the parameter regime. By Theorem $\\ref{teo2}$ we are able to determine the initial condition\n\t$\\psi(0)$ at the point $(\\omega_0,A_0)$. To do so, we multiply equation (\\ref{psi1}) by $\\bar{y}$, where\n\t$\\bar{y}$ is given in (\\ref{y}), and integrate the\n\tfirst term twice. We get\n\t$$\n\t- \\int_0^{L}  \\phi_{(\\omega_0,A_0)} \\;\n\t\\bar{y}\\; dx = \\psi(L) \\bar{y}'(L)=\\psi(0) \\bar{y}'(L).\n\t$$\n\tSimilarly, from $(\\ref{eta1})$ one has\n\t$$\n\t\\int_0^{L}\\bar{y}\\; dx =  \\eta(L) \\bar{y}'(L)=\\eta(0) \\bar{y}'(L).\n\t$$\n\t\n\t\n\t\n\tSince $\\theta \\neq 0$ we conclude that $\\bar{y}'(L) \\neq 0$ and then\n\t$\\psi(x)$ and $\\eta(x)$ are obtained by solving, respectively, the\n\tfollowing initial value problems \\begin{equation} \\left\\{\n\t\\begin{array}{lllllllllllll}\n\t\t- \\psi'' + \\omega_0 \\psi -g'(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)})\\psi= -\n\t\t\\phi_{(\\omega_0,A_0)}  \\\\\n\t\t\\psi(0) = - \\frac{1}{\\bar{y}'(L)}  \\int_0^{L}  \\phi_{(\\omega_0,A_0)} \\; \\bar{y}\\; dx \\\\\n\t\t\\psi'(0)=0,\n\t\\end{array} \\right.\\\n\t\\left\\{\n\t\\begin{array}{l}- \\eta'' + \\omega_0 \\eta -g'(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)})\\eta=\n\t\t1  \\\\\n\t\t\\eta(0) =  \\frac{1}{\\bar{y}'(L)}  \\int_0^{L}  \\bar{y}\\; dx \\\\\n\t\t\\eta'(0)=0.\n\t\\end{array} \\right.\n\t\\label{psi2}\n\t\\end{equation}\n\tBoth initial value problems are very useful to determine $\\mathcal{I}$ and $\\mathcal{D}$ given in Section 2. \n\t\\section{Applications - Spectral stability of periodic waves}\n\t\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\t\\subsection{Case $g(s)=s^{p+1}$ - Existence of periodic waves using variational methods.} Using a variational method, we establish the existence of periodic waves for the equation $(\\ref{odekdv})$. The main advantage of the approach presented here is that the quantity of negative eigenvalues of $\\mathcal{L}$ in $(\\ref{linop})$ defined for periodic waves $\\phi$ in a single point $(\\omega_0,A_0)\\in\\mathcal{P}$ is precisely determined. Thus, in this specific case, Remark $\\ref{obsiso}$ can be used to deduce the quantity and multiplicity of negative eigenvalues for all $(\\omega,A)$.\n\t\n\t\n\t\\indent Let $L>0$ be fixed. For each $\\gamma>0$, we define the set\n\t$$Y_{\\gamma}=\\left\\{u\\in H_{per}^1([0,L]);\\ \\int_0^{L}u^{p+2}dx=\\gamma\\right\\},$$\n\twhere $p$ is an even integer. Our first goal is to find a minimizer of the constrained minimization problem\n\t\\begin{equation}\n\t\t\\label{infB}\n\t\tm=m_{\\omega}=\\inf_{ u\\in Y_{\\gamma}}\\mathcal{B}_{\\omega}(u),\n\t\\end{equation}\n\twhere for each $\\omega>0$, $\\mathcal{B}_{\\omega}$ is given by\n\t\\begin{equation}\\label{Bfunctional}\n\t\t\\mathcal{B}_{\\omega}(u)=\\frac{1}{2}\\int_{0}^{L}u'^2+\\omega u^2dx.\n\t\\end{equation}\n\tWe observe that $\\mathcal{B}_{\\omega}$ is a smooth functional on $H_{per}^1([0,L])$.\n\t\\medskip\n\t\n\t\\begin{lema}\\label{minlem}\n\t\tThe minimization problem \\eqref{infB} has at least one nontrivial solution, that is, there exists $\\phi\\in Y_{\\gamma}$ satisfying\n\t\t\\begin{equation}\\label{minBfunc}\n\t\t\t\\mathcal{B}_{\\omega}(\\phi)=\\inf_{ u\\in Y_{\\gamma}}\\mathcal{B}_{\\omega}(u).\n\t\t\\end{equation}\n\t\\end{lema}\n\t\\begin{proof}\n\t\tSince $m\\geq0$ and $\\mathcal{B}_{\\omega}$ is a smooth functional, we are enabled to consider $\\{u_n\\}=\\{u_{n,\\omega}\\}$ as a minimizing sequence for \\eqref{infB}, that is, a sequence in $Y_\\gamma$ satisfying\n\t\t$\\displaystyle \\mathcal{B}_{\\omega}(u_n)\\rightarrow\\inf_{u\\in Y_{\\gamma}} \\mathcal{B}_{\\omega}(u)=m, \\ \\mbox{as} \\ n\\rightarrow \\infty.$\n\t\t\n\t\t\\indent The fact that $\\omega>0$ enables us to conclude $\\{u_{n}\\}$ as a bounded set in $H_{per}^1([0,L])$. Thus, modulus a subsequence, there exists\n\t\t$\\phi=\\phi_{\\omega}\\in H_{per}^1([0,L])$  such that\n\t\t$u_n\\rightharpoonup \\phi \\ \\mbox{weakly in} \\ H_{per}^1([0,L]),  \\ \\  \\mbox{as} \\ n\\rightarrow \\infty.$\n\t\t\n\t\t\n\t\tNow, since the energy space $H_{per}^1([0,L])$ is compactly embedded in $L_{per}^{p+2}([0,L])\\hookrightarrow L_{per}^2([0,L])$, we have for $n\\rightarrow +\\infty$ that\n\t\t$u_n\\rightarrow \\phi \\ \\mbox{in} \\ L^{p+2}_{per}([0,L]),$\n\t\tthat is, $\\int_0^{L}\\phi^{p+2} dx=\\gamma$.\n\t\t\n\t\tMoreover, the weak lower semi-continuity of $\\mathcal{B}_{\\omega}$ gives us that\n\t\t$\n\t\t\\mathcal{B}_{\\omega}(\\phi)\\leq\\liminf_{n\\rightarrow \\infty} \\mathcal{B}_{\\omega}(u_n)=m.\n\t\t$\n\t\tThe lemma is now proved.\n\t\\end{proof}\n\t\n\t\n\tBy Lemma \\ref{minlem} and  Lagrange's Multiplier Theorem, we guarantee the existence of $C_1$ such that\n\t\\begin{equation}\\label{lagrange}\n\t\t-\\phi''+\\omega\\phi=C_1\\phi^{p+1}.\n\t\\end{equation}\n\tWe note that $\\phi$ is nontrivial because $\\gamma>0$ and a standard rescaling argument enables us to deduce that the Lagrange Multiplier $C_1$ can be chosen as $C_1=1$. Now, let $L>0$ be fixed as before. Since the minimization problem $(\\ref{infB})$ can be solved for any $\\omega>0$, we guarantee by arguments of smooth dependence in terms of the parameters for standard ODE (see for instance, \\cite[Chapter I, Theorem 3.3]{hale}), the existence of a convenient open interval $I$ and a smooth curve $\\omega\\in I\\mapsto\\phi\\in H_{per}^n([0,L])$, $n\\in\\mathbb{N}$, satisfying the equation\n\t\\begin{equation}\\label{ode-wave13}\n\t\t-\\phi''+\\omega\\phi-\\phi^{p+1}=0.\n\t\\end{equation}\n\t\n\tIn this setting, the existence of a smooth curve of periodic waves depending on $\\omega$ enables us to conclude by Proposition $\\ref{propsimp}$ that $\\ker(\\mathcal{L})$ is simple. Concerning $n(\\mathcal{L})$, we see that $\\phi$ is a minimizer of $\\mathcal{B}_{\\omega}$ with one constraint. Since $\\langle\\mathcal{L}\\phi,\\phi\\rangle<0$, we obtain by Courant's Min-Max Principle that $n(\\mathcal{L})=1$.\\\\\n\t\\indent Analysis above gives us the following sentence: for $\\phi$  solution of $(\\ref{ode-wave13})$ with $\\omega_0>0$ and the corresponding single point $(\\omega_0,0)$ in the parameter regime, we have that $n(\\mathcal{L}_{(\\omega_0,0)})=1$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$. Therefore, by Theorem $\\ref{teo1}$ and Remark $\\ref{obsiso}$ one has that $\\theta\\neq0$ for all $(\\omega,A)$ in an open subset $\\mathcal{O}\\subset\\mathcal{P}$. This means that the solution $\\phi$ of $(\\ref{odekdv})$ satisfies $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$ for all $(\\omega,A)\\in \\mathcal{O}$.\n\t\n\t\\begin{obs}\\label{remN1}\n\t\tSolutions $\\phi$ which are minimizers of the problem $(\\ref{minBfunc})$ are well determined by the analysis above. In fact, they are rounding the center point(s) in the phase portrait and have the homoclinic as a limit for large periods. The corresponding solution $\\phi$ enjoys the same property. As we have already mentioned before, the parameter regime is the maximal set constituted of pairs $(\\omega,A)$ such that all periodic waves round the center point(s) in the phase portrait. The analysis above gives us that  $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$ in an open subset contained in the parameter regime.\t\n\t\\end{obs}\n\t\n\t\n\t\\indent As before, let us consider a fixed $L>0$. Again, for a fixed $\\gamma>0$, we define \n\t$$Z_{\\gamma}=\\left\\{u\\in H_{per,odd}^1([0,L]);\\ \\int_0^{L}u^{p+2}dx=\\gamma\\right\\},$$\n\twhere $p$ is an even integer. Now, we need to find a minimizer of the constrained minimization problem\n\t\\begin{equation}\n\t\t\\label{infB1}\n\t\tr=r_{\\omega}=\\inf_{ u\\in Z_{\\gamma}}\\mathcal{B}_{\\omega}(u),\n\t\\end{equation}\n\twhere for each $\\omega>0$, $\\mathcal{B}_{\\omega}$ is given by $(\\ref{Bfunctional})$. We have the following result for the existence of odd periodic waves.