Existence of Solutions to a Phase-Field Model of 3D Grain Boundary Motion Governed by a Regularized 1-Harmonic Type Flow

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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-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 an invariance principle for the orientation variable.