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+{"text":"\\section{INTRODUCTION}\nAstrophysics and cosmology observations reveal that the dominant matter component in the universe is dark matter (DM), but the particle nature of DM remains unknown~\\cite{Komatsu:2008hk,Ade:2015xua}.\nThe existence of DM cannot be explained within the framework of the standard model (SM), and thus provides a hint of the physics beyond the SM. Great efforts have been devoted to DM researches, including collider detection, direct detection, and indirect detection experiments.\n\nDM particles can be traced by cosmic ray (CR) experiments through their annihilation products from the Galaxy halo.\nThe Alpha Magnetic Spectrometer (AMS-02), launched in 2011, is able to measure CR spectra with an unprecedented precision~\\cite{Aguilar:2013qda}. The precise results released by AMS-02 have confirmed the CR $e^\\pm$ excess above $\\sim 10$ GeV, which indicates the existence of exotic $e^\\pm$ sources. Many astrophysical explanations have been proposed for this excess, such as primary sources like pulsars~\\cite{Hooper:2008kg,Yuksel:2008rf,Profumo:2008ms}, or the CR interactions occurring around CR acceleration sources~\\cite{Blasi:2009hv,Hu:2009zzb,Fujita:2009wk}.\nInterestingly, this excess can also be explained by DM annihilations\/decays to charged leptons~\\cite{Bergstrom:2008gr,Barger:2008su,Cirelli:2008pk,Yin:2008bs,Zhang:2008tb,Bergstrom:2009fa,Lin:2014vja}.\n\nOn the other hand, DM particles would also generate high energy photons associated with charged leptons.\nThe related gamma-ray signatures can be significant in systems with high DM densities and low baryon densities, such as dwarf galaxies. However, the Fermi-LAT observations do not find such signatures, and set strong constraints on the DM annihilation cross section~\\cite{Ackermann:2013yva,Ackermann:2015zua,Fermi-LAT:2016uux}.\nSince the large annihilation cross section required by the CR $e^\\pm$ excess seems not to be allowed by the Fermi-LAT constraints~\\cite{Lin:2014vja}, the DM annihilation explanation is strongly disfavored.\n\n\nMoreover, the electromagnetically interacting particles generated by DM annihilations at recombination could affect cosmic microwave background (CMB)~\\cite{Chen:2003gz,Padmanabhan:2005es,Slatyer:2015kla,Slatyer:2015jla,Galli:2009zc}.\nPrecise measurements performed by WMAP~\\cite{Komatsu:2008hk} and recently by Planck~\\cite{Ade:2015xua} have been used to set constraints on the DM energy injections and the DM annihilation cross sections for specified final states.\nCompared to the results from CR and gamm-ray observations, these constraints are more stringent, and are free of some astrophysical uncertainties, which arise from the large-scale structure formation, DM density files and so on~\\cite{Slatyer:2015jla}.\n\nApparently, the results from the Fermi-LAT and Planck observations strongly disfavor the large DM annihilation cross sections required by  the CR $e^\\pm$ excess~\\cite{Lin:2014vja}. However, note that DM particles have very different relative velocities in different circumstances. For the DM particles potentially impacting on the CR $e^\\pm$, dwarf galaxy gamma-ray, and CMB observations, the typical relative velocities are $v\\sim 10^{-3}$, $10^{-4}$, and $\\ll 10^{-6}$, respectively. Therefore, the inconsistence between the DM explanations for different experimental results can be relaxed or even avoided by a velocity dependent annihilation cross section. In fact, the velocity dependent DM annihilation models, such as the Sommerfeld~\\cite{Hisano:2003ec,Hisano:2006nn,Cirelli:2007xd,ArkaniHamed:2008qn,Feng:2009hw, Feng:2010zp, Cirelli:2016rnw} and Breit-Wigner mechanisms~\\cite{Feldman:2008xs, Ibe:2008ye,Guo:2009aj,Bi:2009uj,Bi:2011qm,Bai:2017fav}, have been widely used to simultaneously explain the thermal DM relic density and the CR $e^\\pm$ excess. In these models, DM particles have a much larger annihilation cross section in the Galaxy with $v\\sim 10^{-3}$ than that in the early Universe for explaining the relic density with $v\\sim 10^{-1}$.\n\n\nIn this paper, we explain the AMS-02 $e^\\pm$ excess in an annihilating DM scenario with the Breit-Wigner mechanism. The DM relic density and the constraints from the Fermi-LAT and Planck observations are also taken into account. In this scenario, two DM particles resonantly annihilate via the s-channel exchange of a heavy mediator. The typical form of the DM annihilation cross section is characterized by two parameters, namely $\\gamma \\equiv \\Gamma_{Z'}\/m_{Z'}$ and $\\delta \\equiv 1- m_{Z'}^2 \/ 4 m_{\\chi}^2$, where $\\Gamma_{Z'}$, $m_{Z'}$, and $m_{\\chi}$ are the mediator decay width, the mediator mass, and the DM mass, respectively. The assumptions of $\\delta >0$ and $\\delta <0$ correspond to the cases with an unphysical pole and a physical pole, respectively. As shown in\nRef.~\\cite{Feldman:2008xs,Ibe:2008ye,Guo:2009aj}, both these two cases can simultaneously explain the high energy positron excess observed by PAMELA and the DM relic density. In our analysis, we perform a fitting to the AMS-02 $e^\\pm$ data with the DM contribution, and derive the corresponding DM annihilation cross sections for $\\mu^+\\mu^-$ and $\\tau^+\\tau^-$ final states. Then we adjust the parameters $\\gamma$ and $\\delta$ to obtain suitable DM annihilation cross sections with different relative velocities. We find that there exists a parameter region with $\\delta<0$, simultaneously accounting for the AMS-02 $e^\\pm$ excess and DM relic density, which is also allowed by the Fermi-LAT dwarf galaxy gamma-ray and the Planck CMB observations.\n\n\nThis paper is organized as follows.\nIn Sec.~\\ref{sec:fit} we perform a fitting to the AMS-02 data, and derive the corresponding DM annihilation cross sections for $\\mu^+\\mu^-$ and $\\tau^+\\tau^-$ final states.\nIn Sec.~\\ref{sec:enhance} we briefly introduce the Breit-Wigner scenario.\nIn Sec.~\\ref{sec:results} we show how to relax the tension between DM explanations for the AMS-02, Fermi, and Planck observations, and obtain the correct DM relic density.\nSec.~\\ref{sec:conclusion} is our conclusions and discussions.\n\n\n\\section{Fit to the AMS-02 data}\n\\label{sec:fit}\nThe complicated CR propagation process can be described by a propagation equation involving some free parameters. In order to predict the CR $e^\\pm$ background, some additional parameters describing the primary and secondary CR injections are needed. In principle, these parameters are determined by available CR observations. In this work, we use the package GALPROP~\\cite{Strong:1998pw,Moskalenko:1997gh} to resolve the propagation equation, and perform a Markov chain Monte Carlo fitting to the AMS-02 data in the high dimensional parameter space.\n\nThe propagation parameters are dominantly determined by a fitting to the measured secondary-to-primary ratios \\cite{Lin:2014vja}, including the B\/C data from ACE~\\cite{2000AIPC..528.....M} and AMS-02~\\cite{2013ICRC-AMS02}, and the $^{10}\\mathrm{Be}\/^{9}\\mathrm{Be}$ data from several experiments. Two kinds\nof propagation models, namely the diffusion-convection (DR) model and the diffusion-reacceleration (DR) model, are taken into account in \\cite{Lin:2014vja}.\nThe injection spectrum of the primary electron background is assumed to be a three-piece broken power law with two breaks.\nComparing to the spectrum with only one break at the low energy, we find that the spectrum with an additional break around $60~\\GeV$ can provide a better fit to the AMS-02 data.\nThe nucleon injection parameters are constrained by fitting the proton flux of AMS-02~\\cite{2013ICRC-AMS02}.\nAfter deriving the propagated proton spectrum, the injection of the secondary $e^\\pm$ backgrounds is calculated by using the parameterized cross section presented in Ref.~\\cite{Kamae:2006bf}.\n\nFor the DM signature, we assume that DM particles purely annihilate to $\\mu^+\\mu^-$ or $\\tau^+\\tau^-$. The initial $e^\\pm$ spectra from DM annihilation are calculated by PPPC 4 DM ID~\\cite{Cirelli:2010xx}, which includes the electroweak corrections~\\cite{Ciafaloni:2010ti}. The DM density profile is taken to be the NFW profile~\\cite{Navarro:1996gj} defined by $\\rho(r)=\\rho_s r_s\/r(1+r\/r_s)^{2}$, with a characteristic halo radius $\\rho_s = 20~\\mathrm{kpc}$ and a characteristic halo density $\\rho_s = 0.26~\\mathrm{GeV cm^{-3}}$.\n\n\\begin{figure}[!htbp]\n\t\\subfigure[$\\mu^+\\mu^-$ channel.\\label{fig:ams02_a}]\n\t{\\includegraphics[width=.45\\textwidth]{positron_mu.eps}}\n\t\\subfigure[$\\tau^+\\tau^-$ channel.\\label{fig:ams02_b}]\n\t{\\includegraphics[width=.45\\textwidth]{positron_tau.eps}}\n  \\caption{Fittings to the positron flux measured by AMS-02 for DM annihilations to $\\mu^+\\mu^-$ (left panel) and $\\tau^+\\tau^-$ (right panel),respectively. The pink bands indicate the contributions from DM annihilation within $2\\sigma$ uncertainty. The blue lines represent the secondary CR positron flux. Total positron fluxes are shown as green bands.\n  \\label{fig:ams02}\n}\n\\end{figure}\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{c|c|c|c|c}\n\\hline\\hline\nChannels    & $m_\\chi (\\TeV)$ &  AMS-02 ($2\\sigma$)    & Fermi limits    & Planck limits\\\\\n\\hline\n$\\mu^+ \\mu^-$ & 0.89 & $3.79 \\times 10^{-24} < \\langle \\sigma_{\\mathrm{ann}} v \\rangle < 6.48 \\times 10^{-24}$\n               &  $2.95 \\times 10^{-24}$\n\t\t\t\t\t&  $2.58 \\times 10^{-24}$\n\\\\\n$\\tau^+ \\tau-$ & 3.89  & $5.29 \\times 10^{-23} < \\langle \\sigma_{\\mathrm{ann}} v \\rangle < 1.06 \\times 10^{-22}$\n               &  $ 1.25 \\times 10^{-23}$\n\t\t\t\t\t&  $ 1.06 \\times 10^{-23}$\n\\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{The best-fit values of DM masses $m_\\chi$ and corresponding thermally averaged annihilation cross sections $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ (in units of $\\mathrm{cm}^3\\mathrm{s}^{-1}$) given by the fitting to the AMS-02 data with the DR propagation model. The corresponding limits from the Fermi-LAT and Planck observations are also shown.\n}\n\\label{tb:limits}\n\\end{table}\n\nCombining the contributions of primary CR $e^-$, secondary CR $e^\\pm$, and $e^\\pm$ from DM annihilation, we perform a fit to the latest AMS-02 $e^\\pm$ data, including the positron fraction $\\frac{e^+}{e^+ + e^-}$ and the fluxes of $e^+$, $e^-$, and $e^+ + e^-$~\\cite{AMS02-posi-2014,AMS02-elec-2014,AMS02-tot-2014}.\nWe provide the fitting results to the observed $e^+$ flux with the DR prorogation model in Fig.~\\ref{fig:ams02}; the bands representing $2\\sigma$ uncertainties are also shown. The best-fit values of $m_\\chi$ and related $2\\sigma$ regions of $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ (in $\\mathrm{cm}^3\\mathrm{s}^{-1}$) are listed in Table.~\\ref{tb:limits}.\nThe corresponding exclusion limits derived from the Fermi-LAT dwarf galaxy gamma-ray~\\cite{Ackermann:2015zua} and Planck CMB~\\cite{Slatyer:2015jla} observations are also given.\nIt is obvious that the parameter regions of $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ for explaining the AMS-02 $e^\\pm$ excess are excluded by other two kinds of observations.\nCompared to the $\\mu^+\\mu^-$ channel, the tension in the $\\tau^+\\tau^-$ channel is severer due to tremendous photons from the hadronic decays of $\\tau$.\n\n\n\\section{BREIT-WIGNER ENHANCEMENT}\n\\label{sec:enhance}\n\nIn the Breit-Wigner scenario, the DM annihilation cross section has a typical form of\n\\begin{equation}\n  \\begin{split}\n  \\sigma v \\propto \\frac{1}{16 \\pi m_\\chi^2}\\frac{1}{(\\delta+v^2\/4)^2+\\gamma^2 }\n\\end{split}\n\\label{eq:sigmav_s}\n\\end{equation}\nThis form is valid in the non-relativistic limit with $v^2 << 1$ and $\\delta<< 1$ at the center-of-mass energy $\\sqrt{s} \\sim \\sqrt{4 m_\\chi^2 + m_\\chi^2 v^2} $.\n\nAs an example, we consider a simple leptophilic fermionic DM model, where DM particles interact with charged leptons through a vector mediator $Z'$ \\cite{Bi:2009uj}. The corresponding lagrangian is\n\\begin{equation}\n  \\mathscr{L}_{int}  \\supset -g(a \\bar{\\chi} \\gamma^\\mu \\chi +\n    \\bar{l_i} \\gamma^\\mu l_i\n    ) Z_\\mu',\n\\label{eq:l}\n\\end{equation}\nwhere $l_i$ represents the species of leptons, $g$ and $a g$ are the couplings of $Z'$ to the leptons and DM particles, respectively.\nThis model can easily avoid the constraints from DM direct detection and collider experiments due to its leptophilic property.\n\nThe DM annihilation cross section in this model is given by\n\\begin{equation} \\label{eq:sv}\n  \\sigma v = \\frac{1}{ 6 \\pi}\\frac{a^2 g^4 s}{(s-m_{Z'}^2)^2+ m_{Z'}^2 \\Gamma_{Z'}^2}\n  (1+\\frac{ 2m_\\chi^2}{s}),\n\\end{equation}\nwhere $m_\\chi$, $m_{Z'}$ and $\\Gamma_{Z'}$ are the DM mass, the $Z'$ mass, and the decay width of $Z'$, respectively,\n$v$ is the relative velocity between two incident DM particles. Note that the lepton mass has been neglected in Eq.~\\ref{eq:sv}\ndue to the large $\\sqrt{s}$ considered in our analysis. The decay width of $Z'$ can be expressed as\n\\begin{equation}\n  \\Gamma_{Z'} = \\frac{m_{Z'}}{12 \\pi} a^2 g^2 \\xi_\\chi^3 \\Theta(m_{Z'}-2 m_{\\chi})\n  + \\frac{m_{Z'}}{12 \\pi} g^2  \\xi_{l_i}^3,\n\\end{equation}\nwhere $\\xi_\\chi \\equiv \\sqrt{1-4 m_\\chi^2 \/ m_{Z'}^2}$, $\\xi_{l_i} \\equiv \\sqrt{1-4 m_{l_i}^2 \/ m_{Z'}^2}$, and $\\Theta(x)$ is the unit step function.\nFor $m_{Z'} \\sim 2 m_\\chi$, $Z'$ dominantly decays to leptons with the decay width given by $\\sim g^2 m_{Z'}\/{12\\pi^2}$.\n\nThen we calculate the thermally averaged DM annihilation cross section through the formula of ~\\cite{Gondolo:1990dk}\n\\begin{equation}\n  \\langle \\sigma_{\\mathrm{ann}} v \\rangle =\\frac{1}{n_{EQ}^2} \\frac{m_\\chi}{ 64 \\pi^4 x}\n  \\int_{4 m_\\chi^2}^{\\infty} \\hat{\\sigma}(s) \\sqrt{s} K_1(\\frac{x\\sqrt{s}}{m_\\chi}) ds,\n  \\label{eq:sigmav}\n\\end{equation}\nwith\n\\begin{equation}\n  \\begin{split}\n    n_{EQ} & = \\frac{g_i}{2 \\pi^2}\\frac{m_\\chi^3}{x} K_2(x),\\\\\n    \\hat{\\sigma}(s) & = 2 g_i^2 m_\\chi \\sqrt{s- 4 m_\\chi^2} \\sigma v,\n  \\end{split}\n\\end{equation}\nwhere $K_i(x)$ is the modified Bessel function of order\t$i$, $g_i$ is the internal degree of freedom of the DM particle, which equals 4 in this model.\n\nThe evolution of the DM density is determined by numerically solving the Boltzmann equation\n\\begin{equation}\n  \\begin{split}\n    \\frac{d Y}{d x} =  - \\frac{ s(x)}{Hx} \\langle \\sigma_{\\mathrm{ann}} v \\rangle (Y^2 -Y_{eq}^2),\n  \\end{split}\n\\end{equation}\n where $Y \\equiv n\/s$, $n$ is the DM number density, $s = \\frac{2 \\pi^2 }{45} g_{\\ast s}\\frac{m^3}{x^3}$ is the Universe entropy density, $H= \\sqrt{\\frac{4 \\pi^3 g_{\\ast}}{45 m_{pl}^2}} \\frac{m^2}{x^2}$ is the Hubble parameter, and $g_{\\ast s}$ and $g_\\ast$ are the effective degrees of freedom defined by the entropy density and the radiation density, respectively.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\begin{figure}[!htbp]\n\t\\subfigure[$\\delta < 0$.\\label{fig:efactor_a}]\n\t{\\includegraphics[width=.45\\textwidth, trim={40 20 40 30},clip]{efactor_m.eps}}\n\t\\subfigure[$\\delta > 0$.\\label{fig:efactor_b}]\n\t{\\includegraphics[width=.45\\textwidth, trim={40 20 40 30},clip]{efactor.eps}}\n  \\caption{The scaling factor $S\\equiv \\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}\/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ in the $\\delta-\\gamma$ plane, where $\\sigma_\\mathrm{D}$ and $\\sigma_\\mathrm{G}$ denote the annihilation cross sections in dwarf galaxies with $v=10^{-4}$ and near the solar system in the Galaxy with $v=10^{-3}$, respectively. The left and right panels represent the physical pole case with $\\delta<0$ and unphysical pole case with $\\delta>0$, respectively.\n  \\label{fig:efactor}\n}\n\\end{figure}\n\nIn principle, we can accommodate the DM explanations for observations with different DM relative velocities. Only the DM particles located in the Galaxy within a range of $\\sim 1$ kpc around the Solar system could provide significant contributions to the observed high energy CR $e^\\pm$, because of the prorogation effects. The typical relative velocities of these particles are $\\sim 10^{-3}$, while the typical relative velocities of DM particles in dwarf galaxies are $\\sim 10^{4}$. Their annihilation cross sections may be very different in the velocity dependent annihilation models. In order to obtain the constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$, the constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$ from the Fermi-LAT observation should be rescaled by a factor of $1\/S \\equiv \\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G} \/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$, where $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D} $ and $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ are the thermally averaged DM annihilation cross sections in dwarf galaxies and near the solar system in the Galaxy, respectively. In order to relax the tension between the DM explanations for the Fermi-LAT and AMS-02 observations, the $S$ factor should be smaller than 1.\n\nWe show the $S$ factor in Fig.~\\ref{fig:efactor}, and find that a parameter region with $10^{-8} < \\gamma < 10^{-6}$ and $-4 \\times 10^{-6} < \\delta < -10^{-7}$ can satisfy our requirement with $S \\ll 1$. For the cases of $\\delta > 0$ corresponding to an unphysical pole, there is no parameter region with $S<1$ as shown in Fig.~\\ref{fig:efactor_b}. This can be understood by Eq.~\\ref{eq:sigmav_s}: the DM annihilation cross section always increases with decreasing relative velocity for $\\delta >0$. Therefore, only the cases of $\\delta < 0$ can be used to relax the tension between different observations.\n\n\n\\begin{figure}[!h]\n\t\\subfigure[$\\mu^+\\mu^-$ channel.]\n\t{\\includegraphics[width=.45\\textwidth]{fermi_channel_mu.eps}}\n\t\\subfigure[$\\tau^+\\tau^-$ channel.]\n\t{\\includegraphics[width=.45\\textwidth]{fermi_channel_tau.eps}}\n  \\caption{Contour regions represent the parameter regions accounting for the AMS-02 results in the DC and DR propagation models.\n  Solid lines are the constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ from the Fermi-LAT dwarf galaxy gamma-ray observation for different parameter sets of $\\delta$ and $\\gamma$. The original the Fermi-LAT limits on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$ are also shown. The left and right panels represent the cases of DM annihilation to $\\mu^+\\mu^-$ and $\\tau^+\\tau^-$, respectively.\n  }\n\\label{fig:fermi}\n\\end{figure}\n\nIn Fig.~\\ref{fig:fermi}, we compare the parameter regions accounting for the AMS-02 $e^\\pm$ excess with the dwarf galaxy gamma-ray constraints, which are obtained by rescaling the limits given by the Fermi-LAT collaboration \\cite{Ackermann:2015zua}. It is shown that the cases with a negative tiny $-\\delta \\leq 10^{-6}$ can evade the dwarf galaxy constraints. As $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ is almost proportional to $1\/m_\\chi^2$ as can be seen from Eq.~\\ref{eq:sigmav}, the ratio of $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G} \/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$ is independent of the DM mass. Therefore, the modified dwarf galaxy gamma-ray constraints for different parameter sets of $\\delta$ and $\\gamma$ are parallel in Fig.~\\ref{fig:fermi}. Note that in the above analysis, we fix the DM relative velocity in dwarf galaxies to be $v=10^{-4}$. Strictly speaking, since DM particles in dwarf galaxies have different typical relative velocities with an order of $\\mathcal{O}(10^{-4})$, the total constraint should be obtained by combining the individual constraints specified for each dwarf galaxy with a large J factor. A detailed discussion can be found in Ref.~\\cite{Zhao:2016xie}.\n\n\n\\begin{figure}[!h]\n\t\\subfigure[$\\mu^+\\mu^-$ channel.]\n\t{\\includegraphics[width=.45\\textwidth]{planck_channel_mu.eps}}\n\t\\subfigure[$\\tau^+\\tau^-$ channel.]\n\t{\\includegraphics[width=.45\\textwidth]{planck_channel_tau.eps}}\n\t\\caption{The same as Fig.~\\ref{fig:fermi} but the constraints are derived from the Planck CMB observation.\n  }\n\\label{fig:planck}\n\\end{figure}\n\nThe above analysis can be directly applied to reconcile the tension between the DM explanations for the AMS-02 $e^\\pm$ and Planck CMB observations. In order to derive the constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ from CMB observations, we define a rescaling factor of $1\/S' \\equiv \\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}\/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{z_r}$, where $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{z_r}$ is the thermally averaged annihilation cross section of DM particles affecting CMB at recombination with $v\\ll 10^{-6}$. In fact, the Breit-Wigner effect would saturate for DM particles with such a small $v$. In Fig.~\\ref{fig:fermi}, we compare the parameter regions accounting for the AMS-02 $e^\\pm$ excess with the CMB constraints, which are obtained by rescaling the limits given by Ref.~\\cite{Slatyer:2015kla}. We find that the cases with a negative tiny $\\delta \\sim -10^{-6}$ can also evade the CMB constraints.\n\n\n\\begin{figure}[!h]\n\t\\subfigure[$\\mu^+\\mu^-$ channel.\\label{fig:enhance_m_a}]\n\t{\\includegraphics[width=.45\\textwidth]{mu_enhancement_m.eps}}\n\t\\subfigure[$\\tau^+ \\tau^-$ channel.\\label{fig:enhance_m_b}]\n\t{\\includegraphics[width=.45\\textwidth]{tau_enhancement_m.eps}}\n  \\caption{Parameter regions accounting for various observations in the $\\delta-\\gamma$ plane with $\\delta<0$ for DM annihilation to $\\mu^+\\mu^-$ (left panel) and $\\tau^+\\tau^-$ (right panel), respectively. The DM mass is taken to be the value given in  Table.~\\ref{tb:limits}. In each parameter point, $a$ and $ag^2$ are derived by requiring the correct relic density $\\Omega h^2 =0.1188$; then $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$, $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$, and $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{Z_r}$ can also be obtained. Red shaded region are the parameter regions corresponding to $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ given in Tab.~\\ref{tb:limits} which can explain the AMS-02 results. The gray and cyan regions denote parameter regions excluded by the Planck and Fermi-LAT observations, respectively. The green and blue solid lines are the isolines of $a$ and $ag^2$, respectively.}\n\\label{fig:enhance_m}\n\\end{figure}\n\n\nFor each point in the $\\delta-\\gamma$ plane with $\\delta<0$, we determine $ag^2$ and $a$ through the correct relic density $\\Omega h^2=0.1188$ \\cite{Ade:2015xua} by resolving the Bolzmann equation, and derive corresponding $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$, $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{D}$, and $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{Z_r}$. In Fig.~\\ref{fig:enhance_m}, the red bands represent the parameter regions simultaneously accounting for the AMS-02 CR $e^\\pm$ excess and the correct relic density. Here we only consider $m_\\chi$ and $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ derived with the DR propagation model as given in Table.~\\ref{tb:limits}. The parameter regions excluded by the Fermi-LAT and Planck limits are also shown in Fig.~\\ref{fig:enhance_m}. We find that there exists a parameter region with $\\gamma \\uwave{<}  10^{-7}$ and $\\delta \\sim -10^{-6}$, which can accommodate all the observations.\n\nWe also show the isolines of $a g^2$ and $a$ satisfying the correct DM relic density in Fig.~\\ref{fig:enhance_m}. The behavior of these lines can be understood as follows. Roughly speaking, the thermal relic density $\\Omega h^2$ in the usual DM models is determined by the freeze-out temperature $x_f \\sim \\mathcal{O}(10)$ (corresponding to $v^2 \\sim 10^{-1}$) and $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{f}$ as $\\Omega h^2 \\propto x_f \/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{f}$. For the resonant case, since the annihilation cross section would increase with dropping temperature, the annihilation process may be significant until the Breit-Wigner effect almost saturates at a temperature of $x_b$. $x_b$ can be roughly determined by $|\\delta|^{-1}$ for $\\delta< 0$. This is because that there are not enough DM particles with velocities of $v\\sim |\\delta|^{\\frac{1}{2}}$ for sufficient resonant annihilation when $x\\gg 1\/|\\delta|$. Using the approximated form of $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{b} \\propto a^2g^4 |\\delta|^{\\frac{1}{2}} x_b^{\\frac{3}{2}} \/\\gamma$ by integrating out the pole~\\cite{Bai:2017fav}, we get $\\Omega h^2 \\propto x_b \/\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{b} \\propto \\gamma\/a^2g^4$. Therefore, the correct relic density can be easily obtained by adjusting $ag^2$ with $\\gamma^{\\frac{1}{2}}$ as shown in Fig.~\\ref{fig:enhance_m}.\n\n\n\nAn important issue that should be addressed is the kinetic decoupling effect. In the parameter regions discussed above, since the scatterings between DM particles and SM radiations are not sufficient due to the t-channel exchange of a heavy mediator, the kinetic decoupling would occur at a high temperature of $T > \\mathcal{O} (1)$ GeV. The velocities of DM particles drop as $\\sim R^{-1}$ after the kinetic decoupling rather than $\\sim R^{-\\frac{1}{2}}$ before the kinetic decoupling, where $R$ is the scale factor of the Universe. Then the Breit-Wigner mechanism would significantly enhance the DM annihilation cross section at the freeze-out epoch and drastically reduce the DM relic density. As discussed in Ref.~\\cite{Bi:2011qm}, it is difficult to simultaneously explain the CR $e^\\pm$ excess and the relic density with such a significant kinetic decoupling effect. Moreover, after the kinetic decoupling, the velocity distribution of DM particles would depart from the thermal distribution and is difficult to deal with in the calculation of the relic density. A solution is introducing some additional mediators, which can enhance the scattering rate between DM particles and SM radiations and\/or the DM self scattering rate. The detailed discussions can be found in Ref.~\\cite{Feng:2010zp,Bai:2017fav}.\n\n\n\\section{CONCLUSIONS and DISCUSSIONS}\n\\label{sec:conclusion}\n\nIn this work we show that the DM annihilation througth the Breit-Wigner mechanism can reconcile the tension between the DM explanation for the AMS-02 CR $e^\\pm$ excess and the constraints from Fermi-LAT dwarf galaxy gamma-ray and Planck CMB observations. Since DM particles affecting these observations have different relative velocities, their annihilation cross sections are different for interpretating the experimental results. In order to check whether the DM explanation for the AMS02 results is excluded by other observations, we should translate all the limits into those on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ for DM particles with a typical relative velocity $v\\sim 10^{-3}$.\n\nWe take a leptophilic $Z'$ model as a benchmark model. This kind of leptophilic model is not constrained by the results of current direct detection and collider experiments. For the tiny values of the mediator decay width and the mass deviation from the pole, $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ would be sensitive to the relative velocity. For the unphysical pole case with $\\delta > 0$, $\\langle \\sigma_{\\mathrm{ann}} v \\rangle$ increases with deceasing velocity. Thus the enhanced constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ from the dwarf galaxy gamma-ray and CMB observations exclude the explanation for the CR $e^\\pm$ excess in this case.\n\n\nFor the physical pole case with $\\delta \\sim -10^{-6}$, DM particles accounting for the CR $e^\\pm$ excess with $v\\sim \\mathcal{O}(10^{-3})$ have the largest annihilation cross section close to the pole. On the other hand, the DM annihilation cross section is suppressed for DM particles with smaller relative velocities in dwarf galaxies and at recombination, which may impact on the gamma-ray and CMB observations, respectively. Therefore, the constraints on $\\langle \\sigma_{\\mathrm{ann}} v \\rangle_\\mathrm{G}$ from these observations are weaken. We find that a parameter region with $\\delta \\sim -10^{-6}$ and $\\gamma \\lesssim 10^{-7}$ can simultaneously account for the AMS-02, Fermi-LAT dwarf galaxy gamma-ray, and Planck CMB observations, and the relic density.\n\nFrom the perspective of model building, a question is how to naturally realize the tiny values of $\\delta$ and $\\gamma$ derived in above analysis. Here we consider the benchmark point with $\\gamma=7.1 \\times 10^{-8}$ and $\\delta= - 1.5 \\times 10^{-6}$ marked in the right panel of Fig.~\\ref{fig:enhance_m}. For the small decay width of the mediator, we get $g\\sim 1.8\\times 10^{-3}$ and $ag\\sim 1.8 \\times 10^{-2}$. These values are easy to realize in a realistic model. The problem is how to achieve a tiny $\\delta \\sim -10^{-6}$, which seems to require a significant fine-tuning. A solution is given by Ref.~\\cite{Bai:2017fav} through the nontrivial flavour symmetry-breaking in the dark sector. By assigning a particular symmetry-breaking mode, a resonance with a mass of almost $2 m_\\chi$ can be realized. The tiny mass deviation of $\\delta$ is naturally induced by loop effects.\n\n\n\n\\section*{Acknowledgment}\nThis work is supported by the National Key Program for Research and Development (No.\n2016YFA0400200) and by the National Natural Science Foundation of China\nunder Grants No. 11475189 and 11475191.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-0.15cm}\n\nAdversarial training (AT) has been one of the most effective defense strategies against adversarial attacks~\\citep{biggio2013evasion,Szegedy2013,Goodfellow2014}. Based on the primary AT frameworks like PGD-AT~\\citep{madry2018towards}, many improvements have been proposed from different perspectives, and demonstrate promising results (detailed in Sec.~\\ref{sec2}). However, the recent benchmarks~\\citep{croce2020reliable,chen2020rays} find that simply early stopping the training procedure of PGD-AT~\\citep{rice2020overfitting} can attain the gains from almost all the previously proposed improvements, including the state-of-the-art TRADES~\\citep{zhang2019theoretically}.