\n\t\n\t\\begin{lema}\\label{minlem1}\n\t\tThe minimization problem \\eqref{infB1} has at least one odd nontrivial solution, that is, there exists $\\phi\\in Z_{\\gamma}$ satisfying\n\t\t\\begin{equation}\\label{minBfunc1}\n\t\t\t\\mathcal{B}_{\\omega}(\\phi)=\\inf_{ u\\in Z_{\\gamma}}\\mathcal{B}_{\\omega}(u)=r.\n\t\t\\end{equation}\n\t\\end{lema}\n\t\\begin{proof}\n\t\tThe proof of this result is similar to the proof of Lemma $\\ref{minlem}$.\n\t\\end{proof}\n\t\n\tAs before, by Lemma \\ref{minlem1} and  Lagrange's Multiplier Theorem, we guarantee the existence of $C_2$ such that\n\t\\begin{equation}\\label{lagrange1}\n\t\t-\\phi''+\\omega\\phi=C_2\\phi^{p+1}.\n\t\\end{equation}\n\tSince $\\phi$ is nontrivial, a standard rescaling argument gives us that the Lagrange Multiplier  can be chosen as $C_2=1$. By using similar arguments as determined above, we guarantee the existence of a convenient open interval $I$ and a smooth curve $\\omega\\in I\\mapsto\\phi\\in H_{per}^n([0,L])$, $n\\in\\mathbb{N}$, satisfying the equation\n\t\\begin{equation}\\label{ode-wave134}\n\t\t-\\phi''+\\omega\\phi-\\phi^{p+1}=0.\n\t\\end{equation}\n\t\n\t\\begin{lema}\\label{lemnLodd} Let $\\phi$ be the solution obtained by Lemma $\\ref{minlem1}$. We have that $\\ker(\\mathcal{L})=[\\phi']$ and $n(\\mathcal{L})=2$.\n\t\\end{lema}\n\t\\begin{proof} The existence of a smooth curve of odd periodic waves depending on $\\omega$ gives us by Proposition $\\ref{propsimp}$ that $\\ker(\\mathcal{L})$ is simple. \\\\\n\t\t\\indent We determine $n(\\mathcal{L})$. In fact, we see that $\\phi$ is a minimizer of $\\mathcal{B}_{\\omega}$ with one constraint in the Sobolev space $H_{per,odd}^1([0,L])$ constituted by odd functions. Since $\\langle\\mathcal{L}|_{odd}\\phi,\\phi\\rangle<0$, we obtain by Courant's Min-Max Principle that $n(\\mathcal{L}|_{odd})=1$. Next, solution $\\phi'$ is even having two zeros in the interval $[0,L)$. Using the standard Floquet theory in \\cite{Magnus}, we see that zero is not the first eigenvalue of $\\mathcal{L}|_{even}$, so that $n(\\mathcal{L}|_{even})\\geq1$. Again from the Floquet theory, since $\\phi'$ has only two zeros in the interval $[0,L)$, we see that $n(\\mathcal{L})\\leq2$. Therefore, the only possibility is that $n(\\mathcal{L})=n(\\mathcal{L}|_{odd})+n(\\mathcal{L}|_{even})=2$.\n\t\\end{proof}\n\t\n\t\\indent Analysis above gives us the following sentence: for $\\phi$  solution of $(\\ref{ode-wave13})$ with $\\omega_0>0$ and the corresponding single point $(\\omega_0,0)$ in the parameter regime, we have that $n(\\mathcal{L}_{(\\omega_0,0)})=2$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$. Therefore, by Theorem $\\ref{teo1}$ and Remark $\\ref{obsiso}$ one has that $\\theta\\neq0$ for all $(\\omega,A)$ in an open subset $\\mathcal{O}\\subset\\mathcal{P}$. This means that the solution $\\phi$ of $(\\ref{odekdv})$ satisfies $n(\\mathcal{L})=2$ and $\\ker(\\mathcal{L})=[\\phi']$ for all $(\\omega,A)\\in \\mathcal{O}$.\n\t\\begin{obs} Defining $\\varphi:=\\phi(\\cdot-L\/4)$, we can consider the solution obtained by Lemma $\\ref{minlem1}$ as being even and satisfying the mean zero condition $\\int_0^{L}\\varphi(x)dx=0$. In our paper and in order to avoid dubiety of notation, we keep the notation $\\phi$ instead of $\\varphi$ to indicate an even zero mean periodic wave satisfying equation $(\\ref{ode-wave134})$. \n\t\\end{obs}\n\t\\subsection{Positive periodic waves for the critical KdV - Proof of Theorem $\\ref{teoest}$-a)} We start our examples studying the spectral stability of periodic waves  for $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. According with \\cite{AN2}, it is possible to determine a positive periodic wave with dnoidal profile as\n\t\\begin{equation}\\label{dn4kdv}\n\t\t\\phi(x)=\\frac{a{\\rm dn}\\left(\\frac{2K(k)}{L}x,k\\right)}{\\sqrt{1-b{\\rm sn}^2\\left(\\frac{2K(k)}{L}x,k\\right)}},\n\t\\end{equation}\n\twhere $K(k)=\\int_0^1\\frac{dt}{\\sqrt{(1-t^2)(1-k^2t^2)}}$ is the complete elliptic integral of the first kind. Parameters $a$, $b$ in $(\\ref{dn4kdv})$ and the wave speed $\\omega$ in $(\\ref{ode-wave134})$ depend smoothly on the modulus $k\\in(0,1)$ and they are given by\n\t\\begin{equation}\\label{apos}\t\t\n\t\ta=\\frac{[4(2k^2-1+2(1-k^2+k^4)^{1\/2})K(k)^2L^2]^{1\/4}}{L},\n\t\t\\ \\ \\ \n\t\tb=1-k^2-\\sqrt{k^4-k^2+1},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{wkpos}\n\t\t\\omega=\\frac{4K(k)^2\\sqrt{k^4-k^2+1}}{L^2}.\n\t\\end{equation}\n\t\\indent By Remark $\\ref{remN1}$, one has that $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$. Therefore, by Theorem $\\ref{teo2}$ we guarantee the existence of a smooth \n\tsurface $(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^n([0,L]),\\ n\\gg1,$ of periodic waves, all of them\n\twith the same period $L>0$.\\\\\n\t\\indent Let $L>0$ be fixed. According with the tables below (see also Figure 1), we obtain that $\\mathcal{I}>0$ and  $\\det(\\mathcal{D})<0$ for all $k\\in(0,1)$. Then, one has the spectral stability of the periodic wave $\\phi$ in an open neighbourhood of $(\\omega,0)$ where $\\omega>\\frac{\\pi^2}{L^2}$ (see Corollary $\\ref{coroest}$).\n\t\n\t\\begin{obs}\n\t\tLet $L>0$ be fixed. Since $\\omega$ in $(\\ref{wkpos})$ is a strictly increasing function depending smoothly on the modulus $k\\in(0,1)$, we see for $k\\rightarrow 0^+$ and the fact $K(0)=\\frac{\\pi}{2}$ that $\\omega\\rightarrow \\frac{\\pi^2}{L^2}^{+}$. Therefore, we have the basic estimate $\\omega>\\frac{\\pi^2}{L^2}$ for the existence of periodic waves with dnoidal profile in $(\\ref{dn4kdv})$. In addition, if $\\omega>\\frac{\\pi^2}{L^2}$, the standard ODE theory enables us to conclude that $\\phi$ in $(\\ref{dn4kdv})$ is the unique positive solution which solves equation $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. Therefore, $\\phi$ in $(\\ref{dn4kdv})$ satisfies the minimization problem $(\\ref{infB})$\n\t\\end{obs}\n\t\n\t\n\t\\begin{obs} Another important fact: for a fixed $L>0$, we see that $\\det(\\mathcal{D})$ goes to zero when $k\\rightarrow 1^{-}$ (in our tables, this fact also occurs for different values of $L$ by taking $k$ closer to $1$). Thus, we ``recover\" the property $\\frac{d}{d\\omega}\\int_{-\\infty}^{\\infty}Q^2dx=0$, where $Q$ is the hyperbolic secant profile for the critical KdV equation with wave speed $\\omega>0$.