Communications in Mathematical Sciences
Volume 6 (2008)
The Willmore functional and instabilities in the Cahn-Hilliard equation
Pages: 309 – 329
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 equation; transition solutions; Willmore functional; asymptotics; stability
2010 Mathematics Subject Classification