Attractors for gradient flows of non convex functionals and applications

TL;DR

This paper investigates the long-term dynamics of gradient flows for non-convex functionals in Hilbert spaces, establishing conditions for global attractors and applying these to phase field models.

Contribution

It introduces new sufficient conditions for the existence of global attractors for non-convex gradient flows, extending the understanding of their long-time behavior.

Findings

Existence of global attractors for certain non-convex gradient flows

Application to phase field models demonstrating long-time stability

Framework based on generalized semiflows for analyzing asymptotic behavior

Abstract

This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor.