Time relaxation of a phase-field model with entropy balance

TL;DR

This paper analyzes a coupled phase-field model with entropy balance, proving well-posedness and examining the limit as the relaxation parameter approaches zero, leading to a stationary phase equation.

Contribution

It introduces a time-relaxation term into a phase-field model with entropy balance and studies its limit behavior, which was not previously addressed.

Findings

Proved well-posedness of the coupled system.

Demonstrated convergence to a stationary phase equation as relaxation tends to zero.

Established mathematical framework for phase transition models with entropy considerations.

Abstract

We deal with a system of two coupled differential equations, describing the evolution of a first order phase transition. In particular, we have two non-linear parabolic equations: the first one is deduced from a balance law for entropy and it describes the evolution of the absolute temperature; the other one is an equilibrium equation for microforces and it regulates the behaviour of a scalar phase parameter. Moreover, the second equation shows a time-relaxation coefficient related to the time-derivative of the phase parameter. We prove well-posedness of solution to the given system, using a standard method of approximating problems; afterwards, we study the behaviour of the system as the time-relaxation coefficient tends to zero: as a result, we find out that the original problem converges to a new problem, with a stationary phase equation.