On the viscous Cahn-Hilliard equation with singular potential and inertial term

TL;DR

This paper studies a modified viscous Cahn-Hilliard equation with an inertial term and a singular potential, establishing a framework for weak solutions and proving their existence.

Contribution

It introduces a novel approach to handle the inertial term combined with a singular potential in the viscous Cahn-Hilliard equation, proving existence of weak solutions.

Findings

Existence of weak solutions for the equation.

Development of a duality-based solution concept.

Handling of the singular potential in the mathematical analysis.

Abstract

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term~$u_{tt}$. The equation also contains a semilinear term $f(u)$ of "singular" type. Namely, the function $f$ is defined only on a bounded interval of ${\mathbb R}$ corresponding to the physically admissible values of the unknown $u$, and diverges as $u$ approaches the extrema of that interval. In view of its interaction with the inertial term $u_{tt}$, the term $f(u)$ is difficult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.