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 $L^2$-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.