\n\n\n\n\n\nThis fact is somewhat striking since TRADES also executes early stopping (one epoch after decaying the learning rate) in the code implementation\\footnote{https:\/\/github.com\/yaodongyu\/TRADES}, as clarified in~\\citet{rice2020overfitting}. Besides, the reported robustness of PGD-AT in \\citet{rice2020overfitting} is much higher than it in \\citet{madry2018towards}, even without early-stopping. These motivate us to check the implementation details of \\citet{rice2020overfitting}\\footnote{https:\/\/github.com\/locuslab\/robust\\_overfitting}, and compare the basic hyperparameter settings with TRADES. We find that TRADES uses weight decay of $2\\times 10^{-4}$ and eval mode of batch normalization (BN) when crafting adversarial examples, while \\citet{rice2020overfitting} use weight decay of $5\\times 10^{-4}$ and train mode of BN to generate adversarial examples. In our experiments (e.g., Table~\\ref{table9}), we show that the two slightly different settings can differ the robust accuracy by $\\sim 4\\%$, which is significant according to the reported benchmarks.\n\n\n\nTo have a comprehensive study, we further investigate the implementation details of tens of papers working on the AT methods, some of which are summarized in Table~\\ref{table1}. We find that even using the same model architectures, the basic hyperparameter settings (e.g., weight decay, learning rate schedule, etc.) used in these papers are highly inconsistent and customized, which could affect the model performance and may override the gains from the methods themselves. Under this situation, if we directly benchmark these methods using their released code or checkpoints, some actually effective improvements would be under-estimated due to the improper hyperparameter settings. \n\n\n\\textbf{Our contributions.} We evaluate the effects of a wide range of basic training tricks (e.g., warmup, early stopping, weight decay, batch size, BN mode, etc.) on the adversarially trained models. Our empirical results suggest that improper training settings can largely degenerate the model performance, while this degeneration may be mistakenly ascribed to the methods themselves. We provide a baseline recipe for PGD-AT on CIFAR-10 as an example, and demonstrate the generality of the recipe on training other frameworks like TRADES. As seen in Table~\\ref{tableappendix2}, the retrained TRADES achieve new state-of-the-art performance on the AutoAttack benchmark~\\citep{croce2020reliable}.\n\n\n\nAlthough our empirical conclusions may not generalize to other datasets or tasks, we reveal the facts that adversarially trained models could be sensitive to certain training settings, which are usually neglected in previous work. These results also encourage the community to re-implement the previously proposed defenses with fine-tuned training settings to better explore their potentials.\n\n\n\n\n\n\n\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.35cm}\n  \\caption{Hyperparameter settings and tricks used to implement different AT methods on CIFAR-10. We convert the training steps into epochs, and provide code links for reference in Table~\\ref{tableappendix1}. Compared to the model architectures, the listed settings are easy to be neglected and paid less attention to unify.}\n  \\vspace{-0.15cm}\n    \\begin{tabular}{l|c|c|c|c|c|c}\n    \\hline\n   \\;\\;\\;\\;\\;\\;\\;\\;\\; \\multirow{2}{*}{Method} \\!\\! & \\!\\!  \\multirow{2}{*}{l.r.} \\!\\! & \\!\\!  Total epoch \\!\\! & \\!\\!\\!  Batch \\!\\!\\! & \\!\\!  Weight \\!\\! & \\!\\!\\!\\!  Early stop \\!\\!\\!\\! & \\!\\!\\!\\!  Warm-up \\!\\!\\!\\! \\\\\n    \\!\\! & \\!\\!  \\!\\! & \\!\\!  (l.r. decay) \\!\\! & \\!\\!\\!  size \\!\\!\\! & \\!\\!  decay \\!\\! & \\!\\!\\!\\!  (train \/ attack) \\!\\!\\!\\! & \\!\\!\\!\\!  (l.r. \/ pertub.) \\!\\!\\!\\! \\\\\n     \\hline\n     \n     \\!\\!\\!\\citet{madry2018towards} \\!\\!\\!\\! \\!\\! & \\!\\! 0.1 \\!\\! & \\!\\! 200 (100, 150) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n     \n       \\!\\!\\!\\citet{cai2018curriculum} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  300 (150, 250) \\!\\! & \\!\\!  200 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ Yes\\\\\n     \n      \\!\\!\\!\\citet{zhang2019theoretically} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\! 76 (75) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  Yes \/ No \\!\\! & \\!\\!  No \/ No\\\\\n     \n    \\!\\!\\!\\citet{wang2019convergence} \\!\\!\\!\\! \\!\\! & \\!\\!  0.01 \\!\\! & \\!\\! 120 (60, 100) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $1\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ Yes \\!\\! & \\!\\!  No \/ No\\\\\n           \n      \\!\\!\\!\\citet{qin2019adversarial} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1  \\!\\! & \\!\\! 110 (100, 105) \\!\\! & \\!\\!  256 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ Yes\\\\\n     \n     \\!\\!\\!\\citet{mao2019metric} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1  \\!\\! & \\!\\! 80 (50, 60) \\!\\! & \\!\\!  50 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No \\\\\n     \n      \\!\\!\\!\\citet{carmon2019unlabeled} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  100 (cosine anneal) \\!\\! & \\!\\!  256 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n      \n      \\!\\!\\!\\citet{alayrac2019labels} \\!\\!\\!\\! \\!\\! & \\!\\!  0.2 \\!\\! & \\!\\!  64 (38, 46, 51) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n        \n        \\!\\!\\!\\citet{shafahi2019adversarial} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (100, 150) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n        \n       \\!\\!\\!\\citet{zhang2019you} \\!\\!\\!\\! \\!\\! & \\!\\!  0.05 \\!\\! & \\!\\!  105 (79, 90, 100) \\!\\! & \\!\\!  256 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n      \n       \\!\\!\\!\\citet{zhang2019defense} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (60, 90) \\!\\! & \\!\\!  60 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n      \n      \\!\\!\\!\\citet{atzmon2019controlling} \\!\\!\\!\\! \\!\\! & \\!\\!  0.01 \\!\\! & \\!\\!  100 (50) \\!\\! & \\!\\!  32 \\!\\! & \\!\\!  $1\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n      \n       \\!\\!\\!\\citet{wong2020fast} \\!\\!\\!\\! \\!\\! & \\!\\!  0$\\sim$0.2\\!\\! & \\!\\!  30 (one cycle) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  Yes \/ No\\\\\n    \n     \\!\\!\\!\\citet{rice2020overfitting} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (100, 150) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  Yes \/ No \\!\\! & \\!\\!  No \/ No\\\\\n     \n     \\!\\!\\!\\citet{ding2019mma} \\!\\!\\!\\! \\!\\! & \\!\\!  0.3 \\!\\! & \\!\\!  128 (51, 77, 102) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n     \n      \\!\\!\\!\\citet{pang2019rethinking} \\!\\!\\!\\! \\!\\! & \\!\\!  0.01 \\!\\! & \\!\\!  200 (100, 150) \\!\\! & \\!\\!  50 \\!\\! & \\!\\!  $1\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n      \n      \\!\\!\\!\\citet{zhang2020attacks} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  120 (60, 90, 110) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ Yes \\!\\! & \\!\\!  No \/ No\\\\\n     \n     \n      \\!\\!\\!\\citet{huang2020self} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (cosine anneal) \\!\\! & \\!\\!  256 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  Yes \/ No\\\\\n    \n     \\!\\!\\!\\citet{cheng2020cat} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (80, 140, 180) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $5\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n     \n     \\!\\!\\!\\citet{lee2020adversarial} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  200 (100, 150) \\!\\! & \\!\\!  128 \\!\\! & \\!\\!  $2\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n    \n     \\!\\!\\!\\citet{xu2020exploring} \\!\\!\\!\\! \\!\\! & \\!\\!  0.1 \\!\\! & \\!\\!  120 (60, 90) \\!\\! & \\!\\!  256 \\!\\! & \\!\\!  $1\\times 10^{-4}$ \\!\\! & \\!\\!  No \/ No \\!\\! & \\!\\!  No \/ No\\\\\n    \n    \\hline\n    \\end{tabular}%\n    \\label{table1}\n\\end{table}%\n\n\n\\vspace{-0.15cm}\n\\section{Related work}\n\\vspace{-0.15cm}\n\\label{sec2}\nIn this section, we introduce related work on the adversarial defenses and recent benchmarks. We detail on the adversarial attacks in Appendix~\\ref{advattacks}.\n\n\n\\vspace{-0.1cm}\n\\subsection{Adversarial defenses}\n\\vspace{-0.1cm}\nTo alleviate the adversarial vulnerability of deep learning models, a large number of defense strategies have been proposed, but most of them can eventually be evaded by adaptive attacks~\\citep{carlini2017adversarial,athalye2018obfuscated}. Other more theoretically guaranteed routines include training provably robust networks~\\citep{dvijotham2018training,dvijotham2018dual,hein2017formal,wong2018provable} and obtaining certified models via randomized smoothing~\\citep{cohen2019certified}. While these methods are exciting, they cannot match the state-of-the-art robustness under empirical evaluations.\n\n\n\n\nThe idea of adversarial training (AT) stems from the seminal work of \\citet{Goodfellow2014}, while other AT frameworks like PGD-AT~\\citep{madry2018towards} and TRADES~\\citep{zhang2019theoretically} occupied the winner solutions in the adversarial competitions~\\citep{kurakin2018competation,brendel2020adversarial}. Based on these primary AT frameworks, many improvements have been proposed via encoding the mechanisms inspired from other domains, including ensemble learning~\\citep{tramer2017ensemble,pang2019improving}, metric learning~\\citep{mao2019metric,li2019improving,pang2020boosting}, generative modeling~\\citep{jiang2018learning,pang2018max,wang2019direct,deng2020adversarial}, semi-supervised learning~\\citep{carmon2019unlabeled,alayrac2019labels,zhai2019adversarially}, and self-supervised learning~\\citep{hendrycks2019using,chen2020self,chen2020adversarial,naseer2020self}. On the other hand, due to the high computational cost of AT, many efforts are devoted to accelerating the training procedure via reusing the computations~\\citep{shafahi2019adversarial,zhang2019you}, adaptive adversarial steps~\\citep{wang2019convergence,zhang2020attacks} or one-step training~\\citep{wong2020fast,liu2020using,vivek2020single}. The following works try to solve the side effects (e.g., catastrophic overfitting) caused by these fast AT methods~\\citep{andriushchenko2020understanding,li2020towards}.\n\n\\vspace{-0.1cm}\n\\subsection{Adversarial benchmarks}\n\\vspace{-0.1cm}\n\n\nDue to the large number of proposed defenses, several benchmarks have been developed to rank the adversarial robustness of existing methods. \\citet{dong2019benchmarking} perform large-scale experiments to generate robustness curves, which are used for evaluating typical defenses. \\citet{croce2020reliable} propose AutoAttack, which is an ensemble of four selected attacks. They apply AutoAttack on tens of previous defenses and provide a comprehensive leader board. \\citet{chen2020rays} propose the black-box RayS attack, and establish a similar leader board for defenses. In this paper, we mainly apply PGD attack and AutoAttack as two common ways to evaluate the models. \n\n\nExcept for the adversarial robustness, there are other efforts that introduce augmented datasets for accessing the robustness against general corruptions or perturbations. \\citet{mu2019mnist} introduce MNIST-C with a suite of 15 corruptions applied to the MNIST test set, while \\citet{hendrycks2019benchmarking} introduce ImageNet-C and ImageNet-P with common corruptions and perturbations on natural images. Evaluating robustness on these datasets can reflect the generality of the proposed defenses, and avoid overfitting to certain attacking patterns~\\citep{engstrom2017rotation,tramer2019adversarial}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-0.15cm}\n\\section{Bag of tricks}\n\\vspace{-0.15cm}\nOur overarching goal is to investigate how the implementation details affect the performance of the adversarially trained models. Our experiments are done on CIFAR-10~\\citep{Krizhevsky2012} under the $\\ell_{\\infty}$ threat model of maximal perturbation $\\epsilon=8\/255$, without accessibility to additional data. We evaluate the models under 10-steps PGD attack (\\textbf{PGD-10})~\\citep{madry2018towards} and AutoAttack (\\textbf{AA})~\\citep{croce2020reliable}. We consider some basic training tricks and perform ablation studies on each of them, based on the default training setting as described below:\n\n\n\\textbf{Default setting.} Following \\citet{rice2020overfitting}, in the default setting, we apply the primary PGD-AT framework and the hyperparameters including batch size $128$; SGD momentum optimizer with the initial learning rate of $0.1$; weight decay $5\\times 10^{-4}$; ReLU activation function and no label smoothing; train mode for batch normalization when crafting adversarial examples. All the models are trained for $110$ epochs with the learning rate decays by a factor of $0.1$ at $100$ and $105$ epochs, respectively. \\emph{We report the results on the checkpoint with the best PGD-10 accuracy}. \n\n\n\nNote that our empirical observations and conclusions may not always generalize to other datasets or AT frameworks, but we emphasize the importance of using consistent implementation details (not only the same model architectures) to enable fair comparisons among different AT methods.  \n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.4cm}\n  \\caption{Test accuracy (\\%) under different \\textbf{early stopping} and \\textbf{warmup} on CIFAR-10. The model is ResNet-18. For early stopping on attack iterations, we denote, e.g., 40 \/ 70 as the epochs to increase the tolerance step by one~\\citep{zhang2020attacks}. For warmup, the learning rate (l.r.) and the maximal perturbation (perturb.) linearly increase from zero to the preset value in the first 10 \/ 15 \/ 20 epochs.}\n  \\vspace{-0.2cm}\n  \\renewcommand*{\\arraystretch}{1.3}\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\hline\n    & \\multirow{2}{*}{Base} & \\multicolumn{3}{c|}{\\!\\!\\!\\!\\textbf{Early stopping attack iter.}\\!\\!\\!\\!} & \\multicolumn{3}{c|}{\\textbf{Warmup on l.r.}} &\\multicolumn{3}{c}{\\textbf{Warmup on perturb.}}  \\\\\n    & & \\!\\! 40 \/ 70 \\!\\! & \\!\\! 40 \/ 100 \\!\\! & \\!\\! 60 \/ 100 \\!\\!  & 10 & 15 & 20 & 10 & 15 & 20 \\\\\n    \\hline\n   \\!\\!\\!\\! Clean \\!\\!\\!\\! & 82.52 & 86.52 & 86.56 & 85.67& 82.45 & 82.64 & 82.31 & 82.64 & 82.75 & 82.78\\\\\n   \\!\\!\\!\\! PGD-10 \\!\\!\\!\\! & 53.58 & 52.65 & 53.22 & 52.90 & 53.43 & 53.29 & 53.35 & 53.65 & 53.27 & 53.62\\\\\n   \\!\\!\\!\\! AA \\!\\!\\!\\! & 48.51 & 46.6 & 46.04 & 45.96 & 48.26 & 48.12 & 48.37 & 48.44 & 48.17 & 48.48 \\\\\n    \\hline\n    \\end{tabular}%\n    \\vspace{-0.25cm}\n    \\label{table5}\n\\end{table}%\n\n\\vspace{-0.1cm}\n\\subsection{Early stopping and warmup}\n\\vspace{-0.1cm}\n\\textbf{Early stopping training epoch.} The trick of early stopping w.r.t. the training epoch was first applied in the code implementation of TRADES~\\citep{zhang2019theoretically}, where the learning rate decays at $75$ epoch and the training is stopped at $76$ epoch. Later~\\citet{rice2020overfitting} provide a comprehensive study on the overfitting phenomenon in AT, and advocate early stopping the training epoch as a general strategy for preventing adversarial overfitting, which could be triggered according to the PGD accuracy on a split validation set. Due to its effectiveness, we regard this trick as a default choice.\n\n\n\\textbf{Early stopping adversarial intensity.} Another level of early stopping happens on the adversarial intensity, e.g., early stopping PGD steps when crafting adversarial examples for training. This trick was first applied by the runner-up of the defense track in NeurIPS 2018 adversarial vision challenge~\\citep{brendel2020adversarial}. Later efforts are devoted to formalizing this early stopping mechanism with different trigger rules~\\citep{wang2019convergence,zhang2020attacks}. \\citet{balaji2019instance} early stop the adversarial perturbation, which has a similar effect on the adversarial intensity. In the left part of Table~\\ref{table5}, we evaluate the method proposed by \\citet{zhang2020attacks} due to its simplicity. As seen, this kind of early stopping can improve the performance on clean data while keeping comparable accuracy under PGD-10. However, the performance under the stronger AutoAttack is degraded.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.4cm}\n\\begin{minipage}[t]{.5\\linewidth}\n\\captionof{table}{Test accuracy (\\%) under different \\textbf{batch size} and \\textbf{learning rate} (l.r.) on CIFAR-10. The basic l.r. is $0.1$, while the scaled l.r. is, e.g., $0.2$ for batch size $256$, and $0.05$ for batch size $64$.}\n\\vspace{-0.35cm}\n  \\begin{center}\n  \\renewcommand*{\\arraystretch}{1.2}\n \n  \\begin{tabular}{c|c|c|c|c}\n   \\hline\n   \\multicolumn{5}{c}{\\textbf{ResNet-18}} \\\\\n   \\hline\n      Batch & \\multicolumn{2}{c|}{Basic l.r.}  & \\multicolumn{2}{c}{Scaled l.r.}\\\\\n      size & Clean & PGD-10 & Clean & PGD-10\\\\\n     \\hline\n      64 & 80.08 & 51.31 & 82.44 & 52.48 \\\\\n   \n    128 & 82.52 & \\textbf{53.58} & - & -\\\\\n   \n    256 & 83.33 & 52.20 & 82.24 & 52.52\\\\\n    \n    512 & 83.40 & 50.69 & 82.16 & 53.36\\\\\n    \\hline\n      \\end{tabular}\n  \\begin{tabular}{c|c|c|c|c}\n   \\multicolumn{5}{c}{\\textbf{WRN-34-10}} \\\\\n   \\hline\n      Batch & \\multicolumn{2}{c|}{Basic l.r.}  & \\multicolumn{2}{c}{Scaled l.r.}\\\\\n      size & Clean & PGD-10 & Clean & PGD-10\\\\\n     \\hline\n      64 &  84.20 & 54.69 & 85.40 & 54.86 \\\\\n   \n    128 & 86.07 & \\textbf{56.60} & - & -\\\\\n   \n    256 & 86.21 & 52.90 & 85.89 & 56.09 \\\\\n    \n    512 & 86.29 & 50.17 & 86.47 & 55.49\\\\\n    \\hline\n      \\end{tabular}\n      \\label{table2}\n\n  \\end{center}\n  \\end{minipage}\n  \\hspace{0.55cm}\n\\begin{minipage}[t]{.42\\linewidth}\n\\captionof{table}{Test accuracy (\\%) under different degrees of \\textbf{label smoothing} (LS) on CIFAR-10. We evaluate the models under AutoAttack to avoid gradient obfuscation.}\n\\vspace{-0.35cm}\n  \\begin{center}\n  \\renewcommand*{\\arraystretch}{1.2}\n \n  \\begin{tabular}{c|c|c|c}\n   \\hline\n   \\multicolumn{4}{c}{\\textbf{ResNet-18}} \\\\\n   \\hline\n      LS & Clean & PGD-10 & \\!\\!\\!\\! AutoAttack \\!\\!\\!\\! \\\\\n     \\hline\n   \n    0 & 82.52 & 53.58 & 48.51 \\\\\n    \n    0.1 & 82.69 & 54.04 & 48.76\\\\\n    \n    0.2 & 82.73 & 54.22 & 49.20\\\\\n    \n    0.3 & 82.51 & 54.34 & \\textbf{49.24}\\\\\n    \n    0.4 & 82.39 & 54.13 & 48.83\\\\\n    \\hline\n      \\end{tabular}\n     \\begin{tabular}{c|c|c|c}\n   \\multicolumn{4}{c}{\\textbf{WRN-34-10}} \\\\\n   \\hline\n      LS & Clean & PGD-10 & \\!\\!\\!\\! AutoAttack \\!\\!\\!\\! \\\\\n     \\hline\n   \n    \n    0 & 86.07 & 56.60 & 52.19 \\\\\n    \n    0.1 & 85.96 & 56.88 & 52.74\\\\\n    \n    0.2 & 86.09 & 57.31 & \\textbf{53.00} \\\\\n    \n    0.3 & 85.99 & 57.55 & 52.70\\\\\n    \n    0.4 & 86.19 & 57.63 & 52.71 \\\\\n    \\hline\n      \\end{tabular}\n      \\label{table3}\n\n  \\end{center}\n\\end{minipage}\n  \\vspace{-0.1cm}\n  \\end{figure}\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.15cm}\n  \\caption{Test accuracy (\\%) using different \\textbf{optimizers} on CIFAR-10. The model is ResNet-18. The initial learning rate for Adam and AdamW is $0.0001$, while for other optimizers is $0.1$.}\n  \\vspace{-0.15cm}\n  \\renewcommand*{\\arraystretch}{1.2}\n    \\begin{tabular}{c|c|c|c|c|c|c}\n    \\hline\n    & Mom & Nesterov & Adam & AdamW & SGD-GC & SGD-GCC \\\\\n    \\hline\n    Clean & 82.52 & 82.83 & 83.20 & 81.68 & 82.77 & 82.93 \\\\\n    PGD-10 & 53.58 & 53.78 & 48.87 & 46.58 & 53.62 & 53.40 \\\\\n    AutoAttack & 48.51 & 48.22 & 44.04 & 42.39 & 48.33 & 48.51 \\\\\n    \\hline\n    \\end{tabular}%\n    \\vspace{-0.15cm}\n    \\label{table15}\n\\end{table}%\n\n\n\n\n\\textbf{Warmup w.r.t. learning rate.} Warmup w.r.t. learning rate is a general trick for training deep learning models~\\citep{Goodfellow-et-al2016}. In the adversarial setting, \\citet{wong2020fast} show that the one cycle learning rate schedule is one of the critical ingredients for the success of FastAT. Thus, we evaluate the effect of this trick for the piecewise learning rate schedule and PGD-AT framework. We linearly increase the learning rate from zero to the preset value in the first $10$ \/ $15$ \/ $20$ epochs. As shown in the middle part of Table~\\ref{table5}, the effect of warming up learning rate is marginal.\n\n\n\n\\textbf{Warmup w.r.t. adversarial intensity.} In the AT procedure, warmup can also be executed w.r.t. the adversarial intensity. \\citet{cai2018curriculum} propose the curriculum AT process to gradually increase the adversarial intensity and monitor the overfitting trend. \\citet{qin2019adversarial} increase the maximal perturbation $\\epsilon$ from zero to $8\/255$ in the first $15$ epochs. In the right part of Table~\\ref{table5}, we linearly increase the maximal perturbation in the first $10$ \/ $15$ \/ $20$ epochs, while the effect is still limited.\n\n\n\n\n\n\n\n\n\n\\vspace{-0.1cm}\n\\subsection{Training hyperparameters}\n\\vspace{-0.1cm}\n\\textbf{Batch size.} On the large-scale datasets like ImageNet~\\citep{deng2009imagenet}, it has been recognized that the mini-batch size is an important factor influencing the model performance~\\citep{goyal2017accurate}, where larger batch size traverses the dataset faster but requires more memory usage. In the adversarial setting, \\citet{xie2018feature} use a batch size of $4096$ to train a robust model on ImageNet, which achieves state-of-the-art performance under adversarial attacks. As to the defenses reported on the CIFAR-10 dataset, the mini-batch sizes are usually chosen between $128$ and $256$, as shown in Table~\\ref{table1}. To evaluate the effect, we test on two model architectures and four values of batch size in Table~\\ref{table2}. Since the number of training epochs is fixed to $110$, we also consider applying the linear scaling rule introduced in \\citet{goyal2017accurate}, i.e., when the mini-batch size is multiplied by $k$, multiply the learning rate by $k$. We treat the batch size of $128$ and the learning rate of $0.1$ as a basic setting to obtain the factor $k$. We can observe that the batch size of $128$ works well on CIFAR-10, while the linear scaling rule can benefit the cases with other batch sizes.\n\n\n\n\n\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.3cm}\n  \\caption{Test accuracy (\\%) under different \\textbf{non-linear activation function} on CIFAR-10. The model is ResNet-18. We apply the hyperparameters recommended by \\citet{xie2020smooth} on ImageNet for the activation function. Here the notation $^\\ddagger$ indicates using weight decay of $5\\times 10^{-5}$, where applying weight decay of $5\\times 10^{-4}$ with these activations will lead to much worse model performance.}\n  \\vspace{-0.15cm}\n  \\renewcommand*{\\arraystretch}{1.3}\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c}\n    \\hline\n    & ReLU & Leaky. & ELU $^\\ddagger$ & CELU $^\\ddagger$ & SELU $^\\ddagger$ & GELU & Softplus & Tanh $^\\ddagger$  \\\\\n    \\hline\n   Clean & 82.52 & 82.11 & 82.17 & 81.37 & 78.88 & 80.42 & \\textbf{82.80} & 80.13 \\\\\n   PGD-10 & 53.58 & 53.25 & 52.08 & 51.37 & 49.53 & 52.21 & \\textbf{54.30} & 49.12 \\\\\n    \\hline\n    \\end{tabular}%\n    \\label{table4}\n\\end{table\n\n\n\\begin{figure}\n\\vspace{-0.cm}\n\\centering\n\\includegraphics[width=.95\\textwidth]{figures\/wd_plot.pdf}\n\\vspace{-0.2cm}\n\\caption{\\textbf{(a)} Test accuracy w.r.t. different values of \\textbf{weight decay}. The reported checkpoints correspond to the best PGD-10 accuracy~\\citep{rice2020overfitting}. We test on two model architectures, and highlight (with red circles) three most commonly used weight decays in previous work; \\textbf{(b)} Curves of test accuracy w.r.t. training epochs, where the model is WRN-34-10. We set weight decay be $1\\times 10^{-4}$, $2\\times 10^{-4}$, and $5\\times 10^{-4}$, respectively. We can observe that smaller weight decay can learn faster but also more tend to overfit w.r.t. the robust accuracy. In Fig.~\\ref{fig:wd_appendix}, we early decay the learning rate before the models overfitting, but weight decay of $5\\times 10^{-4}$ still achieve better robustness.}\n\\label{fig:wd}\n\\end{figure}\n\n\n\n\\textbf{Label smoothing.} \\citet{shafahi2019label} propose to utilize label smoothing to mimic the AT procedure. \\citet{pang2019improving} also find that imposing label smoothing on the ensemble prediction can alleviate the adversarial transferability among individual members, and thus promote robustness of the ensemble model. Beyond previous observations, we further evaluate the effect of label smoothing on AT under stronger AutoAttack, which can rule out gradient obfuscation. As shown in Table~\\ref{table3}, label smoothing can improve $\\sim 1\\%$ accuracy under PGD-10 and AutoAttack, without affecting the clean performance. This can be regarded as the effect induced by calibrating the confidence~\\citep{stutz2019confidence} of adversarially trained models ($80\\%\\sim 85\\%$ accuracy on clean data). However, excessive label smoothing could degrade the robustness (e.g., $\\textrm{LS}=0.3$ vs. $\\textrm{LS}=0.4$ on ResNet-18 shown in Table~\\ref{table3}), which is consistent with the recent observations in~\\citet{jiang2020imbalanced} (they use $\\textrm{LS}=0.5$).\n\n\n\n\n\\textbf{Optimizer.} Most of the AT methods apply SGD with momentum as the optimizer. The momentum factor is usually set to be $0.9$ with zero dampening. In other cases, \\citet{carmon2019unlabeled} apply SGD with Nesterov, and \\citet{rice2020overfitting} apply Adam for cyclic learning rate schedule. We test some commonly used optimizers in Table~\\ref{table15}, as well as the decoupled AdamW~\\citep{loshchilov2018decoupled} and the recently proposed gradient centralization trick SGD-GC \/ SGD-GCC~\\citep{yong2020gradient}. We can find that SGD-based optimizers (e.g., Mom, Nesterov, SGD-GC \/ SGD-GCC) have similar performance, while Adam \/ AdamW performs worse for piecewise learning rate schedule.\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.35cm}\n\\caption{Test accuracy (\\%) under different \\textbf{BN modes} on CIFAR-10. We evaluate across several model architectures, since the BN layers have different positions in different models.}\n  \\vspace{-0.15cm}\n  \\renewcommand*{\\arraystretch}{1.2}\n    \\begin{tabular}{c|c|c|c|c|c|c|c}\n    \\hline\n    & BN &\\multicolumn{6}{c}{Model architecture}  \\\\\n    & mode & \\!\\! ResNet-18 \\!\\! & \\!\\! SENet-18 \\!\\! & \\!\\!\\! DenseNet-121 \\!\\!\\! & \\!\\!\\! GoogleNet \\!\\!\\! & DPN26 & \\!\\!\\! WRN-34-10 \\!\\!\\! \\\\\n    \\hline\n   \\multirow{3}{*}{Clean} & train & 82.52 & 82.20 & 85.38 & 83.97 & 83.67 & 86.07 \\\\\n   & eval & 83.48 & 84.11 & 86.33 & 85.26 & 84.56 & 87.38 \\\\\n   & - & {\\color{red}+0.96} & {\\color{red}+1.91} & {\\color{red}+0.95} & {\\color{red}+1.29} & {\\color{red}+0.89} & {\\color{red}+1.31} \\\\\n   \\hline\n   \\multirow{3}{*}{PGD-10} & train & 53.58 & 54.01 & 56.22 & 53.76 & 53.88 & 56.60 \\\\\n   & eval & 53.64 & 53.90 & 56.11 & 53.77 & 53.41 & 56.04 \\\\\n   & - & {\\color{red}+0.06} & {\\color{mydarkgreen}-0.11} & {\\color{mydarkgreen}-0.11} & {\\color{red}+0.01} & {\\color{mydarkgreen}-0.47} & {\\color{mydarkgreen}-0.56}\\\\\n   \\hline\n   \\multirow{3}{*}{AA} & train & 48.51 & 48.72 & 51.58 & 48.73 & 48.50 & 52.19 \\\\\n   & eval & 48.75 & 48.95 & 51.24 & 48.83 & 48.30 & 51.93 \\\\\n   & - & {\\color{red}+0.24} & {\\color{red}+0.23} & {\\color{mydarkgreen}-0.34} & {\\color{red}+0.10} & {\\color{mydarkgreen}-0.20} & {\\color{mydarkgreen}-0.26} \\\\\n    \\hline\n    \\end{tabular}%\n    \\label{table8}\n\\end{table}%\n\n\n\n\\begin{figure}\n\\vspace{-0.25cm}\n\\centering\n\\includegraphics[width=.6\\textwidth]{figures\/archi_plot.pdf}\n\\vspace{-0.25cm}\n\\caption{Clean accuracy vs. PGD-10 accuracy for different \\textbf{model architectures}. The circle sizes are proportional to the number of parameters that specified in Table~\\ref{tableappendix3}.}\n\\label{fig:archi}\n\\end{figure}\n\n\n\n\\textbf{Weight decay.} As observed in Table~\\ref{table1}, the weight decay in previous work almost falls in three values: $1\\times10^{-4}$, $2\\times10^{-4}$, and $5\\times10^{-4}$. While $5\\times10^{-4}$ is a fairly widely used value for\nweight decay in deep learning, the prevalence of the value $2\\times10^{-4}$ should stem from \\citet{madry2018towards} in the adversarial setting. In Fig.~\\ref{fig:wd}(a), we report the best test accuracy under different values of weight decay\\footnote{Note that \\citet{rice2020overfitting} also investigate the effect of different weight decay (i.e., $\\ell_{2}$ regularization), but they focus on a coarse value range of $\\{5\\times 10^{k}\\}$, where $k\\in\\{-4,-3,-2,-1,0\\}$.}. We can see that the gap of robust accuracy can be significant due to slightly different values of weight decay (e.g., up to $\\sim 7\\%$ for $1\\times10^{-4}$ vs. $5\\times10^{-4}$). Besides, in Fig.~\\ref{fig:wd}(b) we plot the learning curves of test accuracy w.r.t. training epochs. Note that smaller values of weight decay make the model learn faster in the initial phase, but the overfitting phenomenon also appears earlier. As a result, weight decay is a critical and usually neglected ingredient that largely influences the robust accuracy of adversarially trained models. In contrast, the clean accuracy is much less sensitive to the choice of weight decay, for both adversarially and standardly (shown in Fig.~\\ref{fig:wd_standard}) trained models.\n\n\n\n\n\n\\textbf{Activation function.} Most of the previous AT methods apply ReLU as the non-linear activation function in their models, while \\citet{xie2020smooth} empirically demonstrate that smooth activation functions can better improve model robustness on ImageNet. Following their settings, we test if a similar conclusion holds on CIFAR-10. By comparing the results on ReLU and Softplus in Table~\\ref{table4} (for PGD-AT) and Table~\\ref{tableappendix4} (for TRADES), we confirm that smooth activation indeed benefits model robustness for ResNet-18. However, as shown in Table~\\ref{table9} (for PGD-AT) and Table~\\ref{table20} (for TRADES), this benefit is less significant on larger models like WRN. Thus we deduce that smaller model capacity can benefit more from the smoothness of activation function. Besides, as shown in Table~\\ref{table4}, models trained on CIFAR-10 seem to prefer activation function $\\sigma(x)$ with zero truncation, i.e., $\\sigma(x)\\geq 0$. Those with negative return values like ELU, LeakyReLU, Tanh have worse performance than ReLU.\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.35cm}\n  \\caption{The default hyperparameters include batch size $128$ and SGD momentum optimizer. The AT framework is \\textbf{PGD-AT}. We highlight the setting used by the implementation in \\citet{rice2020overfitting}.