On the 2D Cahn-Hilliard equation with inertial term

TL;DR

This paper investigates the mathematical properties of a modified 2D Cahn-Hilliard equation with inertial effects, establishing well-posedness, long-term behavior, and the existence of attractors in bounded domains.

Contribution

It provides new results on the existence of global and exponential attractors for the 2D Cahn-Hilliard equation with inertial term, addressing regularity and stability issues.

Findings

Existence of bounded absorbing sets in phase spaces

Proof of global attractor existence

Demonstration of exponential attractor presence

Abstract

P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.