Mosco convergence framework for singular limits of gradient flows on Hilbert spaces with applications

TL;DR

This paper develops a framework based on Mosco convergence for analyzing the singular limits of gradient flows across different Hilbert spaces, with applications to various complex systems.

Contribution

It introduces connecting operators to extend Mosco convergence to gradient flows on varying Hilbert spaces, providing a new theoretical tool for singular limit analysis.

Findings

Established a generalized Mosco convergence framework for gradient flows

Proved convergence results for flows in thin domains and discrete-to-continuum limits

Provided multiple examples demonstrating the applicability of the framework

Abstract

We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of Mosco convergence of functionals to our present setting, and state a desired convergence result for gradient flows, which we then prove. We present a variety of examples, including thin domains, dynamic boundary conditions, and discrete-to-continuum limits.