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 (P)ϵ is introduced, and then, a variational approach and a fixed-point argument are employed to prove existence of strong solutions to (P)ϵ. More precisely, we introduce a family of functionals (defined over entire trajectories) parametrized by a small parameter ϵ, 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 uϵ to the elliptic-in-time regularized equations (P)ϵ. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of uϵ as ϵ → 0. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.
|Journal of Convex Analysis
|Published - 2018