An existence result and evolutionary \(\varGamma \)-convergence for perturbed gradient systems

Abstract

The initial-value problem for the perturbed gradient flow

$$\begin{aligned} B(t,u(t)) \in \partial \varPsi _{u(t)}(u'(t))+\partial {\mathcal {E}}_t(u(t)) \ \text { for a.a. } t\in (0,T),\qquad u(0)=u_0, \end{aligned}$$

with a perturbation B in a Banach space V is investigated, where the dissipation potential \(\varPsi _u: V\rightarrow [0,+\infty )\) and the energy functional \({\mathcal {E}}_t:V\rightarrow (-\infty ,+\infty ]\) are non-smooth and supposed to be convex and nonconvex, respectively. The perturbation \(B:[0,T]\times V \rightarrow V^*,\ (t,v)\mapsto B(t,v)\) is assumed to be continuous and satisfies a growth condition. Under suitable assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique. Moreover, for perturbed gradient systems \((V,{\mathcal {E}}^\varepsilon ,\varPsi ^\varepsilon ,B^\varepsilon )\) depending on a small parameter \(\varepsilon >0\), we develop a theory of evolutionary \(\varGamma \)-convergence in terms of the suitable convergences of \({\mathcal {E}}^\varepsilon \), \(\varPsi ^\varepsilon \), and \(B^\varepsilon \) to the limit system \((V,{\mathcal {E}}^0, \varPsi ^0,B^0)\).

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Acknowledgements

This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through the grant SFB 910 Control of self-organizing nonlinear systems, Project A5 “Pattern formation in coupled parabolic systems” (for A.M.) and Project A8 “Nonlinear evolution equations: model hierarchies and complex fluids” (for A.B. and E.E.).

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Correspondence to Alexander Mielke.

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The research was partially supported by DFG via SFB 910, subprojects A5 and A8.

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Bacho, A., Emmrich, E. & Mielke, A. An existence result and evolutionary \(\varGamma \)-convergence for perturbed gradient systems. J. Evol. Equ. 19, 479–522 (2019). https://doi.org/10.1007/s00028-019-00484-x

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Keywords

  • Doubly nonlinear equations
  • Generalized and perturbed gradient flows
  • Evolutionary Gamma convergence
  • Homogenization of reaction-diffusion systems

Mathematics Subject Classification

  • 35A15
  • 35K50
  • 35K85
  • 49Q20