Convergence of a one dimensional Cahn-Hilliard equation with degenerate mobility

TL;DR

This paper proves the convergence of a regularized one-dimensional Cahn-Hilliard equation with degenerate mobility to a degenerate parabolic limit as the regularization parameter tends to zero, using gradient flow techniques.

Contribution

It establishes the rigorous convergence of the regularized Cahn-Hilliard equation to its degenerate limit leveraging the gradient flow framework and detailed energetic analysis.

Findings

Convergence of solutions as regularization parameter approaches zero

Identification of the limit as a well-posed degenerate parabolic equation

Insights into energetic properties of solutions to related thin-film equations

Abstract

We consider a one dimensional periodic forward-backward parabolic equation, regularized by a non-linear fourth order term of order $\epsilon^2\ll 1$. This equation is known in the literature as Cahn-Hilliard equation with degenerate mobility. Under the hypothesis of the initial data being well prepared, we prove that as $\epsilon\to0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in $\mathcal{W}^2$ and utilizes the framework of convergence of gradient flows developed by Sandier-Serfaty. As an incidental, we study fine energetic properties of solutions to the Thin-film equation $\partial_t\nu=(\nu\nu_{xxx})_x$.