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Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

Patrik Knopf, Andrea Poiatti, Jonas Stange, Sema Yayla

Abstract

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as $t\to\infty$ to a single steady state. The proof of this convergence relies on the Łojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.

Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

Abstract

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as to a single steady state. The proof of this convergence relies on the Łojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.
Paper Structure (16 sections, 18 theorems, 49 equations)

This paper contains 16 sections, 18 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Main results, difficulties, and novelties of this paper.
  3. Notation, assumptions, and preliminaries
  4. Notation
  5. Assumptions
  6. Preliminaries
  7. Useful inequalities
  8. Well-posedness and instantaneous regularization
  9. Well-posedness and energy estimates
  10. Instantaneous regularization
  11. Existence of a minimal pullback attractor
  12. Formulation as a process
  13. Existence of a unique pullback attractor
  14. Existence of a pullback absorbing set
  15. Existence of a minimal pullback attractor
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $K\in[0,\infty)$ and $\alpha,\beta\in\mathbb{R}$ with $\alpha\beta| \Omega | + | \Gamma | \neq 0$ be arbitrary. Then there exists a constant $C_P >0$ depending only on $K, \alpha, \beta$ and $\Omega$ such that for all $\left( \phi , \psi \right)\in\mathcal{H}^1_{K,\alpha}$ satisfying $\beta| \Omega |{\langle \phi \rangle}_{\Omega} + | \Gamma |{\langle \psi \rangle}_{\Gamma} = 0$.

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1: Weak solution of system \ref{['conCH*']}
  • Lemma 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Corollary 3.6
  • Definition 4.1: Continuous process
  • ...and 18 more