Kobayashi--Warren--Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

TL;DR

This paper analyzes a PDE system modeling grain boundary motion in polycrystals with fixed boundary orientations, establishing global solutions, steady states, and characterizing long-term behavior in 1D and 2D cases.

Contribution

It provides the first global existence results for energy-dissipative solutions of the Kobayashi--Warren--Carter system with nonhomogeneous boundary data.

Findings

Global existence of energy-dissipative solutions

Characterization of steady states as asymptotic limits

Complete description of the omega-limit set in 1D

Abstract

In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the $\omega$-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the $\omega$-limit set belong to thee space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constants solutions.