Analysis of a nonisothermal and conserved phase field system with inertial term

TL;DR

This paper investigates a conserved phase field system with inertial effects, analyzing existence and convergence of solutions, and extending classical models to include inertial and viscous terms for rapid phase transformations.

Contribution

It introduces a mathematical analysis of a nonisothermal, inertial phase field system, proving existence of solutions and their convergence as inertial effects diminish.

Findings

Proved existence of global solutions for the inertial phase field system.

Established convergence results as inertial coefficient approaches zero.

Extended analysis to viscous variants of the system.

Abstract

This paper deals with a conserved phase field system that couples the energy balance equation with a Cahn--Hilliard type system including temperature and the inertial term for the order parameter. In the case without inertial term, the system under study was introduced by Caginalp. The inertial term is motivated by the occurrence of rapid phase transformation processes in nonequilibrium dynamics. A double-well potential is well chosen and the related nonlinearity governing the evolution is assumed to satisfy a suitable growth condition. The viscous variant of the Cahn--Hilliard system is also considered along with the inertial term. The existence of a global solution is proved via the analysis of some approximate problems with Yosida regularizations, and the use of the Cauchy--Lipschitz--Picard theorem in an abstract setting. Moreover, we study the convergence of the system, with or without the viscous term, as the inertial coefficient tends to zero.