New results for the Cahn-Hilliard equation with non-degenerate mobility: well-posedness and longtime behavior

TL;DR

This paper establishes well-posedness, regularity propagation, and long-term stability of solutions to the 2D Cahn-Hilliard equation with non-degenerate mobility and logarithmic potential, advancing previous results.

Contribution

It proves uniqueness, regularity propagation, and convergence to equilibrium for the equation, improving upon prior work by Barrett and Blowey.

Findings

Weak solutions are unique.

Solutions exhibit uniform-in-time regularity.

Solutions stabilize to equilibrium states.

Abstract

We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.