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}.
@article{RLIN_2004_9_15_3-4_183_0,
author = {Riccarda Rossi and Giuseppe Savar\'e},
title = {Existence and approximation results for gradient flows},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {15},
year = {2004},
pages = {183-196},
zbl = {1162.34343},
mrnumber = {2148878},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_2004_9_15_3-4_183_0}
}
Rossi, Riccarda; Savaré, Giuseppe. Existence and approximation results for gradient flows. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004) pp. 183-196. http://gdmltest.u-ga.fr/item/RLIN_2004_9_15_3-4_183_0/
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