}\n  \\vspace{-0.15cm}\n    \\begin{tabular}{c|cccc|ccc}\n    \\hline\n    \\multirow{2}{*}{Architecture} & Label & Weight & Activation & BN & \\multicolumn{3}{c}{Accuracy} \\\\\n      &  smooth &  decay & function & mode & Clean & PGD-10 & AA \\\\\n     \\hline\n     \\multirow{8}{*}{WRN-34-10} & 0 & $1 \\times 10^{-4}$ & ReLU & train & 85.87 & 49.45 & 46.43 \\\\\n    & 0 & $2 \\times 10^{-4}$ & ReLU & train & 86.14 & 52.08 & 48.72 \\\\\n   & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{0} & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{$5 \\times 10^{-4}$} & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{ReLU} & \\multicolumn{1}{>{\\columncolor{mycyan}}c|}{train} & 86.07 & 56.60 & 52.19 \\\\\n   & 0 & $5 \\times 10^{-4}$ & ReLU & eval & \\textbf{87.38} & 56.04 & 51.93 \\\\\n   & 0 & $5 \\times 10^{-4}$ & Softplus & train & 86.60 & 56.44 & 52.70\\\\\n   &   0.1 & $5 \\times 10^{-4}$ & Softplus & train & 86.42 & {57.22} & \\textbf{53.01} \\\\\n   & 0.1 & $5 \\times 10^{-4}$ & Softplus & eval & 86.34 & 56.38 & 52.21 \\\\ \n   & 0.2 & $5 \\times 10^{-4}$ & Softplus & train & 86.10 & 56.55 & 52.91 \\\\\n   & 0.2 & $5 \\times 10^{-4}$ & Softplus & eval & 86.98 & 56.21 & 52.10 \\\\ \n   \\hline\n     \\multirow{8}{*}{WRN-34-20} & 0 & $1 \\times 10^{-4}$ & ReLU & train & 86.21 & 49.74 & 47.58 \\\\\n    & 0 & $2 \\times 10^{-4}$ & ReLU & train & 86.73 & 51.39 & 49.03 \\\\\n  & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{0} & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{$5 \\times 10^{-4}$} & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{ReLU} & \\multicolumn{1}{>{\\columncolor{mycyan}}c|}{train} & 86.97 & 57.57 & 53.26 \\\\\n   & 0 & $5 \\times 10^{-4}$ & ReLU & eval & 87.62 & 57.04 & 53.14 \\\\\n   & 0 & $5 \\times 10^{-4}$ & Softplus & train & 85.80 & 57.84 & 53.64 \\\\\n   & 0.1 & $5 \\times 10^{-4}$ & Softplus & train & 85.69 & 57.86 & \\textbf{53.66}\\\\\n   & 0.1 & $5 \\times 10^{-4}$ & Softplus & eval & \\textbf{87.86} & 57.33 & 53.23 \\\\ \n   & 0.2 & $5 \\times 10^{-4}$ & Softplus & train & 84.82 & {57.93} & 53.39 \\\\\n   & 0.2 & $5 \\times 10^{-4}$ & Softplus & eval & 87.58 & 57.19 & 53.26\\\\ \n    \\hline\n    \\end{tabular}%\n    \\vspace{-0.1cm}\n    \\label{table9}\n\\end{table}%\n\n\n\\textbf{Model architecture.} \\citet{Su_2018_ECCV} provide a comprehensive study on the robustness of standardly trained models, using different model architectures. For the adversarially trained models, it has been generally recognized that larger model capacity can usually lead to better robustness~\\citep{madry2018towards}. Recently, \\citet{guo2020meets} blend in the technique of AutoML to explore robust architectures. In Fig.~\\ref{fig:archi}, we perform similar experiments on more hand-crafted model architectures. The selected models have comparable numbers of parameters. We can observe that DenseNet can achieve both the best clean and robust accuracy, while being memory-efficient (but may require longer inference time). This is consistent with the observation in \\citet{guo2020meets} that residual connections can benefit the AT procedure. Interestingly, \\citet{wu2020skip} demonstrate that residual connections allow easier generation of highly transferable adversarial examples, while in our case this weakness for the standardly trained models may turn out to strengthen the adversarially trained models.\n\n\n\n\n\n\\textbf{Batch normalization (BN) mode.} When crafting adversarial examples in the training procedure, \\citet{zhang2019theoretically} use eval mode for BN, while \\citet{rice2020overfitting} and \\citet{madry2018towards} use train mode for BN. Since the parameters in the BN layers are not updated in this progress, the difference between these two modes is mainly on the recorded moving average BN mean and variance used in the test phase. As pointed out in \\citet{Xie2020intriguing}, properly dealing with BN layers is critical to obtain a well-performed adversarially trained model. Thus in Table~\\ref{table8}, we employ the train or eval mode of BN for crafting adversarial examples during training, and report the results on different model architectures to dig out general rules. As seen, using eval mode for BN can increase clean accuracy, while keeping comparable robustness. We also advocate for the eval mode, because if we apply train mode for multi-step PGD attack, the BN mean and variance will be recorded for every intermediate step, which could blur the adversarial distribution used by BN layers during inference.  \n\n\n\\vspace{-0.5cm}\n\\begin{center}\n    \\doublebox{\n    \\parbox{1.\\textwidth}\n    {\n\\textbf{Takeaways:}\\\\\n\\textbf{(\\romannumeral 1)} Slightly different values of weight decay could largely affect the robustness of trained models;\\\\\n\\textbf{(\\romannumeral 2)} Moderate label smoothing and linear scaling rule on l.r. for different batch sizes are beneficial;\\\\\n\\textbf{(\\romannumeral 3)} Applying eval BN mode to craft training adversarial examples can avoid blurring the distribution;\\\\\n\\textbf{(\\romannumeral 4)} Early stopping the adversarial steps or perturbation may degenerate worst-case robustness;\\\\\n\\textbf{(\\romannumeral 5)} Smooth activation benefits more when the model capacity is not enough for adversarial training.\n}\n}\n\\end{center}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-0.15cm}\n\\subsection{Combination of tricks}\n\\vspace{-0.15cm}\nIn the above, we separately evaluate the effect of each training trick in the AT procedure. Now we investigate combining the selected useful tricks, which involve label smoothing, weight decay, activation function and BN mode. As demonstrated in Table~\\ref{table9}, the improvements are not ideally additive by combining different tricks, while label smoothing and smooth activation function are helpful, but not significant, especially when we apply model architectures with a larger capacity.\n\nWe also find that the high performance of the models trained by \\citet{rice2020overfitting} partially comes from its reasonable training settings, compared to previous work. Based on these, we provide a trick list for training robust models on CIFAR-10 for reference.\n\n\n\n\n\n\\vspace{-0.4cm}\n\\begin{center}\n    \\doublebox{\n    \\parbox{1.\\textwidth}\n    {\n\\textbf{Baseline setting (CIFAR-10):}\\\\\nBatch size $128$; initial learning rate $0.1$ (decay factor 10 at 100 and 105 epochs, totally 110 epochs); SGD momentum optimizer; weight decay $5\\times 10^{-4}$; eval mode BN for generating adversarial examples; warmups are not necessary; label smoothing and smooth activation function are optional.\n}\n}\n\\end{center}\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\vspace{-0.35cm}\n  \\caption{Test accuracy (\\%). The AT framework is \\textbf{TRADES}. We highlight the setting used by the original implementation in \\citet{zhang2019theoretically}. As listed in Table~\\ref{tableappendix2}, our retrained TRADES models can achieve state-of-the-art performance in the AutoAttack benchmark.}\n  \\vspace{-0.2cm}\n  \\renewcommand*{\\arraystretch}{1.1}\n    \\begin{tabular}{c|ccc|ccc}\n    \\hline\n    \\multicolumn{7}{c}{\\emph{Threat model: $\\ell_\\infty$ constraint, $\\epsilon=0.031$}}\\\\\n    \\hline\n   Architecture  & Weight decay & BN mode & Activation & Clean & PGD-10 & AA \\\\\n   \\hline\n   \\multirow{4}{*}{WRN-34-10} & $2\\times10^{-4}$ & train & ReLU & 83.86 &54.96 & 51.52\\\\\n& \\multicolumn{1}{>{\\columncolor{mycyan}}c}{$2 \\times 10^{-4}$} &  \\multicolumn{1}{>{\\columncolor{mycyan}}c}{eval} & \\multicolumn{1}{>{\\columncolor{mycyan}}c|}{ReLU} & 85.17 &55.10 &51.85\\\\\n   & $5\\times10^{-4}$ & train & ReLU & 84.17 & 57.34 &53.51\\\\\n   & $5\\times10^{-4}$ & eval & ReLU & \\textbf{85.34} & {58.54} & \\textbf{54.64}\\\\\n   & $5\\times10^{-4}$ & eval & Softplus & 84.66 &58.05 &54.20\\\\\n   \\hline\n   \\multirow{2}{*}{WRN-34-20} & $5\\times10^{-4}$ & eval & ReLU & \\textbf{86.93} & 57.93 & \\textbf{54.42} \\\\\n   & $5\\times10^{-4}$ & eval & Softplus & 85.43 & 57.94 & 54.32\\\\\n   \\hline\n    \\multicolumn{7}{c}{\\emph{Threat model: $\\ell_\\infty$ constraint, $\\epsilon=8\/255$}}\\\\\n    \\hline\n   Architecture  & Weight decay & BN mode & Activation & Clean & PGD-10 & AA \\\\\n   \\hline\n   \\multirow{5}{*}{WRN-34-10} & $2\\times10^{-4}$ & train & ReLU & 84.50 &54.60 &50.94\\\\\n   & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{$2 \\times 10^{-4}$} &  \\multicolumn{1}{>{\\columncolor{mycyan}}c}{eval} & \\multicolumn{1}{>{\\columncolor{mycyan}}c|}{ReLU} & 85.17 &54.58 &51.54\\\\\n   & $5\\times10^{-4}$ & train & ReLU & 84.04 &57.41 &{53.83}\\\\\n   & $5\\times10^{-4}$ & eval & ReLU & \\textbf{85.48} & {57.45} &53.80\\\\\n   & $5\\times10^{-4}$ & eval & Softplus & 84.24 & 57.59 &\\textbf{53.88}\\\\\n   \\hline\n   \\multirow{5}{*}{WRN-34-20} & $2\\times10^{-4}$ & train & ReLU & 84.50 &53.86\t&51.18\\\\\n   & \\multicolumn{1}{>{\\columncolor{mycyan}}c}{$2 \\times 10^{-4}$} &  \\multicolumn{1}{>{\\columncolor{mycyan}}c}{eval} & \\multicolumn{1}{>{\\columncolor{mycyan}}c|}{ReLU} & 85.48 &53.21\t&50.59\\\\\n   & $5\\times10^{-4}$ & train & ReLU & 85.87  &57.40 &54.22\\\\\n   & $5\\times10^{-4}$ & eval & ReLU & \\textbf{86.43}  & {57.91} &\\textbf{54.39} \\\\\n   & $5\\times10^{-4}$ & eval & Softplus & 85.51 & 57.50 & 54.21 \\\\\n   \\hline\n    \\end{tabular}%\n    \\vspace{-0.cm}\n    \\label{table20}\n\\end{table}%\n\n\n\n\n\\vspace{-0.15cm}\n\\subsection{Re-implementation of TRADES}\n\\vspace{-0.15cm}\nAs a sanity check, we re-implement TRADES to see if our conclusions derived from PGD-AT can generalize and provide the results in Table~\\ref{table20}. We can observe that after simply changing the weight decay from $2\\times 10^{-4}$ to $5\\times 10^{-4}$, the clean accuracy of TRADES improves by $\\sim 1\\%$ and the AA accuracy improves by $\\sim 4\\%$, which make the trained model surpass the previously state-of-the-art models reported by the AutoAttack benchmark, as listed in Table~\\ref{tableappendix2}. This fact highlights the importance of employing a standardized training setting for fair comparisons of different AT methods.\n\n\n\n\n\n\n\\vspace{-0.2cm}\n\\section{Conclusion}\n\\vspace{-0.2cm}\nIn this work, we take a step in examining how the usually neglected implementation details impact the performance of adversarially trained models. Our empirical results suggest that compared to clean accuracy, robustness is more sensitive to some seemingly unimportant differences in training settings. Thus when building AT methods, we should more carefully fine-tune the training settings (on validation sets), or follow certain long-tested setup in the adversarial setting.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\bibliographystyle{iclr2020_conference_arxiv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe aim of this study is to compute multiprecision solutions of the Lane--Emden equation of stellar hydrodynamics by a code implementing the Runge--Kutta--Fehlberg method of fourth and fifth order (see e.g. \\citep{EU96}, Sec. 17.3.4.4; see also \\citep{GV12}, Sec.~2), working in the multiprecision environment of the ``Fortran--90 Multiprecision System'', abbreviated MPFUN90, written by D.~H.~Bailey (\\citep{B95a, B95b} and references therein) --- available in http:\/\/crd-legacy.lbl.gov\/ $\\sim$dhbailey\/mpdist\/, and licensed under the Berkeley Software Distribution License found in that site. Such highly accurate solutions can be used for checking other numerical codes and prescribing a measure of their accuracy.\n\nSince however multiprecision computations are time-consuming, we proceed with applying parallel programming techniques, appropriate for multicore machines. The Open Multi-Processing (OpenMP; http\/\/openmp.org\/) is an Application Program Interface (API) supporting shared-memory parallel programming in C\/C++ and Fortran. Several compilers implement the OpenMP API. The GNU Compiler Collection (GCC; http:\/\/gcc.gnu.org\/) includes a Fortran--95 compiler, so-called ``gfortran'' (http:\/\/gcc.gnu.org\/fortran\/), licensed under the GNU General Public License (GPL; http:\/\/www.gnu.org\/licenses\/gpl.html); the official manual of gfortran \\citep{GF} will hereinafter be referred to by the abbreviation ``GF-M''. The GCC 4.7.x releases, including corresponding gfortran 4.7.x releases, support the OpenMP API Version 3.1; the official manual of this version \\citep{OMP} will hereinafter be referred to by the abbreviation ``OMP-M''. This OpenMP version is used here with gfortran. \n     \n\n\n\\section{Stellar Polytropic Models}\n\nPolytropic models have been widely applied to the study of nondistorted (hence spherical) stars. Such stars obey the equations of hydrostatic equilibrium,\n\\begin{equation}\n\\frac{dP}{dr}=-\\frac{Gm\\rho}{r^2}, \\qquad \\qquad \n\\frac{dm}{dr}=4\\pi r^2\\rho,\n\\label{eq1}\n\\end{equation}\nwhere $P(r)$ is the pressure, $m(r)$ the mass inside a sphere of radius $r$, and $\\rho(r)$ the density. As ``equation of state\" (EOS) in the polytropic models of stars is used the well-known ``polytropic'' EOS (\\citep{CH39}, Chapter~IV, Eq.~(1))\n\\begin{equation}\nP=K\\rho^{1+(1\/n)} .\n\\label{eq2}\n\\end{equation}\nWe can introduce the so-called ``normalization equations'' (\\citep{CH39}, Chapter~IV, Eqs.~(8a) and (10a), respectively)\n\\begin{equation}\n\\rho = \\lambda \\, \\theta^n, \\qquad r = \\alpha \\, \\xi,\n\\label{eq3}\n\\end{equation}\nwhere $\\lambda$ is the central density of the star, $\\lambda = \\rho_\\mathrm{c}$, and $\\alpha$ a length defined by (\\citep{CH39}, Chapter~IV, Eq.~(10b))\n\\begin{equation}\n\\alpha=\\left[\\frac{(1+n)K\\lambda^{(1\/n)-1}}{4\\pi G}\\right]^{1\/2}.\n\\label{eq4}\n\\end{equation}\nAccordingly, using $\\lambda$ as the ``polytropic unit of density'' and $\\alpha$ as the ``polytropic unit of length'', we verify that, in such ``polytropic units'', $\\theta^n$ is the measure of density and $\\xi$ the measure of length.\n \nWe then insert Eqs.~\\eqref{eq2} and \\eqref{eq3} into Eqs.~(\\ref{eq1}a,\\,b) and, after some algebra, we obtain the so-called ``Lane--Emden equation\" (cf. \\citep{CH39}, Chapter~IV, Eqs.~(11) and (12))\n\\begin{equation}\n\\theta'' + \\, \\frac{2}{\\xi} \\, \\theta' = \\, -\\theta^n, \n\\label{eq5a}\n\\end{equation}\nwhich, when integrated along a prescribed integration interval,\n\\begin{equation}\n\\xi \\in [\\xi_\\mathrm{start} = 0, \\, \\xi_\\mathrm{end}] = \n\\mathbb{I}_\\xi \\subset \\mathbb{R},\n\\label{eq5I}\n\\end{equation}\nwith initial conditions\n\\begin{equation}\n\\theta(\\xi_\\mathrm{start}) = 1, \\qquad \\qquad \\theta'(\\xi_\\mathrm{start})=0\n\\label{eq5bc}\n\\end{equation}\nwhere primes denote derivatives with respect to $\\xi$, gives as solution the so-called ``Lane--Emden function\" $\\theta[\\mathbb{I}_{\\xi}\\subset\\mathbb{R}$]. \n\nThe second-order differential equation~\\eqref{eq5a} together with the initial conditions~\\eqref{eq5bc} establish an ``initial value problem'' (IVP). Our aim in this study is to compute multiprecision solutions $\\theta[\\mathbb{I}_\\xi]$ of this IVP.\nThere are, however, two problems regarding Eq.~\\eqref{eq5a}. First, to remove the indeterminate form $\\theta'\/\\xi$ at the origin, appearing in the left-hand side, we modify the denominator by adding a tiny quantity, $\\xi_0$, to it, i.e. $\\theta'\/(\\xi+\\xi_0)$. Since $\\xi_0$ is small, the initial conditions~\\eqref{eq5bc} are valid at the starting point $\\xi_\\mathrm{start} + \\xi_0 = \\xi_0$ as well. Accordingly, the integration interval becomes\n\\begin{equation}\n\\xi \\in [\\xi_0, \\, \\xi_\\mathrm{end}] = \n\\mathbb{I}_{\\xi0} \\subset \\mathbb{R}.\n\\label{eq5Imod}\n\\end{equation}\n\nThe second problem is that for values $\\xi$ greater than the first root $\\xi_1$ of $\\theta(\\xi)$, $\\xi>\\xi_1$, the Lane--Emden function changes sign, $\\theta(\\xi)<0$, and thus the term $\\theta^n$ in Eq.~\\eqref{eq5a} becomes undefined --- raising a negative real number to a real power, e.g. $(-2.7)^{1.5}$, is not defined in $\\mathbb{R}$. This undefined issue can be removed by taking instead the real power of the absolute value of $\\theta$, $|\\theta|^n$. Note however that this ``numerical trick\" is appropriate only when interested in finding the first root $\\xi_1$ of the function $\\theta$; while it becomes unreliable when searching for higher roots of $\\theta$ (i.e. $\\xi_2$, $\\xi_3$, $\\xi_4$, \\dots). To compute such higher roots --- involved, e.g., in models of planetary systems --- we need to apply the so called  ``complex--plane strategy\"; full details of this method can be found in \\citep{GK14}.\n\nAs it is usual in numerical analysis, we proceed by transforming Eq.~\\eqref{eq5a} into a system of two first-order differential equations, with the IVP under consideration having then the form\n\\begin{equation}\n\\theta'=\\eta, \\quad \\eta'= - \\, \\frac{2}{(\\xi+\\xi_0)} \\, \\eta - |\\theta|^n, \\quad \\xi\\in\\mathbb{I}_{\\xi}, \\quad \\theta(0)=1, \\quad \\eta(0)=0.  \n\\label{eq7}\n\\end{equation}\nBy solving this IVP, we can compute several significant physical characteristics of a stellar model with finite radius (i.e. $n \\in [0,\\ 5)$, since the model with polytropic index $n=5$ has an infinite radius). If the radius $R$, the mass $M$, and an appropriate polytropic index $n$ are known for a stellar model, then the polytropic units $\\alpha$ and $\\lambda$ are given by \n\\begin{equation}\n\\alpha = \\frac{R}{\\xi_1(n)}, \\qquad \\qquad \n\\lambda=-\\left[\\frac{\\xi_1(n)}\n        {4\\pi\\theta'(\\xi_1(n))}\\right]\\frac{M}{R^3},  \n\\label{eq8}\n\\end{equation}\nwhere the symbol $\\xi_1(n)$ shows that the root $\\xi_1$ varies with $n$.\n\nThe ratio of the central density, $\\lambda = \\rho_\\mathrm{c}$, to the mean density, $\\bar{\\rho}$, is given by (cf. \\citep{CH39}, Chapter~IV, Eq.~(78))\n\\begin{equation}\n\\frac{\\lambda}{\\bar{\\rho}} = \n- \\, \\frac{\\xi_1(n)}{3\\, \\theta'(\\xi_1(n))} \\ .  \n\\label{eq12}\n\\end{equation}\n\nFurthermore, it can be proved that the coefficient $N_n$ appearing in the mass-radius relation (\\citep{CH39}, Chapter~IV, Eqs.~(72) and (75)) is equal to\n\\begin{equation}\nN_n = \\, \\frac{1}{n + 1} \\, \\left[ \\, \\frac{4 \\, \\pi}\n{_0\\omega_n^{n-1}} \\, \\right]^{1\/n},  \n\\label{eq13}\n\\end{equation}\nwhere the coefficient $_0\\omega_n$ is defined by (\\citep{CH39}, Chapter~IV, Eq.~(73)) \n\\begin{equation}\n_0\\omega_{n}= \\,  -\\xi_1(n)^{(n+1)\/(n-1)} \\, \\theta'(\\xi_1(n)).  \n\\label{eq14}\n\\end{equation}\n\nFinally, the coefficient $W_n$ appearing in the central-pressure relation \n(\\citep{CH39}, Chapter~IV, Eq.~(80)) is defined by (\\citep{CH39}, Chapter~IV, Eq.~(81))\n\\begin{equation}\nW_{n}= \\, \\frac{1}{4 \\, \\pi \\, (n+1) \\left[ \\theta'(\\xi_1(n)) \\right]^2}.  \n\\label{eq15}\n\\end{equation}\n\n\n\\section{Parallel Programming}\n\\label{PCCE}\nOur computer comprises an Intel\\textsuperscript{\\textregistered} Core\\texttrademark \\, i7--950 processor with four physical cores. This processor  possesses the Intel\\textsuperscript{\\textregistered} Hyper-Threading Technology, which delivers two processing threads per physical core. gfortran has been installed in this computer by the TDM-GCC ``Compiler Suite for Windows'' (http:\/\/tdm-gcc.tdragon.net\/; free software distributed under the terms of the GPL).\n\nAccording to GF-M (Sec.~6.1.16), to enable the processing of the OpenMP directive sentinel \\texttt{!\\$OMP} (OMP-M, Sec.~2.1), gfortran is invoked by the ``-fopenmp'' option. Then all lines beginning with the sentinels \\texttt{!\\$OMP} and \\texttt{!\\$} (OMP-M, Sec.~2.2) are processed by gfortran.  \n\nWe develop here a parallel programming code for solving the polytropic IVP~(\\ref{eq7}). The code consists of two parts. The task of the first part is to provide all the available computer cores with the required variables. The second part performs numerical computations in parallel for all the polytropic indices given. We use the work-sharing constructs of OpenMP package (OMP-M, Sec.~2.5) in order to share the numerical work and to activate the available computer cores. \n\nA critical step in parallel programming is the demarcation of the shared memory by using data-sharing attribute clauses (OMP-M, Sec.~2.9.3) like \\texttt{SHARED}, \\texttt{PRIVATE} and \\texttt{FIRSTPRIVATE} (for an indicative list of variables categorized this way, see Table~\\ref{ta1}). Variables with values to be shared among all threads are declared as \\texttt{SHARED}. On the other hand, variables are declared as \\texttt{PRIVATE} when each thread has to work with its own copy of values of such variables. Furthermore, variables initialized according to values assigned prior to entry into parallel processing are declared as \\texttt{FIRSTPRIVATE}. The computations are distributed over all the threads by the \\texttt{SCHEDULE(DYNAMIC)} clause (OMP-M, Sec.~2.5.1, Table~2-1). Thus, once a particular thread finishes its allocated iteration, it returns to get a next one from the iterations that are left to be processed. The worksharing construct \\texttt{DO} (OMP-M, Sec.~2.5.1) is used in order for the computer to share among its threads the work of calculating the first root for each polytropic index. The user has to provide a value to an integer variable \\texttt{NMODEL}, equal to the number of threads available in his\/her computer; thus the number of the models, which are to be processed in parallel, must be less or equal to \\texttt{NMODEL}. \n\n\n\n\n\\section{Multiprecision Environment and \\\\ Modified \nRunge--Kutta--Fehlberg 5(4) Code}\n\\label{RKF}\nThe computations are performed in the high precision environment of the package MPFUN90 \\citep{B95a,B95b}. This package supports a flexible, arbitrarily high level of numeric precision --- with hundreds or thousands of decimal digits. MPFUN90 is written in Fortran--90. An intermediate translating program is not required, since translation of the code in multiprecision is accomplished by merely utilizing advanced features of Fortran--90. In this study, our computations are carried out with a presicion of 64 decimal digits. \n\nThe code \\texttt{DCRKF54}, developed and used in \\citep{GV12}, can solve complex IVPs. We have modified properly this code in order for the modified one, so-called \\texttt{DDRKF54}, to solve real IVPs in the multiprecision environment of MPFUN90. The header of the subroutine \\texttt{DDRKF54} has the form\n\\begin{verbatim}\n! Part #[000]: Header of DDRKF54\n      SUBROUTINE DDRKF54(IMDL,A,B,N,Y,DEQS,H,HMIN,HMAX,\n     &                                     ATOL,RTOL,QLBD,NFLAG)\n\\end{verbatim}\nWe assign to these arguments the values $\\mathtt{IMDL} = n$ (integer), $\\mathtt{A} = r_\\mathrm{start}$ (real of proper kind), $\\mathtt{B} = r_\\mathrm{end}$ (real of proper kind), $\\mathtt{N} = 2$ (integer), $\\mathtt{Y(1)} = \\theta(\\xi_\\mathrm{start})$ (real of proper kind), $\\mathtt{Y(2)} = \\theta'(\\xi_\\mathrm{start})$ (real of proper kind). The remaining arguments are discussed in \\citep{GV12} (Appendix).\n\nWe compute the first root of the Lane--Emden function $\\theta$ by combining \\texttt{DDRKF54} with a code that mimics the well-known bisection algorithm. \nThe basic idea is that, when the density changes sign from plus to minus, then the code returns to the last computed point, reduces the stepsize to its half, and repeats the computation at the new point. If the density is again negative, then the process is repeated.  \n\n\n\n\\section{Numerical Results}\nIntergration takes place along two successive intervals. The first interval is a very short one, $\\mathbb{I}_1 = [10^{-26}, \\ 30\\times10^{-26}]$, in order for the code \\texttt{DDRKF54} to be accurately initiated. The stepsize along this interval is kept constant and very small, of order  $10^{-29}$ (Table \\ref{ta2}).  \n\nThe second integration interval, $\\mathbb{I}_2$, extends from the end of $\\mathbb{I}_1$ up to a value $\\sim \\frac{3}{2} \\, \\xi_1$. In this study, we resolve 9 polytropic models with indices $n=$ 0.00, 1.00, 1.50, 2.00, 2.45, 2.50, 3.00, 3.23, 3.25, and 3.50.  \n\nThe multiprecision results of this study are given in Tables \\ref{ta3}--\\ref{ta5}. A polytropic index appropriate for verifying the accuracy of our code is $n=1$, since, as it is well-known, this case has an analytical solution and the corresponding first root of the Lane--Emden function is $\\xi_1 = \\pi$ (\\citep{CH39}, Chapter~IV). \n\n\n\n\\section*{Acknowledgments} \nThe authors acknowledge the use of the Fortran--90 Multiprecision System \\citep{B95a,B95b}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThe \\textbf{QCD interaction} \n can be studied in light quark ($u,d,s$) hadrons as well as \nheavy quark ($c,b$) hadrons. \nIn contrast to light quarks, heavy quark states are narrow\nand do not mix with the states of other quarks. \nThis is illustrated in Fig.~1(left) for charmonium.\nAlso, the effective coupling constant and relativistic problems are far \nmore tractable. Thus, the  \nspectra of charmonium and bottomonium are easier to characterize and study. \n\n\n\\section{CLEO Data for Charmonium and Bottomonium Spectroscopy}\n\n\nThe world's largest pre--BESIII sample of 26 million $\\psi(2S)$ \ncomes from CLEO. These $\\psi(2S)$ data have been used to study \nthe spectroscopy of \n$\\chi_{cJ}(^{3}P_{J})$ and $h_{c}(^{1}P_{1})$.\nUsing  $\\pi\\pi$ tag in the decay \n $\\psi(2S)\\to \\pi^{+}\\pi^{-}J\/\\psi$ ($B$=35\\%), the spectroscopy of\n$J\/\\psi$ is also studied.\n\nCLEO collected a sample of 21 million $\\Upsilon(1S)$, 9 million \n$\\Upsilon(2S)$, and 6 million $\\Upsilon(3S)$. Besides bottomonium \nspectroscopy, the $\\Upsilon$ data are used for charmonium \nspectroscopy using two-photon fusion reactions.\n\nMy talk contains two parts: (a)\n CLEO measurements of the masses of charmonium and\nbottomonium singlet states $\\eta_c'(2S)$, $h_{c}(1P)$,\nand $\\eta_{b}(1S)$, and  their implications for the $q\\bar{q}$\n hyperfine interaction; (b) CLEO measurements for the decay \nbranching fractions of charmonium and bottomonium states.\n\n\\begin{figure}[!tb]\n\\includegraphics[width=3.in]{ccbar_spect.eps}\n\\includegraphics[width=2.2in]{qqbar_potential-1p.eps}\n\\caption{(left) Spectra of the states of Charmonium.\n (right) Schematic of the QCD $q\\bar{q}$ potential (solid line), \nand its Coulombic and confinement parts (dotted lines).\nThe vertical lines show the approximate location of the \n$\\left|c\\bar{c}\\right>$ charmonium and $\\left|b\\bar{b}\\right>$ \nbottomonium bound states. }\n\\end{figure}\n\n\n\n\\section{The $q\\bar{q}$ Hyperfine Interaction}\n\nIn the quark model the hyperfine spin--spin interaction determines  \nthe \\textbf{ground-state masses} of the hadrons. The\n mass of a pseudoscalar or vector $q\\bar{q}$ meson is\n$$M(q_1\\bar{q}_2) = m_1(q_1) + m_2(q_2) + A \\left[ \\frac{\\vec{s}_1\\cdot\\vec{s}_2}{m_1m_2} \\right].$$\nThe $\\vec{s_1}\\cdot\\vec{s_2}$ spin--spin, or \n\\textbf{hyperfine interaction} gives\n rise to the hyperfine, or spin-singlet\/spin-triplet splitting\nin quarkonium spectrum,\n$$\\Delta M_{hf}(nL) \\equiv  M(n^3L_J) - M(n^1L_{J=L}).$$\n\nThe hyperfine interaction is not well understood because \nuntil recently there were not enough experimental data to provide\nthe required constraints for the theory.\nFor thirty years after the discovery of $J\/\\psi$, \nthe only hyperfine splitting measured in a hidden flavor meson was\n$\\bm{\\Delta M_{hf}(1S)_{c\\bar{c}}\\equiv M(J\/\\psi)-M(\\eta_c) = 116.4\\pm1.2~\\mathrm{MeV}}$~\\cite{pdg}.\nNo other singlet states, \n$\\bm{\\eta_c'(2^1S_0)_{c\\bar{c}},~h_c(1^1P_1)_{c\\bar{c}},~\\mathrm{or}~\\eta_b(1^1S_0)_{b\\bar{b}}}$\n were identified, and none of the \nimportant questions about the hyperfine interaction could be answered. \n\\textbf{This has changed in the last few years.} \n\n\n\\subsection{$\\eta_c'(2S)$, Hyperfine Splitting in a Radial Excitation}\n\nIn 2002, Belle claimed identification of $\\eta_c'$ in the decay of 45 \nmillion $B$ mesons, $B\\to K(K_SK\\pi)$ and reported \n$M(\\eta_c')=3654\\pm10$ MeV, which would correspond to \n$\\Delta M_{hf}(2S)=32\\pm10$ MeV~\\cite{belle-etacpb}, a factor \\textbf{two smaller} than \nexpected and a factor \\textbf{four} smaller than $\\Delta M_{hf}(1S)$.\nIt became important to confirm this result.\n\nThere are two important ways $2S$ states differ from $1S$ states.  \n$1S$ states, with $r\\approx0.4~\\mathrm{f}$, lie in the Coulombic \nregion ($\\sim1\/r$) of the $q\\bar{q}$ potential, $V=A\/r+Br$, \nwhereas the $2S$ states, with $r\\approx0.8~\\mathrm{f}$, lie in \nthe confinement part ($\\sim{r}$) of the potential (see Fig.~1, right).  \nThe spin--spin potential in the two regions could be different.  \nThe second difference is that the $2S$ states, particularly $\\psi(2S)$, \nlie close to the $D\\bar{D}$ breakup threshold at 3730 MeV, and can \nbe expected to mix with the continuum as well as \nhigher $1^{--}$ states.  All in all, it is important to nail \ndown $\\eta_c'$ experimentally, and measure its mass accurately.\n\nThis was successfully done by  CLEO~\\cite{cleo-etacp} and \nBaBar~\\cite{babar-etacp} in 2004 by observing $\\eta_c'$ in two--photon \nfusion, $\\gamma\\gamma\\to\\eta_c'\\to K_SK\\pi$.  \nThe two observations are shown in Fig.~2.  The average of all \nmeasurements is $M(\\eta_c')=3637\\pm4$ MeV~\\cite{pdg}, which leads \nto $\\bm{\\Delta M_{hf}(2S)=49\\pm4}$ MeV, which is almost \na factor \\textbf{2.5 smaller} than $\\Delta M_{hf}(1S)$.  \nExplaining this large difference is a challenge to the theory.  \nThe challenge for the experimentalists lies in completing the \nspectroscopy of $\\eta_c'$, now that its mass is known.\nIn particular, it is important to measure its width.\n\n\\begin{figure}[!tb]\n\\includegraphics[width=2.5in]{etacp_cleo_sum.eps}\n\\includegraphics[width=2.4in]{etacp_babar.eps}\n\\caption{The invariant mass $M(K_SK\\pi)$ spectra from two--photon fusion measurements by CLEO (left) and BaBar (right).  The $\\eta_c(2S)$ peak is prominent in both spectra.}\n\\end{figure}\n\n\\subsection{$h_c(1^1P_1)$, Hyperfine Interaction in $P$--wave}\n\nIn this case, we have a very simple, and provocative theoretical expectation, namely\n\\begin{equation}\n\\Delta M_{hf}(1P) \\equiv M(^3P) - M(^1P) = 0.\n\\end{equation}\nThis arises from the fact that a non-relativistic reduction of the Bethe-Salpeter equation makes the hyperfine interaction a \\textbf{contact interaction}.  Since only S--wave states have finite wave function at the origin,\n\\begin{equation}\n\\Delta M_{hf}(L\\ne0)=0.