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Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility

Jonas Stange

Abstract

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility

Abstract

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

Paper Structure

This paper contains 14 sections, 16 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Functional framework, assumptions and preliminaries
  3. Notation and Function Spaces.
  4. Important tools
  5. Main Assumptions.
  6. Main results
  7. Well-posedness of weak solutions
  8. Propagation of regularity and instantaneous separation property.
  9. Long-time behavior.
  10. Elliptic bulk-surface system with non-constant coefficients
  11. Existence and Uniqueness of Weak Solutions
  12. Propagation of Regularity and Instantaneous Separation Property
  13. Convergence to Equilibrium
  14. A bulk-surface chain rule

Key Result

Lemma 2.1

Let $K\in[0,\infty)$ and $\alpha,\beta\in\mathbb R$ such that $\alpha\beta\left| \Omega \right| + \left| \Gamma \right|\neq 0$. Then, there exists a constant $C_P > 0$, depending only on $K,\alpha,\beta$ and $\Omega$ such that for all pairs $(\zeta,\xi)\in\mathcal{H}^1_K$ satisfying $\textnormal{mean} \left( \zeta , \xi \right) = 0$.

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 22 more