Weak Solution to KWC Systems of Pseudo-Parabolic Type

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a class of pseudo-parabolic PDE systems modeling grain boundary motion, even with less regular initial data, and analyzes their continuous dependence on parameters.

Contribution

It extends prior work by proving weak solution existence and uniqueness under weaker initial data regularity and studies parameter dependence.

Findings

Proved existence of weak solutions.

Established uniqueness of weak solutions.

Analyzed continuous dependence on parameters.

Abstract

In this paper, a class of systems of pseudo-parabolic PDEs is considered. These systems (S)$_\varepsilon$ are derived as a pseudo-parabolic dissipation system of Kobayashi--Warren--Carter energy, proposed by [Kobayashi et al., Physica D, 140, 141--150 (2000)], to describe planar grain boundary motion. These systems have been studied in [arXiv:2402.10413], and solvability, uniqueness and strong regularity of the solution have been reported under the setting that the initial data is sufficiently smooth. Meanwhile, in this paper, we impose weaker regularity on the initial data, and work on the weak formulation of the systems. In this light, we set our goal of this paper to prove two Main Theorems, concerned with: the existence and the uniqueness of weak solution to (S)$_\varepsilon$, and the continuous dependence with respect to the index $\varepsilon$, initial data and forcings.