Solvability and asymptotic analysis of a generalization of the Caginalp phase field system

TL;DR

This paper analyzes a generalized phase field model incorporating thermal displacement and a non-smooth graph, proving existence, uniqueness, regularity, and asymptotic convergence to the classical Caginalp model.

Contribution

It extends the Caginalp phase field system by including a diffusive term for thermal displacement and handles non-smooth graphs, providing comprehensive mathematical analysis.

Findings

Proved existence and uniqueness of weak solutions.

Established regularity and boundedness of solutions.

Showed convergence to the classical Caginalp system as diffusion coefficient tends to zero.

Abstract

We study a diffusion model of phase field type, which consists of a system of two partial differential equations involving as variables the thermal displacement, that is basically the time integration of temperature, and the order parameter. Our analysis covers the case of a non-smooth (maximal monotone) graph along with a smooth anti-monotone function in the phase equation. Thus, the system turns out a generalization of the well-known Caginalp phase field model for phase transitions when including a diffusive term for the thermal displacement in the balance equation. Systems of this kind have been extensively studied by Miranville and Quintanilla. We prove existence and uniqueness of a weak solution to the initial-boundary value problem, as well as various regularity results ensuring that the solution is strong and with bounded components. Then we investigate the asymptotic behaviour of the solutions as the coefficient of the diffusive term for the thermal displacement tends to 0 and prove convergence to the Caginalp phase field system as well as error estimates for the difference of the solutions.