Kobayashi-Warren-Carter System of Singular Type under Dynamic Boundary Condition

TL;DR

This paper studies a coupled Allen--Cahn and singular diffusion system modeling grain boundary motion, focusing on dynamic boundary conditions and proving existence of solutions amid conflicting conditions.

Contribution

It establishes the existence of solutions for the Kobayashi--Warren--Carter system with dynamic boundary conditions, addressing the conflict between transmission continuity and diffusion discontinuity.

Findings

Proved existence of solutions with energy dissipation.

Provided a key lemma interpreting the conflicting boundary conditions.

Analyzed the mathematical structure of the singular diffusion system.

Abstract

In this paper, we consider a coupled system, known as Kobayashi--Warren--Carter system, abbreviated as the KWC system. KWC system consists of an Allen--Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140, 141--150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation.