The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

TL;DR

This paper studies the Cahn-Hilliard and Allen-Cahn equations on manifolds with conical singularities, proving short-term solvability, regularity preservation, and asymptotic behavior near singularities using advanced singular analysis techniques.

Contribution

It establishes short-time solvability and regularity results for these equations on singular manifolds, extending existing analysis to conical singularities.

Findings

Existence of bounded imaginary powers for the bilaplacian extensions.

Short-time solvability of the Cahn-Hilliard and Allen-Cahn equations.

Asymptotic behavior of solutions near conical points.

Abstract

We consider the Cahn-Hilliard equation on a manifold with conical singularities and show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clement and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.