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

Aras Bacho, Alexander Mielke, Etienne Emmrich

Abstract:

The initial value problem for the perturbed gradient flow
\begin{align*}
\label{eq:I.1}
\begin{cases}\quad
B(t,u(t)) \in \partial\Psi_{u(t)}(u'(t))+\partial \calE_t(u(t)) &
\text{ for a.a. } t\in (0,T),\\
\quad u(0)=u_0, &
\end{cases}
\end{align*}
with a perturbation $B$ in a \textsc{Banach} space $V$ is
investigated, where the dissipation potential $\Psi_u: V\rightarrow
[0,+\infty)$ and the energy functional $\calE_t:V\rightarrow
(-\infty,+\infty]$ 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.