Existence and approximation results for gradient flows

Rossi, Riccarda   ; Savaré, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 183-196

Résumé

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$ $$ \begin{cases} u^{\prime}(t) + \partial \phi(u(t)) \ni 0 \quad \text{a.e. in} \, (0,T),\\ u(0) = u_{0}, \end{cases} $$ where $\phi : H \rightarrow (-\infty , +\infty \,]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial \phi$ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs have a \textit{common gradient flow structure}. In particular, when quasi-stationary models are considered, highly non-convex functionals naturally arise. We will present some existence results for the solution of the gradient flow equation by exploiting a variational \textit{approximation} technique, featuring some ideas from the theory of \textit{Minimizing Movements}.