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Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential

Maurizio Grasselli, Andrea Poiatti

TL;DR

This work proves that weak solutions to the Cahn–Hilliard equation with non-degenerate mobility and singular (logarithmic) potential converge to a single equilibrium in dimensions two and three, under only the assumption of a global weak solution satisfying an energy inequality. The authors introduce a robust strategy that uses a weak ω-limit in H^1, a De Giorgi-based separation on a set of times, and a Łojasiewicz–Simon framework to derive convergence without requiring finite-time regularization. They also extend the result to the Abels–Garcke–Grün CH–Navier–Stokes system with unmatched densities and viscosities, under the same minimal mobility condition, illustrating the method’s versatility. The approach yields convergence to equilibrium under analytic nonlinearity, provides uniform separation from pure phases, and delivers convergence in H^s for all s∈(0,1), offering a new tool for long-time analysis of CH-type models and related systems.

Abstract

We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely.

Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential

TL;DR

This work proves that weak solutions to the Cahn–Hilliard equation with non-degenerate mobility and singular (logarithmic) potential converge to a single equilibrium in dimensions two and three, under only the assumption of a global weak solution satisfying an energy inequality. The authors introduce a robust strategy that uses a weak ω-limit in H^1, a De Giorgi-based separation on a set of times, and a Łojasiewicz–Simon framework to derive convergence without requiring finite-time regularization. They also extend the result to the Abels–Garcke–Grün CH–Navier–Stokes system with unmatched densities and viscosities, under the same minimal mobility condition, illustrating the method’s versatility. The approach yields convergence to equilibrium under analytic nonlinearity, provides uniform separation from pure phases, and delivers convergence in H^s for all s∈(0,1), offering a new tool for long-time analysis of CH-type models and related systems.

Abstract

We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely.
Paper Structure (17 sections, 11 theorems, 176 equations)

This paper contains 17 sections, 11 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and functional setting
  4. Main assumptions
  5. Existence of a global weak solution
  6. Main results
  7. The Cahn--Hilliard equation with non-degenerate mobility
  8. The Abels-Garcke-Grün system with non-degenerate mobility
  9. Proof of Lemma \ref{['convaaa']}
  10. Proof of Lemma \ref{['twoparts']}
  11. Proof of Theorem \ref{['uniqueeq']}
  12. The Abels-Garcke-Grün system with non-degenerate mobility
  13. Proof of Lemma \ref{['lemmauconv']}
  14. Proof of Theorem \ref{['uniqueeq1']}
  15. Some technical lemmas
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $\Omega\subset R^d$, $d=2,3$, be a bounded domain of class $C^3$, and let assumptions ASS:0-ASS:S1 be satisfied. If $\varphi_0 \in H^1(\Omega)$ is such that $\left\vert \varphi_0 \right\vert\leq 1$ and $\overline\varphi_0\in(-1,1)$, then there exists a global weak solution $(\varphi,\mu)$ to pro and together with $\partial_{\mathbf{n}}\varphi=0$ on $\partial\Omega\times(0,\infty)$. Additional

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 2.2: Existence of global weak solutions BB
  • Remark 2.3
  • Theorem 2.4: Weak existence of global solutions to AGG system ADG
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 11 more