\n\t\\end{obs}\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 2\\pi$} \\\\\\hline\n\t\t\t$\\kappa_0$ &  $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\t\n\t\t\n\t\t\n\t\t\t$0.1$  & $ 5.4972$ &$-0.2692$ \\\\\\hline\n\t\t\t$ 0.3$  & $5.4932$ & $-0.2690$ \\\\\\hline\n\t\t\t$ 0.5$ & $ 5.4568$ & $-0.2669$\\\\\\hline\n\t\t\t$0.7$  & $5.2936$ & $-0.2573$\\\\\\hline\n\t\t\t$ 0.9$  &  $4.6183 $& $-0.2148$\\\\\\hline\n\t\t\t$ 0.9999$ &  $1.4287$& $-0.0269$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\\multicolumn{3}{|c|}{$L= 20$} \\\\\\hline\n\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\n\t\t$0.1$ & $177.272 $ & $-2.7286$\\\\\\hline\n\t\t$ 0.3$  & $177.167$& $-2.7257$ \\\\\\hline\n\t\t$ 0.5$ & $175.993$ & $-2.7046$\\\\\\hline\n\t\t$0.7$  &$170.729$ & $-2.6074$\\\\\\hline\n\t\t$0.9$  &  $148.949$ & $-2.1768$\\\\\\hline\n\t\t$0.9999$  &  $46.0816$ & $-0.2772$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\end{table}\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 50$} \\\\\\hline\n\t\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\t\n\t\t\n\t\t\t$0.1$  & $2770.18$ & $-17.0531$\\\\\\hline\n\t\t\t$ 0.3$  & $2768.24$ & $-17.0361$\\\\\\hline\n\t\t\t$ 0.5$ & $2749.89 $ & $-16.9039$\\\\\\hline\n\t\t\t$0.7$  &$2667.64$ & $-16.2963$ \\\\\\hline\n\t\t\t$ 0.9$ &  $2327.33$ & $-13.6055$\\\\\\hline\n\t\t\t$ 0.9999$ &  $720.16$ & $-1.7242$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 100$} \\\\\\hline\n\t\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$\\\\\\hline\n\t\t\n\t\t\n\t\t\t$0.1$  &$22159.9 $ & $-68.2148$\\\\\\hline\n\t\t\t$ 0.3$  & $22145.9 $ & $-68.1444$\\\\\\hline\n\t\t\t$ 0.5$ & $21999.1 $ & $-67.6155$\\\\\\hline\n\t\t\t$0.7$  &$21341.1$ & $-65.1854$\\\\\\hline\n\t\t\t$ 0.9$ &  $18618.7$ & $-54.4218$\\\\\\hline\n\t\t\t$ 0.9999$  &  $5761.48$ & $-6.8758$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\n\t\\end{table}\n\n\t\\indent Using Maple program, we can plot the behavior of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ in terms of the modulus $k\\in(0,1)$ for the case $L=20$. \n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.3]{spline-DN.jpg}\\label{Fig2}\n\t\t\t\\caption{\\small  Graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for $L=20$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\\subsection{Zero mean periodic waves for the critical KdV - Proof of Theorem $\\ref{teoest}$-b)} In what follows, we still consider $A=0$ and $g(s)=s^{5}$ in equation $(\\ref{odekdv})$. Our intention is to give a complete scenario for the spectral stability in this case.\\\\\n\t\\indent Let $L>0$ be fixed. An explicit even periodic wave satisfying the minimization problem in $(\\ref{infB1})$ is given by\n\t\\begin{equation}\\label{cnoid4kdv}\n\t\t\\phi(x)=\\frac{a{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right)}{\\sqrt{1-b{\\rm sn}^2\\left(\\frac{4K(k)}{L}x,k\\right)}},\n\t\\end{equation}\n\twhere $a$, $b$ and $\\omega$ also depend smoothly on the elliptic modulus $k\\in(0,1)$ as\n\t\\begin{equation}\\label{ak}\n\t\ta=\\frac{2[(2-k^2+2\\sqrt{k^4-k^2+1})K(k)^2L^2]^{\\frac{1}{4}}}{L},\n\t\\end{equation}\n\t\\begin{equation}\\label{bk}\n\t\tb=-1+k^2-\\sqrt{k^4-k^2+1},\n\t\\end{equation}\n\tand\n\t\n\t\\begin{equation}\\label{wk}\n\t\t\\omega=\\frac{16K(k)^2\\sqrt{k^4-k^2+1}}{L^2}.\n\t\\end{equation}\n\n\\begin{obs}\n\tLet $L>0$ be fixed. Similarly as in the case of positive solutions, we also have the basic estimate $\\omega>\\frac{4\\pi^2}{L^2}$ for the existence of periodic waves with cnoidal profile in $(\\ref{cnoid4kdv})$. Moreover, when  $\\omega>\\frac{4\\pi^2}{L^2}$, we obtain by the standard ODE theory that $\\phi$ in $(\\ref{cnoid4kdv})$ is the unique mean zero solution which solves equation $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. The uniqueness of solutions gives us that $\\phi$ in $(\\ref{cnoid4kdv})$ satisfies the minimization problem $(\\ref{infB1})$.\n\\end{obs}\n\tFor a fixed $\\omega_0>0$, we have already determined in the last subsection that the associated linearized operator around the periodic wave $\\phi$ given by $(\\ref{cnoid4kdv})$ satisfies $n(\\mathcal{L}_{(\\omega_0,0)})=2$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$ (see Theorem $\\ref{teo1}$ and Proposition $\\ref{propsimp}$). Therefore, we obtain the same spectral properties for all $(\\omega,A)$ in an open subset contained in the parameter regime. From Theorem $\\ref{teo2}$ we guarantee the existence of a smooth\n\tsurface $$(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^n([0,L]),\\ \\ \\ \\ n\\gg1,$$ of periodic waves, all of them\n\twith the same period $L>0$.\\\\\n\t\\indent Since we have constructed the smooth surface\n\t$(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^2([0,L])$ of even periodic waves which solves the\n\tnonlinear differential equation $(\\ref{travkdv})$ with $p=4$, the\n\tnext step is to calculate $n(\\mathcal{I})$ and $n(\\mathcal{D})$. Table below shows us the behavior of the quantity $\\mathcal{I}$ for some (fixed) values of $L>0$.\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\t\\multicolumn{2}{|c|}{$L= 2\\pi$} \\\\\\hline\n\t\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ &   $ -0.47290$\\\\\\hline\n\t\t\t$0.1$ &  $ -0.47296$\\\\\\hline\n\t\t\t$ 0.3$ &  $ -0.4643$\\\\\\hline\n\t\t\t$ 0.5$ &$ -0.3886$\\\\\\hline\n\t\t\t$0.7$ &  $-0.0985$ \\\\\\hline\n\t\t\t$0.739$ &  $-0.0024$ \\\\\\hline\n\t\t\t$0.746$ &  $0.0001$ \\\\\\hline\n\t\t\t$ 0.9$ &  $ 0.4919 $\\\\\\hline\n\t\t\t$ 0.9999$ &  $ 0.9958 $\\\\\\hline\n\t\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\t\\multicolumn{2}{|c|}{$L= 20$} \\\\\\hline\n\t\t\t$\\kappa_0$& $ \\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ &   $-16.2331$\\\\\\hline\n\t\t\t$0.1$ & $-16.2298 $\\\\\\hline\n\t\t\t$ 0.3$ & $-15.9064$\\\\\\hline\n\t\t\t$ 0.5$ &  $ -13.3279 $\\\\\\hline\n\t\t\t$ 0.7$ &  $-3.6809 $\\\\\\hline\n\t\t\t$ 0.744$ &  $-0.070 $\\\\\\hline\n\t\t\t$0.7449$ & $0.0095$\\\\\\hline\n\t\t\t$ 0.9$ &  $15.7314$\\\\\\hline\n\t\t\t$ 0.9999$ &   $44.3814$\\\\\\hline\t\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\\multicolumn{2}{|c|}{$L= 50$} \\\\\\hline\n\t\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ &  $ -255.07$\\\\\\hline\n\t\t\t$0.1$ &  $-255.01 $\\\\\\hline\n\t\t\t$ 0.3$ &  $-249.89$\\\\\\hline\n\t\t\t$ 0.5$ & $  -209.40 $\\\\\\hline\n\t\t\t$0.7$ & $-58.24$\\\\\\hline\n\t\t\t$0.74521$ & $-0.0263$\\\\\\hline\n\t\t\t$0.74523$ & $0.0017$\\\\\\hline\n\t\t\t$ 0.9$ &   $ 245.608$\\\\\\hline\n\t\t\t$ 0.9999$ &   $ 71.1702$\\\\\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\\multicolumn{2}{|c|}{$L= 100$} \\\\\\hline\n\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t$ 0.0001$ &  $-2042.21$\\\\\\hline\n\t\t$0.1$ & $ -2041.74 $\\\\\\hline\n\t\t$ 0.3$ &  $-2000.67 $\\\\\\hline\n\t\t$ 0.5$ & $-1676.52 $\\\\\\hline\n\t\t$0.7$ & $-466.79$\\\\\\hline\n\t\t$0.74528$ & $-0.1066$\\\\\\hline\n\t\t$0.74529$ & $0.0045$\\\\\\hline\n\t\t$ 0.9$ &  $ 1964.63$\\\\\\hline\n\t\t$ 0.9999$ &  $813.37$\\\\\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\n\t\\end{table}\n\n\t\n\t\n\t\n\tNow, we plot the graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for the case $L=2\\pi$.\n\t\\newpage\n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.3]{spline.jpg}\\label{Fig3}\n\t\t\t\\caption{\\small  Graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for $L=2\\pi$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\n\tIt remains to calculate $\\det(\\mathcal{D})$. For the case $(\\omega,A)=(\\omega,0),$ we see that $\\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx=0$. If $\\mathcal{I}\\neq0$, we can use $(\\ref{Dmatrix})$ to obtain $$\\det(\\mathcal{D})=-\\frac{1}{2}\\ \\frac{d}{d\\omega}\\int_0^L\\phi^2dx.$$\n\t\n\t\\indent Formula (411.03) in \\cite{byrd} gives us that\n\t\\begin{eqnarray}\\label{norm}\\int_0^L\\phi^2 dx&=&\\frac{a^2L}{K(k)}\\int_0^{K(k)}\\frac{\\textrm{cn}^2(u,k)}{1-b\\ \\textrm{sn}^2(u,k)}du\\nonumber\\\\\n\t\t\\nonumber\\\\\n\t\t&=&\\frac{a^2L}{K(k)}\\ \\frac{\\pi(1-b)\\ [1-\\Lambda_0(\\beta,k)]}{2\\sqrt{b(1-b)(b-k^2)}}\\\\\n\t\t\\nonumber\\\\\n\t\t&=&\\frac{2\\pi\\sqrt{(k^2-2b)(1-b)}\\ [1-\\Lambda_0(\\beta,k)]}{\\sqrt{b(b-k^2)}}:=\\tau(k).\\nonumber\\end{eqnarray}\n\tHere, $\\Lambda_0$ indicates de Lambda Heumann function defined by\n\t$$\\Lambda_0(\\beta,k)=\\frac{2}{\\pi}[E(k)F(\\beta,k')+K(k)E(\\beta,k')-K(k)F(\\beta,k')],$$  \n\twhere $\\beta=\\arcsin\\displaystyle\\left(\\frac{1}{\\sqrt{1-b}}\\right)$ and $ k'=\\sqrt{1-k^2}.