The Willmore functional and instabilities in the Cahn-Hilliard equation

Abstract

Abstract. In this paper we are interested in the finite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. We show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighbourhood of transition solutions local instabilities occur in finite time. We also show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time.

Cahn‐Hilliard inpainting and the Willmore functional

“…We are interested in the properties of local stability, instability and asymptotics in time of smooth solutions of the Cahn-Hilliard equation in a neighborhood of a stationary solution u 0 . In [1] the Willmore functional is proposed to study stability properties of solutions of the Cahn-Hilliard equation. The Willmore functional of the Cahn-Hilliard equation (1) is given by…”

“…In [1] we could prove: Theorem 1.1 Let u be the solution of Cahn-Hilliard equation with initial data u 0 (x), Ω is a bounded domain with Neumann boundary conditions. Then…”

“…On the other hand a linear stability analysis in [1] for a perturbation v =ũ+δv 0 , with v 0 is the eigenvector to the eigenvalue λ gives for the Willmore functional…”

Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition

“…where we used Lipschitz continuity (3b) in the first step and the Poincaré-Friedrichs inequality (17) in the second step since e u ∈H 1 ( ); see (11). Next consider η ∈H 1 ( ):…”

“…Many aspects of the Cahn-Hilliard equation have been explored, for example, its dynamical properties [13] as well as its limit to certain free-boundary problems [14][15][16]. For a recent investigation of linear instability, we refer to Burger et al [17]. the chemical potential.…”