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Existence of global attractors and convergence of solutions for 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 and prove existence of global attractors in higher order Mellin-Sobolev spaces with asymptotics. We also show convergence of solutions in the same spaces to an equilibrium point and provide asymptotic behavior of the equilibrium near the conical tips in terms of the local geometry.

Existence of global attractors and convergence of solutions for the Cahn-Hilliard equation on manifolds with conical singularities

Abstract

We consider the Cahn-Hilliard equation on manifolds with conical singularities and prove existence of global attractors in higher order Mellin-Sobolev spaces with asymptotics. We also show convergence of solutions in the same spaces to an equilibrium point and provide asymptotic behavior of the equilibrium near the conical tips in terms of the local geometry.
Paper Structure (7 sections, 25 theorems, 115 equations)

This paper contains 7 sections, 25 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Sobolev immersions
  4. Realizations of the Laplacian and bi-Laplacian
  5. Existence and regularity of the global attractors
  6. Convergence to the equilibrium
  7. Acknowledgements

Key Result

Theorem 1

Let $s\ge0$, $\gamma$ be as gamma and $\mathcal{D}(\Delta_{s}^{2})$ be the bi-Laplacian domain described in eq:defbilap-bilapintro4. (i) (Global attractor) The semiflow $T:[0,\infty)\times X_{1,0}^{s}\to X_{1,0}^{s}$ has an $s$-independent global attractor $\mathcal{A}\subset\cap_{r>0}\mathcal{D}(\D (ii) (Convergence to equilibrium) If $u_{0}\in X_{1,0}^{0}$, then there exists a $u_{\infty}\in\cap

Theorems & Definitions (54)

  • Theorem 1
  • Definition 2: Mellin-Sobolev spaces
  • Remark 3
  • Definition 4
  • Remark 5
  • Proposition 6
  • proof
  • Lemma 7: Poincaré-Wirtinger inequality
  • proof
  • Definition 8
  • ...and 44 more