\n\\end{equation}\nWe can test this prediction in charmonium by\n\\begin{itemize}\n\\item identifying the singlet--P state $h_c(1^1P_1)$, and\n\\item by estimating $M(^3P)$, given the masses of the triplet--P states $\\chi_{0,1,2}~(^3P_{0,1,2})$.\n\\end{itemize}\n\n\n\\begin{figure}\n\\includegraphics[width=2.0in]{psi2s_pwave_transition-3.eps}\n\\caption{Comparing allowed E1 transitions from $\\psi'(^3S_1)$ to $\\chi_{cJ}(^3P_J)$ states of charmonium with the isospin forbidden $\\pi^0$ transition to the singlet P--state $h_c(^1P_1)$.}\n\\end{figure}\n\nThe experimental identification of $h_c(1^1P_1)$ is even more difficult than that of $\\eta_c'$.  The centroid of the $^3P_J$ states is at \n$3525.30\\pm0.04$~MeV~\\cite{pdg}. If Eq.~1 is true, $M(h_c)\\approx3525$~MeV, i.e., $\\sim160$~MeV below the $\\psi(2S)$ state from which it must be fed.  Unfortunately, populating $h_c$ has several problems.\n\\begin{itemize}\n\\item The radiative transition $\\psi(2S)(1^{--})\\to\\gamma h_c(1^{+-})$ is forbidden by \\textbf{charge conjugation} invariance.  \n\\item The only other alternative is to populate $h_c$ in the reaction \n$\\psi(2S)\\to\\pi^0h_c$.\nBut that is not easy, because a $\\pi^0$ transition ($M(\\pi^0)=139$~MeV) has very little phase space, and further, the reaction is forbidden by \\textbf{isospin conservation}. Nevertheless, this is the only possible way of populating $h_c$, and we at CLEO had to valiantly go for it.\nAn illustration of the allowed E1 transitions from $\\psi(2S)(^3S_1)$ to \n$\\chi_{cJ}(^3P_J)$ states and the isospin forbidden $\\pi^0$ transition to the singlet P--state $h_c(^1P_1)$ is shown in Fig.~3.\n\\end{itemize}\n\nIn 2005, we at CLEO made the first firm identification (significance$>6\\sigma$) of $h_c$ in the reaction \n$$\\psi(2S)\\to\\pi^0h_c,~~h_c\\to\\gamma\\eta_c,$$ \nin an analysis of 3.08~million $\\psi(2S)$ decays~\\cite{cleo-hc}.\n\n\\begin{figure}\n\\includegraphics[width=2.5in]{3850208-003.eps}\n\\includegraphics[width=2.5in]{3850208-004.eps}\n\\caption{The recoil mass of $\\pi^0$ in the decay $\\psi(2S)\\to\\pi^0h_c$.  (left) Full and background subtracted spectra for inclusive analysis.  (right) Spectrum of exclusive analysis.}\n\\end{figure}\n\n\nIn 2008, we repeated our measurement with 8 times larger luminosity, and  24.5 million $\\psi(2S)$~\\cite{cleo-hc-new}.  As before, data were analyzed in two ways.  In the inclusive analysis, the photon energy, $E_\\gamma$, was loosely constrained, but the decay products of $\\eta_c$ were not identified. In the exclusive analysis, instead of constraining $E_\\gamma$ fifteen hadronic decay channels of $\\eta_c$ were measured. \nAs shown in Fig.~4, $h_c$ was observed with significance~~$>13\\sigma$.\nThe total number of events was $N(h_c)=1146\\pm118$ from inclusive \nanalysis, and $N(h_c)=136\\pm14$ from exclusive analysis. \nThe results from inclusive and exclusive analyses were consistent.\nThe precision results were \n\\begin{align}\nM(h_c) & =3525.28\\pm0.19\\pm0.12~\\mathrm{MeV},\\\\\n\\nonumber\\mathcal{B}_1\\times\\mathcal{B}_2 & =(4.19\\pm0.32\\pm0.45)\\times10^{-4}.\n\\end{align}\nThus, $h_c(^1P_1)$ \\textbf{is now firmly established.} \n\n\nIf it is assumed that $M(^3P)$ is identical to the centroid of the triplet--P states, $\\left<M(^3P_J)\\right>=[5M(\\chi_{c2})+3M(\\chi_{c1})+M(\\chi_{c0})]\/9=3525.30\\pm0.04$~MeV, then the above $M(h_c)$ leads to the hyperfine splitting,\n\\begin{equation}\n\\Delta M_{hf}(1P)_{c\\bar{c}}=\\left<M(^3P_J)\\right>-M(^1P_1)=0.02\\pm0.23~\\mathrm{MeV,}\n\\end{equation}\nbut, $\\left<M(^3P_J)\\right>_{0,1,2} \\ne M(^3P)$!\n\nThe centroid  $\\left<M(^3P_J)\\right>$ is a good measure of $M(^3P)$ only if the spin--orbit splitting between the states $^3P_2$, $^3P_1$, and $^3P_0$ is perturbatively small.  It is obviously not so.\nThe splitting, $M(^3P_2)-M(^3P_0)=142$~MeV, is not small.\nFurther, the perturbative prediction is that \n\\begin{align}\nM(^3P_1)-M(^3P_0)& =\\frac{5}{2}\\left[M(^3P_2)-M(^3P_1)\\right]=114~\\mathrm{MeV},\n\\end{align}\nwhile the experimental value is\n\\begin{equation}\nM(^3P_1)-M(^3P_0)=96\\pm1~\\mathrm{MeV}.\n\\end{equation}\nThis is a 18 MeV difference! So we are obviously not in the \nperturbative regime.\n\nThis leads to serious questions.\n\\begin{itemize}\n\\item What mysterious cancellations are responsible for the wrong estimate of $M(^3P)$ giving the expected answer that\n$$\\Delta M_{hf}(1P)=0.$$\n\\item Or, is it possible that the expectation is wrong?  Is it possible that the hyperfine interaction is not entirely a \\textbf{contact interaction}?  \n\\item Potential model calculations are not of much help because they smear the potential at the origin in order to be able to do a Schr\\\"odinger equation calculation.\n\\item Can Lattice help? So far we have no lattice predictions with\nsufficient precision.\n\\end{itemize}\n\n\n\\subsection{$\\eta_b(^1S_0)$, Hyperfine Interaction Between $b$--Quarks}\n\n\nThe $b\\bar{b}$ bottomonium system is, in principle, the best one\n to study the fundamental aspects of the hyperfine interaction \nbetween quarks.   Unfortunately, until last year we had no knowledge \nof the hyperfine interaction between $b$--quarks.  \nThe spin--triplet $\\Upsilon(1^3S_1)$ state of bottomonium was \ndiscovered in 1977, but its partner, the \nspin--singlet $\\eta_b(1^1S_0)$ ground state of bottomonium, was not \nidentified for thirty years, mainly because of  \nthe difficulty in observing weak M1 radiative \ntransitions.  There were many pQCD based theoretical predictions  \nwhich varied all over the map, with $\\Delta M_{hf}(1S)_b=35-100$~MeV, \nand $\\mathcal{B}(\\Upsilon(3S)\\to\\gamma\\eta_b)=(0.05-25)\\times10^{-4}$.  \n\nThis has changed now.  The $\\eta_b(1^1S_0)$ ground state of the\n$\\left|b\\bar{b}\\right>$ Upsilon family \\textbf{has been finally identified!}\n\nIn July 2008, BaBar announced the identification of $\\eta_b$~\\cite{ups-babar}.  They analyzed the inclusive photon spectrum of\n\\begin{equation}\n\\Upsilon(3S)\\to\\gamma\\eta_b(1S)\n\\end{equation}\nin their data for \\textbf{120 million} $\\Upsilon(3S)$ (28~fb$^{-1}~e^+e^-$). \n BaBar's success owed to their very large data set and a clever way of \nreducing the continuum background, a cut on the so--called thrust angle, \nthe angle between the signal photon and the thrust vector of the \nrest of the event,  $|\\cos\\theta_{Thrust}|<0.7$. BaBar's results were:\n\\begin{gather}\nM(\\eta_b)=9388.9^{+3.1}_{-2.3}\\pm2.7~\\mathrm{MeV},\\\\\n\\nonumber \\Delta M_{hf}(1S)_b=71.4^{+3.1}_{-2.3}\\pm2.7~\\mathrm{MeV}, \\\\\n\\nonumber \\mathcal{B}(\\Upsilon(3S)\\to\\gamma\\eta_b) = (4.8\\pm0.5\\pm0.6)\\times10^{-4}.\n\\end{gather}\n\nThe significance of $\\eta_b$ observation was $>$10$\\sigma$. \nRecently, BaBar has also reported a 3.0$\\sigma$ identification of\n$\\eta_b$ in $\\Upsilon(2S)\\to\\gamma\\eta_b$~\\cite{ups-babar}.\n\nAny important discovery requires confirmation by \nan \\textbf{independent} experiment. At CLEO we \nhad data for only \\textbf{5.9~million} $\\Upsilon(3S)$, i.e., \nabout 20~times less than BaBar.    \nBut we have better photon energy resolution, and we have been able \nto improve on BaBar's analysis technique.  We make three improvements.  \nWe make very detailed analysis of the large continuum background under \nthe resonance photon peaks.  We determine photon peak \nshapes by analyzing background from peaks in background--free radiative \nBhabhas and in exclusive $\\chi_{b1}$ decays.  And we make a joint \nfit of the full data in three bins of $|\\cos\\theta_T|$, covering \nthe full range $|\\cos\\theta_T|=0-1.0$ (see Fig.~5).  \nMonte-Carlo simulations show that the joint fit procedure leads to  \nan average increase of the significance of an $\\eta_b$ signal by $\\sim20\\%$\nover accepting only events with  $|\\cos\\theta_{Thrust}|<0.7$.\nSo, despite our \npoorer statistics, we have succeeded in confirming BaBar's discovery with\nsignificance level $\\sim$4$\\sigma$.  The results have been submitted for \npublication~\\cite{ups-cleo-new}. \n\nOur results are:\n\\begin{gather}\nM(\\eta_b)=9391.8\\pm6.6\\pm2.0~\\mathrm{MeV},\\\\\n\\nonumber \\Delta M_{hf}(1S)_b = 68.5\\pm6.6\\pm2.0~\\mathrm{MeV},\\\\\n\\nonumber \\mathcal{B}(\\Upsilon(3S)\\to\\gamma\\eta_b)=(7.1\\pm1.8\\pm1.3)\\times10^{-4}.\n\\end{gather}\n\n\n\n\\begin{figure}\n\\includegraphics[width=3.0in]{3850809-007a-label.eps}\n\\caption{Illustrating CLEO results for the identification of $\\eta_b$ in a joint fit of data in three bins of the thrust angle,  I:~$|\\cos\\theta_T|=0-0.3$,  II:~$|\\cos\\theta_T|=0.3-0.7$,  III:~$|\\cos\\theta_T|=0.7-1.0$.}\n\\end{figure}\n\n\n\n The results agree with those of BaBar. The average of our and BaBar's \nresults for the hyperfine splitting is\n$$\\left<\\Delta M_{hf}(1S)_b\\right> \\equiv M(\\Upsilon(1S))-M(\\eta_b) = 69.4\\pm2.8~\\mathrm{MeV}.$$\nA recent unquenched lattice calculation predicts \n(NRQCD with $u,d,s$ sea quarks) $\\Delta M_{hf}(1S)_b=61\\pm14$~MeV.  \nA quenched lattice calculation (chiral symmetry and $s,c$ sea quarks) \npredicts $\\Delta M_{hf}(1S)_b=70\\pm5$~MeV.\nThus, as far as the hyperfine splitting for the $\\left|b\\bar{b}\\right>$ \nis concerned, lattice calculations appear to be on the right \ntrack~\\cite{lqcd}.\n\nFor more details on $\\eta_b$ analysis by CLEO see the talk by\nS. Dobbs in the parallel session 7C~\\cite{dobbs}.\n\n\n\\subsection{Hyperfine Splittings Measurements}\n\nTo summarize, we now have well--measured experimental results for \nseveral hyperfine splittings, with significant contributions \nfrom CLEO measurements.\\\\\\\\\n\\begin{tabular}{ll}\n$\\left|c\\bar{c}\\right>$ Charmonium: &   $\\Delta M_{hf}(1S)=116.4\\pm1.2$~MeV, \\\\[2pt]\n &    $\\Delta M_{hf}(2S)=49\\pm4$~MeV, \\\\[2pt]\n &    $\\Delta M_{hf}(1P)=0.02\\pm0.23$~MeV,  \\\\[4pt]\n$\\left|b\\bar{b}\\right>$ Bottomonium: &   $\\Delta M_{hf}(1S)=69.4\\pm2.8$~MeV. \\\\[6pt]\n\\end{tabular}\n\n\nIn charmonium, we do not have satisfactory understanding of the variation of hyperfine splitting for the S--wave radial states, and for P--wave state.\n\\begin{itemize}\n\\item For charmonium, we do not have any unquenched lattice predictions, at present.\n\\item For bottomonium, lattice predictions are available, and they appear to be on the right track.\n\\item For neither charmonium nor bottomonium there are any reliable predictions of transitions strength, particularly for forbidden M1 transitions.\n\\end{itemize}\n\nMuch remains to be done.  On the experimental  front it is very important to identify for bottomonium the allowed M1 transition, $\\Upsilon(1S)\\to\\gamma\\eta_b(1S)$, and to identify the bottomonium singlet P--state, $h_b(^1P_1)$.  On the theoretical front one would like to see unquenched lattice calculations for charmonium singlets, and, of course, for transition strengths.\n\n\n\n\\section{Measurements of the decay branching fractions of charmonium and bottomonium states}\n\n\\subsection{ Search for Exclusive Decays of $\\eta_c'(2S)$}\n\nRecently, CLEO has performed a search for the decay \n$\\psi(2S) \\rightarrow \\gamma \\eta_c'(2S)$ in a sample \nof 26 million $\\psi(2S)$ events~\\cite{cleo-etacp1}. \nExpected $E_{\\gamma}=48$ MeV. \nEleven exclusive decay modes, \n$\\eta_c'(2S)\\to\\mathrm{hadrons},~(\\pi,~K,~\\eta,~\\eta')$ with up \nto 6 particles (charged and neutrals) were reconstructed, but\nno signals of $\\eta_c'(2S)$ were observed in any of \nthe decay modes, or in their sum. \n\nThe product branching fraction upper limits were determined for the individual\nmodes, and they are at the level of (4--15)$\\times 10^{-6}$. \nThese upper limits are an order of magnitude smaller than expected by \nassuming that the partial widths for $\\eta_c'(2S)$ decays are the same \nas for  $\\eta_c(1S)$.\n\n Thus, so far $K_SK\\pi$ is the only decay mode in which\n$\\eta_c'(2S)$ has been identified.\n\n\\subsection{Evidence for Exclusive Decay of $h_c(1P)$ to Multipions}\n\nNow that $h_c$ has been discovered, CLEO has searched \nfor hadronic decays of $h_c$ in multipion channels~\\cite{hc-je}. \nOf the three decays investigated, only one,  the five pion decay \n$h_c\\to 2(\\pi^+\\pi^-)\\pi^0$,\nis found to have a statistically significance signal, with \n$B(\\psi(2S)\\to h_c)\\times B(h_c \\to 2(\\pi^+\\pi^-)\\pi^0)=\n(1.9^{+0.7}_{-0.5})\\times10^{-5}$ (see Table I). \nThis is $\\sim5\\%$ of \n$B(\\psi(2S)\\to h_c)\\times B(h_c \\to \\gamma\\eta_c)=(4.19\\pm0.32\\pm0.45)\\times10^{-4}$. \\\\\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c}\n\\hline\nMode  & Efficiency (\\%) & Yield & \n $B(\\psi(2S)\\to h_c)\\times B(h_c \\to n(\\pi^+\\pi^-)\\pi^0)\\times10^5$ \\\\\n\\hline\n$\\pi^+\\pi^-\\pi^0$ &27.0\\% & $1.6^{+6.7}_{-5.9}$ &$ <0.2$ (90\\%)  \\\\\n$2(\\pi^+\\pi^-)\\pi^0$ & 18.8\\% & $92^{+23}_{-22}$ & $1.88^{+0.48+0.47}_{-0.42-0.16}$~(significance $\\sim4\\sigma)$\\\\ \n$3(\\pi^+\\pi^-)\\pi^0$ & 11.5\\% & $35\\pm26$ & $<2.5$ (90\\%) \\\\\n\\hline\n\\end{tabular}\n\\caption{Results for exclusive decays of $h_c(1P)$ to multipions.}\n\\end{table}\n\n\\subsection{Observation of $J\/\\psi\\to 3\\gamma$}\n\n No $3\\gamma$ decay of a meson has been observed before.\n In the lowest order, $3\\gamma$ decay   \nis a QED process, and the predicted ratio \n$\\mathcal{B}(J\/\\psi\\to3\\gamma)\/\\mathcal{B}(J\/\\psi\\to e^+e^-)=5.3\\times10^{-4}$, which is independent of charm quark mass and wave function, leads to\n$\\bm{\\mathcal{B}(J\/\\psi\\to3\\gamma)=3.2\\times10^{-5}}$.\nQCD radiative corrections, which are not reliably known, may modify \nthe prediction.\n\nTo search for $3\\gamma$ decay of $J\/\\psi$,  \nCLEO has used a QED background free sample of \n9.6 million $J\/\\psi$ obtained by $\\pi^{+}\\pi^{-}$ tagging\nin the decay $\\psi(2S)\\to (\\pi^{+}\\pi^{-})J\/\\psi$~\\cite{gamma3}.\nKinematting fitting of the data leads to the result,\n$\\bm{\\mathcal{B}(J\/\\psi\\to3\\gamma)=(1.2\\pm 0.3 \\pm 0.2)\\times10^{-5}~\n(\\textbf{Significance}\\sim6\\sigma}).$\nFig.~6 shows background subtracted data and signal Monte-Carlo\ndistributions. \n\n\\begin{figure}\n\\includegraphics[width=2.1in]{2541009-005.eps}\n\\caption{Observation of $J\/\\psi\\to 3\\gamma$. \nBackground subtracted data and signal Monte-Carlo distributions\nfor variable $\\chi^2\/dof$ of kinemattic fit. The signal and background \nnormalization ragions are shown by full horizontal, \n $\\chi^2\/dof=0-3$(signal), $\\chi^2\/dof=5-20$(background) lines.}\n\\end{figure}\n\n\\subsection{Precision Measurements of Branching Fractions}\n\n\n Using the data set of \\textbf{26 million} $\\bm{\\psi(2S)}$, CLEO has\n made precision measurements of decays of $\\bm{\\psi(2S)}$,~$\\bm{J\/\\psi(1S)}$,\nand $\\bm{\\chi_{cJ}(1P)}$~\\cite{all}.  Among the decays measured are:\\\\\n\\begin{tabular}{ll}\n$\\psi(2S), J\/\\psi\\to \\gamma h$  ($h=\\pi^0,\\eta,\\eta'$) & \\\\\n$\\psi(2S), J\/\\psi\\to \\gamma gg$  & \\\\\n$\\chi_{cJ}\\to {X}\\overline{X}$ ($X=p,\\Lambda,\\Sigma,\\Xi$)~~~(6 decay modes) & \\\\\n$\\chi_{cJ}\\to h^+h^-,h^0h^0,~h^+h^-h^0h^0$ ($h^\\pm=\\pi^\\pm,K^\\pm,h^0=\\pi^0,\\eta,\\eta',K^0$)~~~(12 decay modes)&  \\\\\n$\\chi_{cJ}\\to\\gamma\\gamma$,~$\\gamma V$ ($V=\\rho,~\\omega,~\\phi)$ & \\\\\n\\end{tabular}\n\nMany of these decays have been measured for the first time, and others have greater precision than the results in the literature.  \nSome of the interesting theoretical problems that the branching fractions pose are:\n\\begin{itemize}\n\\item The ratio\n$\\bm{\\mathcal{B}(\\psi(nS)\\to\\gamma\\eta)\/\\mathcal{B}(\\psi(nS)\\to\\gamma\\eta')}$\nis expected to be $\\sim$equal for 1S and 2S states; CLEO measured\nan order of magnitude difference between the two,\n(21.1$\\pm$0.9)\\% for 1S, and $<$1.8\\% for 2S. \n\\item The measured rates \n$\\bm{\\mathcal{B}(\\chi_{c1}\\to\\gamma\\rho)}$ and\n$\\bm{\\mathcal{B}(\\chi_{c1}\\to\\gamma\\omega)}$ are significantly \nhigher than those predicted by pQCD.\n\\item The ratio \n$\\bm{\\mathcal{B}(\\chi_{c0}\\to\\gamma\\gamma)\/\\mathcal{B}(\\chi_{c2}\\to\\gamma\\gamma)}$ disagrees with pQCD expectations. This result provides experimental\nconfirmation of the inadequacy of the present first order radiative \ncorrections.\n\\end{itemize}\n\n\n\\subsection{Hadronic Decays of $\\chi_{bJ}(1P,2P)$, and  Inclusive $\\chi_{bJ}(1P,2P)$ Decays to Open Charm}\n\nNo hadronic decays of $\\chi_{bJ}(1P)$ have been measured before.  \nFor $\\chi_{bJ}(2P)$ the only hadronic decays measured so far \nwere $\\chi_{bJ}(2P)\\to\\pi\\pi\\chi_{bJ}(1P)$ and \n$\\chi_{bJ}(2P)_{b1,2}\\to\\omega\\Upsilon(1S)$.\n\nAt CLEO we have made the first measurements of 14 different decays \nof $\\chi_{bJ}(1P,2P)$ to light hadrons~\\cite{chibj}. Up to 12 particles were detected.\nThe branching fractions for the corresponding decays of $\\chi_{b1,2}(1P)$ \nand $\\chi_{b1,2}(2P)$ were found to be nearly equal. \nThe ratios between decays to $n$ charged pions and $(n-2)$ charged +2 \nneutral pions were found to approximately follow the expectations based \non combinatorics.\n\nCLEO also measured the inclusive decays of   \n$\\chi_{bJ}(nP)\\to D^0+X$~\\cite{d0x}.\nThe enhanced rates for $\\chi_{b1}(1P,2P)\\to D^0+X$ were found to be\nconsistent with NRQCD predictions.\n\n\\section{Summary}\n\n\nCLEO data at  $\\psi(2S)$ and $\\Upsilon(1S,2S,3S)$ resonances\nwere analyzed. The prominent results are the following.\n\n\\begin{itemize}\n\\item Observation of $\\bm{\\eta_c'(2S)}$ in  $\\gamma\\gamma$ fusion, and \nits mass measurement. Search for  $\\eta_c'(2S)$ in exclusive decays, \nand upper limit measurements for decay branching fractions.\n\n\\item Observation of $\\bm{h_c(1P)}$ in $\\psi(2S)\\to \\pi^{0}h_c$, \nand precision measurement of its mass.\nEvidence of $h_c$ decay in multi-pion exclusive final state.\n\n\\item Confirmation of $\\bm{\\eta_b(1S)}$ observation, and measurement of \nmass of $\\eta_b$ and decay branching fraction \n$\\mathcal{B}(\\Upsilon(3S)\\to\\gamma\\eta_b)$.\n\n\\item Observation of decay $\\bm{J\/\\psi\\to 3\\gamma}$ (first observation of\nmeson decay in $3\\gamma$).\n\n\\item  Precision measurements of decay branching fractions of\n  $\\bm{\\psi(2S)}$,~$\\bm{J\/\\psi(1S)}$, and $\\bm{\\chi_{cJ}(1P)}$ \ncharmonium states, and $\\bm{\\chi_{bJ}(1P,2P)}$ bottomonium states. \n Many of these decays have been measured for the first time, and \nothers have much greater precision than the results in the literature.\n\n\\end{itemize}\n\n There is a rich program of hadronic physics at CLEO; too extensive\nto cover it all in one talk. There are also quite a few analyses in\na preliminary stage. Expect new results in the coming years.\n\n\n\n\n\n\n\n\\bibliographystyle{aipproc}  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{se:Intro}\n\nThe aim of this paper is to provide existence results for the\ninitial-value problem for the doubly nonlinear evolution inclusion\n\\begin{align}\n  B(t,u(t)) \\in \\partial\\Psi_{u(t)}(u'(t))+\\partial \\calE_t(u(t))\n  &\\quad \\text{in } V^* \\text{ for a.a. } t\\in (0,T),\n\\end{align} \nwith a continuous perturbation $B$ in the separable and reflexive real\n\\textsc{Banach} space $(V,\\Vert\\cdot\\Vert)$, where $\\partial \\Psi_u$\nand $\\partial \\calE_t$ denote the subdifferential of $\\Psi_u$ and\n$\\calE_t$, respectively. The functional $\\Psi_u$ is supposed to be a\ndissipation potential for all $u\\in \\operatorname{dom}(\\calE_t)$, i.e., it is proper,\nlower semicontinuous and convex with $\\Psi_u(0)=0$ for all $u\\in\n\\operatorname{dom}(\\calE_t)$. If the functionals $\\Psi_u$ and $\\calE_t$ are\n\\textsc{Fr\\'echet} differentiable, the differential inclusion (1.1)\nbecomes the abstract evolution equation (also called doubly nonlinear\nequation in \\cite{ColVis90CDNE,Coll92DNEE})\n\\begin{align*}\n \\rmD\\Psi_{u(t)}(u'(t))=-\\rmD\\calE_t(u(t))+B(t,u(t)) &\\quad \\text{in } \nV^* \\text{ a.e. in } (0,T),\n\\end{align*} \nwhere $\\rmD\\Psi_u$ and $\\rmD\\calE_t$ denote the \\textsc{Fr\\'echet} derivative\nof $\\Psi_u$ and $\\calE_t$ respectively. The question arises why it is\ninteresting to study perturbed gradient systems. First of all, to\nconsider perturbed systems is sometimes important in order to describe\nphysical systems near or far from equilibrium properly. There are many\nways to incorporate the perturbation in the equation.\n\nThe most frequently used method is to consider an $\\varepsilon$-family\nof equations, where the occurring terms depend on the parameter\n$\\varepsilon$, and then to pass to the limit as\n$\\varepsilon\\rightarrow \\infty$, where the limit equation corresponds\nto the unperturbed system. Another way to treat perturbed systems is\nto use an additional term in the equations like the term $B_t$ in\n\\eqref{eq:I.1} or even a combination of both as in \\cite{Miel16DEMM},\nwhere the author considered the family of equations\n\\begin{align*}\n \\rmD\\Psi^\\varepsilon_{u(t)}(u'(t))=-\\rmD\\calE^\\varepsilon_t(u(t))+B^\\varepsilon(t,u(t)) \n\\end{align*} to derive results on the so-called evolutionary\n$\\Gamma$-convergence.\n\nSecond, \\cite [p.\\,235]{Miel16DEMM} highlights with an example that in\nsome cases it can be easier to treat a system with a nontrivial but\nexact gradient structure $(X,\\wt\\calE, \\wt \\Psi)$ perturbed gradient\nsystem $(V,\\calE,\\Psi,B)$ with a simpler energy $\\calE$ and simpler\ndissipation potentials $\\Psi_u$.\n\nWhile Section \\ref{se:ExistResult} provides the main existence result\nin Theorem \\ref{th:MainExist}, we devote Section \\ref{se:EGC} to the\nquestion of evolutionary $\\Gamma$-convergence of families \n$(V,\\calE^\\eps,\\Psi^\\eps,B^\\eps)$ of perturbed gradient systems. This\nprovides a generalization of the results developed in\n\\cite{SanSer04GCGF, Serf11GCGF, Miel16EGCG} for exact gradient flows,\ni.e.\\ the case where $B^\\eps \\equiv 0$. Following the ideas in\n\\cite[Thm.\\,4.8]{MiRoSa13NADN}, our Theorem \\ref{th:EGC.main} shows that\nunder suitable technical assumptions, including convexity of\n$\\calE^\\eps$, it is enough to establish $\\calE_t^\\eps \\xrightarrow{\\Gamma} \\calE^0_t$\n(strong $\\Gamma$-convergence in $V$) and $\\Psi^\\eps_{u_\\eps} \\xrightarrow{\\,\\rmM\\,}\n\\Psi^0_{u_0}$ in $V$, where \\textsc{Mosco} convergence means weak and strong\n$\\Gamma$-convergence.  \n\nIn Section \\ref{se:Example} we show that the abstract result on\nevolutionary $\\Gamma$-convergence can be used for the homogenization\nof quasilinear parabolic systems. For that application the \\textsc{Mosco}\nconvergence $\\Psi^\\eps_{u_\\eps} \\xrightarrow{\\,\\rmM\\,} \\Psi^0_{u_0}$ is too restrictive,\nsuch that it is necessary to generalize it to situations where the\nstrong $\\Gamma$-convergence $\\Psi^\\eps_{u_\\eps} \\xrightarrow{\\Gamma} \\Psi^0_{u_0}$ is\nsufficient, see Corollary \\ref{co:StrongGa}. Here we rely on an \nnovel argument from \\textsc{Liero-Reichelt} \\cite{LieRei15?HCHT},\nwhere the weak convergence of $u_\\eps\\weak u_0$ in $\\rmW^{1,1}(0,T;V)$ is\ncircumvented by exploiting the strong convergence of the piecewise\naffine interpolants $\\wh u^\\tau_\\eps \\to \\wh u^\\tau_0$ in\n$\\rmW^{1,1}(0,T;V)$ for $\\eps \\to 0$ and $\\tau>0$ fixed. \n \nThe general structure is that we provide a full and detailed proof of\nthe existence result in Section \\ref{se:ExistResult}, where we use\n\\textsc{De Giorgi}'s minimization scheme using variational\ninterpolators. The result on the evolutionary $\\Gamma$-convergence in\nSection \\ref{se:EGC} follows the same lines but is considerably\nsimpler as existence of solutions is assumed to be shown. Hence, for\ngetting an overview of the strategy in Section \\ref{se:ExistResult} it\nmight be helpful to browse through the more compact proof of Theorem\n\\ref{th:EGC.main} first. This will facilitate the subsequent reading\nof the full details in Section \\ref{se:ExistResult}. In particular,\nthe elaborate time-discretization using \\textsc{De Giorgi}'s\nvariational interpolants is only needed there. \n\n\n\n\\section{The main existence result}\n\\label{se:ExistResult}\n\nBefore making all the assumptions concerning the dissipation\npotential, the energy functional and the perturbation, we need some\nbasic tools from convex analysis.\n\n \n\n\\subsection{Preliminaries and notation}\n\\label{su:PrelimNot}\nIn this section we collect some important notions and results on\nconvex analysis and $\\Gamma$-convergence, which we need later on for\nthe existence result. First of all, we introduce the so-called\n\\textsc{Legendre-Fenchel} transform (or conjugate) $\\Psi^*$ of a\nproper, lower semicontinuous and convex functional $\\Psi:V\\rightarrow\n(-\\infty,+\\infty]$ that is defined by\n\\begin{align*}\n  \\Psi^*(\\xi):=\\sup_{u\\in V}\\left \\lbrace \\langle \\xi,u\\rangle -\n    \\Psi(u)\\right \\rbrace, \\quad \\xi\\in V^*,\n\\end{align*} where $\\langle\\cdot,\\cdot\\rangle$ denotes the duality\npairing between the \\textsc{Banach} space $V$ and it's topological\ndual space $V^*$. From the definition, the \\textsc{Fenchel-Young}\ninequality\n\\begin{align*}\n \\langle \\xi,u\\rangle \\leq  \\Psi(u)+\\Psi^*(\\xi), \\quad v\\in V, \\xi\\in V^*,\n\\end{align*} \nimmediately follows. It is also easy to check that the conjugate\nitself is proper, lower semicontinuous and convex, see for example\n\\textsc{Ekeland} and \\textsc{T\\'emam} \\cite{EkeTem74ACPV}. If, in\naddition, $\\Psi(0)=0$, then $\\Psi^*(0)=0$ holds too. For a proper\nfunctional $F:V\\rightarrow (-\\infty,+\\infty]$, the\n$($\\textsc{Fr\\'echet}$)$-subdifferential of $F$ is given by the\nmultivalued map $\\partial F:V\\rightarrow 2^{V^*}$ with\n\\begin{align*}\n\\partial F(u):=\\left\\lbrace \\xi\\in V^*: \\ \\liminf_{v\\rightarrow u}\n  \\frac{F(v)-F(u)-\\langle \\xi,v-u\\rangle}{\\Vert v-u\\Vert} \\geq 0\\right\n\\rbrace \n\\end{align*} \nfor all elements $u$ in the effective domain $\\operatorname{dom}(F):=\\lbrace v\\in V\n\\mid F(v)<+\\infty \\rbrace$ of $F$.  For convex and proper functions\n$F$, it follows by simple calculations that the subdifferential of $F$\nis given by\n\\begin{align*}\n\\partial F(u)=\\left\\lbrace \\xi\\in V^*: \\ F(u)\\leq F(v)+\\langle\n  \\xi,u-v\\rangle \\quad \\text{ for all  } v\\in V \\right\\rbrace. \n\\end{align*}\nThe following lemma gives a relation between the subdifferential of a\nfunctional and it's \\textsc{Legendre-Fenchel} transform.\n\n\n\\begin{lem}\\label{le:Leg.Fen}\n  Let $\\Psi:V\\rightarrow (-\\infty,+\\infty]$ be a proper, lower\n  semicontinuous and convex functional and let $\\Psi^*:V^*\\rightarrow\n  (-\\infty,+\\infty]$ be the \\textsc{Legendre-Fenchel} transform of\n  $\\Psi$. Then for all $(u,\\xi)\\in V\\times V^*$ the following\n  assertions are equivalent:\n\\begin{itemize}\n\\item[$i)$]\\quad $\\xi\\in \\partial \\Psi(u) \\quad \\text{in } V^*;$\n\\item[$ii)$]\\quad  $u\\in \\partial \\Psi^*(\\xi)\\quad \\text{in } V;$\n\\item[$iii)$]\\quad $\\langle \\xi, u\\rangle=\\Psi(u)+\\Psi^*(\\xi) \\quad \\text{in\n  } \\mathbb{ R}.$\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n  \\textsc{Ekeland} and \\textsc{T\\'emam} \\cite[Prop.\\,5.1 and\n  Cor.\\,5.2 on pp.\\,21]{EkeTem74ACPV}.\n\\end{proof}\n\nFor the dissipation potentials $\\Psi_u$ we need the notion of\n$\\Gamma$-convergence, see \\cite{Dalm93IGC, Brai02GCB, Brai06HGC} (also called\nepigraph convergence in \\cite{Atto84VCFO}). We consider a functional\n$\\Psi:V\\to (-\\infty,\\infty]$ and a sequence $(\\Psi_n)_{n\\in \\N}$ of\nfunctionals all of which are lower semicontinuous convex functionals.\nThe (strong) $\\Gamma$-convergence $\\Psi_n \\xrightarrow{\\Gamma} \\Psi$ in $V$ is defined via \n\\[\n\\Psi_n\\xrightarrow{\\Gamma} \\Psi\\ \\Longleftrightarrow \\ \\left\\{ \\ba{cl} \n\\text{(a) }&\\ds\\ v_n \\to v \\  \\Longrightarrow \\ \\Psi(v) \\leq\n               \\liminf_{n\\to \\infty} \\Psi_n(v_n),\\\\[0.4em]\n\\text{(b) }&\\ \\forall\\, \\wh v\\in V\\ \\exists\\, (\\wh v_n)_{n\\in \\N}: \\ \\\n\\wh v_n\\to \\wh v \\text{ and }\\Psi(v) \\geq\n               \\limsup_{n\\to \\infty} \\Psi_n(v_n).\\\\\n  \\ea \\right. \n\\]\nHere (a) is called the (strong) liminf estimate, while (b) is called\nthe (strong) limsup estimate or the existence of recovery\nsequences. Similarly we define the (sequential) weak\n$\\Gamma$-convergence $\\Psi_n \\overset{\\Gamma}{\\weak} \\Psi$ in $V$ via \n\\[\n\\Psi_n\\overset{\\Gamma}{\\weak} \\Psi\\ \\Longleftrightarrow \\ \\left\\{ \\ba{cl} \n\\text{(a) }&\\ds\\ v_n \\weak v \\  \\Longrightarrow \\ \\Psi(v) \\leq\n               \\liminf_{n\\to \\infty} \\Psi_n(v_n),\\\\[0.4em]\n\\text{(b) }&\\ \\forall\\, \\wh v\\in V\\ \\exists\\, (\\wh v_n)_{n\\in \\N}: \\ \\\n\\wh v_n\\weak \\wh v \\text{ and }\\Psi(v) \\geq\n               \\limsup_{n\\to \\infty} \\Psi_n(v_n).\\\\\n  \\ea \\right. \n\\] \nIf both convergences hold, then we say that $\\Psi_n$ \\textsc{Mosco}\nconverges to $\\Psi$ and write $\\Psi_n \\xrightarrow{\\,\\rmM\\,} \\Psi$. In \\cite[pp.\\,271]{Atto84VCFO}\nthe following fundamental relation between $\\Gamma$-convergence and\nthe \\textsc{Legendre-Fenchel} transform was established:\n\\begin{equation}\n \\label{eq:Gcvg.sw*}\n  \\Psi_n \\xrightarrow{\\Gamma} \\Psi \\quad \\Longleftrightarrow \\quad \\Psi_n^* \\overset{\\Gamma}{\\weak} \\Psi^*,\n\\end{equation}\nwhich always holds on reflexive Banach spaces $V$ if all $\\Psi_n$ and\n$\\Psi_n^*$ are nonnegative (as for our dissipation potentials). \n\n\n\\subsection{Semi-implicit variational approximation scheme}\n\\label{su:VarApprox}\nThe basic idea to show the existence of strong solutions to \\eqref{eq:I.1} with an initial condition $u=u_0\\in V$ is to construct a solution via a\nparticular discretization scheme, more precisely, with a semi-implicit\n\\textsc{Euler} method. The usual implicit Euler method does not work\nsince the equation \\eqref{eq:I.1} does not possess the gradient flow structure due to the nonpotential perturbation. With our approach, it is\npossible to construct time-discrete solutions via a variational\napproximation scheme. To illustrate this let for $N\\in\n\\mathbb{N}\\backslash \\lbrace 0\\rbrace$\n\\begin{align}\n\\label{eq:II.1}\n I_\\tau=\\lbrace 0=t_0<t_1<\\cdots< t_n=n\\tau<\\cdots t_N=T \\rbrace\n\\end{align} be an equidistant partition of the time interval $[0,T]$ with step size $\\tau:=T\/N$, where we omit the dependence of $t_n$ on the step size $\\tau$ for simplicity. The approximation of \\eqref{eq:I.1} is then given by\n\\begin{align}\n\\label{eq:II.2}\nB(t_{n-1},U_\\tau^{n-1})\\in \\partial \\Psi_{U_\\tau^{n-1}}\\left(\\frac{U_\\tau^{n}-U_\\tau^{n-1}}{\\tau}\\right)+\\partial \\calE_{t_n}(U_\\tau^{n}), \\quad n=1,\\cdots,N,\n\\end{align} where the values $U_\\tau^{n}\\approx u(t_n)$, which shall\napproximate the exact solution of \\eqref{eq:I.1} at $t_n$, are to determine. If both the dissipation potential and the energy functional are\n\\textsc{Fr\\'echet}-differentiable the inclusion \\eqref{eq:II.2} becomes the\nequation\n\\begin{align}\n\\label{eq:II.3}\nB(t_{n-1},U_\\tau^{n-1})=\n\\rmD\\Psi_{U_\\tau^{n-1}}\\left(\\frac{U_\\tau^{n}-U_\\tau^{n-1}}{\\tau}\\right)+\\rmD\n\\calE_{t_n}(U_\\tau^{n}), \\quad n=1,\\cdots,N.