Convergence properties for a generalization of the Caginalp phase field system

TL;DR

This paper investigates the asymptotic behavior of a generalized phase field system, extending the Caginalp model, as the thermal diffusion coefficient approaches zero, providing convergence results and rates under minimal assumptions.

Contribution

It extends previous work by analyzing the asymptotic behavior and convergence rates of a generalized phase field system with thermal diffusion tending to zero.

Findings

Proves convergence of solutions as the thermal diffusion coefficient tends to zero.

Provides uniform regularity estimates and discusses convergence rates.

Addresses minimal assumptions on data for the convergence analysis.

Abstract

We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diffusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by proving well-posedness and regularity results, as well as convergence of the problem as the coefficient of the diffusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behaviour of the problem as the coefficient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the difference of the solutions in suitable norms.