Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

TL;DR

This paper establishes the existence of weak solutions, local strong solutions, and weak-strong uniqueness for a thermodynamically consistent phase-field model in two and three dimensions, advancing the mathematical understanding of such models.

Contribution

It introduces a new framework for proving existence and uniqueness of solutions for a phase-field model using energy and entropy inequalities.

Findings

Existence of weak solutions in 2D and 3D

Existence of local-in-time strong solutions

Weak-strong uniqueness of solutions

Abstract

In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.