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Smoothness and long time existence for solutions of the Cahn-Hilliard equation on manifolds with conical singularities

Pedro T. P. Lopes, Nikolaos Roidos

Abstract

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data, we show that the solution exists in the maximal $L^q$-regularity space for all times and becomes instantaneously smooth in space and time, where the maximal $L^q$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.

Smoothness and long time existence for solutions of the Cahn-Hilliard equation on manifolds with conical singularities

Abstract

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data, we show that the solution exists in the maximal -regularity space for all times and becomes instantaneously smooth in space and time, where the maximal -regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.

Paper Structure

This paper contains 13 sections, 18 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Maximal $L^{q}$-regularity theory for semilinear parabolic problems
  3. Existence and uniqueness
  4. Smoothing
  5. Manifolds with conical singularities
  6. Cone differential operators and Mellin-Sobolev spaces.
  7. Realizations of the Laplacian on conic manifolds
  8. The Cahn-Hilliard equation on conic manifolds
  9. Short time solution and smoothness
  10. Long time solution
  11. A priori estimate in $\mathcal{H}^{1,1+\gamma}_{2}(\mathbb{B})$
  12. A priori estimate in $\mathcal{D}((c-\underline{\Delta}_{0})^{\frac{3+\nu}{2}})$
  13. Acknowledgment

Key Result

Theorem 1.1

Let $p\in(1,\infty)$, $s\ge0$, $s+2>\frac{n+1}{p}$, where $\lambda_{1}$ is the greatest non-zero eigenvalue of the boundary Laplacian $\Delta_{h(0)}$ on $(\partial\mathcal{B},h(0))$. Moreover, denote by $\mathbb{C}_{\omega}$ the space of smooth functions on $\mathbb{B}$ that are locally constant close to the singularities, see Definition constfunt. T there exists a $T>0$ and a unique solving CH1

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1: Sectorial operators
  • Definition 2.2: Bounded imaginary powers
  • Definition 2.3
  • Theorem 2.4: Dore and Venni, DV
  • Theorem 2.5: Clément and Li
  • Remark 2.6
  • Corollary 2.7: Global existence
  • proof
  • Theorem 2.8: Prüss and Simonett
  • ...and 30 more