$\n\t\n\t\\indent Next, for all $k\\in(0,1)$ we have \n\t\\begin{equation}\\label{dwdk}\\frac{d\\omega}{dk}=-\\frac{16 K(k)\\ [K(k)\\ (k^4-3k^2+2)-2E(k)\\ (k^4-k^2+1)]}{L^2k(1-k^2)\\sqrt{k^4-k^2+1}}>0.\\end{equation}\n\tThus, \n\t\n\t\\begin{equation}\\label{dwnorn}\n\t\t\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\displaystyle\\frac{\\frac{d}{dk}\\int_0^L\\phi^2 dx}{\\frac{d\\omega}{dk}}=\\frac{\\tau'(k)}{\\frac{d\\omega}{dk}}.\n\t\\end{equation}\n\t\n\t\\indent We can plot the behavior of $\\tau'$ in terms of the modulus $k\\in(0,1)$ to conclude from $(\\ref{dwdk})$ and $(\\ref{dwnorn})$ that $\\frac{d}{d\\omega}\\int_0^L\\phi^2\\ dx>0.$ \n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.35]{Fig1.jpg}\\label{Fig1}\n\t\t\t\\caption{\\small  Graphic of $\\tau'(k)$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\indent Let $L>0$ be fixed. One sees that $\\det(\\mathcal{D})<0$ for all $\\omega>\\frac{4\\pi^2}{L^2}$ and $A\\approx 0$. Previous tables give us a threshold value $k_0\\approx 0.745 $ satisfying \n\t$\\mathcal{I}=\\langle\\mathcal{L}^{-1}1,1\\rangle=0$ at $k=k_0$ with $\\mathcal{I}<0$ if $k\\in(0,k_0)$ and $\\mathcal{I}>0$ if $k\\in(k_0,1)$. Therefore, we can apply Corollary $\\ref{coroest}$ to conclude that $\\phi$ is spectrally stable if $k\\in(0,k_0)$ and spectrally unstable if $k\\in(k_0,1)$.\n\t\n\t\n\t\n\t\n\t\\subsection{The Gardner equation - Proof of Theorem $\\ref{teoestG}$.} Now, we apply the arguments in the previous sections to study the spectral stability for the Gardner equation expressed in a general form as\n\t\n\t\\begin{equation}\\label{gardnereq1}\n\t\tu_t+\\alpha_1 (u^2)_x+\\alpha_2 (u^3)_x+u_{xxx}=0,\n\t\\end{equation}\n\twhere $\\alpha_1$ and $\\alpha_2$ are non-negative constants satisfying $\\alpha_1^2+\\alpha_2^2\\neq0$. When $\\alpha_2=0$, equation $(\\ref{gardnereq1})$ reduces to the well KdV equation while $\\alpha_1=0$, the same equation provides us the modified KdV equation. Results of spectral\/orbital stability of periodic waves in both cases have been discussed in \\cite{AN1}, \\cite{AN2}, \\cite{DK}, and references therein. \\\\\n\t\\indent Let $L>0$ be fixed and consider $\\alpha_1=\\alpha_2=1$. Explicit periodic waves $\\phi$ for this equation can be determined as\n\t\\begin{equation}\\label{cngardner}\n\t\t\\phi(x)=-\\frac{1}{3}+b{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right),\n\t\\end{equation}\n\twhere $b$, $\\omega$ and $A$ are given by\n\t\\begin{equation}\\label{ak1}\n\t\tb=\\frac{4\\sqrt{2}kK(k)}{L},\n\t\\end{equation}\n\t\n\t\\begin{equation}\\label{wk1}\n\t\t\\omega=-\\frac{1}{3}-\\frac{16K(k)^2(1-2k^2)}{L^2},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{Ak1}\n\t\tA=\\frac{1}{27}+\\frac{144K(k)^2(1-2k^2)}{27L^2}.\n\t\\end{equation}\n\tNow, let us define $\\varphi(x)=\\frac{1}{3}+\\phi(x)=b{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right)$. We see that $\\varphi$ solves the equation\n\t\\begin{equation}\\label{mkdv9}\n\t\t-\\varphi''+\\left(\\omega+\\frac{1}{3}\\right)\\varphi-\\varphi^3=0,\n\t\\end{equation}\n\tthat is, $\\varphi$ is a periodic wave with cnoidal profile for the modified KdV equation.\\\\\n\t\\indent Let $\\mathcal{L}_{\\varphi}$ be the linearized operator around $\\varphi$ for the equation $(\\ref{mkdv9})$ with $g(s)=s^3$. It is a surprising fact that\n\t$$\\mathcal{L}_{\\varphi}=-\\partial_x^2+\\omega+\\frac{1}{3}-3\\varphi^2=-\\partial_x^2+\\omega-2\\phi-3\\phi^2=\\mathcal{L},$$\n\twhere $\\mathcal{L}$ is the linearized operator around $\\phi$ for the Gardner equation $(\\ref{gardnereq1})$. Using the arguments in \\cite{AN1} and \\cite{DK}, we see that $n(\\mathcal{L})=n(\\mathcal{L}_{\\varphi})=2$ and $\\ker(\\mathcal{L})=\\ker(\\mathcal{L}_{\\varphi})=[\\phi']$.\\\\\n\t\\indent Now, since $\\mathcal{L}_{\\varphi}=\\mathcal{L}$, one sees that $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$. In addition, we obtain similarly as determined in \\cite{AN1} and \\cite{DK} that $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ for $k\\in (0,k_0)$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$ for $k\\in(k_0,1)$, where $k_0\\approx 0.909$.\\\\\n\t\\indent It remains to calculate $\\mathcal{D}$ in this specific case. First, we deal with\n\t\\begin{equation}\\label{norma1}\\int_0^L\\phi^2 dx=\\frac{L}{9}-\\frac{2b}{3}\\int_0^L\\textrm{cn}\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx+b^2\\int_0^L\\textrm{cn}^2\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx. \\end{equation}\n\t\n\t\\indent The middle term on the right-hand side of $(\\ref{norma1})$ has zero mean since\n\t\\begin{equation}\\label{int01}\\int_0^L\\textrm{cn}\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx=\\frac{L}{2K(k)}\\int_0^{2K(k)}\\textrm{cn}(u,k)\\ du=0.\\end{equation} \n\tIn addition, the last term containing the quadratic power can be simplified as\n\t\\begin{equation}\\label{int02}\\int_0^L\\textrm{cn}^2\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx=\\frac{L}{K(k)}\\int_0^{K(k)}\\textrm{cn}^2(u,k)\\ du=\\frac{L[E(k)-(1-k^2)K(k)]}{k^2K(k)}.\\end{equation}\n\t\n\t\\indent Collecting the arguments contained in (\\ref{ak1}), (\\ref{norma1}), (\\ref{int01}), and (\\ref{int02}), we conclude that\n\t\\begin{equation}\\label{norma2}\\int_0^L\\phi^2 dx=\\frac{L}{9}+\\frac{32 K(k)[E(k)-(1-k^2)K(k)]}{L}. \\end{equation}\n\t\n\t\\indent In order to calculate $\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$, we need to observe first that for all $k\\in(0,1)$ we have\n\t\\indent  \\begin{equation}\\label{dwdk2}\\frac{d\\omega}{dk}=-\\frac{32K(k)[(1-2k^2)E(k)-(1-k^2)K(k)]}{k(1-k^2)L^2}>0.\\end{equation}\n\tThus, we are in position to give a convenient expression for $\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$ using the chain rule as \n\t\\begin{equation}\\label{derphi2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\frac{\\frac{d}{dk}\\int_0^L\\phi^2dx}{\\frac{d\\omega}{dk}}.\\end{equation}\n\tBy (\\ref{norma2}), we obtain for all $k\\in (0,1)$ that \\begin{equation}\\label{dnorm2}\\frac{d}{dk}\\int_0^L\\phi^2 dx=-\\frac{32[(1-k^2)K(k)(2E(k)-K(k))-E(k)^2]}{k(1-k^2)L}>0.\\end{equation}\n\t\n\t\n\t\\indent Finally, by (\\ref{dnorm2}) and (\\ref{dwdk2}) we deduce \\begin{equation}\\label{positivephi2}\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\frac{\\frac{d}{dk}\\int_0^L\\phi^2 dx}{2\\frac{d\\omega}{dk}}>0.\\end{equation}\n\t\n\t\\indent We are in position to calculate $\\mathcal{D}$. We have already determined  that $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$. Similarly, since $\\varphi$ has zero mean it follows that $\\langle\\mathcal{L}_{\\varphi}^{-1}\\varphi,1\\rangle=\\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx=0$. In addition, a straightforward calculation also gives us that $\\langle\\mathcal{L}_{\\varphi}^{-1}\\varphi,\\varphi\\rangle=\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$. Thus, for $k\\neq k_0\\approx 0.909$ one has from $(\\ref{positivephi2})$ that $\\det(\\mathcal{D})$ is \n\t$$\\det(\\mathcal{D})=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx<0.$$ Corollary $\\ref{coroest}$ can be applied to deduce the spectral stability of $\\phi$ when $k\\in(0,k_0)$ and the spectral instability when $k\\in (k_0,1)$. \n\t\n\t\n\t\n\t\\section*{Acknowledgments} S.A. was supported by CAPES.  F.N. is partially supported by Funda\\c{c}\\~ao Arauc\\'aria 002\/2017, CNPq 304240\/2018-4 and CAPES MathAmSud 88881.520205\/2020-01.