A variational principle for nonpotential perturbations of gradient flows of nonconvex energies

TL;DR

This paper develops a variational framework for analyzing nonpotential perturbations of gradient flows associated with nonconvex energies, proving existence and convergence of solutions in Hilbert spaces.

Contribution

It introduces a novel variational approach combining minimization and fixed-point methods to handle nonpotential perturbations in gradient flows of nonconvex energies.

Findings

Existence of solutions to elliptic-in-time regularizations established.

Regularized solutions converge to true gradient flow solutions as regularization vanishes.

Application demonstrated in nonlinear reaction-diffusion systems.

Abstract

We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the minimization of a parameter-dependent functional over entire trajectories and a fixed-point argument. These regularized solutions converge up to subsequence to solutions of the gradient flow as the regularization parameter goes to zero. Applications of the abstract theory to nonlinear reaction-diffusion systems are presented.