Gradient flows of non convex functionals in Hilbert spaces and applications

Rossi, Riccarda; Savaré, Giusepp

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 564-614.

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathscr{H}$

$$\left\{\begin{array}{cc}{u}^{\text{'}}\left(t\right)+{\partial }_{\ell }\phi \left(u\left(t\right)\right)\ni f\left(t\right)\hfill & \text{a.e.}\text{in}(0,T),\hfill \\ u\left(0\right)=u_0,\hfill \end{array}\right.$$

where $\phi :\mathscr{H}\to (-\infty ,+\infty ]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ${\partial }_{\ell }\phi$ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures. Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.

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