\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nClustering in metal alloys is known as the very early stage of first-order transformations within a bulk crystal, which largely influences the mechanical properties of quenched\/aged materials. Coherent with the matrix, small clusters of solute atoms have a significantly lower nucleation barrier than the terminal second-phase of a completely different crystal structure. Early clusters nucleate and grow in size forming the so-called GP-zones known to be a metastable precursor of the equilibrium phase~\\cite{christian}. The formation and growth mechanisms of the early clusters are poorly understood presently due to the lack of direct atomistic observations of structural changes during the transition process. However, inspired by the observations made on the quenched structures using transmission electron microscopy (TEM)~\\cite{marceau10,marceau10-2,nutyen67,ozawa70}, 3D atom probe~\\cite{sato04,esmaeili07,marceau10,marceau10-2,biswas11}, and positron annihilation~\\cite{somoza02,somoza10,marceau10-2} techniques, the formation of early clusters has been empirically associated with so-called quenched-in defects. Formed within the bulk crystal upon quenching, excess vacancies and\/or dislocations loops have been presumed to decrease the energy barrier for nucleation facilitating cluster formation~\\cite{esmaeili07,ozawa70,somoza10,marceau10,marceau10-2}.\n\nUnderstanding solute clustering mechanisms is of crucial importance to design an effective age-hardening process producing desired mechanical properties in alloys~\\cite{christian}. To our knowledge, no systematic study of the clustering mechanisms has been done using atomic-scale simulation methods such as molecular dynamics (MD) and Monte Carlo (MC) simulations, due to their main restrictions of accessing the relevant time scales of diffusional transformations. Dynamical calculations using classical density functional theory (CDFT) are also inefficient due to the high spatial resolution required to resolve the sharp density spikes in solid phases~\\cite{jaatinen09}.\n\nThe recently developed phase field crystal (PFC) method~\\cite{elder04,elder07,wu10,greenwood10,greenwood11} has shown promise for simulating structural transformations on diffusive time scales. This new formalism carries the essential physics of CDFT without the need to resolve the sharp atomic density peaks. In the most recent PFC formalism developed by Greenwood et al.~\\cite{greenwood10,greenwood11,greenwood11-2}, various crystal symmetries can be easily stabilized by construction of relevant correlation kernels. This approach preserves the numerical efficiency of the original PFC model and is able to dynamically simulate the precipitation of solid phases within a parent phase of different crystal symmetry~\\cite{greenwood10} and\/or chemical composition~\\cite{greenwood11-2}.\n\nThis letter proposes a new approach to study the clustering phenomenon that relies on atomic-scale simulations using the previously developed alloy PFC model of ref.~\\cite{greenwood11-2}. We explore the formation and growth mechanisms of early clusters in a quenched bulk lattice of a supersaturated Al-Cu alloy initially containing quenched-in defects such as dislocations.\n\n\\section{Model Structure}\nWe start with the free energy functional in the binary PFC model~\\cite{greenwood11-2},\n\\begin{align}\n\\frac{\\Delta\\freedm}{kT\\rho^o} = \\int f dr &=\n\\int \\bigg\\{\\frac {n^2}{2}-\\eta\\frac {n^3}{6}+\\chi\\frac {n^4}{12}+\\nline(n+1)\\Delta\\freedm _{mix} &-\\frac 1{2}n \\int dr'C^n_{eff}(|r-r'|)n'+\\alpha|\\vec{\\nabla}c|^2\\bigg\\} dr\n\\label{PFCalEnrgy2}\n\\end{align}\nwhere $n$ and $c$ represent reduced dimensionless atomic number density and solute concentration fields, respectively. $\\eta$ and $\\chi$ are coefficients added to fit the ideal energy to a polynomial expansion ($\\eta=\\chi=1$ describes a Taylor series expansion of the bulk free energy around the reference density) and\n\\begin{align}\n\\Delta\\freedm_{mix}=\\omega\\{c\\ln(\\frac{c}{c_o})+(1-c)\\ln(\\frac{1-c}{1-c_o})\\}\n\\label{Fmix}\n\\end{align}\nrepresents the energy density associated with the entropy of mixing. The coefficient $\\omega$ is introduced to fit the entropic energy away from the reference composition $c_0$. The parameter $\\alpha$ is a coefficient (taken as 1 in this study). These parameters are discussed further in ref.~\\cite{greenwood11-2}.\n\nFor a binary alloy, Greenwood et al.~\\cite{greenwood11-2} introduced the correlation function\n\\begin{align}\nC^n_{eff}=X_1(c)C^{AA}_2 + X_2(c)C^{BB}_2\n\\label{CorrEff}\n\\end{align}\n, where $X_1(c)=1-3c^2+2c^3$ and $X_2(c)=1-3(1-c)^2+2(1-c)^3$. $C^{AA}_2$ and $C^{BB}_2$ are correlation functions representing, respectively, contributions to the excess free energy for the situations where A atoms are in the preferred crystalline network of B atoms and B atoms which are in a structure preferred by A atoms. The correlation functions $\\hat{C}^{ii}_2(\\vec{k})$ are defined to have reciprocal space peaks (i.e. $k_j$, corresponding to the inverse of interplanar spacings) determined by the main families of planes in the equilibrium crystal unit cell structure for the $i^{th}$ component. Each peak is represented by the following Gaussian form of width $\\alpha_j$, modulated for temperature $\u03c3$ by a Debye-Waller prefactor which accounts for an effective transition temperature $\\sigma_{Mj}$~\\cite{greenwood11-2}.\n\\begin{align}\n\\hat{C}^{ii}_{2j}=e^{-\\frac{\\sigma^2}{\\sigma^2_{Mj}}}e^{-\\frac{(k-k_j)^2}{2\\alpha^2_j}} \n\\label{CorrF}\n\\end{align}\n\nThe equations of motion of the total density and concentration fields follow dissipative dynamics~\\cite{archer05}. The total mass density and total reference density per unit volume are defined as $\\rho=\\rho_A+\\rho_B$ and $\\rho^o=\\rho_A^o+\\rho_B^o$, respectively. Thus, the equations of motion can be written for $n$($=\\rho\/\\rho^o-1$) and $c$($=\\rho_B\/\\rho$) as $\\pxpy{n}{t}=\\vec{\\nabla}.\\{M_n\\vec\\nabla(\\frac{\\delta \\Delta F}{\\delta n})\\}+\\eta_n(\\sigma,t)$ and $\\pxpy{c}{t}=\\vec{\\nabla}.\\{M_c\\vec\\nabla(\\frac{\\delta \\Delta F}{\\delta c})\\}+\\eta_c(\\sigma,t)$, respectively~\\cite{greenwood11-2}. $M_n$ and $M_c$ are dimensionless kinetic mobility parameters (equal to 1 in this study). $\\eta_n(\\sigma,t)$ and $\\eta_c(\\sigma,t)$ are stochastic noise variables subsuming the role of fast atomic vibrations in density and concentration fields, respectively.\n\n\\section{Results}\n\\subsection{Phase diagram reconstruction}\nTo examine the equilibrium properties of this binary PFC model for a 2D Al-Cu system, we construct the phase diagram for the coexistence of two square phases. The coexistence lines between the respective phases are obtained by a common tangent construction of the free energy curves of solid and liquid at the reference density ($\\bar{n}=0$). Following Greenwood et al.~\\cite{greenwood11-2}, the free energy curves of the square phases are calculated using the two-mode approximation of the density fields which is defined by \n\\begin{align}\nn_i(\\vec{r})=\\sum_{j=1}^{N_i}A_j\\sum_{l=1}^{N_j}e^{2\\pi\\mathbf{i}\\vec{k}_{l,j}.\\vec{r}\/a_i}\n\\label{Density}\n\\end{align}\n, where the subscript $i$ denotes a particular solid phase with a lattice spacing $a_i$, and the index $j$ counts the $N_i$ modes of the $i$-phase. $A_j$ is the amplitude of mode $j$ and $l$ is the index over the group of reciprocal space peaks corresponding to mode $j$, $N_j$. Accordingly, $\\vec{k}_{l,j}$ is the reciprocal lattice vector normalized to a lattice spacing of 1, corresponding to each index $l$ in the family $j$. The free energy curve for each phase can be calculated as a function of the composition $c$ by substituting the above density field approximation into Eq.