Existence of solutions to the Kobayashi–Warren–Carter system

Abstract

In this paper, a coupled system of two parabolic type initial-boundary value problems is considered. The system is known as the Kobayashi–Warren–Carter model of grain boundary motion in a polycrystal. Kobayashi–Warren–Carter model is derived as a gradient descent flow of an energy functional, which is called “free-energy”, with respect to two unknown variables and it involves a weighted-unknown dependent total variation term. The main goal of this paper is to obtain existence of solutions to this system. We solve the problem by means of a time-discretization of a relaxed system and a highly non-trivial passage to the limit. We point out that our time-discretization method is effective not only for the original Kobayashi–Warren–Carter system but also for its relaxed versions. Therefore, we provide a uniform approach for obtaining solutions to systems associated with this model.