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

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 ${(\mathrm{P})}_{\epsilon }$ 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 $\epsilon$) 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 $u_{\epsilon }$ to the elliptic-in-time regularized equations ${(\mathrm{P})}_{\epsilon }$. Here, the minimization problem mentioned above defines an operator $S$ whose fixed point corresponds to a solution $u_{\epsilon }$ of ${(\mathrm{P})}_{\epsilon }$. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of $u_{\epsilon }$ as $\epsilon \to 0$. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.