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

TL;DR

This paper establishes the existence of solutions for a class of perturbed gradient flow equations in Banach spaces, using a semi-implicit discretization and variational approximation, extending the understanding of nonsmooth, nonconvex systems.

Contribution

It introduces an existence result for perturbed gradient systems with nonsmooth, nonconvex energy and dissipation, employing a novel semi-implicit discretization and variational approach.

Findings

Proved existence of strong solutions under certain conditions.

Developed a semi-implicit discretization scheme for the problem.

Extended gradient flow analysis to nonsmooth, nonconvex settings.

Abstract

The initial-value problem for the perturbed gradient flow \[ B(t,u(t)) \in \partial\Psi_{u(t)}(u'(t))+\partial \mathcal E_t(u(t)) \text{ for a.a. } t\in (0,T),\qquad u(0)=u_0 \] with a perturbation $B$ in a Banach space $V$ is investigated, where the dissipation potential $\Psi_u: V\rightarrow [0,+\infty)$ and the energy functional $\mathcal E_t:V\rightarrow (-\infty,+\infty]$ are nonsmooth 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 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.