\n\\end{align} \nIt is now simple to see that the value $U_\\tau^{n}$ can be\ncharacterized as a solution of the \\textsc{Euler-Lagrange} equation\nassociated to the map\n\\begin{align*}\nv\\mapsto \\Upphi(\\tau,t_{t_{n-1}},{U_\\tau^{n-1}},B(t_{n-1},U_\\tau^{n-1});v),\n\\end{align*} where \n\\begin{align}\n\\label{eq:II.4}\n  \\Upphi(r,t,u,w;v)=\n  r\\Psi_{u}\\left(\\frac{v-u}{r}\\right)+\\calE_{t+r}(v)-\\langle w,v\\rangle\n\\end{align}\nfor $ r\\in \\mathbb{R}^{>0},t\\in[0,T)$ with $r+t\\in[0,T]$,  $u,v\\in V$,\nand $ w\\in V^*$. In fact, we determine the value $U_\\tau^{n}$ by\nminimizing the functional $\\Upphi$ in the variable $v\\in V$ under\nsuitable conditions on the dissipation potential and the energy\nfunctional. To assure that the value $U_\\tau^{n}$ satisfies the\ninclusion \\eqref{eq:II.2} also in the nonsmooth case, which is in general not\ntrue, we make an assumption to enforce property. \n\n\n\\subsection{Assumptions for the main existence result}\n\\label{su:AssumpExistRes}\n\nWe now collect the assumption on the perturbed gradient system\n$\\mathrm{PG}=(V,\\calE,\\Psi,B)$ for our existence result. They will be denoted\nin via (2.En), (2.$\\Uppsi$m), and (2.Bk). \n\nThe assumptions for the energy functional are the following. \n\n\\begin{enumerate}[label=(\\thesection .E\\alph*),\n leftmargin=3.2em]\n\\item \\label{eq:cond.E.1}\n \\textbf{Constant domain.} For all $t\\in[0,T]$, the\n  functional $\\calE_t:V \\rightarrow (-\\infty,+\\infty]$ is proper and lower\n  semicontinuous with the time-independent effective domain $D\\equiv\n  \\operatorname{dom}(\\calE_t)\\subset V$ for all $t\\in [0,T]$.\n\\item \\label{eq:cond.E.2} \\textbf{Compactness of sublevels.}  There exists\n  $t^*\\in [0,T]$ such that the functional $E_{t^*}$ has compact\n  sublevels in $V$.\n\\item \\label{eq:cond.E.3} \\textbf{Energetic control of power.} For all $u\\in D$,\n  the power map $t\\mapsto \\calE_t(u)$ is continuous on $[0,T]$ and\n  differentiable in $(0,T)$ and its derivative $\\partial_t\\calE_t$ is\n  controlled by the function $\\calE_t$, i.e., there exist $C>0$ such that\n\\begin{align*}\n  \\vert \\partial_t \\calE_t(u)\\vert \\leq C \\calE_t(u)\\quad \\text{for all } t\\in\n  (0,T) \\text{ and } u\\in D.\n\\end{align*} \n\\item \\label{eq:cond.E.4}\\textbf{Chain rule.} For every absolutely continuous\n  curve $v\\in \\mathrm{AC}([0,T];V)$ and every \\textsc{Bochner} integrable\n  functions $\\xi \\in \\rmL^1(0,T;V^*)$ such that\n\\begin{align*}\n  \\sup_{t\\in[0,T]}\\vert \\calE_t(u(t))\\vert <+\\infty, \\quad \\xi(t)\\in \\partial \\calE_t(u(t)) \\quad \\text{ a.e. in } (0,T),\\\\\n  \\int_0^T\\Psi_{u(t)}(u'(t))\\dd t <+\\infty \\quad \\text{and} \\quad\n  \\int_0^T\\Psi^*_{u(t)}(\\xi(t))\\dd t <+\\infty,\n\\end{align*} the map $t\\mapsto \\calE_t(u(t))$ is absolutely continuous on\n$[0,T]$ and\n\\begin{align*}\n  \\frac{d}{\\dd t}\\calE_t(u(t))\\geq \\langle \\xi(t),u'(t)\\rangle + \\partial_t\n  \\calE_t(u(t))\\quad \\text{a.e. in }(0,T).\n\\end{align*}\n\\item \\label{eq:cond.E.5} \\textbf{Strong-weak closedness.} For all $t\\in[0,T]$ and all sequences $(u_n,\\xi_n)_{n\\in \\mathbb{N}} \n     \\subset V\\ti V^*$ with $\\xi_n\\in \\partial \\calE^{\\eps_n}_t(u_n)$ such that \n\\begin{align*}\n  u_n \\rightarrow u\\in V, \\quad \\xi_n \\rightharpoonup \\xi\\in V^*,\n  \\quad \\calE_t(u_n)\\rightarrow \\mathcal{E}\\in \\mathbb{R} \\quad \\text{and}\n  \\quad \\partial_t \\calE_t(u_n)\\rightarrow \\calP \\in\n  \\mathbb{R}\n\\end{align*} as $n\\rightarrow\\infty$, we have the relations\n\\begin{align*} \n\\xi \\in \\partial \\calE_t(u), \\quad \\calP \\leq \\partial_t \\calE_t(u)\\quad\n\\text{and} \\quad \\mathcal{E}=\\calE_t(u). \n\\end{align*}\n\\end{enumerate}\nWe first give a few relevant comments on these assumptions that will\nbe important below. \n\n\\begin{rem}\\label{re:Assump.E} \\mbox{}\\vspace{-0.6em}\n\n\\begin{itemize}\n\n\\item[$i)$] From Assumption \\ref{eq:cond.E.3} we deduce with\n  \\textsc{Gronwall}'s lemma the chain of inequalities\n\\begin{align}\n\\label{eq:II.5}\n\\ee^{-C\\vert t-s\\vert}\\calE_s(u)\\leq \\calE_t(u)\\leq \\ee^{C\\vert\n  t-s\\vert}\\calE_s(u) \\quad \\text{ for all }s,t\\in [0,T].\n\\end{align} \nIn particular there exists a constant $C_1>0$ such that\n\\begin{align}\n  \\label{eq:II.6}\n  G(u)=\\sup_{t\\in [0,T]}\\calE_t(u)\\leq C_1 \\inf_{t\\in[0,T]} \\calE_t(u)\n  \\quad \\text{for all } u\\in D.\n\\end{align}\n\n\\item[$ii)$] From Assumptions \\ref{eq:cond.E.2} and \\ref{eq:cond.E.3}\n  we deduce the existence of a real number $S$ which bounds the energy\n  functional from below, i.e.,\n\\begin{align}\n\\label{eq:II.7}\n\\calE_t(u)\\geq S \\quad \\text{for all }u\\in V,\\, t\\in [0,T].\n\\end{align}\n\n\\item[$iii)$] From the strong-weak closedness property of the graph of\n  $\\partial E$ in \\ref{eq:cond.E.5} and \\textsc{Mordukhovich}\n  \\cite[Lem.\\,2.32, p.\\,214]{Mord06VAGD1} one can argue as in\n  \\cite[Prop.\\,4.2, p.\\,273]{MiRoSa13NADN}, in order to show the\n  following variational sum rule:\n  \\\\\n  If for $u_0\\in V,\\ r>0$, and $t\\in[0,T]$ the point $u\\in V$ is a\n  global minimizer of $\\Upphi(\\tau,t,u_0,w;\\cdot)$, then \n\\begin{align}\n\\label{eq:II.8}\n\\exists\\, \\xi \\in \\partial \\calE_t(u):\\quad w-\\xi \\in \\partial\n\\Psi_{u_0}\\left(\\frac{u-u_0}{r}\\right); \n\\end{align} \nor equivalently $\\ds w\\in \\partial\n\\Psi_{u_0}\\left(\\frac{u-u_0}{r}\\right)+ \\partial \\calE_{t+r}(u)$. \n\n\\item[$iv)$] Assumption \\ref{eq:cond.E.2} and point $i)$ in this\n  remark yields immediately that the functional $\\calE_t$ has compact\n  sublevels for all $t\\in[0,T]$.\n\\item[$v)$] It is possible to relax Assumption \\ref{eq:cond.E.3} by\n  assuming not the time differentiability but a kind of\n  \\textsc{Lipschitz } continuity and a conditioned one-sided time\n  differentiability of the map $t\\mapsto \\calE_t(u)$, see\n  \\cite{MiRoSa13NADN}. We shall confine ourselves to Assumption\n  \\ref{eq:cond.E.3} just to simplify the proofs.\n \\end{itemize} \n\\end{rem}\n\nNow, we collect the assumptions concerning the dissipation potential $\\Psi$.\n\n\\begin{enumerate}[label=(\\thesection.$\\Uppsi$\\alph*), leftmargin=3.2em]\n\\item \\label{eq:Psi.1} \\textbf{Dissipation potential.} For all $u\\in\n  V$ the functional $\\Psi_u: V\\rightarrow [0,+\\infty)$ is lower\n  semicontinuous and convex with $\\Psi(0)=0$. Furthermore if $w_1,w_2\n  \\in \\partial \\Psi_u(v)$ for any $v \\in V$ then\n  $\\Psi_u^*(w_1)=\\Psi_u^*(w_2)$.\n\n\\item \\label{eq:Psi.2} \\textbf{Superlinearity.} The functionals\n  $\\Psi_u$ and $\\Psi_u^*$ are coercive uniformly with respect to $u\\in\n  V$ in sublevels of $E$, i.e., for all $R>0$ there hold\n  \\begin{align*}\n    \\lim_{\\Vert \\xi\\Vert_*\\rightarrow +\\infty}\\frac{1}{\\Vert\n      \\xi\\Vert_*}\\Big(\\inf_{\\overset{u\\in V}{G(u)\\leq R}\n    }\\Psi^*_u(\\xi)\\Big)=\\infty,\\quad \\lim_{\\Vert v\\Vert\\rightarrow\n      +\\infty}\\frac{1}{\\Vert v\\Vert}\\Big(\\inf_{\\overset{u\\in\n        V}{G(u)\\leq R} }\\Psi_u(v)\\Big)=\\infty,\n  \\end{align*} \n  where $G(u):=\\sup_{t\\in[0,T]}\\calE_t(u)$ for all $u\\in V$.\n\n\\item \\label{eq:Psi.3} \\textbf{State-dependence is \\textsc{Mosco}\n    continuous.} The functional $\\Psi$ is continuous in the sense of\n  \\textsc{Mosco}-convergence, i.e., for all $R>0$ and sequences\n  $(u_n)_{n\\in\\mathbb{N}}\\subset V$ with $u_n\\rightarrow u\\in V$ as\n  $n\\rightarrow\\infty$ and $\\sup_{n\\in\\mathbb{N}}G(u_n)\\leq R$, we\n  have $\\Psi_{u_n} \\xrightarrow{\\,\\rmM\\,} \\Psi_u$.\n\\end{enumerate}\n\n\\begin{rem}\\label{re:Assump.Psi} \\mbox{} \\vspace{-0.6em}\n\\begin{itemize}\n\\item[$i)$] Since $\\operatorname{dom}(\\Psi_u)=V$ for all $u\\in V$, the lower\n  semicontinuity and convexity of $\\Psi_u$ yields the continuity of\n  $\\Psi_u$ and $\\partial \\Psi_u(v)\\neq \\emptyset$ for all $u\\in V,\n  u\\in D$. Together with Assumption \\ref{eq:Psi.2}, this implies that the\n  \\textsc{Legendre-Fenchel} conjugate $\\Psi^*$ is everywhere finite,\n  i.e., $\\operatorname{dom}(\\Psi^*)=V^*$, and the operator $\\partial \\Psi_u:\n  V\\rightarrow 2^{V^*}$ is for all $u\\in D$ bounded, i.e., it maps\n  bounded subsets of $V$ into bounded subsets of $V^*$. The former in\n  turn entail the same properties for $\\Psi^*_u$ for all $u\\in V$.\n\n\\item[$ii)$] The \\textsc{Mosco} convergence of $\\Psi_{u_n} \\xrightarrow{\\,\\rmM\\,} \\Psi_u$\n  from Assumption \\ref{eq:Psi.3} implies \\textsc{Mosco} convergence of the dual\n  potentials, namely $\\Psi^*_{u_n} \\xrightarrow{\\,\\rmM\\,} \\Psi^*_n$, see\n  \\eqref{eq:Gcvg.sw*}. In particular, this implies that for all\n  $R>0$, all \nsequences $(u_n)_{n\\in\\mathbb{N}}\\subset V$ with $u_n\\rightarrow u\\in\nV$ and $\\sup_{n\\in \\N} G(u_n)\\leq R$, and all sequences $ (\\xi_n\n)_{n\\in\\mathbb{N}}\\subset V^*$ with  $\\xi_n\\rightharpoonup \\xi\\in V^*$\nwe have \n\\begin{align}\n\\label{eq:Psi*liminf}\n\\Psi^*_u(\\xi)\\leq  \\liminf_{n\\rightarrow\\infty}\\Psi^*_{u_n}(\\xi_n).\n\\end{align}\n\\end{itemize}\n\\end{rem}\n\nFinally, we make the following assumptions on the non-variational perturbation $B$.\n\\begin{enumerate}[label= (\\thesection.B\\alph*), leftmargin=3.2em]\n\n\\item \\label{eq:B.1} \\textbf{Continuity.} The map $(t,u) \\mapsto\n  B(t,u):[0,T]\\times V \\rightarrow V^*$ is continuous on sublevels of\n  $G$, i.e.\\ $(t_n,u_n)\\to (t,u)$ in $[0,T]\\ti V$ and\n  $\\sup_{n\\in\\mathbb{N}}G(u_n)\\leq R$ implies $B(t_n,u_n)\\to B(t,u)$\n  in $V^*$.\n  \n\\item \\label{eq:B.2} \\textbf{Control of $B$ by the energy.} There\n  exist  $\\beta>0$ and $c\\in(0,1)$ such that\n  \\begin{align*}\n    c\\,\\Psi^*_u\\left(\\frac1c B(t,u)\\right)\\leq \\beta \\big(1+\\calE_t(u)\\big)\n    \\quad \\text{ for all } u\\in D, \\, t\\in[0,T].\n  \\end{align*}\n\\end{enumerate}\n\n\n\\begin{rem}\\label{re:Assump.B} \n  We note that Assumption \\ref{eq:B.1} ensures that the\n  \\textsc{Nemytskij} operator associated to $B$ maps strongly\n  measurable functions contained in sublevels of $G$ into strongly measurable functions, i.e., for\n  all strongly measurable functions $u$ with $\\sup_{t\\in[0,T]}G(u(t))\\leq R$, the map $t\\mapsto B(t,u(t))$\n  is strongly measurable.\n\\end{rem}\n\n\\subsection{Statement of the existence result}\n\\label{su:StateExistRes}\n\nBefore we state the main result, we say that $u\\in \\mathrm{AC}([0,T];V)$ is a\nsolution to \\eqref{eq:I.1} with the initial datum $u_0\\in D$ if $u$ satisfies the differential inclusion \\eqref{eq:I.1} with $u(0)=u_0$.\n\n\\begin{thm}[Main existence result for $\\mathrm{PG}=(V,\\calE,\\Psi,B)$]\n  \\label{th:MainExist}  Let the perturbed\n  gradient system $(V,\\calE,\\Psi,B)$ satisfy the Assumptions\n  \\textnormal{(2.E), (2.$\\Uppsi$)}, and \n  \\textnormal{(2.B)}.  Then for every $u_0\\in D$ there exists a solution\n  $u\\in \\mathrm{AC}([0,T];V)$ to \\eqref{eq:I.1} with $u(0)=u_0$ and an\n  integrable function $\\xi \\in \\rmL^1(0,T;V)$ with $\\xi(t)\\in \\partial\n  \\calE_t(u(t))$ for a.a. $t\\in(0,T)$ such that the following\n  energy-dissipation balance holds:\n\\begin{align}\n\\label{eq:EDB}\n\\begin{split}\n  & \\calE_t(u(t))+ \\int_s^t \\left( \\Psi_{u(r)}(u'(r)) +\n    \\Psi_{u(r)}^*\\big(B(r,u(r))-\\xi(r)\\big) \\right) \\dd r \n\\\\ \n  &= \\calE_s(u(s))+\\int_s^t \\partial_r \\calE_r(u(r))\\dd r +\\int_s^t \\langle\n  B(r,u(r)),u'(r)\\rangle \\dd r \\quad \\text{for all } s,t\\in [0,T].\n\\end{split}\n\\end{align}\n\\end{thm} \n\n It is clear that every solution of \\eqref{eq:EDB} is already a\nsolution for the perturbed gradient system $\\mathrm{PG}=(V,\\calE,\\Psi,B)$,\nsince by the chain rule can and the \\textsc{Legendre-Fenchel} theory we easily\nrecover \\eqref{eq:I.1}, see e.g.\\ \\cite{AmGiSa05GFMS,RosSav06GFNC}. \n\nOur proof will be done by time discretization and solving variational\nproblems for each time interval $(t_n,t_{n+1}]$. To obtain a useful\ndiscrete counterpart of the energy-dissipation balance proper we\nemploy \\textsc{De Giorgi}'s variational interpolant, see\n\\cite[Lem.\\,2.5]{Ambr95MM} or\n\\cite[Sec.\\,4.2]{RosSav06GFNC}. We then follow the ideas in\n\\cite{MiRoSa13NADN}, but need to generalize to the case of a\nnontrivial perturbation $B$, which only satisfies our mild assumptions\n\\ref{eq:B.1} and \\ref{eq:B.2}. The proof will be completed in Section\n\\ref{su:Proof}. \n\n\n\\subsection{Estimates on the {\\mdseries\\scshape Moreau-Yosida} regularization}\n\\label{su:MoreauYosida}\n\nIn order to prove the existence result, we need to show some\nproperties of the $\\Psi$-\\textsc{Moreau-Yosida} regularization\n\\begin{align*}\n\\Phi_{r,t}(w;u):= \\inf_{v\\in V} \\Upphi(r,t,u,w;v)\n\\end{align*} for $r>0, t\\in [0,T)$ with $r+t\\in [0,T]$ and $u\\in D$ as\nwell as $w\\in V^*$. Therefore, we have to ensure that the resolvent\nset $J_{r,t}(w;u):= \\argmin_{v\\in V} \\Upphi(r,t,u,w;v)$ is not empty.\n\n\\begin{lem}\\label{le:Exist.Min} Let the perturbed\n  gradient system $(V,\\calE,\\Psi,B)$ satisfy the Assumptions\n  \\textnormal{\\ref{eq:cond.E.1}-\\ref{eq:cond.E.2}} and\n  \\textnormal{\\ref{eq:Psi.1}}. Then for all $r>0$, $t\\in [0,T)$ with\n  $t+r\\leq T$, $u\\in D $, and $w\\in V^*$, the resolvent set\n  $J_{r,t}(w;u)$ is nonempty.\n\\end{lem}\n\\begin{proof} Let $u\\in D, w\\in V^*$ and $r>0, t\\in [0,T)$ with\n  $r+t\\in [0,T]$ be given.  First of all, we see with the\n  \\textsc{Fenchel-Young} inequality and with \\eqref{eq:II.7} that\n\\begin{align}\n\\label{eq:II.11}\n  \\Upphi(r,t,u,w;v)&= r\\Psi_{u}\\left(\\frac{v-u}{r}\\right)+\\calE_{t+r}(v)-\\langle w,v\\rangle \\notag \\\\\n  &\\geq -r\\Psi_u^*(w)+\\calE_{t+r}(v)-\\langle w,u\\rangle\\\\\n  &\\geq -r\\Psi_u^*(w) +S -\\langle w,u\\rangle \\notag.\n\\end{align} \nThis implies $\\Phi_{r,t}(w;u)>-\\infty$. On the other hand, we observe that\n\\begin{align}\n\\label{eq:II.12}\n  \\inf_{v\\in V}\\Big\\lbrace\n  r\\Psi_{u}\\left(\\frac{v-u}{r}\\right)+\\calE_{t+r}(v)-\\langle\n  w,v\\rangle \\Big\\rbrace \\leq \\calE_{t+r}(u)-\\langle w,u\\rangle,\n\\end{align} \nso that we also have $\\Phi_{r,t}(w;u)<+\\infty$.  Let now $(v_n)_{n \\in\n  \\mathbb{N}}\\subset V$ be a minimizing sequence for\n$\\Upphi(r,t,u,w;\\cdot)$. From \\eqref{eq:II.11}, we deduce with \\eqref{eq:II.5} that\n$(v_n)_{n \\in \\mathbb{N}}\\subset V$ is contained in a sublevel set of\n$\\calE_t$. Thus, by Assumption \\ref{eq:cond.E.2} and Remark \\ref{re:Assump.E} $iv)$ there exists\na subsequence (not relabeled) which converges strongly in $V$ towards a\nlimit $ v\\in V$. Together with the lower semicontinuity of the map\n$v\\mapsto \\Upphi(r,t,u,w;v)$, we have\n\\begin{align*}\n  \\Upphi(r,t,u,w;v)\\leq \\liminf_{n\\rightarrow\n    \\infty}\\Upphi(r,t,u,w;v_n)=\\inf_{\\tilde{v}\\in V}\n  \\Upphi(r,t,u,w;\\tilde{v})\n\\end{align*} and therefore $v\\in J_{r,t}(w;u)\\neq \\emptyset$ from what $v\\in D$ follows.\n\\end{proof}\nLemma \\ref{le:Exist.Min} is important for justifying the existence of a sequence of approximate values\n$(U_\\tau^n)_{n=1}^N\\subset D$ that complies with\n\\begin{align}\n\\label{eq:II.13}\n  U_\\tau^n \\in J_{\\tau,t_{n-1}}(B(t_{n-1},U_\\tau^{n-1}),U_\\tau^{n-1})\n  \\quad \\text{for all } n=1, \\cdots, N,\n\\end{align} \nin order to construct discrete solutions of \\eqref{eq:II.2}, where\n$U_\\tau^0:=u_0$ and the time $t\\in [0,T)$ as well as the time step\n$\\tau\\in (0,T-t)$ are fixed. \n\nThe following lemma is crucial in order to proof the existence result\nand in particular to derive a priori estimates for the interpolation\nfunctions we define later on. The result is an adaptation to the\ncase $w\\neq 0$ of \\cite[Lem.\\,4.2]{RosSav06GFNC} and\n\\cite[Lem.\\,6.1]{MiRoSa13NADN}.\n\n\n\\begin{lem}\\label{le:Main.Lem} Let the perturbed\n  gradient system $(V,\\calE,\\Psi,B)$ satisfy the Assumptions\n  \\textnormal{(2.E), (2.$\\Uppsi$)}, and \n  \\textnormal{(2.B)}. Then for every $\\mathrm{t}\\in [0,T), u\\in D$ and \n  $w\\in V^*$ there exists a measurable selection $r\\mapsto u_r:\n  (0,T-\\mathrm{t})\\rightarrow J_{r,\\mathrm{t}}(w;u)$ such that\n\\begin{align}\n\\label{eq:II.14}\nw\\in \\partial \\Psi_u \\left(\\frac{u_r-u}{r}\\right) + \\partial \\calE_{\\mathrm{t}+r}(u) \n\\end{align} and there exists a constant $\\widetilde{C}>0$ such that\n\\begin{align}\n\\label{eq:II.15}\nG(u_r)\\leq \\widetilde{C} ( G(u)+r\\Psi_u^*( w))\\quad \\text{for all } r\\in(0,T-\\mathrm{t})\n\\end{align} Furthermore, there holds\n\\begin{align}\n\\label{eq:II.16}\n\\lim_{r\\rightarrow 0}\\sup_{u_r\\in J_{r,\\mathrm{t}}(w;u)}\\Vert u_r-u\\Vert=0 \\quad \\text{and}\\quad \\lim_{r\\rightarrow 0} \\Phi_{r,\\mathrm{t}}(w;u)= \\calE_\\mathrm{t}(u)-\\langle w,u\\rangle\n\\end{align}\n for all $\\mathrm{t}\\in [0,T), u\\in D$ and $w\\in V^*$. Finally the map $r\\mapsto \\Phi_{r,\\mathrm{t}}(w;u)$ is almost everywhere differentiable in $(0,T-\\mathrm{t})$ and for every $r_0\\in (0,T-\\mathrm{t})$ and every measurable selection $r\\mapsto u_r: (0,r_0)\\rightarrow J_{r,\\mathrm{t}}(w;u)$ there exists a measurable selection $r\\mapsto \\xi_r: (0,T-\\mathrm{t}) \\rightarrow  \\partial \\calE_{\\mathrm{t}+r}(u)$ with $w-\\xi_r\\in \\partial \\Psi_u \\left(\\frac{u_r-u}{r}\\right)$ such that\n\\begin{align}\n\\label{eq:II.17}\n\\begin{split}\nE_{\\mathrm{t}+r_0}(u_{r_0}) + r_0\\Psi_u\\left(\\frac{u_{r_0}-u}{r_0}\\right)\n+&\\int_0^{r_0} \\Psi_u^*(w-\\xi_r)\\dd r \\\\ \n&\\leq \\calE_t(u)+\\int_0^{r_0}\\partial_r \\calE_{\\mathrm{t}+r}(u_r)\\dd r +\\langle w,u_{r_0}-u\\rangle.\n\\end{split}\n\\end{align} \n\\end{lem}\n\\begin{proof}\n  Let $\\mathrm{t}\\in [0,T), u\\in D$ and $w\\in V^*$ be given. The\n  non-emptiness of the resolvent set $J_{r,\\mathrm{t}}(w;u)$ for all\n  $r\\in(0,T-\\mathrm{t})$ is guaranteed by Lemma \\ref{le:Main.Lem}. The existence of a\n  measurable selection $r \\mapsto u_r: (0,T-\\mathrm{t})\\rightarrow\n  J_{r,\\mathrm{t}}(w;u)$ is provided by \\textsc{Castaing} and\n  \\textsc{Valadier} \\cite[Cor.\\,III.3, Prop.\\,III.4, Thm.\\,III.6,\n  pp.\\,63]{CasVal77CAMM}. The inclusion \\eqref{eq:II.14} follows then by the\n  variational sum rule \\eqref{eq:II.8}. Further, we obtain from \\eqref{eq:II.11} for\n  $v=u_r, r\\in(0,T-\\mathrm{t})$ and \\eqref{eq:II.12} the inequality\n\\begin{align*}\n\\calE_{\\mathrm{t}+r}(u_r)\\leq \\calE_{\\mathrm{t}+r}(u)+r\\Psi_u^*(w),\n\\end{align*} so that together with the estimate \\eqref{eq:II.6} it follows the inequality \\eqref{eq:II.15} with $\\widetilde{C}=C_1$, where $C_1>0$ is the constant in \\eqref{eq:II.6}. In order to show the convergences in \\eqref{eq:II.16}, we note that Assumption \\ref{eq:Psi.2} implies: For all $R>0$ and $\\gamma>0$, there exists $K>0$ such that \n\\begin{align*}\n\\Psi_u(v)\\geq \\gamma\\Vert v\\Vert\n\\end{align*} for all $u\\in D$ with $G(u)\\leq R$ and all $v\\in V$ with $\\Vert v\\Vert\\leq K$.\n Based on this fact, we infer\n\\begin{align}\n\\label{eq:II.18}\n\\gamma \\left \\Vert \\frac{u_r-u}{r} \\right\\Vert &\\leq  \\Psi_u\\left(\\frac{u_r-u}{r}\\right)+\\gamma K \\quad \\text{ for every }r>0.\n\\end{align} Together with \\eqref{eq:II.7}, \\eqref{eq:II.11} and \\eqref{eq:II.12}, we obtain\n\\begin{align*}\n\\gamma \\Vert u_r-u \\Vert &\\leq \\langle w, u_r-u\\rangle +\\calE_{\\mathrm{t}+r}(u)-\\calE_{\\mathrm{t}+r}(u_r) +r\\gamma K\\\\\n&\\leq \\Vert w \\Vert \\Vert u_r-u\\Vert +\\calE_{\\mathrm{t}+r}(u)-S+r\\gamma K.\n\\end{align*} \nThis implies the estimate\n\\begin{align*}\n(\\gamma-\\Vert w\\Vert_*) \\Vert u_r{-} u \\Vert &\\leq\n\\calE_{\\mathrm{t}+r}(u)-S+r\\gamma K\n\\leq \\ee^{CT}\\calE_0(u)-S+r\\gamma K \n\\end{align*} for all $\\gamma>0, r\\in (0,T-\\mathrm{t})$ and $u_r\\in J_{r,\\mathrm{t}}(w;u)$, where we used again \\eqref{eq:II.5}. By taking the supremum over all $u_r\\in J_{r,\\mathrm{t}}(w;u)$ and taking the limes superior as $r\\rightarrow 0$, we finally obtain\n\\begin{align*}\n(\\gamma-\\Vert w\\Vert_*) \\limsup_{r\\rightarrow 0} \\sup_{u_r\\in J_{r,\\mathrm{t}}(w;u)} \\Vert u_r-u \\Vert \\leq  \\ee^{CT} \\calE_0(u)-S\\quad \\text{for every }\\gamma>\\Vert w\\Vert_*.\n\\end{align*} By choosing $\\gamma>0$ sufficiently large, we conclude \n\\begin{align*}\n\\limsup_{r\\rightarrow 0} \\sup_{u_r\\in J_{r,\\mathrm{t}}(w;u)} \\Vert u_r-u \\Vert=0,\n\\end{align*} \nwhich shows the first convergence in \\eqref{eq:II.16}. We now use \nthe lower semicontinuity and the time continuity of the energy\nfunctional, the estimate\n\\begin{align*}\n&\\calE_{\\mathrm{t}+r}(u_r)-\\langle w,u_r\\rangle \\leq \\Phi_{r,\\mathrm{t}}(w;u)\\\\\n&\\quad =  r\\Psi_{u}\\left(\\frac{u_r-u}{r}\\right)+\\calE_{\\mathrm{t}+r}(u_r)-\\langle\nw,u_r\\rangle \n\\leq \\calE_{\\mathrm{t}+r}(u)-\\langle w,u\\rangle,\n\\end{align*} \nand the fact that $\\liminf_{r\\rightarrow\n  0}\\calE_{\\mathrm{t}+r}(u_r)=\\liminf_{r\\rightarrow 0}\\calE_{\\mathrm{t}}(u_r)$,\nwhich follows from \\eqref{eq:II.5}. Hence, the second convergence in\n\\eqref{eq:II.16} follows from the estimate \n\\begin{align*}\n&\\hspace*{-1em}\\calE_\\mathrm{t}(u)-\\langle w, u\\rangle\\leq \\liminf_{r\\rightarrow 0} \\left( \\calE_{\\mathrm{t}+r}(u_r)-\\langle w, u_r\\rangle\\right) \\\\\n&\\leq \\liminf_{r\\rightarrow 0} \\Phi_{r,t}(w;u) \n\\ \\leq \\ \\limsup_{r\\rightarrow 0}\\Phi_{r,\\mathrm{t}}(w;u)\\\\\n&\\leq \\limsup_{r\\rightarrow 0} \\left( \\calE_{\\mathrm{t}+r}(u)-\\langle\n  w,u\\rangle \\right)\n\\ = \\ \\calE_\\mathrm{t}(u)-\\langle w,u\\rangle.\n\\end{align*} \n\nIn order to show the last assertion of this lemma, let $u_{r_i}\\in J_{r,\\mathrm{t}}(w;u), i=1,2,$ with $0<r_1<r_2<T-\\mathrm{t}$. Then we have\n\\begin{align}\n&\\Phi_{r_2,\\mathrm{t}}(w;u)-\\Phi_{r_1,\\mathrm{t}}(w;u) - \\left( \\calE_{\\mathrm{t}+r_2}(u_{r_1}) - \\calE_{\\mathrm{t}+r_1}(u_{r_1})\\right) \\notag \\\\\n&\\leq r_2\\Psi_u\\left(\\frac{u_{r_1}-u}{r_2}\\right)-r_1\\Psi_u\\left(\\frac{u_{r_1}-u}{r_1}\\right)\\notag \\\\\n&= (r_2-r_1)\\Psi_u \\left(\\frac{u_{r_1}-u}{r_2} \\right)+r_1 \\left( \\Psi_u \\left(\\frac{u_{r_1}-u}{r_2}\\right)-\\Psi_u \\left(\\frac{u_{r_1}-u}{r_1} \\right) \\right) \\notag\\\\\n\\label{eq:II.19}\n&\\leq (r_2-r_1)\\left( \\Psi_u \\left(\\frac{u_{r_1}-u}{r_2} \\right)-\\left\\langle w_2^1,\\frac{u_{r_1}-u}{r_2} \\right\\rangle \\right)\\\\\n\\label{eq:II.20}\n&=-(r_2-r_1)\\Psi_u^*(w_2^1)\\leq 0,\n\\end{align} \nwhere we used in \\eqref{eq:II.19} the fact from Remark\n\\ref{re:Assump.Psi} $i)$ which states $w_2^1\\in \\partial \\Psi_u \\left\n  (\\frac{u_{r_1}-u}{r_2}\\right)\\neq \\emptyset$, in \\eqref{eq:II.20}\nthe statement of Lemma \\ref{le:Leg.Fen} and the last inequality the\nfact that by the \\textsc{Fenchel-Young} inequality we have\n$\\Psi_u^*(w)\\geq 0$ for all $w\\in V^*$. Further, we deduce with the\naid of \\ref{eq:cond.E.3}, \\eqref{eq:II.6} and the already proven\ninequality \\eqref{eq:II.15} that\n\\begin{align}\n\\label{eq:II.21}\n&\\Phi_{r_2,\\mathrm{t}}(w;u)\\leq \\Phi_{r_1,\\mathrm{t}}(w;u) + \\left( \\calE_{\\mathrm{t}+r_2}(u_{r_1}) - \\calE_{\\mathrm{t}+r_1}(u_{r_1})\\right) \\notag \\\\\n&= \\Phi_{r_1,\\mathrm{t}}(w;u) - \\int_{r_1}^{r_2} \\partial_r \\calE_{\\mathrm{t}+r}(u_{r_1})\\dd r  \\notag\\\\\n&\\leq \\Phi_{r_1,\\mathrm{t}}(w;u) +(r_2-r_1) CC_1 G(u_{r_1}) \\notag\\\\\n&\\leq \\Phi_{r_1,\\mathrm{t}}(w;u) +(r_2-r_1) CC_1 ( G(u)+r_1\\Psi^*(w)) \\notag\\\\\n&\\leq \\Phi_{r_1,\\mathrm{t}}(w;u) +(r_2-r_1) CC_1 ( G(u)+T\\Psi^*(w)).\n\\end{align} We conclude that the map $r\\mapsto \\Phi_{r,\\mathrm{t}}(w;u) -r\nCC_1 ( G(u)+T\\Psi^*(w))$ is non-increasing on $(0,T-\\mathrm{t})$ and\ntherefore as a real-valued function almost everywhere\ndifferentiable. Since the map $r \\mapsto \\Phi_{r,\\mathrm{t}}(w;u)$ is a\nlinear perturbation of a monotone function, it is also almost\neverywhere differentiable in $(0,T-\\mathrm{t})$. Thus there exists a\nnegligible set $\\mathscr{N}\\subset (0,T-\\mathrm{t})$, such that the map\n$r\\mapsto \\Phi_{r,\\mathrm{t}}(w;u)$ is differentiable on $(0,T-\\mathrm{t})\\backslash\n\\mathscr{N}$. We remark that the negligible set depends on $u$ and\n$w$, that is $\\mathscr{N}=\\mathscr{N}_{u,w}$. Now, to conclude, we\nwant to use the inequality \\eqref{eq:II.20}. For this let\n$r\\in(0,T-\\mathrm{t})\\backslash \\mathscr{N}$ be fixed. Additionally let\n$(h_n)_{n\\in \\mathbb{N}}\\in \\mathbb{R}^{>0}$ be a sequence which\nconverges from above towards zero and whose elements are sufficiently\nsmall. Let also the sequence $(w^r_n)_{n\\in \\mathbb{N}}\\subset V^*$ be\ngiven by $w^r_n\\in \\partial \\Psi_u \\left\n  (\\frac{u_{r}-u}{r+h_n}\\right)$ for all $n\\in \\mathbb{N}$. The\nboundedness of the operator $\\partial\\Psi_u$ according to Remark\n\\ref{re:Assump.Psi} $i)$ implies that the sequence $(w^r_n)_{n\\in\n  \\mathbb{N}}\\subset V^*$ is bounded in $V^*$. Thus there exists a\nsubsequence, labeled as before, and an element $w_r\\in V^* $ such that\n$w^r_n\\rightharpoonup w_r$ weakly in $V^*$. From the strong-weak\nclosedness of the graph of $\\partial \\Psi_u$ in $V\\times V^*$ it\nfollows $w_r \\in \\partial \\Psi_u \\left\n  (\\frac{u_{r}-u}{r}\\right)$. Since the conjugate $\\Psi_u^*$ is convex\nand lower semicontinuous, it is also weakly lower semicontinuous. Then\nwe find with Lemma \\ref{le:Leg.Fen} and the continuity of $\\Psi_u$\nthat\n\\begin{align*}\n  \\Psi_u^*(w_r)&\\leq \\liminf_{n\\rightarrow \\infty} \\Psi_u^*(w^r_n)\n\n  \\leq\\limsup_{n\\rightarrow \\infty} \\Psi_u^*(w^r_n)\\\\\n  &=  \\limsup_{n\\rightarrow \\infty} \\left( \\left\\langle w_n^r,\\frac{u_{r}-u}{r+h_n} \\right\\rangle- \\Psi_u \\left(\\frac{u_{r}-u}{r+h_n} \\right) \\right)\\\\\n  &= \\left\\langle w_r,\\frac{u_{r}-u}{r} \\right\\rangle- \\Psi_u\n  \\left(\\frac{u_{r}-u}{r} \\right)=\\Psi_u^*(w_r)\n\\end{align*} and thus $\\lim_{n\\rightarrow \\infty}\n\\Psi_u^*(w^r_n)=\\Psi_u^*(w_r)$. Due to the inclusion \\eqref{eq:II.14} there\nexist $\\xi_r\\in \\partial \\calE_{\\mathrm{t}+r}(u)$ such that $w-\\xi_r \\in \\partial\n\\Psi_u \\left(\\frac{u_{r}-u}{r} \\right)$. By \\textsc{Aubin} and\n\\textsc{Frankowska} \\cite[Thm.\\,8.2.9, p.\\,315]{AubFra90SVA}, the\nselection $r\\mapsto \\xi_r: (0,T-\\mathrm{t})\\rightarrow \\partial \\calE_{\\mathrm{t}+r}(u)$\ncan be chosen to be measurable. Further, from Assumption \\ref{eq:Psi.1} we\nget $\\Psi_u^*(w_r)= \\Psi_u^*(w-\\xi_r)$. By the differentiability of\nthe map $r \\mapsto\\Phi_{r,\\mathrm{t}}(w;u)$ in $r$, we obtain with \\eqref{eq:II.20}\n\\begin{align}\n & \\frac{\\rmd}{\\rmd r} \\Phi_{r,\\mathrm{t}}(w;u)\\vert_{r=r}+ \\Psi_u^*(w-\\xi_r)\n   \\ = \\ \\lim_{n\\rightarrow \\infty}\\left( \\frac{\\Phi_{r+h_n,\\mathrm{t}}(w;u)- \n  \\Phi_{r,\\mathrm{t}}(w;u)}{h_n} + \\Psi_u^*(w_n^r) \\right)\\notag \\\\\n\\label{eq:II.22}\n  &\\leq \\liminf_{n\\rightarrow \\infty}\t\\left( \n   \\frac{ E_{\\mathrm{t}+r+h_n}(u_{r}) -\n    \\calE_{\\mathrm{t}+r}(u_{r})}{h_n}\\right)\n   \\ = \\ \\partial_t E_{\\mathrm{t} +r}(u_r) \\quad \\text{for a.a. }r\\in(0,T{-}\\mathrm{t}),\n\\end{align} \nwhere we also used the fact that the map $t \\mapsto \\calE_t$ is\ndifferentiable. The claim finally follows by integrating \\eqref{eq:II.22} from\n$r=0$ to $r=r_0$ and by using \\eqref{eq:II.16}.\n\\end{proof}\n\n\n\\subsection{Time discretization and discrete energy-dissipation estimate}\n\\label{su:TimeDiscret}\n\nWith the help of the preceding lemma, we derive in the forthcoming\nresult a priori estimates for the approximate solutions, more\nprecisely for both the piecewise constant interpolation functions\n$\\overline{U}_\\tau $ and $\\underline{U}_\\tau$, and for the piecewise\nlinear interpolation function $\\widehat{U}_\\tau$ as well as for the\nso-called \\textsc{De Giorgi} interpolation function\n$\\widetilde{U}_\\tau$. In order to define the interpolation functions,\nlet the initial value $u_0\\in D$ and the time step $\\tau>0$ be\nfixed. Further let $(U_\\tau^n)_{n=1}^N \\subset D$ be the sequence of\napproximate values, which are defined by the variational approximation\nscheme\n\\begin{align}\n\\label{eq:II.23}\n\\begin{cases} \\quad U_\\tau^0=u_0, \\\\ \\quad \nU_\\tau^n\\in J_\\tau(B(t_{n-1},U_\\tau^{n-1});U_\\tau^{n-1})), \\quad n=1,2,\\dots,N.\n\\end{cases} \n\\end{align}\nThe piecewise constant and linear interpolation functions we define by\n\\begin{align}\n\\label{eq:Approx.tau}\n  &\\overline{U}_\\tau(0)=\\underline{U}_\\tau(0)=\\widehat{U}_\\tau(0):=U_\\tau^0 \\text{ and } \\notag \\\\\n  &\\underline{U}_\\tau(t):=U^{n-1}, \\quad \\widehat{U}_\\tau(t):=\\frac{t_n-t}{\\tau}U_\\tau^{n-1}+\\frac{t-t_{n-1}}{\\tau}U_\\tau^{n} \\quad \\text{for } t\\in[t_{n-1},t_n),\\notag\\\\\n  &\\overline{U}_\\tau(t):=U_\\tau^n \\quad \\text{ for } t\\in(t_{n-1},t_n]\n  \\quad \\text{and all } n=1,\\dots,N.\n\\end{align} \n\nFurthermore, we define by the approximation scheme  \n\\begin{align}\n\\label{eq:II.25}\n\\begin{cases}\\quad\n  \\widetilde{U}_\\tau(0):=U_\\tau^0,  \\\\ \\quad\n  \\widetilde{U}_\\tau(t) \\in J_r(B(t_{n-1},U_\\tau^{n-1});U_\\tau^{n-1}))\n  \\quad \\text{ for }t =t_{n-1}+r \\in (t_{n-1},t_n],\n\\end{cases} \n\\end{align} $n=1,2,\\dots,N,$ the \\textsc{De Giorgi} interpolation\n$\\widetilde{U}_\\tau$. We note that we can assume the measurability of\nthe function $\\widetilde{U}_\\tau$ since by Lemma \\ref{le:Main.Lem} there always\nexists a measurable selection of the \\textsc{De Giorgi}\ninterpolation. Due to the fact that for all $t\\in I_\\tau $ the approximation scheme \\eqref{eq:II.25}\nyields the usual scheme in \\eqref{eq:II.23}, we can assume without loss of\ngenerality that all interpolation functions coincide on the nodes\n$t_n$, i.e.,\n\\begin{align*}\n\\widetilde{U}_\\tau(t_n)=\\overline{U}_\\tau(t_n)=\\underline{U}_\\tau(t_n)=\\widehat{U}_\\tau(t_n)=U_\\tau^n \\quad \\text{for all } n=1,\\cdots,N.\n\\end{align*} Moreover, we denote by $\\widetilde{\\xi_\\tau}$ the\ninterpolation function obtained from Remark \\ref{re:Assump.E}  $iii)$ with the\nvariational sum rule by choosing $t=t_{n-1},\nu_0=\\widetilde{U}_\\tau(t), u=U_\\tau^{n-1}$ and\n$w=B(t_{n-1},U_\\tau^{n-1})$, and which satisfies\n\\begin{align}\n\\label{eq:II.26}\n  \\widetilde{\\xi_\\tau}(t)\\in \\partial\n  \\calE_{t_{n-1}+r}(\\widetilde{U}_\\tau(t)) \\quad \\text{ for }\n  t=t_{n-1}+r\\in (t_{n-1},t_n],\n\\end{align} and \n\\begin{align}\n\\label{eq:II.27}\n  B(t_{n-1},U_\\tau^{n-1})-\\widetilde{\\xi_\\tau}(t) \\in \\partial\n  \\Psi_{U_\\tau^{n-1}}\\left (\\frac{\\widetilde{U_\\tau}(t)-U_\\tau^{n-1}\n    }{t-t_{n-1}}\\right) \\text{ for } t=t_{n-1}+r\\in (t_{n-1},t_n]\n\\end{align} for all $n=1,\\dots,N$. The measurability of the function\n$\\widetilde{\\xi_\\tau}:(0,T)\\rightarrow V^*$ again follows from Lemma \\ref{le:Main.Lem}.