~(\\ref{PFCalEnrgy2}) and integrating over the unit cell. The resulting crystal free energy is then minimized for the amplitudes $A_j$. For the liquid phase, the amplitude $A_j$ is set to zero and the density is considered as the reference density ($\\bar{n}=0$). A more detailed description of this methodology is provided in ref.~\\cite{greenwood11-2}.\n\n\\begin{figure}[htbp]\n\\resizebox{3.3in}{!}{\\includegraphics{PhaseDiagram}}\n\\caption{The constructed phase diagram for a square-square system with the inset showing the Al-rich side of the experimental phase diagram of Al-Cu system taken from Ref.~\\cite{baker}. The parameters for ideal free energy contribution were $\\eta=1.4$ and $\\chi=1$, while $\\omega=0.005$ and $c_0=0.5$ for entropy of mixing. Widths of the correlations peaks are $\\alpha_{11Al}=2.4$, $\\alpha_{10Al}=\\sqrt{2}\\alpha_{11Al}$ (the required ratio to introduce isotropic elastic constants in an square phase~\\cite{greenwood11-2}), $\\alpha_{11\\theta}=2.4$ and $\\alpha_{10\\theta}=\\sqrt{2}\\alpha_{11\\theta}$. The peak positions for pure Al correspond to $k_{11Al}=2\\pi$, $k_{10Al}=\\sqrt{2}k_{11Al}$, $k_{11\\theta}=(81\/38)\\pi$ and $k_{10\\theta}=\\sqrt{2}k_{11\\theta}$. The effective transition temperatures are set to $\\sigma_{M11Al}=0.55$, $\\sigma_{M10Al}=0.55$, $\\sigma_{M11\\theta}=0.55$ and $\\sigma_{M10\\theta}=0.55$; The concentration $c$ is rescaled considering the Cu-content in the $\\theta$-phase.}\n\\label{fig:PhaseDiagram}\n\\end{figure}\n\nIn the Al-rich side of the experimental Al-Cu phase diagram, shown in the inset of Fig.~\\ref{fig:PhaseDiagram}, there is a eutectic transition between the Al-rich $\\alpha$-fcc phase and an intermediate phase $\\theta$ (containing $\\approx 32.5at.\\%$ Cu) with a tetragonal crystal structure. For 2D simulations, in order to approximate these equilibrium properties, we reconstruct the binary phase diagram of Al and $\\theta$, both with a square symmetry but differing in Cu-content. The lattice constant (and thus the reciprocal space peaks) of $\\theta$ is approximated by interpolating between those of Pure Al and Cu. The solid phase free energy is calculated with a variable lattice constant weighted by concentration $c$ using the interpolation functions $X_1$ and $X_2$. The polynomial fitting parameters in Eq.~(\\ref{PFCalEnrgy2}) (namely $\\eta$, $\\chi$ and $\\omega$) and width of various peaks ($\\alpha_j$) in the correlation kernel $\\hat{C}^{ii}_{2j}$ are then chosen so as to obtain the same compositions for $\\alpha$-phase solubility limit and eutectic point as those in the experimental phase diagram. \n\n\\subsection{Simulation of clustering}\nWith the equilibrium properties obtained above, simulations of clustering were performed on a rectangular mesh with grid spacing $dx=0.125$ and time step $dt=1$. Considering the lattice parameter of 1, each atomic spacing is resolved by 8 mesh spacings. The dynamical equations were solved semi-implicitly in Fourier space for higher efficiency. The initial conditions were chosen to study the proposed dominant role of quenched-in dislocation-type defects in the bulk crystal during the early stage precipitation in dilute Al-Cu alloys quenched from a solutionizing temperature~\\cite{nutyen67,ozawa70,somoza02,somoza10,desorbo58}. According to this hypothesis, dislocation loops, generated by excess vacancies, are responsible for local lattice distortions facilitating segregation and diffusion of Cu-atoms, while also driving the system towards a more thermodynamically-stable state~\\cite{nutyen67,ozawa70,somoza10,marceau10,marceau10-2}. Therefore, as initial conditions, we use a crystal lattice of uniform composition distorted by introducing dislocations.\n\n\\begin{figure}[htbp]\n\\resizebox{3.3in}{!}{\\includegraphics{Clustering}}\n\\caption{(colour online) PFC simulation of clustering phenomena on a system of 256$\\times$256 atoms after 225,000 time steps containing clusters with various sizes and concentrations; (a) The developed structure of a long-lived cluster; (b) The initially distorted structure; For graphical illustration, the concentration field is superimposed on the density field, and ranges from dark blue to dark red as the Cu-content increases.}\n\\label{fig:Clustering}\n\\end{figure}\n\nPFC simulation is performed for quench\/aging of Al-2at.$\\%$Cu from the solutionizing temperature of $\\sigma=0.17$ to $\\sigma=0.04$ with the initial conditions shown in Fig.~\\ref{fig:Clustering}(b). During the simulation, first, small clusters form with a slightly higher Cu-content than that of the matrix. As time progresses, some of these clusters shrink in size and concentration and a few get stabilized (e.g. the cluster shown in Fig.~\\ref{fig:Clustering}(a)). In contrast, as expected, quenching the same initial structure from the solutionizing temperature of $\\sigma=0.17$ to a temperature within the single-phase $Sq$-$Al$ region, i.e., $\\sigma=0.16$, leads to complete removal of distortion. \n\n\\section{Discussion}\n\n\\subsection{Evolution of clusters}\nThe dislocation-induced cluster structure shown in Fig.~\\ref{fig:Clustering}(a) is consistent with TEM observations in Al-1.7at.$\\%$Cu~\\cite{nutyen67} and Al-1.1at.$\\%$Cu-0.5at.$\\%$Mg alloys~\\cite{marceau10,marceau10-2}, where dislocation loops appear in the bulk lattice of the quenched structures. Using resistometric measurements and TEM techniques for Al-1.2at.$\\%$Si alloy, Ozawa and Kimura~\\cite{ozawa70,ozawa71} have associated the formation of dislocation (or vacancy) loops upon quenching to the coalescence of excess vacancies. They have further suggested that the solute atoms segregate towards the loops stabilizing them into solute clusters. Also, tracing vacancy clusters by positron annihilation, Somoza et al.~\\cite{somoza02,somoza10} have proposed that vacancy-Cu pairs are present at the quenched-state in Al-1.74at.$\\%$Cu alloy. To our knowledge, our PFC simulations are the first atomic-scale simulations to support the above hypothesis of vacancy\/dislocation-mediated solute clustering and nucleation mechanisms of early stage precipitation.\n\n\\subsection{Analysis of work of formation}\nWe further investigated the above mechanisms of cluster formation and growth by analyzing the system energetics for a long-lived cluster. To avoid possible finite size effects, a test with same conditions as those of the above simulation was performed on a larger system, e.g., 512$\\times$512 atoms. The strain field caused by the dislocations displacement fields is evaluated by\n\\begin{align}\n\\epsilon = \\sum_{i=1}^{N_{tri}} \\sum_{j=1}^{3}\\bigg(\\frac{a_{ij}-a_o}{a_o}\\bigg)\n\\label{Strain}\n\\end{align}\n, which is calculated over triangulated density peaks using the Delaunay Triangulation method. $N_{tri}$ is the number of triangles in the field, $a_o$ is the dimensionless equilibrium lattice parameter (the number of grid points resolving one lattice spacing, i.e., 8), $a_{ij}$ is the length of the $j^{th}$ side of the $i^{th}$ triangle. Small clusters, each accompanied by at least one dislocation, appear to be in local equilibrium with the matrix shown in Fig.~\\ref{fig:StrainField}(a)). During the simulation, following Fig.~\\ref{fig:StrainField}(b) and (c), cluster ``a\" continues to grow while, simultaneously, its accompanying dislocation climbs up towards nearby dislocations, creating larger local strain fields (i.e. 0.001, 0.0016 and 0.014 for cluster ``a\" in Fig.