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On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

Monica Conti, Stefania Gatti, Andrea Giorgini, Giulio Schimperna

TL;DR

This work advances the mathematical theory of the Cahn–Hilliard equation with nonlinear diffusion and non-degenerate mobility by removing the convexity restriction on the gradient energy. The authors develop a nonlinear-diffusion–adapted Lojasiewicz–Simon inequality to analyze long-time behavior, establishing uniqueness and regularization in 2D and local well-posedness plus near-minimizer global existence in 3D, with Lyapunov stability and convergence to equilibria. Key technical innovations include a stationary-problem–based regularity theory, analytic gradient structure under mass conservation, and a systematic treatment of nonlinear diffusion within a gradient-flow framework. The results yield a cohesive well-posedness and asymptotic-dynamics theory for general diffusion and mobility, with explicit separation from pure phases and convergence to energy minimizers, thereby enhancing the modeling and analysis of complex phase-separating systems. These findings have potential implications for understanding crystal and polymer dynamics under nonconvex energetic landscapes where diffusion and mobility depend on the order parameter.

Abstract

We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this work, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions. In two spatial dimensions, we prove uniqueness of weak solutions, their smoothing effect for positive times, and convergence to equilibrium as time tends to infinity. In three dimensions, we show local well-posedness of strong solutions for arbitrary initial data and global existence for data close to energy minimizers, yielding a Lyapunov stability principle. A key ingredient of our analysis is a Lojasiewicz-Simon inequality tailored to the nonlinear diffusion case, which enables us to characterize the longtime dynamics.

On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

TL;DR

This work advances the mathematical theory of the Cahn–Hilliard equation with nonlinear diffusion and non-degenerate mobility by removing the convexity restriction on the gradient energy. The authors develop a nonlinear-diffusion–adapted Lojasiewicz–Simon inequality to analyze long-time behavior, establishing uniqueness and regularization in 2D and local well-posedness plus near-minimizer global existence in 3D, with Lyapunov stability and convergence to equilibria. Key technical innovations include a stationary-problem–based regularity theory, analytic gradient structure under mass conservation, and a systematic treatment of nonlinear diffusion within a gradient-flow framework. The results yield a cohesive well-posedness and asymptotic-dynamics theory for general diffusion and mobility, with explicit separation from pure phases and convergence to energy minimizers, thereby enhancing the modeling and analysis of complex phase-separating systems. These findings have potential implications for understanding crystal and polymer dynamics under nonconvex energetic landscapes where diffusion and mobility depend on the order parameter.

Abstract

We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this work, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions. In two spatial dimensions, we prove uniqueness of weak solutions, their smoothing effect for positive times, and convergence to equilibrium as time tends to infinity. In three dimensions, we show local well-posedness of strong solutions for arbitrary initial data and global existence for data close to energy minimizers, yielding a Lyapunov stability principle. A key ingredient of our analysis is a Lojasiewicz-Simon inequality tailored to the nonlinear diffusion case, which enables us to characterize the longtime dynamics.

Paper Structure

This paper contains 20 sections, 15 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Mathematical setting
  4. Function spaces
  5. Elliptic problems.
  6. The stationary system
  7. A gradient Lojasiewicz-Simon inequality with nonlinear diffusion
  8. A general result
  9. Proof of Theorem \ref{['LS']}.
  10. Proof of Theorem \ref{['LSg-intro']}
  11. Improved estimates for weak solutions
  12. Uniqueness of weak solutions in two dimensions
  13. Global Regularity of weak solutions in two dimensions
  14. Global regularity
  15. Energy equality
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^d$, $d=2,3$, be a bounded smooth domain. Assume that $a$ is real analytic on the open interval $(-1,1)$. Then, for any $\psi\in \mathcal{S}_m$, there exist $\theta\in (0, \frac{1}{2})$, $C_L>0$ and $\beta >0$ such that the following inequality holds for all $u\in H^2(\Omega)$ such that $\overline{u}=m$, $\partial_\textbf{n} u=0$ on $\partial\Omega$ and $\|u-\psi\|_{H^

Theorems & Definitions (26)

  • Theorem 1.1: Lojasiewicz--Simon inequality
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • ...and 16 more