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

Nikolaos Roidos, Elmar Schrohe

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.

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

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.

Paper Structure

This paper contains 17 sections, 18 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminary Results
  3. Bounded imaginary powers
  4. The Laplacian on Mellin-Sobolev spaces over a manifold with conical singularities
  5. Closed extensions of the Laplace operator
  6. The Linearized Problem
  7. The choice of an extension of $\Delta$
  8. The two-dimensional case
  9. The three-dimensional case
  10. Higher dimensions
  11. The domain of $\underline\Delta^2$
  12. Embedding the interpolation space $X_{q}$
  13. Bounded imaginary powers
  14. The Nonlinear Equation in ${\mathcal{H}}^{0,\gamma}_p({\mathbb B})$
  15. The two-dimensional case.
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Clément and Li, CL, Theorem 2.1) Assume that there exists an open neighborhood $U$ of $u_0$ in $X_q$ such that $A(u_0)$ has maximal regularity for $(X_1,X_0)$ and $q$, and that Then there exists a $T>0$ and a unique $u\in L^q(0,T;X_1)\cap W^1_q(0,T;X_0)\cap C([0,T];X_q)$ solving the equation e0.1 on $]0,T[$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 21 more