\n\n For notational convenience, we also introduce the piecewise\nconstant interpolation functions $\\bar{\\mathbf{t}}_\\tau : [0,T]\n\\rightarrow [0,T]$ and $\\underline{\\mathbf{t}}_\\tau:[0,T]\\rightarrow [0,T]$ given by\n\\begin{align*}\n&\\overline{\\mathbf{t}}_{{\\tau}}(0):= 0\\,\\,\\,\\, \\text{ and } \\, \\overline{\\mathbf{t}}_{{\\tau}}(t):=\nt_n\\quad \\, \\, \\, \\, \\, \\text{ for } t\\in (t_{n-1},t_n], \\quad n=1,\\dots,N,\\\\\n&\\underline{\\mathbf{t}}_{{\\tau}}(T):= T \\, \\text{ and } \\, \\underline{\\mathbf{t}}_{{\\tau}}(t):=\nt_{n-1} \\quad \\text{ for } t\\in [t_{n-1},t_n), \\quad n=1,\\dots,N.\n\\end{align*} Obviously, there holds $\\overline{\\mathbf{t}}_\\tau(t)\\rightarrow t$ and\n$\\underline{\\mathbf{t}}_\\tau(t)\\rightarrow t$ as $\\tau\\rightarrow 0$.\n\nWe are now in the position to show a priori estimates for the\napproximate solutions.\n\n\\begin{lem}\\label{le:DUEE} Let the perturbed\n  gradient system $(V,\\calE,\\Psi,B)$ satisfy the Assumptions\n  \\textnormal{(2.E), (2.$\\Uppsi$)}, and \n  \\textnormal{(2.B)}.  Furthermore, let $\\widetilde{U}_\\tau,\n  \\overline{U}_\\tau,\\underline{U}_\\tau, \\widehat{U}_\\tau$ and\n  $\\widetilde{\\xi}_\\tau$ be the interpolation functions defined in\n  \\eqref{eq:Approx.tau}-\\eqref{eq:II.26} associated to a fixed initial datum $u_0\\in D$ and a\n  step size $\\tau>0$.\\\\ Then, the discrete upper energy estimate\n\\begin{align}\n\\label{eq: DUEE}\n  \\calE_{\\overline{\\mathbf{t}}_\\tau(t)}(\\overline{U}_\\tau(t)) + \\int_{\\overline{\\mathbf{t}}_\\tau(s)}^{\\overline{\\mathbf{t}}_\\tau(t)}\\left(\n    \\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) +\n    \\Psi^*_{\\underline{U}_\\tau(r)}\\left(\n      B(\\underline{\\mathbf{t}}_\\tau(r),\\underline{U}_\\tau(r))-\n      \\widetilde{\\xi}_\\tau(r)\\right) \\right) \\dd r\n   \\notag \\\\ \n  \\leq\n  \\calE_{\\overline{\\mathbf{t}}_\\tau(s)}(\\overline{U}_\\tau(s))+\\int_{\\overline{\\mathbf{t}}_\\tau(s)}^{\\overline{\\mathbf{t}}_\\tau(t)} \\partial_r\n  \\calE_r(\\widetilde{U}_\\tau(r))\\dd r+\\int_{\\overline{\\mathbf{t}}_\\tau(s)}^{\\overline{\\mathbf{t}}_\\tau(t)}\n  \\langle B(\\underline{\\mathbf{t}}_\\tau(r),\\underline{U}_\\tau(r)), \\widehat{U}'_\\tau\n  (r) \\rangle \\dd r\n\\end{align} holds for all $0\\leq s< t\\leq T$. Moreover, there exist positive constants $M,\\tau^*>0$ such that the estimates \n\\begin{align}\n\\label{eq:II.29}\n\\sup_{t\\in (0,T)} \\calE_t(\\overline{(U}_\\tau(t)) \\leq M,\\quad \\sup_{t\\in (0,T)} \\calE_t(\\widetilde{(U}_\\tau(t))\\leq M,\\quad \\sup_{t\\in (0,T)}\\vert \\partial_t \\calE_t(\\widetilde{(U}_\\tau(t))\\vert \\leq M \\\\ \n\\label{eq:II.30}\n\\int_0^T \\left( \\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) + \\Psi^*_{\\underline{U}_\\tau(r)}\\left( B(\\underline{\\mathbf{t}}_\\tau(r),\\underline{U}_\\tau(r))- \\widetilde{\\xi}_\\tau(r)\\right) \\right) \\dd r\\leq M\n\\end{align} hold for all $0<\\tau\\leq \\tau^*$. Besides, the families $(\\widehat{U}'_\\tau)_{0<\\tau\\leq \\tau^*}\\subset \\rmL^1(0,T;V)$ as well as $(B(\\underline{\\mathbf{t}}_\\tau,\\underline{U}_\\tau))_{0<\\tau\\leq \\tau^*}\\subset \\rmL^1(0,T;V^*)$ and  $(\\widetilde{\\xi}_\\tau)_{0<\\tau\\leq \\tau^*} \\subset \\rmL^1(0,T;V^*)$ are integrable uniformly with respect to $\\tau$ in the respective spaces. Finally, there holds\n\\begin{align}\n\\label{eq:II.31}\n\\Vert \\underline{U}_\\tau-\\overline{U}_\\tau\\Vert_{\\infty}+\\Vert \\widehat{U}_\\tau-\\overline{U}_\\tau\\Vert_{\\infty}\n+\\Vert \\widetilde{U}_\\tau-\\underline{U}_\\tau\\Vert_{\\infty} \\rightarrow 0\n\\end{align} as $\\tau \\rightarrow 0$.\n\\end{lem}\n\\begin{proof}\nIn order to show the discrete upper energy estimate \\eqref{eq: DUEE}, it is sufficient to restrict ourselves to the case $s=t_{n-1}$ and $t=t_n$ for $n\\in n=1,\\dots, N$. The general case follows by summing up the particular inequalities on the subintervals. But this case follows from \\eqref{eq:II.17} in Lemma \\ref{le:Main.Lem} by choosing $\\mathrm{t}=t_{n-1},u=U_\\tau^{n-1}, r_0=t-t_{n-1}, u_{r_0}=\\widetilde{U}_\\tau(t),  u_{r}=\\widetilde{U}_\\tau(t_{n-1}+r)$ and $\\xi_r=\\widetilde{\\xi}_\\tau(t_{n-1}+r)$, where we chose $t\\in(t_{n-1},t_n]$ to be fixed. Then, we find \n\\begin{align}\n\\label{eq:II.32}\n(t-t_{n-1})\\Psi_{U_\\tau^{n-1}}\\left( \\frac{\\widetilde{U}_\\tau(t)-U_\\tau^{n-1}}{t-t_{n-1}} \\right)+\\int_{t_{n-1}}^t \\Psi_{U_\\tau^{n-1}} \\left( B(t_{n-1},U_\\tau^{n-1})- \\widetilde{\\xi}_\\tau(r)\\right)\\dd r+\\calE_t(\\widetilde{U}_\\tau(t)) \\notag \\\\\n\\leq \\calE_{t_{n-1}}(U_\\tau^{n-1})+\\int_{t_{n-1}}^{t} \\partial_r \\calE_r(\\widetilde{U}_\\tau(r))\\dd r+ \\langle B(t_{n-1},U_\\tau^{n-1}), U_\\tau^n-U_\\tau^{n-1} \\rangle.\n\\end{align} By choosing $t=t_n$, we obtain\n\\begin{align}\n\\label{eq:II.33}\n\\int_{t_{n-1}}^{t_n}\\left( \\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) + \\Psi^*_{\\underline{U}_\\tau(r)}\\left( B(t_{n-1},\\underline{U}_\\tau(r))- \\widetilde{\\xi}_\\tau(r)\\right) \\right) \\dd r +\\calE_{t_n}(\\overline{U}_\\tau(t_n))\\notag \\\\\n\\leq \\calE_{t_{n-1}}(\\underline{U}_\\tau(t_{n-1}))+\\int_{t_{n-1}}^{t_n} \\partial_r \\calE_r(\\widetilde{U}_\\tau(r))\\dd r+\\int_{t_{n-1}}^{t_n}\\langle B(t_{n-1},\\underline{U}_\\tau(r)), \\widehat{U}'_\\tau (r) \\rangle \\dd r\n\\end{align} for all $n=1,\\cdots,N$, which yields the discrete upper energy estimate. Further, we notice that from Assumption \\ref{eq:B.2}, we obtain the estimation\n\\begin{align}\n\\label{eq:II.34}\n&\\int_{t_{n-1}}^{t_n}\\langle B(t_{n-1},\\underline{U}_\\tau(r)), \\widehat{U}'_\\tau (r) \\rangle \\dd r \\notag\\\\\n &\\leq c \\int_{t_{n-1}}^{t_n}\\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) \\dd r+c \\int_{t_{n-1}}^{t_n} \\Psi^*_{\\underline{U}_\\tau(r)}\\left( \\frac{B(t_{n-1},\\underline{U}_\\tau(r))}{c}\\right) \\dd r \\notag \\\\\n&\\leq c \\int_{t_{n-1}}^{t_n}\\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) \\dd r+\\tau \\beta(1+  \\calE_{t_{n-1}}(U_\\tau^{n-1}))\\notag \\\\\n&\\leq c \\int_{t_{n-1}}^{t_n}\\Psi_{\\underline{U}_\\tau(r)}\\left( \\widehat{U}'_\\tau(r)\\right) \\dd r+ \\tau \\beta  (1+G(U_\\tau^{n-1})),\n\\end{align} where we used also the \\textsc{Fenchel-Young} inequality. Since $c\\in(0,1)$, inequality \\eqref{eq:II.33} and \\eqref{eq:II.34} together yield the estimation\n\\begin{align}\n\\calE_{t_n}(U_\\tau^n) \n&\\leq \\calE_{t_{n-1}}(U_\\tau^{n-1})+\\int_{t_{n-1}}^{t_n} \\partial_r \\calE_r(\\widetilde{U}_\\tau(r))\\dd r+ \\tau \\beta(1+ G(U_\\tau^{n-1}))\\notag\\\\\n&\\leq  \\calE_{t_{n-1}}(U_\\tau^{n-1})+ \\tau \\beta \n (1+G(U_\\tau^{n-1}))+C\\widetilde{C}\\int_{t_{n-1}}^{t_n} G(U_\\tau^{n-1})\\dd r \\notag\\\\\n \\label{eq:II.35}\n&+  \\int_{t_{n-1}}^{t_n}(r-t_{n-1})\\Psi^*_{U_\\tau^{n-1}}(B(t_{n-1},U_\\tau^{n-1}))\\dd r \\\\ \n&\\leq  \\calE_{t_{n-1}}(U_\\tau^{n-1})+ \\tau \\beta \n (1+G(U_\\tau^{n-1}))+C\\widetilde{C}\\int_{t_{n-1}}^{t_n} G(U_\\tau^{n-1})\\dd r \\notag\\\\\n \\label{eq:II.36}\n&+  \\int_{t_{n-1}}^{t_n} c \\tau\\Psi^*_{\\underline{U}_\\tau(r)}\\left( \\frac{B(t_{n-1},\\underline{U}_\\tau(r))}{c}\\right) \\dd r  \\\\\n&\\leq  \\calE_{t_{n-1}}(U_\\tau^{n-1})+ \\tau \\beta(1+  G(U_\\tau^{n-1}))+C\\widetilde{C}\\tau G(U_\\tau^{n-1})\\notag \\\\\n&+  \\tau \\beta (1+G(U_\\tau^{n-1}))\\notag \\\\\n\\label{eq:II.37}\n&= \\calE_{t_{n-1}}(U_\\tau^{n-1})+\\tau (2\\beta  +C\\widetilde{C}) G(U_\\tau^{n-1})+2\\tau\\beta\n\\end{align} for all $n=1,\\dots, N$ and $0<\\tau\\leq1 $, where we used in \\eqref{eq:II.35} the inequality \n\\begin{align*}\nG(\\widetilde{U}_\\tau(t))\\leq \\widetilde{C}(G(U_\\tau^{n-1})+(t-t_{n-1})\\Psi_{U_\\tau^{n-1}}^*(B(t_{n-1},U_\\tau^{n-1}))), \\quad t\\in(t_{n-1},t_n],\n\\end{align*}\n from Lemma \\ref{le:Main.Lem} and in \\eqref{eq:II.36} the fact that the map $r\\mapsto  r\\Psi^*_u\\left(\\frac{\\xi}{r}\\right)$ is non-decreasing on $(0,+\\infty)$ for every $\\xi\\in V^*$. Defining $A:=(2\\beta  +C\\widetilde{C})$ and summing up the inequalities \\eqref{eq:II.37},  we obtain\n\\begin{align}\n\\label{eq:II.38}\nG(U_\\tau^n)\\leq C_1 \\calE_{t_n}(U_\\tau^n) \\leq C_1 \\calE_0(u_0)+2C_1T\\beta+ \\tau C_1A\n\\sum_{k=1}^n G(U_\\tau^{k-1}) \n\\end{align}  for all $n=1,\\dots, N$ and $0<\\tau\\leq1 $. Then, applying the discrete version of the \\textsc{Gronwall} Lemma to \\eqref{eq:II.38} yields the uniform boundedness of $G(U_\\tau^n)$ for all $n= 1,\\dots, N$ and $0<\\tau<\\min\\lbrace 1, 1\/(2C_1A)\\rbrace=:\\tau^*$, from what we deduce\n\\begin{align}\n\\label{eq:II.39}\n\\sup_{t\\in(0,T)} \\calE_t(\\overline{U}_\\tau(t)) \\leq C_1\\quad \\text{for all } 0<\\tau<\\tau^*\n\\end{align} for a positive constant $C_1>0$ independent from $\\tau$. Taking into account the inequality \\eqref{eq:II.37} and the Assumptions \\ref{eq:B.2} and \\ref{eq:cond.E.3}, we also obtain the last two inequalities in \\eqref{eq:II.29}. By employing \\eqref{eq:II.33} and \\eqref{eq:II.34}, and arguing as before, we also get \\eqref{eq:II.30}. The constant $M$ can be chosen by the sum of all constants obtained from the shown inequalities of this lemma. Further, the uniform integrability of $(\\widehat{U}'_\\tau)_{0<\\tau\\leq \\tau^*}$ as well as $(B(\\underline{\\mathbf{t}}_\\tau,\\underline{U}_\\tau))_{0<\\tau\\leq \\tau^*}$ and  $(\\widetilde{\\xi}_\\tau)_{0<\\tau\\leq \\tau^*}$ in $\\rmL^1(0,T;V)$ and $\\rmL^1(0,T;V^*)$, respectively, \nfollows from the superlinear growth of $\\Psi_u$ and $\\Psi^*_u$ (Assumption \\ref{eq:Psi.2}), inequality \\eqref{eq:II.30} and the growth condition \\ref{eq:B.2}. To clarify this, let $\\varepsilon>0$ and $\\widetilde{M}:=\\max\\lbrace \\beta(1+M),M\\rbrace$ be given, where $M$ is the constant obtained from the boundedness in \\eqref{eq:II.29} and \\eqref{eq:II.30}. Then, by Assumption \\ref{eq:Psi.2} there exists for $M$ and  $\\widetilde{M}\/\\varepsilon$ positive numbers $K_1,K_2$, such that \n\\begin{align}\n\\label{eq:II.40}\n\\Psi_u(v)\\geq \\frac{\\widetilde{M}}{\\varepsilon}\\Vert v\\Vert \\quad \\text{ and }\\quad \\Psi_{u}^*(\\eta)\\geq \\frac{\\widetilde{M}}{\\varepsilon}\\Vert \\eta \\Vert_* \n\\end{align} for all $v\\in V$ with $\\Vert v\\Vert\\geq K_1$, all $\\eta \\in V^*$ with $\\Vert \\eta \\Vert_*\\geq K_2$ and all $u\\in D$ with $G(u)\\leq M$, \nFor notational convenience, we define $f_\\tau: [0,T]\\rightarrow V$, $g_\\tau: [0,T]\\rightarrow V^*$ and $h_\\tau: [0,T]\\rightarrow V^*$ by $f_\\tau(t):= \\widehat{U}'_\\tau(t)$, $g_\\tau(t):= B(\\underline{\\mathbf{t}}_\\tau(t),\\underline{U}_\\tau(t))$ and $h_\\tau(t):= (B(\\underline{\\mathbf{t}}_\\tau(t),\\underline{U}_\\tau(t))-\\widetilde{\\xi}_\\tau(t))$ for all $t\\in [0,T]$. Then, by \\eqref{eq:II.40}, \\eqref{eq:II.29} and \\eqref{eq:II.30} there hold\n\\begin{align*}\n\\int_{\\lbrace t\\in [0,T] : f_\\tau(t)\\geq K_1 \\rbrace} \\Vert f_\\tau(t) \\Vert \\dd t \\leq \\frac{\\varepsilon}{\\widetilde{M}} \\int_{\\lbrace t\\in [0,T] : f_\\tau(t)\\geq K_1 \\rbrace} \\Psi_{\\underline{U}_\\tau(t)}(f_\\tau(t)) \\dd t\\leq \\varepsilon \\\\\n\\int_{\\lbrace t\\in [0,T] : g_\\tau(t)\\geq K_2 \\rbrace} \\Vert g_\\tau(t) \\Vert_* \\dd t \\leq \\frac{\\varepsilon}{\\widetilde{M}} \\int_{\\lbrace t\\in [0,T] : g_\\tau(t)\\geq K_2 \\rbrace} \\Psi^*_{\\underline{U}_\\tau(t)}(g_\\tau(t)) \\dd t\\leq \\varepsilon\\\\\n\\int_{\\lbrace t\\in [0,T] : h_\\tau(t)\\geq K_2 \\rbrace} \\Vert h_\\tau(t) \\Vert_* \\dd t \\leq \\frac{\\varepsilon}{\\widetilde{M}} \\int_{\\lbrace t\\in [0,T] : h_\\tau(t)\\geq K_2 \\rbrace} \\Psi^*_{\\underline{U}_\\tau(t)}(h_\\tau(t)) \\dd t\\leq \\varepsilon\n\\end{align*} for all $0< \\tau \\leq \\tau^*$, which yields the uniform integrability. Since the sum of two uniformly integrable functions is again uniformly integrable, it follows that $(\\widetilde{\\xi}_\\tau)_{0<\\tau\\leq \\tau^*}$ is also uniformly integrable in $\\rmL^1(0,T;V^*)$ with respect to $\\tau>0$. For the last assertion, we first notice that inequality \\eqref{eq:II.32} considering \\eqref{eq:II.29} and \\eqref{eq:II.30} implies\n\\begin{align*}\n\\sup_{t\\in [0,T]}(t-\\underline{\\mathbf{t}}_\\tau(t))\\Psi_{\\underline{U}_\\tau(t)}\\left( \\frac{\\widetilde{U}_\\tau(t)-\\underline{U}_\\tau(t)}{t-\\underline{\\mathbf{t}}_\\tau(t)} \\right)\\leq C_2 .\n\\end{align*} for a constant $C_2>0$. Then, again Assumption \\ref{eq:Psi.2} implies that for every $R>0$ and $\\gamma>0$ there exists $K>0$ such that\n\\begin{align}\n\\label{eq:II.41}\n\\gamma \\Vert \\widetilde{U}_\\tau(t)-\\underline{U}_\\tau(t) \\Vert &\\leq (t-\\underline{\\mathbf{t}}_\\tau(t))\\Psi_{\\underline{U}_\\tau(t)}\\left( \\frac{\\widetilde{U}_\\tau(t)-\\underline{U}_\\tau(t)}{t-\\underline{\\mathbf{t}}_\\tau(t)} \\right)+ (t-\\underline{\\mathbf{t}}_\\tau(t))\\gamma K \\notag \\\\\n&\\leq M+\\tau \\gamma K \\quad \\text{ for all } t\\in [0,T] \\text{ and all } 0<\\tau<\\tau^*.\n\\end{align} \nTaking the supremum of the left hand side over all\n$t\\in[0,T]$ and taking then the limes superior as $\\tau \\rightarrow\n0$, we obtain\n\\begin{align}\n\\label{eq:II.42}\n  \\gamma \\limsup_{\\tau \\rightarrow 0}\\sup_{t\\in[0,T]}\\Vert\n  \\widetilde{U}_\\tau(t)-\\underline{U}_\\tau(t) \\Vert \\leq M,\n\\end{align} \nfor any $\\gamma>0$, which implies necessarily $\\lim_{\\tau \\rightarrow\n  0}\\sup_{t\\in[0,T]}\\Vert \\widetilde{U}_\\tau(t)-\\underline{U}_\\tau(t)\n\\Vert=0$. Since \\eqref{eq:II.42} holds for every $t\\in [0,T]$, it is\nparticularly satisfied for $t=t_n$, $n=1,\\dots,N$, so that we also get\n$\\lim_{\\tau \\rightarrow 0}\\sup_{t\\in[0,T]}\\Vert\n\\overline{U}_\\tau(t)-\\underline{U}_\\tau(t) \\Vert=0$. The latter\nconvergence in turn implies finally $\\lim_{\\tau \\rightarrow\n  0}\\sup_{t\\in[0,T]}\\Vert \\widehat{U}_\\tau(t)\n-\\overline{U}_\\tau(t)\\Vert=0$ which completes the proof.\n\\end{proof} \n\n\n\\subsection{Limit passage and completion of the proof}\n\\label{su:Proof}\n\nThe next step in constructing a solution to our\n\\textsc{Cauchy}-Problem relies on compactness arguments in order to\nshow the existence of a limit function, which obeys the differential\ninclusion (1.1) and satisfies the initial datum. For this, it is\nnatural to make use of the fact that the interpolation functions are\ncontained in a sublevel set of the energy functional, which by\nhypothesis is compact. We elaborate on this in the following result,\nwhich provides also the characterization of the limit function by\n\\textsc{Young} measures.\n\n\n\\begin{lem} \\label{le:LimitPass} Under the same assumptions of Lemma\n  \\ref{le:Main.Lem}, \n let $u_0 \\in D$ and $(\\tau_n)_{n \\in \\mathbb{N}}$ be a\n  vanishing sequence of positive real numbers. Then, there exists a\n  subsequence $(\\tau_{n_k})_{k \\in \\mathbb{N}}$, a absolutely\n  continuous curve $u\\in \\mathrm{AC}([0,T];V)$ with $u(0)=u_0$, an integrable\n  function $\\widetilde{\\xi}\\in \\rmL^1(0,T;V^*)$, a function\n  $\\mathscr{E}:[0,T]\\rightarrow \\mathbb{R}$ of bounded variation, an\n  essentially bounded function $\\mathscr{P}\\in \\rmL^{\\infty}(0,T)$,\n  and a time-depended \\textsc{Young} measure\n  $\\mathbold{\\mu}=(\\mu_t)_{t\\in[0,T]}\\in \\mathscr{Y}(0,T;V\\times\n  V^*\\times \\mathbb{R})$, such that\n\\begin{subequations}\n\\label{eq:LP.all}\n\\begin{align}\n\\label{eq:LP.u}\n\\overline{U}_{\\tau_{n_k}},\\underline{U}_{\\tau_{n_k}},\\widetilde{U}_{\\tau_{n_k}},\\widehat{U}_{\\tau_{n_k}} \\rightarrow u \\quad \\text{in } &\\rmL^{\\infty}(0,T;V),\\\\\n\\label{eq:LP.ud}\n\\widehat{U}'_{\\tau_{n_k}}\\rightharpoonup u' \\quad \\text{in } &\\rmL^1(0,T;V),\\\\\n\\label{eq:LP.xi}\n\\widetilde{\\xi}_{\\tau_{n_k}}\\rightharpoonup \\widetilde{\\xi} \\quad \\text{in } &\\rmL^1(0,T;V^*),\\\\\n\\label{eq:LP.B}\nB(\\underline{\\mathbf{t}}_{\\tau_{n_k}},\\underline{U}_{\\tau_{n_k}})\\rightarrow B(\\cdot,u(\\cdot))\\quad \\text{in } &\\rmL^{\\infty}(0,T;V^*),\\\\\n\\label{eq:LP.Ed}\n\\partial_t \\calE_t(\\widetilde{U}_{\\tau_{n_k}}(t))\\rightharpoonup^* \\mathscr{P} \\quad \\text{in } &\\rmL^{\\infty}(0,T),\n\\end{align}\n\\end{subequations} \nand \n\\begin{align}\n\\begin{cases}\n\\label{eq:LP.E.ptw}\n\\calE_t(\\overline{U}_{\\tau_{n_k}}(t))\\rightarrow \\mathscr{E}(t) \\quad &\\text{for all }t\\in[0,T], \\quad \\calE_0(u_0)=\\mathscr{E}(0),\\\\\n\\calE_t(u(t))\\leq \\mathscr{E}(t)\\quad &\\text{for all }t\\in [0,T]\\\\\n\\calE_t(u(t))= \\mathscr{E}(t) \\quad &\\text{for a.a. } t\\in (0,T),\n\\end{cases}\n\\end{align} \nas $k\\rightarrow \\infty$. Furthermore, there holds\n\\begin{subequations}\n\\label{eq:YM.all}\n\\begin{align}\n\\label{eq:YM.ud}\nu'(t)&= \\int_{V\\times V^* \\times \\mathbb{R}} v \\, \\dd \\mu_t(v,\\zeta, p) \\quad \\text{ for a.a. }t\\in[0,T],\\\\ \n\\label{eq:YM.xi}\n\\widetilde{\\xi}(t)&= \\int_{V\\times V^* \\times \\mathbb{R}} \\zeta \\, \\dd \\mu_t(v,\\zeta, p) \\quad \\text{ for a.a. }t\\in[0,T],\\\\ \n\\label{eq:YM.Ed}\n\\mathscr{P}(t)&= \\int_{V\\times V^* \\times \\mathbb{R}} p\\, \\dd \\mu_t(v,\\zeta, p)\\leq \\partial_t \\calE_t(u(t))  \\quad \\text{ for a.a. }t\\in[0,T].\n\\end{align}\n\\end{subequations} \nand the following energy inequality\n\\begin{align}\n\\label{eq:EI}\n\\begin{split}\n&\\int_s^t \\left( \\Psi_{u(r)}(u'(r))+\\Psi^*_{u(r)}(B(r,u(r))-\\widetilde{\\xi}(r))  \\right) \\dd r +\\mathscr{E}(t) \\\\\n&\\leq\\int_s^t \\int_{V\\times V^* \\times \\mathbb{R}} \\left( \\Psi_{u(r)}(v)+\\Psi^*_{u(r)}(B(r,u(r))-\\zeta)  \\right) \\dd \\mu_r(v,\\zeta, p) \\, \\dd r +\\mathscr{E}(t) \\\\\n&\\leq \\mathscr{E}(s)+\\int_s^t \\mathscr{P}(r)\\dd r+ \\int_s^t \\langle B(r,u(r)),u'(r) \\rangle \\dd r\\\\\n&\\leq \\mathscr{E}(s)+\\int_s^t \\partial_r \\calE_r(u(r))\\dd r+ \\int_s^t \\langle B(r,u(r)),u'(r) \\rangle \\dd r\n\\end{split}\n\\end{align} for all $0\\leq s<t\\leq T$.\n\\end{lem}\n\\begin{proof} Let the initial datum $u_0 \\in D$ and the sequence\n$(\\tau_n)_{n \\in \\mathbb{N}}$ of vanishing time steps be given, such\nthat $\\tau_n<\\tau^*$ for all $n\\in \\mathbb{N}$. In order to show the\nexistence of an absolutely continuous function, we employ the\n\\textsc{Arzel\\`a-Ascoli} theorem on the family of continuous functions\n$(\\widehat{U}_{\\tau_n})_{n\\in \\mathbb{N}}\\subset C([0,T];V)$. First,\nwe notice that the uniform integrability of\n$(\\widehat{U}'_{\\tau_n})_{n\\in \\mathbb{N}}$ leads to the\nequicontinuity of $(\\widehat{U}_{\\tau_n})_{n\\in \\mathbb{N}}$. Second,\nthe fact that the set $\\lbrace\n\\overline{U}_{\\tau_n}(t)\\rbrace_{t\\in[0,T]}$ belongs to a sublevel set\nof the energy functional $E$ for all $n\\in \\mathbb{N}$, which by\nAssumption \\ref{eq:cond.E.2} are compact, implies by \\textsc{Mazur}'s\nlemma that the set $\\lbrace\n\\overline{U}_{\\tau_n}(t)\\rbrace_{t\\in[0,T]}$ also belongs for all\n$n\\in\\mathbb{N}$ to an compact subset of $V$. Therefore by\n\\textsc{Arzel\\`a-Ascoli}, there exists a continuous function $u\\in\nC([0,T];V)$ such that $\\Vert \\widehat{U}_{\\tau_n}-u\n\\Vert_{C([0,T];V)}\\rightarrow 0$ as $k\\rightarrow \\infty$ so that in\nparticular $u(0)=u_0$. Then, the convergences in \\eqref{eq:LP.u}\nfollows from those in \\eqref{eq:II.31}.\n\nFurther, from the \\textsc{Dunford-Pettis} theorem, see\ne.g. \\textsc{Dunford} and \\textsc{Schwartz} \\cite[Cor.\\,11,\np.\\,294]{DunSch59LO1}, which can be applied since both $V$ and $V^*$\nare reflexive \\textsc{Banach} spaces, we obtain with the uniform\nintegrability of $(\\widehat{U}'_{\\tau_n})_{n\\in \\mathbb{N}}$ and\n$(\\widetilde{\\xi}_{\\tau_n})_{n\\in \\mathbb{N}}$ in $\\rmL^1(0,T;V)$ and\n$\\rmL^1(0,T;V^*)$, respectively, the existence of a subsequence\n(labeled as before) and weak limits $v\\in \\rmL^1(0,T;V)$ and\n$\\widetilde{\\xi} \\in \\rmL^1(0,T;V^*)$ such that\n$\\widehat{U}'_{\\tau_n}\\rightharpoonup v$ weakly in $\\rmL^1(0,T;V)$ and\n$\\widetilde{\\xi}_{\\tau_n}\\rightharpoonup \\widetilde{\\xi}$ weakly in\n$\\rmL^1(0,T,V^*)$ as $n\\rightarrow \\infty$. From a well known\nargument, one can identify $v$ as weak derivative of $u$, i.e., $u'=v$\nin the weak sense which yields $u\\in \\rmW^{1,1}(0,T;V)$ and due to\ncontinuity of $u$, $u\\in \\mathrm{AC}([0,T];V)$.\n\nNow, we shall prove the convergence \\eqref{eq:LP.B} of the\nperturbation. We first note that the functions $t\\mapsto B(t,u(t))$\nand $t\\mapsto B(\\underline{\\mathbf{t}}_{\\tau_{n_k}}(t),\\underline{U}_{\\tau_{n_k}}(t))$\nboth belongs to the space $\\rmL^{\\infty}(0,T;V)$, where the\nmeasurability follows from the continuity of $u$ and $B$, and\nAssumptions \\ref{eq:B.1} together with \\eqref{eq:II.55}, respectively,\nwhereas the (essential) boundedness is a consequence of Assumptions\n\\ref{eq:B.2} and \\ref{eq:Psi.2} as well as the a priori\nestimates. Now, since the interpolation functions are contained in a\nsublevel set of the energy functional, uniformly in $\\tau>0$ and for\nall $t\\in(0,T)$, it is also contained in a compact set of $V$,\nuniformly in $\\tau>0$ and for all $t\\in(0,T)$. Therefore, there exists\na compact set $ \\mathcal{K}\\subset V$ such that by\n\\textsc{Tychonoff's} theorem the set $[0,T]\\times \\mathcal{K}$ is\ncompact with respect to the product topology of $[0,T]\\times V$. This,\nin turn implies with Assumption \\ref{eq:B.1} the uniform continuity of\nthe map $(t,u)\\mapsto B(t,u)$ on $[0,T]\\times \\mathcal{K}$. Together\nwith the convergence of\n$(\\underline{\\mathbf{t}}_{\\tau_{n_k}}(t),\\underline{U}_{\\tau_{n_k}}(t))) \\rightarrow\n(t,u(t))$ uniformly in $t\\in(0,T)$, we obtain\n\\begin{align}\n\\label{eq:II.53}\n  \\lim_{n\\rightarrow \\infty}\\sup_{t\\in(0,T)} \\Vert\n  B(\\underline{\\mathbf{t}}_{\\tau_{n_k}}(t),\\underline{U}_{\\tau_{n_k}}(t)) -B(t,u(t))\\Vert_*\n  \\quad \\quad \\text{as } n\\rightarrow \\infty.\n\\end{align} \n\nIn order to show the convergence in \\eqref{eq:LP.Ed}, we notice that\ndue to \\eqref{eq:II.29} there holds $(\\partial_t\n\\calE_t(\\widetilde{U}_{\\tau_{n_k}}))_{k\\in \\mathbb{N}}\\subset\n\\rmL^{\\infty}(0,T)$. Since the \\textsc{Lebesgue }space\n$\\rmL^{\\infty}(0,T)$ is the dual space of a separable \\textsc{Banach}\nspace $\\rmL^1(0,T)$ there exists a limit $\\mathscr{P}\\in\n\\rmL^{\\infty}(0,T)$ such that (up to a subsequence) $\\partial_t\n\\calE_t(\\widetilde{U}_{\\tau_{n_k}}) \\rightharpoonup^* \\mathscr{P}$\nweakly$^*$ in $\\rmL^{\\infty}(0,T)$ as $k\\rightarrow \\infty$.  \\bigskip\n\nNow, we shall prove \\eqref{eq:LP.E.ptw}. For this, we define\n\\begin{align*}\n  \\eta_\\tau(t):= \\calE_{\\overline{\\mathbf{t}}_\\tau(t)}\n  (\\overline{U}_\\tau(t))-\\int_0^{\\overline{\\mathbf{t}}_\\tau(t)} \\partial_r\n  \\calE_r(\\widetilde{U}_\\tau(r))\\dd r-\\int_0^{\\overline{\\mathbf{t}}_\\tau(t)} \\langle\n  B(\\underline{\\mathbf{t}}_\\tau(r),\\underline{U}_\\tau(r)), \\widehat{U}'_\\tau (r) \\rangle\n  \\dd r\n\\end{align*} \nfor $t\\in[0,T]$ and we deduce from the discrete upper energy estimate\n\\eqref{eq: DUEE} that the map $t\\mapsto \\eta_\\tau(t): [0,T]\\rightarrow\n\\mathbb{R}$ is non-increasing. Then, by \\textsc{Helly}s theorem there\nexists a non-increasing function $\\eta:[0,T]\\rightarrow \\mathbb{R}$\nand a subsequence (labeled as before) such that $\\eta_{\\tau_{n_k}}(t)\n\\rightarrow \\eta(t)$ as $k \\rightarrow \\infty$ for all\n$t\\in[0,T]$. Moreover, we define\n\\begin{align*}\n  \\psi_\\tau(t):= \\int_0^{\\overline{\\mathbf{t}}_\\tau(t)} \\langle\n  B(\\underline{\\mathbf{t}}_\\tau(r),\\underline{U}_\\tau(r)), \\widehat{U}'_\\tau (r) \\rangle\n  \\dd r \\quad \\text{ for } t\\in [0,T].\n\\end{align*} \nSince we have strong convergence of the perturbation\n$B(\\underline{\\mathbf{t}}_\\tau,\\underline{U}_{\\tau_{n_k}})$ in $\\rmL^{\\infty}(0,T;V^*)$\nand weak convergence of the derivative $\\widehat{U}'_{\\tau_{n_k}} $ in\n$\\rmL^1(0,T;V) $ as $k\\rightarrow \\infty$, there holds\n\\begin{align}\n\\label{eq:II.54}\n\\psi_{\\tau_{n_k}}(t)\\rightarrow \\psi(t):=\\int_0^t \\langle B(r,u(r)),\nu'(r) \\rangle \\dd r \\quad \\text{ as } k\\rightarrow \\infty\n\\end{align} \nfor all $t\\in[0,T]$. Considering convergence \\eqref{eq:LP.Ed}, we obtain \n\\begin{align*}\n  \\calE_{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)}(\\overline{U}_{\\tau_{n_k}}(t))\\rightarrow\n  \\mathscr{E}(t):= \\eta(t)+\\int_0^t \\mathscr{P}(r)\\dd r +\\psi(t) \\quad\n  \\text{for all } t\\in[0,T]\n\\end{align*} \nas $k\\rightarrow \\infty$. Since the function $\\eta$ is\nmonotone and both the function $\\psi$ and the map $t\\mapsto \\int_0^t\n\\mathscr{P}(r)\\dd r$ are absolutely continuous, it follows that the\nfunction $\\mathscr{E}$ is of bounded variation. In order to conclude\nthe convergence in \\eqref{eq:LP.E.ptw}, we notice that\n\\begin{align*}\n  \\vert \\calE_{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)}(\\overline{U}_{\\tau_{n_k}}(t)) -\n  \\calE_t(\\overline{U}_{\\tau_{n_k}}(t))\\vert \\rightarrow 0 \\quad\n  \\text{as } k\\rightarrow \\infty\n\\end{align*} \nwhich follows from \\eqref{eq:II.5}, \\eqref{eq:II.29} and the fact that\n$\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)\\rightarrow t$ as $k \\rightarrow \\infty$ for all\n$t\\in[0,T]$. Further, by the lower semicontinuity of the energy\nfunctional, we obtain due to the convergence \\eqref{eq:II.31}\n\\begin{align}\n\\label{eq:II.55}\n\\calE_t(u(t))\\leq \\liminf\n\\calE_t(\\overline{U}_{\\tau_{n_k}}(t))=\\mathscr{E}(t)\\leq M \\quad\n\\text{for all } t\\in[0,T],\n\\end{align} \nwhere the last inequality follows from \\eqref{eq:II.29}. The last\nassertion in \\eqref{eq:LP.E.ptw} follows from Assumption\n\\ref{eq:cond.E.5}.\n\\medskip\n\nWe continue by showing \\eqref{eq:YM.all}. For this\npurpose, we define the (reflexive) \\textsc{Banach} space\n$\\mathcal{V}:=V\\times V^*\\times \\mathbb{R}$ endowed with the product\ntopology space and employ the fundamental theorem of weak topologies\n(Theorem \\ref{th:A2}) applied to the sequence\n$w_k:=(\\widehat{U}'_{\\tau_{n_k}},\n\\widetilde{\\xi}_{\\tau_{n_k}}, \\partial_t \\calE_t\n(\\widetilde{U}_{\\tau_{n_k}}) )_{k\\in \\mathbb{N}}$ which belongs to\n$\\rmL^1(0,T;\\mathcal{V})$ by the a priori estimates, and is uniformly\nintegrable in $\\rmL^1(0,T;\\mathcal{V})$ since every component is in\nthe respective space. Thus, there exists a \\textsc{Young}-measure\n$\\mathbold{\\mu}=(\\mu_t)_{t\\in[0,T]}\\in \\mathscr{Y}(0,T;V\\times\nV^*\\times \\mathbb{R})$ such that $\\mu_t$ is for almost everywhere\n$t\\in(0,T)$ concentrated on the set\n\\begin{align*}\n  \\mathrm{Li}(t):=\\bigcap_{p=1}^\\infty \\mathrm{clos_{weak}}\n  \\big(\\lbrace w_k(t) : k\\geq p \\rbrace \\big)\n\\end{align*}\nof all limit points of $w_k(t)$ with respect to the weak-weak-strong\ntopology of $V\\times V^*\\times \\mathbb{R}$, i.e.\\\n$\\mathrm{sppt}(\\mu_t) \\subset \\mathrm{Li}(t) $. Since the weak limits\nin \\eqref{eq:LP.ud}, \\eqref{eq:LP.xi} and \\eqref{eq:LP.Ed} are unique,\nthe identities in \\eqref{eq:YM.ud} and \\eqref{eq:YM.xi} are direct\nconsequences of the fundamental theorem of weak topologies, whereas\nthe inequality in \\eqref{eq:YM.Ed} is true due to the fact that for\nalmost every $t\\in (0,T)$, there holds\n\\begin{align}\n\\label{eq:II.56}\n\\zeta \\in \\partial \\calE_t(u(t)) \\quad \\text{and} \\quad\np\\leq \\partial_t \\calE_t(u(t)) \\quad \\text{for all } (v,\\zeta,p)\\in\n\\mathrm{Li}(t).\n\\end{align} \nProperty \\eqref{eq:II.56} in turn follows from Assumption\n\\ref{eq:cond.E.5} with the convergences in\n\\eqref{eq:LP.u}{eq:LP.all} and \\eqref{eq:LP.E.ptw} as well as the inclusion\n\\eqref{eq:II.26}: Let $\\mathcal{N}\\subset (0,T)$ a negligible set such\nthat for all $t\\in (0,T)\\backslash \\mathcal{N}$ the set\n$\\mathrm{Li}(t)$ is non-empty. Now let $t\\in t\\in (0,T)\\backslash\n\\mathcal{N}$ and $(v,\\zeta,p)\\in \\mathrm{Li}(t)$, then there exists a\nsubsequence $(k_l)_{l\\in \\mathbb{N}}$ such that\n$\\widehat{U}'_{\\tau_{n_{k_l}}}(t) \\rightharpoonup v,\\,\n\\widetilde{\\xi}_{\\tau_{n_{k_l}}}(t)\\rightharpoonup^* \\zeta$ and\n$\\partial_t \\calE_t(\\widetilde{U}_{\\tau_{n_{k_l}}}(t))\\rightarrow p$\nas $l\\rightarrow \\infty$, where the latter convergence follows from\nthe fact that in finite dimensional spaces the weak topology coincides\nwith the strong topology. In view of convergence \\eqref{eq:LP.u} and\nthe inclusion \\eqref{eq:II.26}, \\eqref{eq:II.56} follows by Assumption\n\\ref{eq:cond.E.5}. Integrating the inequality in \\eqref{eq:II.56} with\nrespect to the \\textsc{Borel} probability measure yields\n\\eqref{eq:YM.Ed}. In order to show the energy inequality\n\\eqref{eq:EI}, we notice first of all that from \\textsc{Jensen}'s\ninequality, we obtain for almost every $t\\in(0,T)$\n\\begin{align}\n\\label{eq:JI.psi}\n\\Psi_{u(t)}(u'(t))&\\leq \\int_{V\\times V^*\\times \\mathbb{R}} \\Psi_{u(t)}(v)\\dd \\mu_t(v,\\zeta,p),\\\\\n\\label{eq:JI.psi*}\n\\Psi^*_{u(t)}(B(t,u(t))-\\widetilde{\\xi}(t)))&\\leq \\int_{V\\times V^*\\times \\mathbb{R}} \\Psi^*_{u(t)}(B(t,u(t))-\\zeta)\\dd \\mu_t(v,\\zeta,p).\n\\end{align} \nThis can also be obtained by integrating the inequalities\n\\begin{align*}\n  \\Psi_{u(t)}(u'(t))&\\leq \\Psi_{u(t)}(v)+\\langle w^*,u'(t)-v\\rangle\n  \\quad \\text{for all }v\\in V \\\\\n  \\Psi^*_{u(t)}(B(t,u(t))-\\widetilde{\\xi}(t))&\\leq\n  \\Psi^*_{u(t)}(B(t,u(t))-\\zeta)+\\langle \\zeta-\\widetilde{\\xi}(t)\n  ,w\\rangle \\quad \\text{for all }\\zeta \\in V^*\n\\end{align*} \nusing the identities in \\eqref{eq:YM.all} as well as\nthe fact that $w^*\\in \\partial \\Psi_{u(t)}(u'(t))\\neq \\emptyset$ and\n$w\\in \\partial \\Psi^*_{u(t)}(B(t,u(t))-\\widetilde{\\xi}(t))\\neq\n\\emptyset$, see Remark \\ref{re:Assump.Psi} i).