~\\ref{fig:StrainField}(a), (b) and (c), respectively). This mechanism of stress relaxation through solute segregation has been shown through phase-field studies by Leonard and Haataja~\\cite{leonard05} to be the main cause of alloy destabilization by structural spinodal decomposition in the presence of dislocations. Also, PFC studies of thin layers deposition by Muralidharan and Haataja~\\cite{muralidharan10} indicated that, due to the above mechanism, some immiscible alloys exhibit miscibility gap around the inter-layer interface in the presence of coherency stresses.\n\n\\begin{figure*}[htbp]\n\\resizebox{5.1in}{!}{\\includegraphics{StrainField}}\n\\caption{(colour online) (a-c) Snapshots taken at 3 different times showing the structural changes during formation of cluster ``a\"; (d) work of formation (evaluated from Eq.(~\\ref{Nucl.En})) vs. $R$ for increasing dislocation strain fields, i.e., increasing $\\Sigma b_i^2$; the dashed curve represents the work of formation for direct homogeneous nucleation of clusters in absence of dislocations, i.e., when $\\Sigma b_i^2=0$; (e) the variation of numerically evaluated total energy, $\\Delta G_{tot}$, and weighted average burger's vectors, $\\Sigma b_i^2$, due to the formation of cluster ``a\" in the above box;}\n\\label{fig:StrainField}\n\\end{figure*}\n\nThe effect of dislocations on the nucleation of clusters is investigated by considering the following form of work of formation:\n\\begin{align}\nW &= 2\\pi R\\gamma + \\pi R^2 (-\\Delta f + \\Delta G_s) - \\Delta G_{sr} + \\Delta G_d\n\\label{Nucl.En}\n\\end{align}\nwhere $R$ is the cluster radius in terms of number of lattice spacings and\n\\begin{align}\n\\gamma = \\frac {\\int_{Area} {\\alpha|\\vec{\\nabla}c|^2 dr}}{L} \n\\label{Surf.En}\n\\end{align}\nis a Cahn-Hilliard type interfacial free energy per unit length of the interface in 2D. $Area$ represents the area of the surface containing the cluster, and $L$ is the circumferential length of a round cluster of radius $R$. Assuming low dislocation density in the system, the interfacial free energy is taken to be solely chemical, neglecting the structural contributions~\\cite{Turnbull}.\n\\begin{align}\n\\Delta f=f^b-\\mu_c^b|_{c^b}(c^b-c^{cl})-f^{cl} \n\\label{DrivingForce}\n\\end{align}\nis the bulk driving force for nucleation of a cluster at a given concentration, where superscripts `$b$' and `$cl$' denote the bulk matrix and cluster ``phase'' quantities, respectively.\n\\begin{align}\n\\Delta G_s = 2 G_A \\delta^2 \\frac{K_B}{K_B+G_A} \n\\label{StrainEnergy}\n\\end{align}\nrepresents the strain energy for a coherent nucleus~\\cite{hoyt}, where $\\delta$ is the misfit strain and $G_A$ and $K_B$ are 2D shear and bulk moduli, respectively, calculated from PFC 2D mode approximation~\\cite{greenwood11-2}. \n\\begin{align}\n\\Delta G_{sr} = \\eta^2\\chi_d E A \\ln(R) \n\\label{StressRelaxE}\n\\end{align}\nis defined as the stress relaxation term due to segregation of solute into dislocations~\\cite{cahn57}, where $A= \\frac{G_A \\Sigma b_i^2}{4 \\pi (1-\\nu)}$, $\\nu = \\frac{E}{2 G_A}-1$, $\\eta=\\frac{1}{a}\\frac{\\partial a}{\\partial c}$ is the linear expansion coefficient with respect to concentration, $E$ is the 2D Young's modulus~\\cite{greenwood11-2}, $\\chi_d=(\\frac{\\partial^2 f}{\\partial c^2})^{-1}$, $\\Sigma b_i^2$ represents a weighted average of the burger's vectors around the dislocations accompanying the cluster and $a$ is the lattice parameter. The prefactor of the logarithm term, $\\eta^2\\chi_d E A$, approximates how strain energy is reduced due to solute segregation around a dislocation~\\cite{larch\u00e985}. \n\\begin{align}\n\\Delta G_d = \\zeta A \n\\label{Disl.E}\n\\end{align}\naccounts for the increase in the total system energy due to presence of dislocations, where $\\zeta$ is a prefactor of order ten giving the average amount of energy per dislocation core~\\cite{Hull}. Fig.~\\ref{fig:StrainField}(d) plots the evaluation of the above form of work of formation (Eq.~(\\ref{Nucl.En})) for cluster ``a\" at different mean concentrations up to that of the largest cluster shown in Fig.~\\ref{fig:StrainField}(c). The mean concentration of each cluster is estimated within a radius of $R$, defined by radially averaging the radius of the concentration field bound by a threshold of $[c^b+\\frac{\\sum^N {c-c^b}}{N}]$. The dashed curve in Fig.~\\ref{fig:StrainField}(d) represents the work of formation for direct homogeneous nucleation of clusters in absence of dislocations, i.e., $\\Sigma b_i^2=0$. The energy barrier for homogeneous nucleation seems to be smaller than that of the dislocation-assisted clustering by a single dislocation, i.e., $\\Sigma b_i^2=1$. However, according to the plots shown in Fig.~\\ref{fig:StrainField}(d), as Cahn~\\cite{cahn57} also pointed out, the barrier for formation of clusters on dislocations can be significantly reduced or even completely eliminated by increasing the magnitude of strain field around the dislocations (i.e. increasing $\\Sigma b_i^2$). Notably, the local minimum also shifts to larger nucleus sizes until it vanishes (i.e., work of formation continuously slops down vs. $R$). \n\nIt is noteworthy that, in the absence of quenched-in defects, nucleation of the second phase requires introduction of a thermally-activated noise to produce fluctuations in both density and concentration fields. Assuming dislocations are present in the bulk matrix of a supersaturated quenched alloy, in this study, we demonstrate how elasticity itself can drive the system into a phase transition. The influence of a thermally-activated noise on the transformation kinetics will be investigated in a future study through use of a well-defined noise algorithm. We have, however, observed in our simulations that in the case of a mismatch between the two species, such as in Al-Cu alloys, introducing a Gaussian noise to both density and concentration fields will not have a major impact on the overall path of the transformation. In other words, the phase transformation is mainly driven by the interactions between the elastic fields of the dislocations and the solute atoms.\n\nThe total work of formation, $\\Delta G_{tot}$, is also estimated numerically by measuring the change in the grand potential within a box engulfing cluster ``a\" during its formation and growth in the bulk matrix, i.e., \n\\begin{align}\n\\Delta G_{tot} = \\int_V\\Omega-\\int_V\\Omega^b =\\nline\\int_V {[f-\\mu_c c - \\mu_n n]} &-\\int_V {[f^b -\\mu_c^b c^b - \\mu_n^b n^b]}\n\\label{TotalE}\n\\end{align}\n. Here, $\\mu_c=\\frac{\\partial f}{\\partial c}$ and $\\mu_n=\\frac{\\partial f}{\\partial n}$ are diffusion potentials of concentration and density fields, respectively, and $V$ is the total volume. The above work of formation has contributions from the interfacial energy and driving force for formation of clusters (i.e., $\\Delta G_{tot}=\\Delta G_{\\gamma}-\\Delta G_v$), both of which include the elastic effects. Since the above box contains only one cluster, the calculated change in the grand potential accounts for the structural and compositional changes during the formation and growth of only cluster ``a\". While the growth of cluster ``a\" raises the local free energy, other parts of the system may undergo a process of annihilation and\/or shrinkage of sub-critical clusters and their accompanying dislocations leading to an overall decrease in the free energy of the system. As can be seen in Fig.