Energy dissipative solutions to the Kobayashi-Warren-Carter system

TL;DR

This paper establishes the existence and properties of energy-dissipative solutions to the Kobayashi-Warren-Carter system, a model for grain boundary motion in polycrystals, using variational and regularization techniques.

Contribution

It introduces a novel approximation method for the energy functional and proves existence, regularity, and convergence results for solutions to the system.

Findings

Existence of energy-dissipative solutions established.

Gamma-convergence results for weighted total variations proved.

Complete characterization of the omega-limit set of solutions.

Abstract

In this paper we study a variational system of two parabolic PDEs, called the Kobayashi-Warren-Carter system, which models the grain boundary motion in a polycrystal. The focus of the study is the existence of solutions to this system which dissipate the associated energy functional. We obtain existence of this type of solutions via a suitable approximation of the energy functional with Laplacians and an extra regularization of the weighted total variation term of the energy. As a byproduct of this result, we also prove some $\Gamma$-convergence results concerning weighted total variations and the corresponding time-dependent cases. Finally, the regularity obtained for the solutions together with the energy dissipation property, permits us to completely characterize the $\omega$-limit set of the solutions.