\n\\bigskip\n\nDefining $ \\mathcal{H}_k:[0,T]\\times \\mathcal{V}\\rightarrow \\mathbb{R}$ by\n\\begin{align*}\n\\mathcal{H}_k(r,w):=\\chi_{[\\overline{\\mathbf{t}}_{\\tau_{n_k}}(s),\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)]} \\Psi_{\\underline{U}_{\\tau_{n_k}}(r)}(v), \\quad (r,v,\\zeta,p)\\in [0,T]\\times \\mathcal{V},\n\\end{align*} together with \\eqref{eq:II.29} and \\eqref{eq:LP.u}, the \\textsc{Mosco} continuity \\ref{eq:Psi.3} leads to \n\\begin{align}\n\\label{eq:II.59}\n\\mathcal{H}(r,w):=\\chi_{[s,t]}\\Psi_{u(r)}(v)\\leq \\liminf_{k\\rightarrow \\infty} \\mathcal{H}_k(r,w_n),\n\\end{align} for all $(r,w)=(r,v,\\zeta,p)\\in [0,T]\\times \\mathcal{V}$ and all weak convergent sequences $w_k\\rightharpoonup w\\in \\mathcal{V}$, where $s,t \\in [0,T]$ with $s\\leq t$ are chosen to be fixed. As the space \\textsc{Banach} space $\\mathcal{V}$ is reflexive, the map\n\\begin{align*}\n(v,\\zeta,p) \\mapsto (\\Vert v \\Vert+\\Vert \\zeta \\Vert_{*}+\\vert p \\vert)\n\\end{align*} has compact sublevel sets with respect to the weak\ntopology of $\\mathcal{V}$. Together with the boundedness of the\nafore-defined sequence $(w_k)_{k\\in \\mathbb{N}}$, which follows from\n\\eqref{eq:LP.all}, we obtain the weak-tightness of $(w_k)_{k\\in\n  \\mathbb{N}}$. Therefore, for a subsequence of $(n_k)_{k\\in\n  \\mathbb{N}}$ (not relabeled), Theorem \\ref{th:A1} provides the\ninequality\n\\begin{align*}\n\\int_0^T \\int_\\mathcal{V} \\mathcal{H}(r,w) \\dd \\mu_r(w)\\dd r \\leq \\liminf_{k\\rightarrow \\infty} \\int_0^T  \\mathcal{H}_k(r,w_k) \\dd r, \n\\end{align*} i.e.,\n\\begin{align}\n\\label{eq:LI.psi}\n\\int_s^t \\int_{\\mathcal{V}}\\Psi_{u(r)}(v)\\dd \\mu(v,\\zeta,p) \\dd r\\leq \\liminf_{k\\rightarrow \\infty} \\int_{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(s)}^{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)} \\Psi_{\\underline{U}_{\\tau_{n_k}}(r)}(\\widehat{U}'_{\\tau_{n_k}}(r))\\dd r<+\\infty,\n\\end{align} where the boundedness follows from the a priori estimate \\eqref{eq:II.30}. Taking into account Remark \\ref{re:Assump.Psi}  $iii)$, then Theorem \\ref{th:A1} applied to the function\n\\begin{align*}\n\\mathcal{H}^*_k(r,w):=\\chi_{[\\overline{\\mathbf{t}}_{\\tau_{n_k}}(s),\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)]} \\Psi^*_{\\underline{U}_{\\tau_{n_k}}(r)}(B(\\underline{\\mathbf{t}}_{\\tau_{n_k}}(r),\\underline{U}_{\\tau_{n_k}}(r))-\\zeta), \\quad (r,v,\\zeta,p)\\in [0,T]\\times \\mathcal{V},\n\\end{align*} yields \n\\begin{align}\n\\label{eq:LI.psi*}\n\\begin{split}\n&\\int_s^t \\int_\\mathcal{V}\\Psi^*_{u(t)}(B(r,u(r))-\\zeta)\\dd \\mu(v,\\zeta,p) \\dd r\\\\\n&\\leq \\liminf_{k\\rightarrow \\infty} \\int_{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(s)}^{\\overline{\\mathbf{t}}_{\\tau_{n_k}}(t)} \\Psi^*_{\\underline{U}_{\\tau_{n_k}}(r)}(B(\\underline{\\mathbf{t}}_{\\tau_{n_k}}(r),\\underline{U}_{\\tau_{n_k}}(r))-\\widetilde{\\xi}_{\\tau_{n_k}}(r))\\dd r<+\\infty,\n\\end{split}\n\\end{align} where again the boundedness follows from \\eqref{eq:II.30}.\nIntegrating \\eqref{eq:JI.psi} and \\eqref{eq:JI.psi*} with respect to\n$t$ yields the first inequality in \\eqref{eq:EI}. The second and third\ninequality follow by passing to the limit in the discrete upper energy\nestimate \\eqref{eq: DUEE} as $k\\rightarrow \\infty$ and considering\n\\eqref{eq:LP.Ed}, \\eqref{eq:LP.E.ptw}, \\eqref{eq:YM.Ed},\n\\eqref{eq:II.54}, \\eqref{eq:II.56} as well as \\eqref{eq:LI.psi} and\n\\eqref{eq:LI.psi*}. This proves Lemma \\ref{le:LimitPass}.\n\\end{proof}\n\n We are now ready to complete the proof of our main existence result in\nTheorem \\ref{th:MainExist}. \n\n\\begin{proof}[Proof of Theorem \\ref{th:MainExist}]\n  In order to show that the absolutely continuous curve $ $ $u\\in\n  \\mathrm{AC}([0,T];V)$ obtained from Lemma \\ref{le:LimitPass} is a solution to\n  the differential inclusion \\eqref{eq:I.1}, we make use of the chain\n  rule for \\textsc{Young} measures in Lemma \\ref{le:A3} which is\n  justified by \\eqref{eq:LP.Ed}, \\eqref{eq:YM.ud}, \\eqref{eq:II.56},\n  \\eqref{eq:LI.psi} and \\eqref{eq:LI.psi*}, where\n  $\\mathbold{\\mu}=(\\mu_t)_{t\\in[0,T]}\\in \\mathscr{Y}(0,T;V\\times\n  V^*\\times \\mathbb{R})$ is to be chosen as in Lemma\n  \\ref{le:LimitPass}. Hence by the chain rule condition, the map $t\n  \\mapsto \\calE_t(u(t))$ is absolutely continuous on $(0,T)$ and there\n  holds\n\\begin{align*}\n  \\frac{\\rmd}{\\rmd t}\\calE_t(u(t))\\geq \\int_{V\\times V^*\\times\n    \\mathbb{R}}\\langle \\zeta,u'(t) \\rangle \\dd\n  \\mu_t(v,\\zeta,p)+\\partial_t \\calE_t(u(t))\\quad \\text{for\n    a.a. }t\\in(0,T).\n\\end{align*} \nThus, together with \\eqref{eq:LP.E.ptw}, \\eqref{eq:YM.Ed} and\n\\eqref{eq:EI}, we obtain with $s=0$\n\\begin{align}\n\\label{eq:II.62}\n\\begin{split}\n&\\int_0^t \\int_{V\\times V^* \\times \\mathbb{R}} \\left( \\Psi_{u(r)}(u'(r))+\\Psi^*_{u(r)}(B(r,u(r))-\\zeta)  \\right) \\dd \\mu_r(v,\\zeta, p) \\, \\dd r +\\calE_t(u(t)) \\\\\n&\\leq \\calE_0(u_0)+\\int_0^t  \\partial_r \\calE_r(u(r)) \\dd r+ \\int_0^t \\langle B(r,u(r)),u'(r) \\rangle \\dd r\\\\\n&\\leq \\calE_t(u(t))-\\int_0^t \\int_{V\\times V^* \\times \\mathbb{R}} \\langle  \\zeta,u'(r) \\rangle \\dd \\mu_r(v,\\zeta,p)\\dd r+\\int_0^t \\langle B(r,u(r)),u'(r) \\rangle \\dd r\\\\\n&= \\calE_t(u(t))+\\int_0^t \\int_{V\\times V^* \\times \\mathbb{R}} \\langle B(r,u(r))-\\zeta,u'(r) \\rangle \\dd \\mu_r(v,\\zeta,p) \\dd r \\quad \\text{for all }t\\in[0,T].\n\\end{split}\n\\end{align} Therefore, there holds\n\\begin{align}\n\\label{eq:II.63}\n\\begin{split}\n\\int_0^t \\int_{V\\times V^* \\times \\mathbb{R}} &( \\Psi_{u(r)}(u'(r)) +\\Psi^*_{u(r)}(B(r,u(r))-\\zeta) \\\\  &-\\langle B(r,u(r))-\\zeta,u'(r) \\rangle ) \\dd \\mu_r(v,\\zeta, p) \\dd r \\leq 0 \\quad \\text{for all }t\\in[0,T].\n\\end{split}\n\\end{align} Then, from the \\textsc{Fenchel-Young} inequality we deduce the non-negativity of the integrand in \\eqref{eq:II.63} and infer therefore\n\\begin{align}\n\\label{eq:II.64}\n& \\int_{V\\times V^* \\times \\mathbb{R}} \\left( \\Psi_{u(t)}(u'(t))+\\Psi^*_{u(t)}(B(t,u(t))-\\zeta) -\\langle B(t,u(t))-\\zeta,u'(t) \\rangle \\right) \\dd \\mu_t(v,\\zeta, p)\\notag \\\\\n&=0 \\quad \\text{for a.a. }t\\in(0,T).\n\\end{align} It follows that all inequalities in \\eqref{eq:II.62} become equalities for all $t\\in[0,T]$, so that we obtain the equation\n\\begin{align}\n\\label{eq:II.65}\n\\begin{split}\n&\\int_s^t \\int_{V\\times V^* \\times \\mathbb{R}} \\left( \\Psi_{u(r)}(u'(r))+\\Psi^*_{u(r)}(B(r,u(r))-\\zeta)  \\right) \\dd \\mu_r(v,\\zeta, p) \\, \\dd r +\\calE_t(u(t)) \\\\\n&= \\calE_s(u(s))+\\int_s^t  \\partial_r \\calE_r(u(r)) \\dd r+ \\int_s^t \\langle B(r,u(r)),u'(r) \\rangle  \\dd r \n\\end{split} \n\\end{align} for all $0\\leq s,t\\leq T$. Defining the marginal $\\mathbold{\\nu}=(\\nu_t)_{t\\in[0,T]}:=\\pi^{2,3}_\\#\\mathbold{\\mu}$ of $\\mathbold{\\mu}$ by $\\nu_t(B):=\\mu_t((\\pi^{2,3})^{-1}(B))$ for all $B\\in \\mathscr{B}(V^*\\times \\mathbb{R})$, where $\\pi^{2,3}:V\\times V^*\\times \\mathbb{R}\\rightarrow V^*\\times \\mathbb{R}$ denotes the canonical projection and $\\mathscr{B}(V^*\\times \\mathbb{R})$ the \\textsc{Borel} $\\sigma$-algebra of $V^*\\times \\mathbb{R}$. Setting \n\\begin{align}\n\\label{eq:II.66}\n\\begin{split}\n\\mathcal{S}(t,u(t),u'(t)):=\\lbrace &(\\zeta,p)\\in V^*\\times \\mathbb{R} \\mid \\zeta \\in \\partial\n \\calE_t(u(t)) \\cap (B(t,u(t))-\\partial\\Psi_{u(t)}(u'(t))\\\\\n & \\text{and } p\\leq \\partial_t\\calE_t(u(t)) \\rbrace\n\\end{split}\n\\end{align} we notice that by \\eqref{eq:II.56} and \\eqref{eq:II.64} it follows that $\\nu_t(\\mathcal{S}(t,u(t),u'(t)))=1$ for a.a. $t\\in(0,T)$ and assumption \\eqref{eq:A7} is fulfilled. Therefore, by Lemma \\ref{le:A4} there exists a measurable selections $\\xi:[0,T]\\rightarrow V^*$ and $p:[0,T]\\rightarrow \\mathbb{R}$ with\n\\begin{align}\n\\label{eq:II.67}\n\\int_0^T \\Psi^*_{u(t)}(B(t,u(t))-\\xi(t))\\dd t<+\\infty,\n\\end{align} such that $(\\xi(t),p(t))\\in \\mathcal{S}(t,u(t),u'(t))$ and there holds\n\\begin{align}\n\\label{eq:II.68}\n\\Psi^*_{u(t)}(B(t,u(t))-\\xi(t))-p(t)=\\min_{(\\zeta,p)\\in \\mathcal{S}(t,u(t),u'(t))} \\Psi^*_{u(t)}(B(t,u(t))-\\zeta)-p\n\\end{align} Since \\eqref{eq:II.67} holds and $B(\\cdot,u(\\cdot))\\in \\rmL^{\\infty}(0,T;V^*)$, we deduce from Assumption from the superlinearity of $\\Psi_u^*$ that $ $  $\\xi\\in \\rmL^1(0,T;V^*)$, so that the pair $(u,\\xi)$ solves the differential inclusion \\eqref{eq:I.1} and $u$ satisfies the initial condition $u(0)=u_0$, where the former follows from \\eqref{eq:II.68} and the latter by Lemma \\ref{le:LimitPass}.\\\\  Furthermore, taking into account property \\eqref{eq:II.56} and equation \\eqref{eq:II.64}, then Lemma \\ref{le:Leg.Fen} yields $\\nu_t(\\mathcal{S}(t,u(t),u'(t))=1$ for almost every $t\\in(0,T)$. Thus from equality \\eqref{eq:II.68} and the definition of $\\mathcal{S}(\\cdot,u(\\cdot).u'(\\cdot))$, there holds\n\\begin{align*}\n&\\int_s^t \\Psi^*_{u(r)}(B(r,u(r)-\\xi(r))\\dd r -\\int_s^t p(r)\\dd r\\\\\n& \\leq \\int_s^t \\int_{V\\times V^*\\times \\mathbb{R}} \\Psi^*_{u(r)}(B(r,u(r))-\\zeta)\\dd \\mu_r(v,\\zeta,p)\\dd r - \\int_s^t p(r) \\dd r\n\\end{align*} Now, by comparison with equation \\eqref{eq:II.65}, we infer\n\\begin{align*}\n\\begin{split}\n&\\int_s^t \\left( \\Psi_{u(r)}(u'(r))+\\Psi^*_{u(r)}(B(r,u(r))-\\xi(r))  \\right)\\dd r +\\calE_t(u(t)) \\\\\n&\\leq  \\calE_s(u(s))+\\int_s^t  \\partial_r \\calE_r(u(r)) \\dd r+ \\int_s^t \\langle B(r,u(r)),u'(r) \\rangle  \\dd r \n\\end{split}\n\\end{align*} for all $0\\leq s\\leq t\\leq T$. On the other hand, applying the chain rule condition \\ref{eq:cond.E.4} to the pair $(u,\\xi)$ yields\n\\begin{align*}\n\\frac{\\rmd}{\\rmd t} \\calE_t(u(t))\\geq \\langle \\xi(t),u'(t)\\rangle + \\partial_t \\calE_t(u(t)) \\quad \\text{ for a.e. }t\\in (0,T). \n\\end{align*} Together with the identity\n\\begin{align*}\n\\Psi_{u(r)}(u'(r))+\\Psi^*_{u(r)}(B(r,u(r))-\\xi(r))  \n= \\langle B(r,u(r))-\\xi(r),u'(r) \\rangle  \\quad \\text{ a.e. in} \\in (0,T),\n\\end{align*} which again follows from Lemma \\ref{le:Leg.Fen} and the definition of $\\mathcal{S}(\\cdot,u(\\cdot).u'(\\cdot))$, we conclude the energy-dissipation balance \\eqref{eq:EDB}.\n\\end{proof}\n\\begin{rem}\nIt is not difficult to prove that for every sequence $(\\tau_n)_{n\\in\\mathbb{N}}$ there exists a subsequence (denoting as before) such that the following convergences holds:\n\\begin{align*}\n  \\calE_t(\\overline{U}_{\\tau_n}(t))&\\rightarrow \\calE_t(u(t))  \\quad \\text{for all }t\\in[0,T],\\\\\n  \\int_s^t\n  \\Psi_{\\underline{U}_{\\tau_n}(r)}(\\widehat{U}_{\\tau_n}'(r))\\dd r &\n  \\rightarrow  \\int_s^t\\Psi_{u(r)}(u'(r)) \\dd r \\quad \\text{and }\\\\\n  \\int_s^t\n  \\Psi^*_{\\underline{U}_{\\tau_n}(r)}\n  (B(\\underline{\\mathbf{t}}_{\\tau_n}(r),\\underline{U}_{\\tau_n}(r)) -\\widetilde{\\xi}_{\\tau_n}(r))\n  \\dd r &\\rightarrow \\int_s^t\\Psi^*_{u(r)}(B(r,u(r))-\\xi(r))\\dd r\n\\end{align*} for all $0\\leq s\\leq t\\leq T$ as $n\\rightarrow \\infty$. Furthermore, if we additionally assume that the dissipation potential $\\Psi_u$ and its conjugate $\\Psi^*_u$ are strictly convex for all $u\\in V$, then there holds $\\pi^1_\\# \\mathbold{\\mu}=\\delta_{u'(t)}$ and $\\pi^2_\\# \\mathbold{\\mu}=\\delta_{\\xi(t)}$, respectively, and there holds\n\\begin{align*}\n\\widehat{U}_{\\tau_n}'(t) \\rightharpoonup u'(t) \\quad \\text{and} \\quad \\widetilde{\\xi}_{\\tau_n}(t)\\rightharpoonup \\xi(t) \\quad \\text{for a.a. }t\\in(0,T).\n\\end{align*} as well as $\\widetilde{\\xi}_{\\tau_n}\\rightharpoonup \\xi$ in $\\rmL^1(0,T;V^*)$ as $n\\rightarrow \\infty$.\n\\end{rem}\n\n\n\n\\section{A result for evolutionary $\\Gamma$-convergence}\n\\label{se:EGC}\n\nIn this section we consider a family of perturbed gradients systems\n$\\mathrm{PG}^\\eps:=(V,\\calE^\\eps, \\Psi^\\eps,B^\\eps)$, where $\\eps\\in [0,1]$ is a small\nparameter. Here the case $\\eps=0$ is the supposed limit equation, also\ncalled effective equation. The major question what type of convergence\nof $\\calE^\\eps$, $\\Psi^\\eps$, and $B^\\eps$ is sufficient to conclude\nthat solutions $u_\\eps:[0,T] \\to V$ for $\\mathrm{PG}^\\eps$ with $\\eps>0$ \nhave subsequences $\\eps_k\\to 0$ that convergence\npointwise in $t\\in [0,T]$ to a limit function $u_0:[0,T]\\to V$ and\nthat $u_0$ is indeed a solution for $\\mathrm{PG}^0$. \n\nThe theory developed here follows \\cite[Thm.\\,4.8]{MiRoSa13NADN},\nwhere the case of pure gradient systems (i.e.\\ $B_\\eps \\equiv 0$) was\nconsidered.  \n\n\n\\subsection{Assumptions and results}\n\\label{su:EGC.Ass.Res}\n\n\nOur assumptions follow closely the assumption for the existence theory\nin Section \\ref{su:AssumpExistRes}, where we need uniformity with\nrespect to $\\eps\\in [0,1]$.  For definiteness we now list the precise\nassumptions on $\\mathrm{PG}^\\eps$. For describing energy functionals\n$\\calE^\\eps$ w define the auxiliary \n\\begin{align*}\n& G^\\eps(u)= \\sup\\bigset{ \\calE^\\eps_t(u)}{ t\\in [0,T] }\\\\\n&\\underline{G}(u):=\\inf\\bigset{ \\calE_t^\\eps(u) }{ t\\in[0,T],\\ \\eps \\in\n  [0,1] }.\n\\end{align*}\nWithout loss of generality we may assume that $\\underline{G}$ is bounded\nfrom below by a positive constant $\\gamma>0$.\n{\\renewcommand{\\theequation}{\\thesection.E$^\\eps$}%\n\\begin{subequations}\\label{eq:cond.E.eps}%\n \\begin{align}\n  \\nonumber\n  &\\text{\\textbf{Constant domains.}} \\quad \\forall \\, t\\in[0,T] \\\n    \\forall\\, \\eps \\in [0,1]:\\\\\n  \\nonumber\n  & \\quad \\calE^\\eps_t:V \\rightarrow (0,\\infty] \\text{ is proper and lower\n   semicontinuous with}\\\\\n  &\\quad \\text{time-independent domain } D^\\eps:= \n   \\mathrm{dom}(\\calE^\\eps_t)\\subset V  \\text{ for all } t\\in [0,T].\n  \\label{eq:cond.E.eps.1}\n \\\\\n  \\nonumber \n  &\\text{\\textbf{Equi-compactness of sublevels.}} \\\\\n  &\\quad\\text{The sublevels of } \\underline{G} \\text{ have compact closure in }  V. \n  \\label{eq:cond.E.eps.2}\n  \\\\\n   \\nonumber\n   &\\text{\\textbf{Uniform energetic control of power.}} \\\\ \n   \\nonumber\n   & \\quad \\forall\\, \\eps \\in [0,1] \\ \\forall \\, u\\in D^\\eps: \\quad \n   t\\mapsto \\calE^\\eps_t(u) \\text{ is differentiable on } (0,T)\n   \\text{and } \\\\\n  \\label{eq:cond.E.eps.3} \n  & \\quad \\exists\\, C_T>0\\ \\forall\\, \\eps \\in [0,1]\\ \\forall\\, t\\in (0,T)\\\n    \\forall\\, u \\in D^\\eps:\\quad   \n    \\vert \\partial_t \\calE^\\eps_t(u)\\vert \\leq C_T \\calE^\\eps_t(u) .\n  \\\\[0.4em]\n  \\label{eq:cond.E.eps.4}\n  &\\text{\\textbf{Chain rule.}} \\quad \\forall\\, \\eps \\in [0,1]:\n    \\quad\\text{the chain rule of \\ref{eq:cond.E.4} holds for }\n    (V,\\calE^\\eps,\\Psi^\\eps). \\hspace*{2em} \n  \\\\[0.4em] \n  &\\label{eq:cond.E.epsGamma}\n   \\text{\\textbf{Liminf estimate.}} \\ (\\eps_k,u_k) \\to (0,u) \\text{\n     implies } \\calE^0_t(u)\\leq \\liminf_{k\\to \\infty} \\calE^{\\eps_k}_t(u_k).\n  \\\\[0.4em] \n  &\\nonumber\n    \\text{\\textbf{Strong-weak closedness in the limit }} \\eps\n     \\to 0. \\quad \\text{For all } t\\in[0,T] \\text{ and}\n  \\\\\n  &\\nonumber\n   \\quad \\text{all sequences } (\\eps_n,u_n,\\xi_n)_{n\\in \\mathbb{N}} \n     \\subset [0,1]\\ti V\\ti V^* \\text{ with }\n     \\xi_n\\in \\partial \\calE^{\\eps_n}_t(u_n) \\text{ and } \n  \\\\\n  &\\nonumber\n    \\qquad\\eps_n\\to 0,\\ \\   u_n \\rightarrow u\\in V, \\ \\  \\xi_n \n       \\rightharpoonup \\xi\\in V^*,\n     \\ \\  \\calE^{\\eps_n}_t(u_n)\\rightarrow \\calE_0,   \n     \\ \\  \\partial_t \\calE^{\\eps_n}_t(u_n)\\rightarrow \\calP\n     \\hspace*{-4em}\\mbox{ }\n  \\\\  \n  &\\nonumber \n   \\quad \\text{for $n\\rightarrow\\infty$, we have the relations }\n  \\\\\n  & \\qquad \\xi \\in \\partial \\calE^0_t(u), \\quad  \\calE^0_t(u)=\\calE_0, \\quad \n     \\text{and} \\quad  \\partial_t \\calE^0_t(u)\\geq \\calP. \n   \\label{eq:cond.E.eps.5}\n \\end{align}\n\\end{subequations}\n\\addtocounter{equation}{-1}}\n As in the existence\ntheory we use a control of the time-derivative, see\n\\eqref{eq:cond.E.eps.3}, which gives $\\calE_t^\\eps(u)\\geq\n\\ee^{-C_T|t{-}s|} \\calE^\\eps_s(u)$. Thus, for all $\\eps\\in [0,1]$ and\n$t\\in [0,T]$ we have the relations \n\\[\n\\underline{G}(u) \\leq G^\\eps(u) \\leq \\ee^{C_T T} \\calE^\\eps_t(u) \\leq  \\ee^{C_T\n  T} G^\\eps(u).\n\\]\nNote that we cannot use a uniform upper bound $G^\\eps(u) \\leq \\ol\nG(u)$ as this would exclude many useful results on\n$\\Gamma$-convergence. \n\nIn the present form of condition \\eqref{eq:cond.E.eps.5} we do not ask for the\nstrong-weak closedness for $\\calE^\\eps_t$ with a given positive\n$\\eps$. However, in our main result we simply assume the existence of\nsolutions $u_\\eps:[0,T]\\to V$ for $\\mathrm{PG}^\\eps$. If we want to show this\nwith the theory of Section \\ref{se:ExistResult}, then one\nhas to impose \\ref{eq:cond.E.5} for all $\\eps>0$ as well (which is the same as\nallowing constant sequences $\\eps_n=\\eps$ in \\eqref{eq:cond.E.eps.5}.\n\nThe closedness condition \\eqref{eq:cond.E.eps.5} looks rather strong,\nhowever in Remark \\ref{re:SWClosGamma}, see after the statement of the main\nconvergence result, we will \nshow that convexity of $\\calE^\\eps_t(\\cdot)$ and strong\n$\\Gamma$-convergence to $\\calE^0_t$ already imply the\ndesired closedness.\n\nThe conditions of the dissipation potentials $\\Psi^\\eps_u:V\\to\n[0,\\infty)$ are the following. \n{\\renewcommand{\\theequation}{\\thesection.$\\Uppsi^\\eps$}%\n\\begin{subequations}\n \\label{eq:Psi.eps}\n \\begin{align}\n  \\nonumber\n   &\\textbf{Dissipation potentials.} \\quad \\forall\\, \\eps\\in [0,1]\\\n   \\forall \\, u\\in V: \\\\ \n  \\label{eq:Psi.eps.1}\n   &\\quad \\Psi^\\eps_u:V\\rightarrow [0,\\infty) \\text{ is lower\n     semicontinuous and convex with } \\Psi^\\eps_u(0)=0.\n   \\\\[0.3em]\n  \\nonumber\n   &\\textbf{Superlinearity.} \\quad \\forall\\ R>0 \\\n     \\exists\\, g_R:[0,\\infty)\\to [0,\\infty) \\text{ superlinear}: \\\\\n   \\nonumber\n   &\\quad \\forall\\, \\eps\\in [0,1]\\ \\forall\\, (v,\\xi)\\in V\\ti V^*\\ \n     \\forall\\, u\\in V \\text{ with } \\underline{G}(u)<R:\\\\\n   & \\label{eq:Psi.eps.2}\n     \\quad\\qquad \\Psi^\\eps_u(v)\\geq g_R(\\|v\\|)\n     \\text{ and } \\Psi^{\\eps,*}_u(\\xi) \\geq g_R(\\|\\xi\\|). \n \\\\\n  \\nonumber\n   & \\text{\\textbf{Mosco convergence.} \\ For all $R>0$ and sequences }\n     (\\eps_n,u_n)_{n\\in \\N} \\subset [0,1]\\ti V\\\\\n  \\label{eq:Psi.eps.3}\n   &\\quad  \\text{ with } \\underline{G}(u_n)\\leq R \\text{ and }\n   (\\eps_n,v_n)\\to (0,v): \\quad \n   \\Psi^{\\eps_n}_{u_n} \\xrightarrow{\\,\\rmM\\,} \\Psi^0_u .  \n\\end{align}\n\\end{subequations}\n\\addtocounter{equation}{-1}}\nAgain we have formulated the \\textsc{Mosco} convergence of the dissipation\npotentials only with the limit $\\eps_n\\to 0$, which is sufficient for\nthe limit passage when solutions $u_\\eps:[0,T]\\to V$ are given. To\nshow the existence of solutions we need \\ref{eq:Psi.3} for all $\\eps\\in\n(0,1]$ as well. \n    \nFinally, we impose the conditions of the non-variational perturbation\n$B^\\eps$, namely \n{\\renewcommand{\\theequation}{\\thesection.B$^\\eps$}%\n\\begin{subequations}\n \\label{eq:Beps}\n  \\begin{align} \n  &\\label{eq:Beps.1\n    \\text{\\textbf{Continuity.}  \\ The map }\\left\\{\\ba{ccc} [0,1]\\ti\n        [0,T]\\ti V & \\to & V^*, \\\\ \\\n    (\\eps,t,u) &\\mapsto& \n    B^\\eps (t,u), \\ea \\right. \\text{ is continuous.}\n  \\\\\n  & \\nonumber\n   \\text{\\textbf{Control of $B^\\eps$ by the energy.}} \\quad \\exists\\,\n   C_B>0\\ \\forall\\, (\\eps, t)\\in [0,1]\\ti [0,T] \\\\\n    &\\quad  \\forall\\, u\\in D^\\eps : \\quad \n     \\Psi^{\\eps,*}_u\\big(B^\\eps(t,u)\\big)\\leq C_B \\calE^\\eps_t(u) .\n  \\label{eq:Beps.2\n\\end{align}\n\\end{subequations} \n\\addtocounter{equation}{-1}}\n\nWe are now ready to formulate our result of evolutionary\n$\\Gamma$-converge. In \\cite{Miel16EGCG} the convergence\nwe will established is called ``pE-convergence'' as we have to impose\nthe well-``p''reparedness of the initial conditions $u^0_\\eps$, viz.\\ \n\\begin{equation}\n  \\label{eq:EGC.well}\n  u^0_\\eps \\to u^0 \\text{ in } V \\ \\text{ and } \\\n  \\calE^\\eps_0(u^0_\\eps) \\to \\calE^0_0(u^0)<\\infty \\quad \\text{ for } \\eps\n  \\to 0.\n\\end{equation}\nMoreover, in the sense of \\cite{LMPR17MOGG,DoFrMi17?EGCW} we even have\nthe much stronger notion of EDP convergence, which means convergence\nin the sense of the energy-dissipation balance. Indeed, as for the\nexistence result in Section \\ref{se:ExistResult} we will also strongly\nrely on the energy-dissipation principle and perform the limit\n$\\eps\\to 0$ in the energy-dissipation balance \\eqref{eq:EDB}. Our proof\nwill be an adaptation of \\cite[Thm.\\,4.8]{MiRoSa13NADN}. \n\n\\begin{thm}[Evolutionary $\\Gamma$-convergence] \\label{th:EGC.main}\nAssume that the family $\\mathrm{PG}^\\eps=(V,\\calE^\\eps,\\Psi^\\eps,B^\\eps)$,\n$\\eps\\in [0,1]$ satisfy the assumptions \\eqref{eq:cond.E.eps},\n\\eqref{eq:Psi.eps}, and \\eqref{eq:Beps}.  Moreover, assume that for\n$\\eps>0$ we have solutions $u_\\eps:[0,T]\\to V$ of $\\mathrm{PG}^\\eps$ such that\nthe initial conditions $u_\\eps(0)=u^0_\\eps$ satisfy\n\\eqref{eq:EGC.well}.\nThen, there exists a subsequence $\\eps_k\\to 0$ and a solution\n$u:[0,T]\\to V$ of the limit system $\\mathrm{PG}^0$ with $u(0)=u^0$ such that\nthe following convergences hold:\n\\begin{subequations}\n \\label{eq:EGC.cvg}\n \\begin{align}\n  & \\label{eq:EGC.cvg.a}\n    u_{\\eps_k}(t)\\ \\to \\ u(t)\\  \\text{ in } \\rmC^0([0,T];V); \\\\\n  & \\label{eq:EGC.cvg.b} \n   \\forall\\, t\\in [0,T]: \\quad \\calE^{\\eps_k}_t(u_{\\eps_k}(t)) \\ \\to \\\n    \\calE^0_t(u(t)); \\\\\n  & \\label{eq:EGC.cvg.c} \n    u'_{\\eps_k}\\  \\weak\\  u' \\ \\text{ in } \\rmL^1(0,T;V); \\\\\n  & \\label{eq:EGC.cvg.d} \n    \\forall\\, r<s:\\ \\int_r^s \\Psi^{\\eps_k}_{u_{\\eps_k}(t)} (u'_{\\eps_k}(t)) \\dd t \\\n    \\to \\int_r^s \\Psi^0_{u(t)} (u'(t)) \\dd t; \\\\\n  & \\label{eq:EGC.cvg.e} \\forall\\, r<s:\\ \\int_r^s \\!\\!\\Psi^{\\eps_k,*}_{u_{\\eps_k}(t)}\n    \\big(B^{\\eps_k}(t,u_{\\eps_k}(t)){-}\\xi_{\\eps_k}(t)\\big) \\dd t \\\n    \\to \\int_r^s \\!\\!\\Psi^{0,*}_{u(t)} \\big(B^0(t,u_0(t)){-}\\xi_0(t)\\big) \\dd t, \n \\end{align}\n\\end{subequations}\nwhere $\\xi_\\eps(t)\\in \\pl\\calE^\\eps_t(u_\\eps(t))$ for $\\eps\\in [0,1]$\nand a.a.\\ $t\\in[0,T]$.   \n\\end{thm}\n\nThe proof of this result is contained in the following two Sections\n\\ref{su:EGC.CvgSubseq} and \\ref{su:EGC.LimitPass}. However, we do not\ngive all the details and refer to the full proof of Theorem\n\\ref{th:MainExist} in Section \\ref{se:ExistResult} for the details. \n\n\\begin{rem}[Strong-weak closedness and\n  $\\Gamma$-convergence] \\label{re:SWClosGamma}\nIt is a well-know fact that the strong-weak closedness in the limit\n$\\eps\\to 0$ as assumed in \\eqref{eq:cond.E.eps.5} often follows from\nthe $\\Gamma$-convergence $\\calE^\\eps_t\\xrightarrow{\\Gamma}\n\\calE^0_t$. For the readers convenience we give the argument for the\nconvex case where $\\pl\\calE^\\eps_t(u)$ is simply the convex\nsubdifferential, i.e.\\ \n\\[\n\\pl\\calE^\\eps_t(u)= \\bigset{\\xi \\in V^*}{ \\forall\\, w\\in V:\\\n  \\calE^\\eps_t(w)\\geq \\calE^\\eps_t(u)+ \\langle \\xi,w{-}v\\rangle }. \n\\]\nThus, having a sequences $u_\\eps \\to u$ and $\\xi_\\eps \\weak \\xi_0$\nwith $ \\xi_\\eps \\in \\pl\\calE^\\eps_t(u_\\eps)$ for $\\eps >0$ and\n$\\calE^\\eps_t(u_\\eps)\\to \\ol e$, we can find, for each $w\\in W$ a\nrecovery sequence $w_\\eps \\to w$ with $\\calE^\\eps_t(w_\\eps)\\to\n\\calE^0_t(w)$. Hence, we obtain\n\\[\n\\calE^\\eps_t(w_\\eps) \\geq \\calE^\\eps_t(u_\\eps) + \\langle \\xi_\\eps ,\nw_\\eps {-}u_\\eps\\rangle \\ \\text{ for }\\eps>0.\n\\]\nPassing to the limit $\\eps\\to 0$ we obtain \n\\begin{equation}\n  \\label{eq:w.u.xi}\n  \\calE^0_t(w) \\geq \\ol e + \\langle \\xi_0  , w{-}u_0\\rangle, \n\\end{equation}\nwhere we used the strong convergence $w_\\eps{-}u_\\eps \\to w{-}u_0$. \nBy $\\calE^\\eps_t \\xrightarrow{\\Gamma}\n\\calE^0_t$ we already know $\\calE^0_t(u_0)\\leq \\ol e$, but choosing\n$w=u_0$ in \\eqref{eq:w.u.xi} gives $\\calE^0_t(u_0)= \\ol e$ as\ndesired. With this, \\eqref{eq:w.u.xi} immediately gives $\\xi_0 \\in\n\\pl\\calE^0_t(u_0)$. \n\\end{rem}\n\nThe above result is only one of many possible versions and several\ngeneralizations are possible. For instance, we may combine time\ndiscretization with time step $\\tau\\to 0$ with the limit $\\eps \\to\n0$. More precisely, if we solve the time discretized problem (see Section \n\\ref{su:TimeDiscret}) for $\\mathrm{PG}^\\eps$  with time step $\\tau$ we obtain\nan approximation $\\wh U_{\\tau_\\eps}$. Then, it can be shown that these\napproximations satisfy good a priori estimates and hence for every\nsequence $(\\tau_n,\\eps_n)\\to (0,0)$ there exists a subsequence and a\nsolution of $\\mathrm{PG}^0$ such that the above convergences hold. We refer to\n\\cite[Thm.\\,4.1]{MiRoSt08GLRR} or\n\\cite[Thm.\\,3.12]{MiRoSa16BVSI} for results of this type.\n\n   \n\n\\subsection{A priori estimates}\n\\label{su:EGC.Apriori}\n\nThe energy-dissipation principle states that every solution $u_\\eps\\in\n\\mathrm{AC}([0,T];V)$ for $\\mathrm{PG}^\\eps$, i.e.\\ \n\\eqref{eq:I.1} is satisfied, also satisfies the energy-dissipation\nbalance in the sense that there exists a measurable selection\n$\\xi_\\eps:(0,T)\\to V^*$ such that $\\xi_\\eps(t)\\in\n\\pl\\calE^\\eps_t(u_\\eps(t))$ a.e.\\ in $(0,T)$ and that  \n\\begin{align}\n \\nonumber\n  &\\calE^\\eps_T(u_\\eps(T))+ \\int_0^T\\!\\! \\Big(\n  \\Psi^\\eps_{u_\\eps(r)}(u'_\\eps(r)) + \\Psi^{\\eps,*}_{u_\\eps(r)}\n  \\big(B^\\eps(r,u_\\eps(r))-\\xi_\\eps(r)\\big) \\Big) \\dd r \n\\\\\n &   \\label{eq:EGC.EDB}\n = \\calE^\\eps_0(u_\\eps(0)) + \\int_0^T \\!\\! \\Big(\\pl_t \\calE^\\eps_r(u_\\eps(r)) + \n  \\big\\langle B^\\eps(t,u_\\eps(r)),u'_\\eps(t) \\big\\rangle \\Big)\\dd r. \n\\end{align}\nEstimating the last term via the \\textsc{Young-Fenchel} inequality and\n\\eqref{eq:Beps.2} we obtain \n\\[\n\\big\\langle B^\\eps(r,u_\\eps(r)),u'_\\eps(r) \\big\\rangle \n\\leq \n\\Psi^\\eps_{u_\\eps(r)}(u'_\\eps(r)) + \\Psi^{\\eps,*}_{u_\\eps(r)}\n  \\big(B^\\eps(r,u_\\eps(r))\\big) \\leq\n  \\Psi^\\eps_{u_\\eps(r)}(u'_\\eps(r))+ C_B \\calE^\\eps_r(u_\\eps(t))\n\\]\nfor the last term. Thus, the terms involving\n$\\Psi^\\eps_{u_\\eps(r)}(u'_\\eps(r))$ and using $\\Psi^{\\eps,*}_{u}\\geq\n0$ and \\eqref{eq:cond.E.eps.2} we arrive at \n\\[\n\\calE^\\eps_T(u_\\eps(T))\\leq \\calE^\\eps_0(u_\\eps(0))+ \\int_0^T\n\\big(C_T{+}C_B\\big) \\calE_r^\\eps(u_\\eps(r)) \\dd r. \n\\]\nWith $u_\\eps(0)=u^0_\\eps$ and the well-preparedness \\eqref{eq:EGC.well}\nthe \\textsc{Gronwall} lemma yields \n\\[\nG^\\eps(u_\\eps(t))\\; \\leq \\;\\calE_t^\\eps(u_\\eps(t)) \\leq\n2\\calE^0_0(u_0^0) \\,\\ee^{(C_T+C_B) t} \\leq \\ol E:=\n2\\calE^0_0(u_0^0)\\,\\ee^{(C_T+C_B) T}.\n\\]\nThus, assumption \\eqref{eq:cond.E.eps.2} guarantees that there exists\na compact set $K\\Subset V$ such that $u_\\eps(t)\\in K$ for all\n$(\\eps,t)\\in (0,1)\\ti[0,T]$.  As $K\\subset B_R(0)\\subset V$ we can\napply the superlinearity \\eqref{eq:Psi.eps.2} and the control \\eqref{eq:Beps.2}\n of $B^\\eps$ to estimate\n\\[\ng_R\\big(B^\\eps(t,u_\\eps(t))\\big) \\leq \\Psi^{\\eps,*}_{u_\\eps(t)}\n\\big(B^\\eps(t,u_\\eps(t))\\big) \\leq C_B \\calE^\\eps_t(u_\\eps(t)) \\leq\nC_B \\ol E.\n\\]\nThis implies the boundedness of the non-variational perturbation,\nviz.\\\n\\begin{equation}\n  \\label{eq:EGC.B.bound}\n  \\exists\\, R^*_B>0\\ \\forall\\, (\\eps,t)\\in (0,1)\\ti[0,T]: \\quad \n \\| B^\\eps(t,u_\\eps(t))\\|_{V^*}\\leq R^*_B. \n\\end{equation}\n\nInserting the bounds for $\\calE^\\eps_t(u_\\eps(t))$ (and hence for\n$\\pl_t \\calE^\\eps_t(u_\\eps(t))$) and for $B^\\eps(t,u_\\eps(t))$ into\n\\eqref{eq:EGC.EDB} we obtain \n\\[\n\\int_0^T \\!\\! \\Big(\n  \\Psi^\\eps_{u_\\eps(r)}(u'_\\eps(r))- R^*_B\\|u'_\\eps(r)\\|_V +\n  \\Psi^{\\eps,*}_{u_\\eps(r)} \n  \\big(B^\\eps(r,u_\\eps(r))-\\xi_\\eps(r)\\big) \\Big) \\dd r \\leq C_E.\n\\]\nUsing that $\\Psi^\\eps$ and $\\Psi^{\\eps,*}$ are bounded from below by the\nsuperlinear function $g_R$ (cf.\\ \\eqref{eq:Psi.eps.2}) and using\n\\eqref{eq:EGC.B.bound} again we arrive at\n\\begin{equation}\n  \\label{eq:EGC.gR.bounds}\n   \\exists\\, C_\\Psi>0\\ \\forall\\, \\eps\\in (0,1]:\\quad \n\\int_0^T \\big( g_R(\\|u'_\\eps(t)\\|_V)+ g_R(\\|\\xi_\\eps\\|_{V^*}) \\big)\n\\dd t \\leq C_\\Psi.\n\\end{equation}\n\n\n\n\\subsection{Convergent subsequences}\n\\label{su:EGC.CvgSubseq}\n\nBy \\eqref{eq:EGC.gR.bounds} and the criterion of \\textsc{de la\nVall\\'ee-Poussin} for uniform integrability, the family\n$u_\\eps:[0,T]\\to V$ is equi-continuous. As all values $u_\\eps(t)$ lie\nin the compact set $K$ the \\textsc{Arzel\\`a-Ascoli} theorem (e.g.\\\n\\cite[Prop.\\,3.3.1]{AmGiSa05GFMS}) gives\na subsequence $\\eps_k\\to 0$ such that the uniform convergence \n\\eqref{eq:EGC.cvg.a}\nholds. Moreover, \\eqref{eq:EGC.gR.bounds} also implies weak\ncompactness, hence we may also assume \n$u'_{\\eps_k} \\weak u'_0$ in $\\rmL^1(0,T;V)$, which is \\eqref{eq:EGC.cvg.c}.\n\nBy the continuity \\eqref{eq:Beps.1} we obtain convergence of the\nnon-variational terms, namely \n\\begin{equation}\n  \\label{eq:EGC.cvg.B}\n  \\forall\\, t\\in [0,T]: \\quad B^{\\eps_k}(t,u_{\\eps_k}(t)) \\to\n  B^0(t,u_0(t)) \\text{ uniformly in } V^*. \n\\end{equation}\n\nUsing the positivity of $\\Psi^\\eps$ and $\\Psi^{\\eps,*}$ we then obtain\nthat $\\ol e^\\eps:t\\mapsto \\calE^\\eps_t(u_\\eps(t))$ are uniformly\nbounded in BV$([0,T])$, such that Helly's selection principle allows\nto extract a subsequence (not relabeled) such that \n\\begin{equation}\n  \\label{eq:EGC.E.cvg}\n\\forall\\ t\\in [0,T]:\\quad \\ol e^{\\eps_k}(t) \\to \\ol e^0(t) \\geq\n\\calE_t^0(u_0(t)),   \n\\end{equation}\nwhere the last estimate follows from \\eqref{eq:cond.E.epsGamma}. \n\nAgain based on the superlinear bounds \\eqref{eq:EGC.gR.bounds} we can\ndefine extract further subsequence (not relabeled) such that $t\\mapsto\n(u'_{\\eps_k}(t),\\xi_{\\eps_k}(t),\\pl_t\\calE^{\\eps_k}_t(u_{\\eps_k}(t)))$\ngenerates a \\textsc{Young} measure $\\mathbold\\mu=(\\mu_t)_{t\\in [0,T]} \\in\n\\calY([0,T];V\\ti V^*\\ti \\R)$ in the sense that\n\\begin{equation}\n  \\label{eq:EGC.F.mu}\n  \\int_0^T F\\big(t,u'_{\\eps_k}(t), \\xi_{\\eps_k}(t),\n\\pl_t\\calE^\\eps_t(u_\\eps(t)) \\big) \\dd t  \\to \\int_0^T \\int_{V\\ti\n  V^*\\ti\\R} F(t, v,\\zeta,p)\\dd \\mu_t(v,\\zeta,p) \\dd t,\n\\end{equation}\nfor all continuous functions $F:[0,T]\\ti V\\ti\n  V^*\\ti\\R \\to \\R$, where $V\\ti V^*$ is equipped with the weak\n  topology, with $F(t,v,\\zeta,p)\\leq C(1{+}\\|v\\|+\\|\\zeta\\|_*)$. We\n  refer to Appendix \\ref{se:Appendix}.  \n\n\n\\subsection{Limit passage and conclusion of the proof of Theorem \\protect\n  {\\ref{th:EGC.main}}}  \n\\label{su:EGC.LimitPass}\n\nWe can now go back to the energy-dissipation balance\n\\eqref{eq:EGC.EDB} and pass to the limit $\\eps_k\\to 0$, where we\nemploy \\textsc{Balder}'s lower semicontinuity result \\cite{Bald84GALS}\nfor weakly normal integrands in the form of\n\\cite[Thm.\\,4.3]{Stef08BEPD}, see Theorem \\ref{th:A1}. The main point\nhere is that for $\\bfalpha=(\\alpha_1,\\alpha_2,\\alpha_3)\\in\n{[0,\\infty)}^3$ the mappings\n\\[\nF^\\bfalpha_k:\\;[0,T]\\ti V\\ti V^*\\ti \\R\\to \\R;\\  (t,v,\\zeta,p) \\;\\mapsto\\;\n\\alpha_1\\Psi_{u_{\\eps_k}(t)}(v) +\n\\alpha_2\\Psi^{\\eps,*}_{u_{\\eps_k}(t)}(\\zeta) \n+ \\alpha_3 p    ,\n\\]\nsatisfy a liminf estimate, namely \n\\[\n(v_k,\\zeta_k,p_k)\\weak (v,\\zeta,p) \\text{ in }V\\ti V^* \\ti \\R \\quad\n\\Longrightarrow \\quad \\liminf_{k\\to \\infty}\nF^\\bfalpha_k(t,v_k,\\zeta_k,p_k) \n\\geq F^\\bfalpha_\\infty(t,v,\\zeta, p),\n\\]\nwhere $F^\\bfalpha_\\infty(t,v,\\zeta,p)= \\alpha_1 \\Psi^0_{u_0(t)}(v)\n+ \\alpha_2 \\Psi^{0,*}_{u_0(t)}(\\zeta) + \\alpha_3 p$.  