Elliptic-regularization of nonpotential perturbations of doubly-nonlinear gradient flows of nonconvex energies: A variational approach

TL;DR

This paper develops a variational method with elliptic-in-time regularization to prove the existence of strong solutions for doubly-nonlinear gradient flows with nonpotential perturbations, extending the analysis to concrete PDE applications.

Contribution

It introduces a novel variational framework and fixed-point approach for doubly-nonlinear flows with nonpotential perturbations, including an elliptic-in-time regularization technique.

Findings

Established existence of strong solutions via regularization and fixed-point methods.

Developed a variational approach applicable to nonconvex energies with perturbations.

Extended the theory to specific PDE models.

Abstract

This paper presents a variational approach to doubly-nonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An elliptic-in-time regularization of the original equation ${\rm (P)}_\varepsilon$ is introduced, and then, a variational approach and a fixed-point argument are employed to prove existence of strong solutions to regularized equations. More precisely, we introduce a functional (defined for each entire trajectory and including a small approximation parameter $\varepsilon$) whose Euler-Lagrange equation corresponds to the elliptic-in-time regularization of an unperturbed (i.e. without nonpotential perturbations) doubly-nonlinear flow. Secondly, due to the presence of nonpotential perturbation, a fixed-point argument is performed to construct strong solutions $u_\varepsilon$ to the elliptic-in-time regularized equations ${\rm (P)}_\varepsilon$. Here, the minimization problem mentioned above defines an operator $S$ whose fixed point corresponds to a solution $u_\varepsilon$ of ${\rm (P)}_\varepsilon$. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of $u_\varepsilon$ as $\varepsilon \to 0$. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.