~\\ref{fig:StrainField}(e), the total work of formation increases with the growth of cluster ``a\" until a maximum value, after which it starts to decrease. Also, as can be seen in this figure, the estimated values of $\\Sigma b_i^2$ at various sizes of cluster ``a\" closely corresponds to its analytical relationship with the cluster size at the local minima mapped on the energy plots of Fig.~\\ref{fig:StrainField}(d). Likewise the work of formation, during formation and growth of cluster ``a\", the value of $\\Sigma b_i^2$ reaches a maximum at the critical size of the cluster. \n\nAccording to our data, cluster ``a\" continuously grows in presence of dislocations implying that, at each sub-critical cluster size, the system is sitting at a local energy minimum. Since cluster ``a\" at each sub-critical size is in a local equilibrium with the matrix we call it a metastable precursor to the cluster ``a\" with a critical size. This is analogous to previous PFC studies of crystals solidification which show that metastable amorphous precursors emerge first due to their lower nucleation barrier than that of a crystalline solid~\\cite{toth11,tegze09}. In our case, the nucleation barrier is lowered by the effect of locally straining a sub-critical cluster (as a result of local accumulation of dislocations burger's vectors, as illustrated in Fig.~\\ref{fig:Clustering}(a)), making it thermodynamically favorable for the cluster to receive more solute atoms from the matrix and grow in size. \n\n\\begin{figure}[htbp]\n\\resizebox{3.4in}{!}{\\includegraphics{StrainELand}}\n\\caption{(colour online) Common tangent construction using mean-field free energy curves of unstrained (solid curve) and strained solid phases (dashed curves).}\n\\label{fig:StrainELand}\n\\end{figure}\n\n\\subsection{Metastable phase coexistence}\nThe metastable coexistence between a sub-critical cluster ``a\" and the matrix at the quench\/aging temperature is elucidated by evaluating the mean field free energy of a system comprising an unstrained matrix phase and strained solid phases with different magnitudes of distortion, i.e., a uniform strain. The free energy-concentration curve of a strained solid phase, at a given temperature, can be achieved by calculating the peaks of correlation kernel $\\hat{C}_{2j}^{ii}$, at locations slightly off those of the equilibrium density peaks, $k_j$, for a square structure. The introduced amount of strain is defined in Fourier space by\n\\begin{align}\n\\epsilon=|k-k_j|\/k_j\n\\label{FourierStrain}\n\\end{align}\n, where index $j$ denotes one family of planes in reciprocal space. As can be inferred from Fig.~\\ref{fig:StrainELand}, increasing the amount of strain from 0.0016 to 0.014 (corresponding approximately to the average strain within the cluster ``a\" shown in Fig.~\\ref{fig:StrainField}(b) and (c), respectively), raises the free energy in the strained solid. The free energy wells also shift to different concentrations of solute. Such a configuration admits a common tangent between the free energy curves of the unstrained matrix (e.g. the solid curve) and the distorted ones (e.g. dashed curves)~\\cite{larch\u00e985}, leading to a (metastable) multiphase coexistence with a lower free energy (as demonstrated in Fig.~\\ref{fig:StrainELand}). In other words, at each level of local strain, there is a thermodynamic driving force for a transformation from a single-phase structure of a strained matrix to a phase-coexistence between a strained cluster and an unstrained matrix. On the other hand, despite the fact that the above transformation is thermodynamically favorable, the configuration of energy plots in Fig.~\\ref{fig:StrainELand} implies that the driving force for nucleation is lower for the strained cluster (i.e., using the definition of $\\Delta f$ in Eq.~(\\ref{Nucl.En})). However, since the energy curves in this figure are derived from a mean-field PFC approximation, the illustrated phase-coexistence does not carry the effect of interfacial energy and only includes a mean-field sense of the misfit strain. These factors have a significant impact on the thermodynamics of phase-coexistence at cluster sizes smaller than that of the critical nucleus. In fact, the previously described stress relaxation term in the definition of work of formation (Eq.~(\\ref{StressRelaxE})), $\\Delta G_{sr}$, overcompensates for the effect of reduced driving force for formation of strained clusters.  \n\n\\subsection{Clustering mechanism}\nBased on our PFC simulations, we propose the following mechanism of clustering: (1) Stress relaxation by segregation of solute atoms into highly-strained areas in the matrix, such as around dislocations, (2) strain-aided nucleation of sub-critical clusters at concentrations higher than that of the matrix and (3) subsequent growth and enrichment of sub-critical clusters into overcritical sizes, only if a sufficient strain field is preserved, to overcome the nucleation barrier. The above mechanism is consistent with the experimentally observed formation and enrichment of highly-strained coherent GP-zones in quenched-aged dilute Al-Cu alloys~\\cite{biswas11}, proposed as the initial step before precipitation of the semi-coherent and incoherent equilibrium $\\theta$-phases~\\cite{christian}. GP-zones in dilute binary Al alloys are normally known as coherent\/semi-coherent particles often with a crystal structure and composition similar to those of the final equilibrium precipitate~\\cite{christian,hoyt}. Our clusters possess the same chemical composition and lattice parameter as those of the equilibrium theta-phase pre-set by the relevant peaks in our correlation functions. Thus, they would represent an early-stage evolution of the so-called GP-zones. An investigation on the transformation of GP-zones into the subsequent metastable and equilibrium precipitates will be followed in a future study in 3D with more complex crystal structures. We expect to observe a gradual loss of coherency as GP-zones grow in size, as dictated by the energy arguments. We also note that we expect our results to hold qualitatively in 3D, since the same type of elastic effects are expected to appear around the dislocations regardless of their dimension and any possible partial splitting of dislocations around the clusters.\n\n\\section{Summary}\nIn summary, we showed that the alloy phase field crystal model of ref.~\\cite{greenwood11-2} which stabilizes different crystal structures can be used to simulate and analyze the mechanisms of clustering phenomenon in bulk lattice of quenched\/aged alloys. In accordance with the existing experimental observations, our simulations suggests that quenched-in defects, such as dislocations, significantly lower the energy barrier for nucleation of clusters. Furthermore, analysis of overall system energy and local energy changes reveal that the formation and growth of sub-critical clusters are thermodynamically favorable in conjunction with quenched-in mobile dislocations. Consistent with existing experiments, our simulations shed significant light on the elusive energetic mechanism of the growth and enrichment of early clusters which are the precursors of bulk precipitation. \n\n\\begin{acknowledgements}\nWe acknowledge the financial support received from National Science and Engineering Research Council of Canada\n(NSERC), Ontario Ministry of Research and Innovation (Early Researcher Award Program) and the Clumeq High Performance Centre.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}