But the latter\nliminf estimate \nfollows easily from the \\textsc{Mosco} convergence condition\n\\eqref{eq:Psi.eps.3}, because we already now $u_{\\eps_k}\\to u_0(t)$\nand $\\calE^{\\eps_k}_t(u_{\\eps_k}(t)) \\leq \\ol E$. In particular, we\nobtain the three liminf estimates\n\\begin{subequations}\n \\label{eq:EGC.liminf}\n\\begin{align}\n  \\label{eq:EGC.liminf.a}\n  &\\int_r^s \\Psi^0_{u_0(t)} (u'_0(t)) \\dd t \\leq \\liminf_{k\\to \\infty} \n   \\int_r^s \\Psi^{\\eps_K}_{u_{\\eps_K}(t)} (u'_{\\eps_k}(t)) \\dd t,\n\\\\\n  \\label{eq:EGC.liminf.b}\n  &\\int_r^s \\Psi^{0,*}_{u_0(t)}\\big(\n            B^{0_k}(t,u_0(t)){-}\\xi_0(t))\\big) \\dd t\n \\leq \\liminf_{k\\to \\infty} \n   \\int_r^s \\Psi^{\\eps_k,*}_{u_{\\eps_k}(t)}\\big(\n  B^{\\eps_k}(t,u_{\\eps_k}(t)){-}\\xi_{\\eps_k}(t))\\big) \\dd t,\n\\\\\n  \\label{eq:EGC.liminf.c}\n& \\int_r^s \\pl_t \\calE^0_t(u_0(t)) \\dd t \\leq \\liminf_{k\\to \\infty} \n   \\int_r^s \\pl_t\\calE^{\\eps_k}_t(u_{\\eps_k}(t)) \\dd t,\n\\end{align}\n\\end{subequations}\nwhere $0\\leq r < s \\leq T$ are arbitrary. \n\nAdding the three inequalities in \\eqref{eq:EGC.liminf} and using the\nlimit $\\ol e^0$ in \\eqref{eq:EGC.E.cvg} we arrive at \n\\begin{align}\n \\nonumber\n &\\ol e^0(T) + \\int_0^T \\!\\!\\int_{V\\ti V^*\\ti \\R} \\!\\!\\Big(\\Psi^0_{u_0(t)}(v)+\n  \\Psi^{0,*}_{u_0(r)} \\big(B^0(r,u_0(r))-\\zeta\\big) - p\\Big) \\dd\n  \\mu_t(v,\\zeta,p) \\dd t \n \\\\\n \\label{eq:EGC.est3}\n &\\leq  \\calE^0_0(u^0_0)+ \\int_0^T \\big\\langle\n  B^0(r,u_0(r)), u'_0(r) \\big\\rangle \\dd r.\n\\end{align}\nHere the convergence of the right-hand side follows from the\nwell-preparedness \\eqref{eq:EGC.well} and the fact that the strong\n$\\rmL^{\\infty}$ convergence \\eqref{eq:EGC.cvg.B} and the weak convergence\n\\eqref{eq:EGC.cvg.c} imply the convergence of the integral. \n \nNow we exploit the main structural property of the \\textsc{Young} measure\n$\\mathbold\\mu$ which states that for a.a.\\ $t\\in [0,T]$ the supports\nof $\\mu_t$ lie in the set of accumulation points of defining\nsequences. More, there is a null set $N\\subset [0,T]$ (i.e.\\ $|N|=0$)\nsuch that \n\\[\n\\forall\\, t\\in [0,T]{\\setminus} N:\\\n \\mathrm{sppt}(\\mu_t)\\subset \\mathrm{Li}(t):= \\bigcap_{m=1}^\\infty\n\\mathrm{clos_{weak}}\\Big(\n\\bigset{(u'_{\\eps_{k}}(t),\\xi_{\\eps_{k}}(t),\\pl_t\\calE^{\\eps_{k}}_t(u_{\\eps_{k}}(t)))\n} { k\\geq m} \\Big) .\n\\]\nHence, the closedness condition \\eqref{eq:cond.E.eps.5} guarantees that\n\\begin{align*}\n&\\forall\\, t\\in [0,T]{\\setminus} N\\ \\forall\\, (v,\\zeta,p)\\in\n\\mathrm{sppt}(\\mu_t): \\  \n\\zeta \\in \\pl\\calE^0_t(u_0(t)), \\ p\\leq \\pl_t\\calE^0_t(u_0(t)),\n\\ \\ol e^0(t)=\\calE^0_t(u_0(t)).\n\\end{align*} \n\nWe can now estimate further in \\eqref{eq:EGC.est3}. By \\eqref{eq:EGC.E.cvg} \nthe first term $\\ol e^0(T)$ is estimated from below by\n$\\calE^0_T(u_0(T))$. The term involving \n$\\Psi^0_u(v)$ can be estimated by the convexity of $\\Psi^0_u(\\cdot)$\nand the fact that $\\mu_t$ is a probability measure with\n$v$-expectation $u'_0$, i.e.\\ \n\\[\n u'_0(t)= \\int_{V\\ti V^*\\ti\\R}  v \\dd \\mu_t(v,\\zeta,p).\n\\]\nThis follows simply by testing \\eqref{eq:EGC.F.mu} by\n$F(t,v,\\zeta,p)=\\langle \\eta(t),v\\rangle$ for all $\\eta \\in\n\\rmL^\\infty(0,T;V^*)$. Thus, we have \n\\[\n\\int_0^T \\Psi_{u_0(t)}(u'_0(t)) \\dd t \\leq \n\\int_0^T \\int_{V\\ti V^*\\ti\\R} \\Psi_{u_0(t)}(v) \\dd \\mu_t(v,\\zeta,p) \\dd t,\n\\]\n\nFor the term involving \n$\\Psi^{0,*}_u(v)$ we cannot apply Jensen's inequality as\n$\\pl\\calE_t^0(u)$ may not be convex. Thus, for $t\\in  [0,T]{\\setminus}\nN$ we select $\\xi_0(t) \\in \\pl\\calE_t^0(u_0(t))$ with \n\\[\n\\Psi^{0,*}_{u_0(r)} \\big(B^0(r,u_0(r))-\\xi_0(t)\\big) = \\min\\bigset{\n\\Psi^{0,*}_{u_0(r)} \\big(B^0(r,u_0(r))-\\zeta\\big)}{ \\zeta \\in\n\\pl\\calE_t^0(u_0(t))}. \n\\]\nSuch a measurable selection exists, see Lemma \\ref{le:A4} in Appendix\n\\ref{se:Appendix}. \n\nFinally using $p\\leq \\pl_t\\calE^0_t(u_0(t))$ on Li$(t)$ the estimate \n\\eqref{eq:EGC.est3} yields, for all $s\\in (0,T]$,\n\\begin{align}\n \\nonumber\n &\\calE^0_s(u_0(s)) + \\int_0^s \\!\\! \\Big(\\Psi^0_{u_0(t)}(u'_0(t)) +\n  \\Psi^{0,*}_{u_0(r)} \\big(B^0(r,u_0(r))-\\xi_0(t)\\big) -\n  \\pl_t\\calE^0_t(u_0(t)) \\Big)  \\dd t \n \\\\\n \\label{eq:EGC.est5}\n &\\qquad \\leq  \\calE^0_0(u^0_0)+ \\int_0^s \\big\\langle\n  B^0(t,u_0(t)), u'_0(t) \\big\\rangle \\dd t.\n\\end{align}\nMoreover, by the \\textsc{Fenchel-Young} inequality and the chain-rule\ninequality \\eqref{eq:cond.E.eps.4}, which \nis used for $\\eps=0$ only, the \nleft-hand side can be estimated from below via\n\\begin{align}\n\\nonumber\n&\\calE^0_s(u_0(s)) + \\int_0^s \\!\\! \\Big(\\Psi^0_{u_0(t)}(u'_0(t)) +\n  \\Psi^{0,*}_{u_0(t)} \\big(B^0(t,u_0(t))-\\xi_0(t)\\big) -\n  \\pl_t\\calE^0_t(u_0(t)) \\Big)  \\dd t \\\\\n\\nonumber\n&\\overset{\\text{FY}}{\\geq} \\calE^0_s(u_0(s)) + \\int_0^s \\!\\!\\Big(\\langle\n   B^0(t,u_0(t)){-}\\xi_0(t) , u'_0(t)\\rangle  -\n  \\pl_t\\calE^0_t(u_0(t))  \\Big) \\dd t \\\\\n\\nonumber\n&\\overset{\\text{chain}}\\geq \\calE^0_s(u_0(s)) + \\int_0^s\\!\\! \\Big(\\langle\nB^0(t,u_0(t)) , u'_0(t)\\rangle -\\frac{\\rmd}{\\rmd t}\\big(\n\\calE_t^0(u_0(t))\\big)  \\Big) \\dd t \n\\\\\n\\label{eq:EGC.est7}\n&= \\calE^0_0(u_0(0)) + \\int_0^s \\big\\langle\n  B^0(t,u_0(t)), u'_0(t) \\big\\rangle \\dd t.\n\\end{align} \n\nThus, we conclude that all inequalities in \\eqref{eq:EGC.est5} and\n\\eqref{eq:EGC.est7} are equalities, which implies the the\n\\textsc{Fenchel-Young} estimate has to hold with equality a.e.\\ in\n$[0,T]$, which gives the desired differential inclusion $\nB^0(t,u_0(t)) - \\xi_0(t) \\in \\pl \\Psi^0_{u_0(t)}(u'_0(t))$ or \n\\[\n B^0(t,u_0(t)) \\in \\pl \\Psi^0_{u_0(t)}(u'_0(t)) + \\pl\\calE^0_t(u_0(t))\n \\qquad \\text{a.e.\\ in } [0,T].\n\\] \n\nAdditionally, we observe that the liminf estimates \n\\begin{align*}\n\\calE^0_t(u_0(t))\n\\leq \\ol e^\\infty(t)=\\lim_{\\eps_k\\to 0} \\calE^{\\eps_k}_t(u_{\\eps_k}(t))\n\\end{align*}\nas well as the liminf estimates in \\eqref{eq:EGC.liminf} are indeed\nequalities as well. Thus, \\eqref{eq:EGC.cvg.b}, \\eqref{eq:EGC.cvg.d},\n and \\eqref{eq:EGC.cvg.b} are established and the proof of Theorem\n \\ref{th:EGC.main} is complete. \n \n\n\\subsection{Improved result for state-independent dissipation}\n \\label{su:EGC.StateIndep}\n\nThe result on evolutionary $\\Gamma$-convergence given in Theorem\n\\ref{th:EGC.main} has a rather strong assumption, namely the \\textsc{Mosco}\nconvergence of $(\\eps,u)\\mapsto \\Phi^\\eps_u(\\cdot)$ in the space $V$. \nThis assumption is too strong for a series of important applications. \nFor instance, for the parabolic equation\n\\[\n\\big( 2+ \\cos(x_1\/\\eps)\\big) u' = \\Div\\big( A(\\tfrac1\\eps x)\\nabla\nu\\big) \\text{ in } \\Omega \\subset \\R^d, \\qquad u=0 \\text{ on } \\pl\\Omega,\n\\]\nwe may choose the gradient structure $(\\bfQ,\\calE^\\eps,\\Psi^\\eps)$\nwith \n\\[\nV=\\rmL^2(\\Omega),\\quad \\calE^\\eps(u)=\\int_\\Omega \\frac12 \\nabla u\n\\cdot A(\\tfrac 1\\eps x) \\nabla u \\dd x, \\quad \n\\Psi^\\eps(v) = \\int_\\Omega \\frac{2{+}\\cos(x_1\/\\eps)}2\\: v(x)^2 \\dd x.\n\\]\nHowever, $\\Psi^\\eps$ $\\Gamma$-converges to $\\Psi_\\text{harm}$ in the\nweak topology of $\\rmL^2(\\Omega)$ while it $\\Gamma$-converges to\n$\\Psi_\\text{arith}$ in the strong topology.\n\nHere we want present a generalized version of \\cite{LieRei15?HCHT}\nwhere evolutionary $\\Gamma$-convergence was established under the\nweaker assumption $\\Psi^\\eps \\xrightarrow{\\Gamma} \\Psi^0$, i.e.\\ $\\Gamma$-convergence\nin the strong topology only.\n\nIf we inspect the proof in the previous subsection, then we see that\nthe weak $\\Gamma$-convergence of $\\Psi^\\eps_{u_\\eps}$ was used only\nonce, namely for deriving the liminf estimate\n\\eqref{eq:EGC.liminf.a}. The point is that we only derived the weak\nconvergence $u'_{\\eps_k} \\weak u'_0$ in $\\rmL^1(0,T;V)$. However, the\n``weak'' convergence may have two origins, namely first due to\noscillations in time and second due to weak convergence of\n$u'_\\eps(t) \\weak u'_0(t)$ in $V$. The idea in \\cite{LieRei15?HCHT} is\nto consider piecewise affine interpolants $u_{\\eps,\\tau}$ of $u_\\eps$\nfor fixed time steps $\\tau>0$. This averages potential oscillations in\ntime as $u'_{\\eps,\\tau}$ is piecewise constant. Moreover, we can use\nthe strong convergence of $u_{\\eps_k}(t)\\to u_0(t)$ which implies that\n$ u'_{\\eps_k,\\tau}(t)\\to u'_{0,\\tau}(t)$ in $V$ for a.a.\\ $[t\\in\n[0,T]$. Finally, the limit $\\tau\\to 0$ is done after the limit $\\eps_k\\to\n0$ is already performed. \n  \nOur precise assumptions, which replace \\eqref{eq:Psi.eps.3}, are the\nfollowing:\n\\begin{subequations}\n\\label{eq:Psi.strongG}\n \\begin{align}\n \\nonumber \n &\\text{{\\bfseries Uniform\n     continuity.} \\quad For all } R>0\\\\\n \\nonumber \n &\\quad \\exists\\; \\text{modulus of continuity } \\omega_R\\  \n   \\forall \\, \\eps\\in [0,1]\\ \\forall\\, u_1,u_2 \\text{ with }G^\\eps(u_j)\\leq R\\\\\n &\\label{eq:Psi.strongG.a}\\quad   \\forall \\, v\\in V: \\quad \n \\big|\\Psi^\\eps_{u_1}(v) - \\Psi^\\eps_{u_2}(v)\\big| \\leq \\omega_R(\\|u_1{-}u_2\\|_V)\n  g_R(\\|v\\|_V) ,\\\\\n \\nonumber \n &\\text{{\\bfseries Strong $\\Gamma$-convergence.} \\quad For all $R>0$\n   we have}\\\\ \n & \\label{eq:Psi.strongG.b}\n  \\qquad u_\\eps \\to u_0\\text{ and } \\sup \\calE^\\eps_t(u_\\eps)\\leq R\n  \\quad \\Longrightarrow \\quad \\Psi_{u_\\eps}^\\eps \\xrightarrow{\\Gamma} \\Psi^0_{u_0},  \n \\end{align}\n\\end{subequations}\nwhere $g_R$ is the coercivity function defined in\n\\eqref{eq:Psi.eps.2}.  \n\n\n\\begin{cor}[Strong $\\Gamma$-convergence for $\\calE^\\eps$ and\n  $\\Psi^\\eps$] \n\\label{co:StrongGa}\nAll results of Theorem \\ref{th:EGC.main} remain true if assumption\n\\eqref{eq:Psi.eps.3} is replaced by \\eqref{eq:Psi.strongG}. \n\\end{cor}\n\\begin{proof}\n  To start with, we recall that the strong $\\Gamma$-convergence of\n  \\eqref{eq:Psi.strongG.b} implies the weak $\\Gamma$-convergence of\n  the \\textsc{Legendre-Fenchel} dual, i.e.\\\n  $\\Psi^{\\eps,*}_{u_\\eps}\\overset{\\Gamma}\\weak \\Psi^{0,*}_u$, see\n  \\eqref{eq:Gcvg.sw*}.  Thus, the liminf estimate\n  \\eqref{eq:EGC.liminf.b} follows exactly as above.\n \nThus, it remains to find a new proof for the liminf estimate\n\\eqref{eq:EGC.liminf.a}. Using the notation\n\\[\nJ^\\eps(u,v) := \\int_0^T \\Psi^\\eps_{u(t)}\\big( v(t)\\big) \\dd t\n\\]\nwe have to show $\\liminf_{k\\to \\infty}\nJ^{\\eps_k}(u_{\\eps_k},u'_{\\eps_k})\\geq J^0(u_0,u'_0)$, where our\nsequence $(u_{\\eps_k})_k$ satisfies\n\\begin{align*}\n\\text{(a) } \\| u_{\\eps_k}{-}u_0\\|_{\\rmC^0([0,T];V)} \\to 0, \n \\text{ (b) } \\|u'_{\\eps_k}{-}u'_0\\|_{\\rmL^1(0,T;V)} \\to 0, \n\\text{ (c) }  \\int_0^T \\!\\!g_R\\big(\\| u'_{\\eps_k}(t)\\|\\big) \\dd t \\leq C_g,\n\\end{align*}\nwhere $R\\geq \\sup\\set{\\|u_{\\eps_k}\\|_\\infty}{ k\\in \\N}$. \n\nFor time steps $\\tau=T\/N>0$ with $N\\in \\N$ we define piecewise\nconstant and piecewise affine interpolants $\\ol u^\\tau_{\\eps_k}$ and $\\wh\nu^\\tau_{\\eps_k}$ as in \\eqref{eq:Approx.tau}. By the uniform convergence (a) we\nhave equi-continuity of the sequence $(u_{\\eps_k})_k$, and hence\n\\[\n\\mu_\\tau := \\sup\\set{\\| u_{\\eps_k}-\\ol u_{\\eps_k}\\|_{\\rmC^0([0,T];V)}\n}{k \\in \\N} \\ \\to \\ 0 \\quad \\text{for }\\tau\\to 0.\n\\]\nWith \\eqref{eq:Psi.strongG.a} and (c) we obtain the lower bound \n\\begin{align*}\nJ^{\\eps_k}(u_{\\eps_k},u'_{\\eps_k}) &\\geq\nJ^{\\eps_k}(\\ol u^\\tau_{\\eps_k},  u'_{\\eps_k}) - \\int_0^T\\!\\!\n\\omega_R\\big(\\|u_{\\eps_k}\\!{-} \\ol u^\\tau_{\\eps_k}\\|\\big)\ng_R\\big(\\|u'_{\\eps_k}\\|\\big) \\dd t \\geq J^{\\eps_k}(\\ol\nu^\\tau_{\\eps_k},  u'_{\\eps_k}) - \\omega_R(\\mu_\\tau) C_g.\n\\end{align*}\nOn the intervals $((n{-}1)\\tau,n\\tau)$ the integrand $\\Psi^{\\eps_k}_{\\ol\nu^\\tau_{\\eps_k}(t)}(\\cdot)$ is independent of $t$ and convex. Hence,\nwe can apply Jensen's inequality and replace\n$v_k(t)=u'_{\\eps_k}(\\cdot)$ by its average over this interval, which\nis exactly \n\\[\n\\frac1\\tau \\int_{(n-1)\\tau}^{n\\tau} u'_{\\eps_k}(r) \\dd r =\n\\frac1\\tau\\big(u_{\\eps_k}(n\\tau)-u_{\\eps_k}((n{-}1)\\tau)\\big)=\n\\wh u^{\\tau}_{\\eps_k}\\!{}'(t) \\quad \\text{for } t\\in ((n{-}1)\\tau,n\\tau).\n\\]\nThus, we have the lower bound $J^{\\eps_k}(u_{\\eps_k},u'_{\\eps_k}) \\geq\nJ^{\\eps_k}(\\ol u^\\tau_{\\eps_k},\\wh u^{\\tau}_{\\eps_k}\\!{}')-\n\\omega_R(\\mu_\\tau)C_g$.\\medskip\n\nFor $k\\to \\infty$ we have $ \\ol u_{\\eps_k}\\to \\ol u_0$ in $V$ and $\n\\wh u^\\tau_{\\eps_k}\\!{}'\\to \\wh u^\\tau_0{}'$ in $V$ a.e.\\ in\n$[0,T]$. Hence, we can exploit the liminf estimate of the strong\n$\\Gamma$-convergence $\\Psi^\\eps_{u_\\eps} \\xrightarrow{\\Gamma} \\Psi^0_{u_0}$. \\textsc{Fatou}'s\nlemma leads to\n\\begin{align*}\n&\\liminf_{k\\to \\infty} J^{\\eps_k}(u_{\\eps_k},u'_{\\eps_k})\\geq \n\\liminf_{k\\to \\infty} J^{\\eps_k}(\\ol u^\\tau_{\\eps_k},\\wh u^{\\tau}_{\\eps_k}\\!{}')-\n\\omega_R(\\mu_\\tau)C_g\\\\\n&\\overset{\\text{Fatou}}\\geq  J^0(\\ol u^\\tau_0,\\wh u^{\\tau}_0{}')-\n\\omega_R(\\mu_\\tau)C_g \\  \\geq \\ J^0(u_0,\\wh\nu^{\\tau}_0{}')-2\\omega_R(\\mu_\\tau)C_g ,\n\\end{align*}\nwhere we used $\\|u_0-\\ol u^\\tau_0\\|_\\infty \\leq \\mu_\\tau$ for the last\nstep. \n\nThus, using $\\omega_R(\\mu_\\tau)\\to 0$ for $\\tau\\to 0$ it remains to\nshow that $L:=\\liminf_{\\tau \\to 0} J^0(u_0, \\wh u^{\\tau}_0{}') \\geq\nJ^0(u_0,u'_0)$. Choose a subsequence $\\tau_m$ such that $ J^0(u_0, \n\\wh u^{\\tau_m}_0{}')\\to L$. We now use the well-known fact that \n$\\wh u^{\\tau_m}_0{}' \\to u'_0$ in $\\rmL^1(0,T;V)$, which implies that \nthere exists a further subsequence (not relabeled) such that \n$\\wh u^{\\tau_m}_0{}'(t) \\to u'_0(t)$ in $V$ a.e.\\ in $[0,T]$. Moreover,\n$\\Psi^0_{u_0(t)}(\\cdot):V\\to [0,\\infty)$ is continuous, because it is\nconvex and bounded from above by the \\textsc{Legendre-Fenchel} dual of $\\xi\n\\mapsto g_R(\\|\\xi\\|_{V^*})$. This gives $\\Psi^0_{u_0(t)}(\\wh\nu^{\\tau_m}_0{}'(t)) \\to \\Psi^0_{u_0(t)}(u'_0(t))$ a.e.\\ in $[0,T]$,\nand \\textsc{Fatou}'s lemma implies $L=\\liminf_{m\\to \\infty} J^0(u_0, \\wh\nu^{\\tau_m}_0{}') \\geq J^0(u_0,u_0')$ as desired.\n \nAltogether we have established $\\liminf_{k\\to \\infty}\nJ^{\\eps_k}(u_{\\eps_k},u'_{\\eps_k})\\geq J^0(u_0,u'_0)$, and thus\nCorollary \\ref{co:StrongGa} is proved. \n\\end{proof}\n\n\n\n\n\\section{Homogenization of reaction-diffusion systems}\n\\label{se:Example}\n\nIn this section we provide a nontrivial example that highlights the\napplicability of our abstract existence theory as well as the theory\nof evolutionary $\\Gamma$-convergence. We refer to\n\\cite{MiReTh14TSHN, Reic16EEEE, Reic17CECI} and the references therein\nfor general homogenization results that are typically for semilinear\nsystems where the leading order terms are decoupled.  \nOur example of a reaction\ndiffusion system is a general quasilinear parabolic system, where the\nleading terms may be coupled but need to have a variational\nstructure. \n\nOur system for the vector $u(t,x)\\in \\R^I$ reads as follows:\n\\begin{align}\n\\nonumber\n \\hspace*{1em}\n A^\\eps(x,u(t,x)) \\pl_t u(t,x) &= \\Div \\Big(\\pl_{\\nabla u}\n  F^\\eps \\!\\,\\big(x,u(t,x),\\nabla u(t,x)\\big) \\Big) \\\\\n\\label{eq:ExaPDE}\n & \\quad - \\pl_u F^\\eps\\!\\,(x,u(t,x),\\nabla\n  u(t,x)) + b^\\eps(x,t,u(t,x)) &\\text{in }&\\Omega,\\\\\n\\nonumber\n 0&=\\pl_{\\nabla u}\n  F^\\eps\\!\\,\\big(x,u(t,x),\\nabla u(t,x)\\big)\\nu(x)&\\text{on }&\\pl\\Omega.\n\\hspace*{1em}\n\\end{align}\nGenerally we assume that $\\Omega\\subset \\R^d$ is a\nbounded domain with Lipschitz boundary $\\pl\\Omega$. For simplicity,\nwe have imposed Neumann boundary conditions only, but more general\nconditions including Dirichlet or Robin boundary conditions could be\nused as well.  \n \nWe first summarize the needed assumptions on the functions $A^\\eps$,\n$F^\\eps$, and $b^\\eps$, then show that these assumptions imply the\nonce needed for the existence theory in Section \\ref{se:ExistResult},\nand finally discuss under which conditions we have evolutionary\n$\\Gamma$-convergence for $\\eps\\to 0$.\n\n\\subsection{The existence result}\n\\label{su:Exa.Exist}\n\nFor the matrix $A^\\eps(x,u)\\in \\R^{I\\ti I}_\\text{sym}:=\\set{A\\in\n  \\R^{I\\ti I}}{A=A^\\top}$  we make the assumption\n\\begin{subequations}\n\\label{eq:Exa.Ass}\n\\begin{align}\n \\label{eq:Exa.Ass.A1} \n &\\forall\\,\\eps \\in [0,1]:\\quad  A^\\eps:\\Omega\\ti \\R^I \\to\n \\R_\\text{sym}^{I\\ti I}\\text{ is a \\textsc{Carath\\'eodory} function}, \\\\ \n \\label{eq:Exa.Ass.A2} \n &\\exists\\, C_A>0\\  \\forall\\,\\eps \\in [0,1]\\ \\forall\\, x\\in\n \\Omega\\ \\forall\\, u,v\\in  \\R^I:\\ \\  \\frac1{C_A} |v|^2 \\leq \\langle\n A^\\eps(x,u)v , v\\rangle \\leq C_A|v|^2. \n\\end{align}\nHere $G:\\Omega\\ti \\R^M\\to \\R^N$ is called a \\textsc{Carath\\'eodory} function,\nif $x\\mapsto G(x,z) $ is measurable for all $z\\in \\R^m$ and  $z\n\\mapsto G(x,z)$ is continuous for a.a.\\ $x\\in \\Omega$. \n\nFor simplicity, we will assume that the functions\n$F^\\eps(x,\\cdot,\\cdot)$ are convex, but much weaker conditions would be\npossible (e.g.\\ $\\lambda$-convexity in $u$ or poly-convexity in $U=\\nabla u$). \n\\begin{align}\n \\label{eq:Exa.Ass.F1} \n &\\forall\\,\\eps \\in [0,1]:\\quad  F^\\eps:\\Omega\\times (\\R^I\\ti \\R^{I\\ti d}) \\to\n \\R\\text{ is a \\textsc{Carath\\'eodory} function}, \\\\ \n \\label{eq:Exa.Ass.F2} \n &\\forall\\,\\eps \\in [0,1]\\ \\forall_\\text{a.a.}x\\in \\Omega:\\quad\n F^\\eps(x,\\cdot,\\cdot):\\R^I\\ti \\R^{I\\ti d} \\to\n \\R\\text{ is convex} , \\\\\n \\nonumber \n &\\exists\\, C_F>0 \\ \\exists\\, p,q>1\\  \\forall\\,\\eps \\in [0,1]\\ \\forall\\,\n (x,u,U) \\in \\Omega\\ti \\R^I\\ti \\R^{I\\ti d}:\\\\\n &\\label{eq:Exa.Ass.F3} \\qquad  F^\\eps (x,u,U)\\geq\n  C_F\\big(1+|u|^q+|U|^p\\big). \n\\end{align}\nFor the non-gradient terms $b^\\eps$ we impose the following conditions:\n\\begin{align}\n \\label{eq:Exa.Ass.B1} \n &\\forall\\,\\eps \\in [0,1]:\\quad  b^\\eps:\\Omega\\times ([0,T]\\ti\n \\R^I) \\to \\R^I \\text{ is a \\textsc{Carath\\'eodory} function}, \\\\ \n \\nonumber\n   &\\exists\\, h\\in \\rmL^2(\\Omega),\\;  C_B>0, \\;  r>1\\\n \\forall\\,(\\eps,t,x,u) \\in [0,1]\\ti \\Omega\\ti [0,T]\\ti\\R^I:\\\\ \n \\label{eq:Exa.Ass.B2} & \\qquad |b^\\eps(x,t,u)| \\leq h(x)+ C_B |u|^r.\n\\end{align}\n\\end{subequations}\n\nWe choose basic space $V= \\rmL^2(\\Omega;\\R^I)$,\nthe energy functionals \n\\[\n \\calE^\\eps(u)= \\left\\{ \\ba{cl}\\ds\\int_\\Omega F^\\eps(x,u(x),\\nabla\n  u(x))\\dd x &\\text{for }u \\in \\rmW^{1,p}(\\Omega;\\R^I),\\\\ \n   \\infty& \\text{otherwise,}\\ea\\right. \n\\]\nand the dissipation potentials \n\\[\n\\Psi^\\eps_u(v):= \\int_\\Omega \\frac12 \\langle A^\\eps(x,u(x)) v(x) ,\nv(x)\\rangle \\dd x.\n\\]\nThus, the perturbed gradient systems $\\mathrm{PG}^\\eps=(V,\\calE^\\eps,\n\\Psi^\\eps,b^\\eps)$ is fully specified, and we want to apply our\nabstract theory. Before doing so, we note that in the our conditions\nthe exponent $q$ appears three times: (i) the first relation in\n\\eqref{eq:Coeff.Rel} below implies $\\rmW^{1,p}(\\Omega) \\subset\n\\rmL^q(\\Omega)$, (ii) the coercivity \\eqref{eq:Exa.Ass.F3} of $F$ asks\nfor the lower bound $C_F|u|^q$, and (iii) the second relation in\n\\eqref{eq:Coeff.Rel} says that $B(\\cdot , u(\\cdot))$ is controlled by\n$C(1{+}\\|u\\|_q^q)$.\n\n\\begin{pro}\\label{pr:Exa.Exist} Let the functions $A^\\eps$, $F^\\eps$,\n  and $b^\\eps$ satisfy the conditions \\eqref{eq:Exa.Ass}, where the\n  coefficients $p$, $q$, and $r$ satisfy the relations\n  \\begin{equation}\n   \\label{eq:Coeff.Rel}\n    1-\\frac dp> -\\frac{d}{q} \\quad \\text{ and } \\quad q \\geq 2r. \n\\end{equation}\n\nThen, for each initial condition\n  $u^0_\\eps\\in \\rmL^2(\\Omega;\\R^I)$ with $\\calE^\\eps(u_\\eps^0)<0$ there is a\n  solution $u_\\eps:[0,T]\\to \\rmL^2(\\Omega;\\R^I)$ of \\eqref{eq:ExaPDE} \n such that $u_\\eps \\in \\rmH^1(0,T;\\rmL^2(\\Omega)) \\cap\n \\rmC^0_\\text{weak}([0,T]; \\rmW^{1,p}(\\Omega))$.  \n\\end{pro}\n\nThe proof is a consequence of our abstract existence result in Theorem\n\\ref{th:MainExist}.\nWe easily find the \\textsc{Legendre-Fenchel} dual $\\Psi^{\\eps,*}_u(\\xi)=\n\\int_\\Omega \\frac12 \\langle \\xi(x),(A^\\eps(x,u(x)))^{-1} \\xi(x)\n\\rangle \\dd x$.  Clearly, \\ref{eq:Psi.1} holds and we have the equi-coercivities\n\\[\n\\Psi^\\eps_u(v)\\geq \\frac1{2C_A}\\|v\\|^2_V \\quad \\text{and} \\quad\n\\Psi^{\\eps,*}_u(\\xi)\\geq \\frac1{2C_A}\\|\\xi\\|^2_{V^*},\n\\] \nwhich imply the desired superlinearities \\ref{eq:Psi.2}. \nFinally, the \\textsc{Mosco} convergence\n$\\Psi^\\eps_{u_n}\\xrightarrow{\\,\\rmM\\,} \\Psi^\\eps_u$ (here $\\eps>0$ is still fixed)\nfollows since $u_n\\to u$ in $V$ implies that $A^\\eps(\\cdot,u_n(\\cdot))\n\\to A^\\eps(\\cdot,u(\\cdot))$ a.e.\\ in $\\Omega$ along suitable\nsubsequences. To see that this is sufficient for\n\\textsc{Mosco}-convergence, we use the \\textsc{Moreau-Yosida}\nregularizations\n\\[\n\\Psi^{\\eps,\\lambda}_{u} (v):= \\inf\\Bigset{\n  \\Psi^\\eps_{u_n}(w)+\\frac\\lambda{2}\\|w{-}v\\|_{\\rmL^2}^2}{ w\\in\n  \\rmL^2(\\Omega;\\R^I)} \n\\]\nwhere $\\lambda>0$. It is easy to see that $\\Psi^{\\eps;\\lambda}_u$ is\nstill quadratic, but now with the matrix $\\lambda\nA^\\eps(A^\\eps{+}\\lambda I)^{-1}$. By \\cite[Thm.\\,3.26]{Atto84VCFO} we\nhave $\\Psi^\\eps_{u_n}\\xrightarrow{\\,\\rmM\\,} \\Psi^\\eps_u$ if and only if for all $v\\in\nV=\\rmL^2(\\Omega;\\R^I)$ and all $\\lambda>0$ we have the pointwise\nconvergence\n$\\Psi^{\\eps,\\lambda}_{u_n}(v)\\to\\Psi^{\\eps,\\lambda}_u(v)$. But this\nfollows immediately by the boundedness of $A^\\eps$ and\n\\textsc{Lebesgue}'s dominated convergence theorem.  Hence,\n\\ref{eq:Psi.3} is shown as well.\n\nThe energy functionals $\\calE^\\eps$ are convex and independent of\ntime. Hence \\ref{eq:cond.E.1} and \\ref{eq:cond.E.3} hold trivially. By the coercivity of\n$F^\\eps$ we obtain the coercivity of $\\calE^\\eps$, namely \n\\begin{equation}\n  \\label{eq:E.coercive}\n  \\calE^\\eps(u) \\geq \\int_\\Omega C_F\\big(1 +|u|^q+|\\nabla u|^p\n\\big) \\dd x \\geq \\wt c \\|u\\|_{\\rmW^{1,p}}^{\\min\\{p,q\\}}\n-\\wt C,  \n\\end{equation}\nsuch that sublevels are bounded in $\\rmW^{1,p}(\\Omega;\\R^I)$. Because\nthis space is compactly embedded in $V=\\rmL^2(\\Omega;\\R^I)$ by\nassumption \\eqref{eq:Coeff.Rel}, we conclude that \\ref{eq:cond.E.2}\nholds.  The chain rule \\ref{eq:cond.E.4} and the weak-strong\nclosedness of the \\textsc{Fr\\'echet} subdifferential (which is the\nsame as the convex subdifferential) follows by convexity, see Remark\n\\ref{re:SWClosGamma} or \\cite{MiRoSa13NADN}.\n\nWe now set $B^\\eps(t,u)(x)=b^\\eps(x,t,u(x))$ and obtain the continuity\n\\ref{eq:B.1} simply from the continuity of $b^\\eps(x,\\cdot,\\cdot)$ and\n$2r\\leq q$. The energy control \\ref{eq:B.2} follows from\n\\eqref{eq:Exa.Ass.F3} and the second condition in\n\\eqref{eq:Coeff.Rel}.  Thus, all the abstract assumptions of Theorem\n\\ref{th:MainExist} are established, and Proposition \\ref{pr:Exa.Exist}\nis established.\n\n\n\\subsection{The homogenization result}\n\\label{su:Exa.Homog}\n\nWe want to apply the evolutionary $\\Gamma$-convergence of Section\n\\ref{se:EGC} for homogenization, i.e.\\ we assume that the\n$x$-dependence of $A^\\eps$, $F^\\eps$, and $b^\\eps$ is of oscillatory\ntype, namely \n\\begin{equation}\n \\label{eq:Coeff.2s}\n  A^\\eps (x,u)=\\bbA({\\ts\\frac{\\ds1}{\\ds\\eps}} x, u),\\quad F^\\eps(x,u,U)=\\bbF({\\ts\\frac{\\ds1}{\\ds\\eps}} x, u,\n  U), \\quad b^\\eps(x,t,u)= \\bbB({\\ts\\frac{\\ds1}{\\ds\\eps}} x , u),\n\\end{equation}\nwhere the functions $\\bbA$, $\\bbF$, and $\\bbB$ are assumed to be\n1-periodic in all directions, i.e.\\ $\\bbG(y{+}k)=\\bbG(y)$ for all\n$y\\in \\R^d$ and $k\\in \\Z^d$.  \n\nFor the quadratic dissipation potentials $\\Psi^\\eps_u$ we have the\nfollowing $\\Gamma$-convergences:\n\\begin{equation}\n  \\label{eq:HomPsiGaCvg}\n (\\eps_n, u_n) \\to (0,u) \\in \\R \\ti \\rmL^2(\\Omega;\\R^n) \\ \n\\Longrightarrow \\ \\Big(\\Psi^{\\eps_n}_{u_n} \\overset{\\Gamma}{\\weak} \\Psi^\\text{harm}_u\n\\text{ and } \\Psi^{\\eps_n}_{u_n} \\xrightarrow{\\Gamma} \\Psi^\\text{aver}_u \\Big), \n\\end{equation}\nwhere the harmonic-mean functional $\\Psi^\\text{harm}_u $ and the\naverage functional $\\Psi^\\text{aver}_u$ are defined via\n\\begin{align*}\n\\Psi^\\text{harm}_u(v) &=\\int_\\Omega \\frac12\\langle A^\\text{harm}(u(x))\nv(x),v(x)\\rangle \\dd x &\\text{with }&\nA^\\text{harm}(u)^{-1}=\\int_{(0,1)^d} \\bbA(y,u)^{-1}\\dd y,\\\\\n\\Psi^\\text{aver}_u(v) &=\\int_\\Omega \\frac12\\langle A^\\text{aver}(u(x))\nv(x),v(x)\\rangle \\dd x &\\text{with }&\nA^\\text{aver}(u)=\\int_{(0,1)^d} \\bbA(y,u)\\dd y.\n\\end{align*} \nThe strong $\\Gamma$-convergence $\\Psi^{\\eps_n}_{u_n} \\xrightarrow{\\Gamma}\n\\Psi^\\text{aver}_u$ follows simply from the pointwise convergence\n$\\Psi^{\\eps_n}_{u_n}(v) \\to \\Psi^\\text{aver}_u(v)$ for all $v$ and the\nequi-\\textsc{Lipschitz} continuity. The weak $\\Gamma$-convergence\n$\\Psi^{\\eps_n}_{u_n} \\overset{\\Gamma}{\\weak} \\Psi^\\text{aver}_u$ follows by\n\\eqref{eq:Gcvg.sw*} and \\textsc{Legendre-Fenchel} transform as\n$\\Psi_u^{\\eps,*}$ is given in terms of $(A^\\eps)^{-1}$, see also\n\\cite[Exa.\\,2.36]{Brai02GCB}.\n\nIn particular, we see that \\textsc{Mosco} convergence only holds for the case\nthat the harmonic and the arithmetic mean are equal, which means that \n$\\bbA(y,u)$ has to be independent of $y$. \n\nFor the energy functional $\\calE^\\eps$ we can rely on the general\ntheory of homogenization as surveyed in\n\\cite{Brai06HGC}. Using the uniform coercivity\n\\eqref{eq:E.coercive} we obtain weak $\\Gamma$-convergence in\n$\\rmW^{1,p}(\\Omega;\\R^I)$ and, by the compact embedding, strong\n$\\Gamma$-convergence in $V=\\rmL^2(\\Omega;\\R^I)$ towards the limit\n\\begin{align*}\n&\\calE^0(u)=\\int_\\Omega F^\\text{hom}(u(x),\\nabla u(x))\\dd x \\text{ with }\\\\\n&F^\\text{hom}(u,U):= \\min\\Bigset{\\int_{(0,1)^d} \\bbF(y,u,U{+}\\nabla\\Phi(y)) \\dd\n  y}{ \\Phi\\in \\rmW^{1,p}_\\text{per}((0,1)^d;\\R^I)},\n\\end{align*} \nsee \\cite[Thm.\\,5.1, pp.\\,135]{Brai06HGC}.  Of course, $\\calE^0:V\\to\n[0,\\infty]$ is a again a convex and lower semicontinuous functional.\nFinally, setting \n\\[\nB^0(t,u): x \\mapsto b^\\text{aver}(t,u(x)) \\quad \\text{ with }\nb^\\text{aver}(t,u)=\\int_{(0,1)^d} \\bbB(y,u)\\dd y   \n\\]\nwe obtain the desired convergence $B^{\\eps_n}(t_n,u_n)\\to B^0(t,u)$ if\n$(\\eps_n,t_n,u_n)\\to (0,t,u)$ in $[0,1]\\ti [0,T]\\ti V$. \n\nHence, we see that Theorem \\ref{th:EGC.main}, which is the\nmain result on evolutionary $\\Gamma$-convergence, is only applicable\nif we have the \\textsc{Mosco} convergence $\\Psi^{\\eps_k}_{u_k}\\xrightarrow{\\,\\rmM\\,} \\Psi^0_u$, which\nmeans  $\\Psi^\\text{harm}_u = \\Psi^\\text{aver}_u$. Thus, we need to\nassume that $\\bbA(y,u)$ does not depend on the microscopic periodicity\nvariable $y \\in \\R^d\/_{\\Z^d}$. In summary we obtain the following\nresult. \n\n\\begin{thm}[Homogenization I] \\label{th:Hom.I}\nConsider the perturbed gradient system $\\mathrm{PG}^\\eps=(\\rmL^2(\\Omega;\\R^I),\n\\calE^\\eps, \\Psi^\\eps, b^\\eps)$ be \ngiven as above. Assume that \\eqref{eq:Exa.Ass} holds and that\n\\eqref{eq:Coeff.2s} holds with $\\bbA$ independent of the variable $y={\\ts\\frac{\\ds1}{\\ds\\eps}} \nx$, the we have evolutionary $\\Gamma$-convergence in the sense of\nTheorem \\ref{th:EGC.main} to the perturbed gradient system $(\\rmL^2(\\Omega;\\R^I), \\calE^0,\n\\Psi^\\text{aver}, b^\\text{aver})$, i.e.\\ solutions $u_\\eps$ of the\nreaction-diffusion system \\eqref{eq:ExaPDE} converge to solutions of\nthe homogenized system \n\\begin{align}\n\\nonumber\n \\hspace*{1em}\n A^\\text{aver}(u) \\pl_t u &= \\Div \\big(\\pl_{\\nabla u}\n  F^\\text{hom} \\!\\,(u,\\nabla u) \\big)\n- \\pl_u F^\\text{hom}\\!\\,(u,\\nabla u) + \n     b^\\text{aver}(t,u) &\\text{in }&\\Omega,\\\\\n\\label{eq:ExaPDE.eff}\n 0&=\\pl_{\\nabla u}\n  F^\\text{hom}\\!\\,(u,\\nabla u)\\nu &\\text{on }&\\pl\\Omega.\n\\hspace*{1em}\n\\end{align}\n\\end{thm}\n\nThe case where $\\bbA(y,u)$ depends on $y\\in \\R^d\/_{\/\\Z^d}$ is more\ndifficult. Under additional assumptions we will be able to use the\nimproved theory developed in Corollary \\ref{co:StrongGa}, as we can\nuse $\\Psi^{\\eps_k}_{u_k}\\xrightarrow{\\Gamma} \\Psi^\\text{aver}_u$, which gives\nassumption \\eqref{eq:Psi.strongG.b}. However, we need to\nestablish the uniform continuity \\eqref{eq:Psi.strongG.a}. For this we\nnote that $G^\\eps(u_j)\\leq R$ implies $\\| u_j\\|_{\\rmW^{1,p}}\\leq\nC_R$. Now, assuming $p>d$ we first observe $\\|u_j\\|_{\\rmL^\\infty}\\leq\n\\wt C_R<\\infty$, and a Gagliardo-Nirenberg estimate yields  \n\\[\n\\|u_1{-}u_2\\|_{\\rmL^\\infty} \\leq C_{\\rmG\\rmN}\n\\|u_1{-}u_2\\|_{\\rmL^2}^\\theta \\|u_1{-}u_2\\|^{1-\\theta}_{\\rmW^{1,p}}\n\\leq  C_{\\rmG\\rmN} (2C_R)^{1-\\theta} \\|u_1{-}u_2\\|_{\\rmL^2}^\\theta.\n\\]\nNow assuming the uniform continuity \n\\begin{equation}\n  \\label{eq:bbAUnifCont}\n\\begin{aligned}\n&\\forall\\, \\rho>0\\ \\exists\\text{ modulus of contin.\\,} \\omega_\\rho\\\n\\forall\\, y\\in (0,1)^d\\ \\forall\\, u_j\\in B_\\rho(0)\\subset \\R^I:\\\\\n&\\qquad \n\\big|  \\bbA(y,u_1)-\\bbA(y,u_2)\\big| \\leq \\omega_\\rho(|u_1{-}u_2|) ,\n\\end{aligned}\n\\end{equation}\nwe can estimate the difference $ \\Psi^\\eps_{u_1}(v)-\n\\Psi^\\eps_{u_2}(v)$ of the dissipation potentials pointwise under the\nintegral and obtain\n\\[\n\\forall\\, u_j\\in V \\text{ with } G^\\eps(u_j)\\leq R: \\quad \n\\big|\\Psi^\\eps_{u_1}(v)- \\Psi^\\eps_{u_2}(v)\\big| \\leq \\omega_{\\wh\n  C_R} \\big( C_{\\rmG\\rmN} (2C_R)^{1-\\theta}\n\\|u_1{-}u_2\\|_{\\rmL^2}^\\theta\\big) \\| v\\|_{\\rmL^2}^2. \n\\]\nThis is exactly the desired uniform continuity \\eqref{eq:Psi.strongG.a}.\nThus, Corollary \\ref{co:StrongGa} is applicable under the additional\nassumption that $p>d$ and that \\eqref{eq:bbAUnifCont} holds, which\ngives our second homogenization result, where $\\bbA$ now may depend\nperiodically on $y={\\ts\\frac{\\ds1}{\\ds\\eps}} x$. \n\n\n\\begin{thm}[Homogenization II] \\label{th:Hom.II}\nConsider the perturbed gradient systems $(\\rmL^2(\\Omega;\\R^I), \\calE^\\eps, \\Psi^\\eps, b^\\eps)$\ngiven as above. Assume that \\eqref{eq:Exa.Ass} holds with $p>d$ and that\n\\eqref{eq:Coeff.2s} together with \\eqref{eq:bbAUnifCont}. Then all the\nconclusions of Theorem \\ref{th:Hom.I} remain true. \n\\end{thm}\n\nIndeed, we conjecture that these two additional conditions (either\n$\\bbA$ independent of $y={\\ts\\frac{\\ds1}{\\ds\\eps}} x$ or \\eqref{eq:bbAUnifCont}) are not\nreally necessary. Using two-scale unfolding as in \\cite{MiReTh14TSHN,\n  Reic16EEEE, Reic17CECI} and   \na suitable version of \\textsc{Ioffe}'s theorem it\nshould be possible to prove the fundamental liminf estimate \n\\[\n\\int_0^T \\Psi^\\text{aver}_{u(t)}(u'(t))\\dd t \\leq \\liminf_{k\\to\n  \\infty} \\int_0^T \\Psi^{\\eps_k}_{u_{\\eps_k}(t)}(u'_{\\eps_k}(t))\\dd t\n\\]\nin much more general cases. \n\\color{black}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}