Existence and boundedness of solutions for a singular phase field system

TL;DR

This paper analyzes a complex thermomechanical PDE model for phase transitions, establishing the existence, boundedness, and uniqueness of solutions despite nonlinear and singular terms.

Contribution

It provides the first rigorous proof of existence and boundedness for solutions to a singular phase field system with thermomechanical coupling.

Findings

Existence of solutions for the PDE system.

Boundedness of phase variable and temperature.

Uniqueness of solutions under Lipschitz conditions.

Abstract

This paper is devoted to the mathematical analysis of a thermomechanical model describing phase transitions in terms of the entropy and order structure balance law. We consider a macroscopic description of the phenomenon and make a presentation of the model. Then, the initial and boundary value problem is addressed for the related PDE system, which contains some nonlinear and singular terms with respect to the temperature variable. Existence of the solution is shown along with the boundedness of both phase variable $\chi$ and absolute temperature $\theta$. Finally, uniqueness is proved in the framework of a source term depending Lipschitz continuously on $\theta$.