Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential

TL;DR

This paper proves that weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential converge to equilibrium, extending results to three dimensions and related coupled systems under minimal assumptions.

Contribution

It establishes convergence to equilibrium for weak solutions of the Cahn--Hilliard equation with singular potential in three dimensions, a previously open problem, using minimal assumptions.

Findings

Weak solutions converge to a single equilibrium.

Results hold in three dimensions.

Applicable to coupled Cahn--Hilliard-Navier--Stokes systems.

Abstract

We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Gr\"un (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely.