Hyperbolic relaxation of the chemical potential in the viscous Cahn-Hilliard equation

TL;DR

This paper introduces a hyperbolic relaxation approach to the viscous Cahn-Hilliard equation, establishing well-posedness, regularity, and convergence results as the relaxation parameter diminishes.

Contribution

It develops a new hyperbolic relaxation model for the viscous Cahn-Hilliard system and proves its well-posedness and convergence to the classical model.

Findings

Proved well-posedness and regularity of the relaxed system

Established convergence to the viscous Cahn-Hilliard system as relaxation parameter tends to zero

Analyzed the asymptotic behavior of the hyperbolic relaxation model

Abstract

In this paper, we study a hyperbolic relaxation of the viscous Cahn--Hilliard system with zero Neumann boundary conditions. In fact, we consider a relaxation term involving the second time derivative of the chemical potential in the first equation of the system. We develop a well-posedness, continuous dependence and regularity theory for the initial-boundary value problem. Moreover, we investigate the asymptotic behavior of the system as the relaxation parameter tends to 0 and prove the convergence to the viscous Cahn--Hilliard system.