Energy dissipative solutions to the Kobayashi–Warren–Carter system

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 on the existence of solutions to this system which dissipate the associated energy functional. We obtain the existence of this type of solution 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 ω-limit set of the solutions.