Energy-dissipation in a coupled system of Allen--Cahn type equation and Kobayashi--Warren--Carter type model of grain boundary motion

Hiroshi Watanabe,Ken Shirakawa

doi:10.22541/au.158359823.36732693

Abstract

In this paper, we consider a system of initial boundary value problems for parabolic equations, as a generalized version of the “φ-η-θ model’‘ of grain boundary motion, proposed by Kobayashi [16]. The system is a coupled system of: an Allen–Cahn type equation as in (1.1) with a given temperature source; and a phase-field model of grain boundary motion, known as “Kobayashi–Warren–Carter type model”. The focus of the study is on a special kind of solution, called energy-dissipative solution, which is to reproduce the energy-dissipation of the governing energy in time. Under suitable assumptions, two Main Theorems, concerned with: the existence of energy-dissipative solution; and the large-time behavior; will be demonstrated as the results of this paper.

Highlights

Let N ∈ N be a constant of spatial dimension, and Ω ⊂ RN be a bounded domain such that Γ := ∂Ω is smooth when N > 1
Let us denote by Q := (0, ∞) × Ω the product space of the time-interval (0, ∞) and the spatial domain Ω, and let us set Σ := (0, ∞) × Γ
For general case of (S)ν, there is only one result for (T1), and there is no result to give mathematical answers for the remaining issues (T2)–(T4), yet. In view of such background, we set the goal of this paper to establish a general mathematical theory that enable a uniform treatment for the issues (T1)–(T4), under various settings of the system (S)ν

Summary

Introduction

Let N ∈ N be a constant of spatial dimension, and Ω ⊂ RN be a bounded domain such that Γ := ∂Ω is smooth when N > 1. We fix a constant ν ≥ 0, and consider the following system of initialboundary value problems of parabolic types, denoted by (S)ν. With regard to the Kobayashi–Warren–Carter type models, the most of mathematical results, obtained in the previous works [20, 21, 24, 25, 26, 27, 28, 29, 32, 33], are classified in the following four issues. In view of such background, we set the goal of this paper to establish a general mathematical theory that enable a uniform treatment for the issues (T1)–(T4), under various settings of the system (S)ν On this basis, the principal discussion will be devoted to the proofs of the following two main theorems. Main Theorem 2: the large-time behavior of energy-dissipative solutions.

Preliminaries

Assumptions and Main Theorems

Approximate problems

Proof of Main Theorem 1

Proof of Main Theorem 2