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

TL;DR

This work analyzes a nonisothermal, conserved phase-field model with an inertial term $\tau \varphi_{tt}$ that couples the energy balance to a Cahn–Hilliard-type equation. The authors develop a multi-layer approximation scheme using Moreau–Yosida regularization in the nonlinear potential and a Yosida regularization of the (Neumann) Laplacian, combined with a Cauchy–Lipschitz–Picard framework to prove global existence of weak solutions for the hyperbolic–parabolic system $(P)_\tau$ and its viscous variant $(P)_{\tau,\eta}$. They obtain uniform energy estimates and perform careful limit passages: $\lambda\to0$, $\varepsilon\to0$, and $\eta\to0$, eventually deriving weak solutions to the original problems and establishing the vanishing inertia limit $\tau\to0$ to connect to the parabolic model. The results provide a rigorous justification for the inertial phase-field system under nonisothermal conditions and offer a robust analytical path for similar thermodynamically consistent models with memory or hyperbolic relaxation.

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.

Theorems & Definitions (51)

• Remark 2.1
• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 2.2
• Remark 3.1
• Remark 3.2
• Definition 3.1
• Theorem 3.1
• Lemma 5.1