Existence of solutions to a phase-field model of 3D-grain boundary motion governed by a regularized 1-harmonic type flow

TL;DR

This paper introduces a quaternion-based formulation for a 3D grain boundary motion model, proving the existence of solutions using approximation techniques and establishing a maximum principle for the orientation variable.

Contribution

It presents a novel quaternion formulation for the 3D Kobayashi–Warren model and proves the existence of solutions via approximation and regularization methods.

Findings

Existence of solutions to the model established.

A maximum principle for the orientation variable proven.

Use of Ginzburg-Landau type approach and nonlinear semigroups.

Abstract

In this paper we propose a quaternion formulation for the orientation variable in the three dimensional Kobayashi--Warren model for the dynamics of polycrystals. We obtain existence of solutions to the $L^2$-gradient descent flow of the constrained energy functional via several approximating problems. In particular, we use a Ginzburg-Landau type approach and some extra regularizations. Existence of solutions to the approximating problems is shown by the use of nonlinear semigroups. Coupled with good a-priori estimates, this leads to successive passages to the limit up to finally showing existence of solutions to the proposed model. Moreover, we also obtain a maximum principle for the orientation variable.