An existence result and evolutionary $\Gamma$-convergence for perturbed gradient systems

Abstract

The initial-value problem for the perturbed gradient flow \[ B(t,u(t)) \in \partialΨ_{u(t)}(u'(t))+\partial \mathcal E_t(u(t)) \text{ for a.a. } t\in (0,T),\qquad u(0)=u_0 \] with a perturbation $B$ in a Banach space $V$ is investigated, where the dissipation potential ${\Psi }_u:V\to [0,+\infty )$ and the energy functional ${\mathcal{E}}_t:V\to (-\infty ,+\infty ]$ are nonsmooth and supposed to be convex and nonconvex, respectively. The perturbation $B:[0,T]\times V\to V^{\ast },(t,v)\mapsto B(t,v)$ is assumed to be continuous and satisfies a growth condition. Under additional 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.