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New results for the Cahn-Hilliard equation with non-degenerate mobility: well-posedness and longtime behavior

Monica Conti, Pietro Galimberti, Stefania Gatti, Andrea Giorgini

TL;DR

The paper addresses well-posedness and long-time behavior of the 2D Cahn-Hilliard equation with a concentration-dependent non-degenerate mobility and a logarithmic potential. It proves uniqueness of weak solutions, establishes propagation of uniform-in-time regularity, and shows convergence to a single equilibrium via a Lojasiewicz-Simon framework, all while employing enhanced energy estimates and elliptic regularity for variable-coefficient operators. The results fill gaps left by prior work and provide a robust analytical framework for diffusion-driven phase-field models with non-constant mobility. These contributions have implications for rigorous analysis of more complex diffuse-interface systems and their long-time dynamics.

Abstract

We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.

New results for the Cahn-Hilliard equation with non-degenerate mobility: well-posedness and longtime behavior

TL;DR

The paper addresses well-posedness and long-time behavior of the 2D Cahn-Hilliard equation with a concentration-dependent non-degenerate mobility and a logarithmic potential. It proves uniqueness of weak solutions, establishes propagation of uniform-in-time regularity, and shows convergence to a single equilibrium via a Lojasiewicz-Simon framework, all while employing enhanced energy estimates and elliptic regularity for variable-coefficient operators. The results fill gaps left by prior work and provide a robust analytical framework for diffusion-driven phase-field models with non-constant mobility. These contributions have implications for rigorous analysis of more complex diffuse-interface systems and their long-time dynamics.

Abstract

We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.

Paper Structure

This paper contains 8 sections, 7 theorems, 200 equations.

Table of Contents

  1. Introduction and main results
  2. Mathematical setting
  3. Notation and function spaces
  4. Elliptic problems.
  5. Proof of Theorem \ref{['Goal_thm']} - (A) - (B): Existence and Uniqueness of Weak Solutions
  6. Proof of Theorem \ref{['Goal_thm']} - (C) : Propagation of regularity
  7. Proof of Theorem \ref{['Goal_thm']} - (D): Convergence to equilibrium
  8. Proof of the inequality \ref{['H2-BB']}

Key Result

Theorem 1.1

Assume that $\Omega \subset \mathbb{R}^d$, with $d=2,3$, is a convex polyhedron or $\partial \Omega \in C^{1,1}$, and the mobility $b$ satisfies m-ndeg. Then, the following holds:

Theorems & Definitions (11)

  • Theorem 1.1: Barrett & Blowey, 1999
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 3.1
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Theorem \ref{['Goal_thm']} - part (C)
  • ...and 1 more