Communications in Mathematical Sciences

The Willmore functional and instabilities in the Cahn-Hilliard equation

M. Burger, S.-Y. Chu, P. A. Markowich, and C. -B Schonlieb

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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.

Article information

Source
Commun. Math. Sci., Volume 6, Number 2 (2008), 309-329.

Dates
First available in Project Euclid: 1 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.cms/1214949925

Mathematical Reviews number (MathSciNet)
MR2433698

Zentralblatt MATH identifier
1154.35009

Subjects
Primary: 35B35: Stability 35K57: Reaction-diffusion equations

Keywords
Cahn-Hilliard equation transition solutions Willmore functional asymptotics stability

Citation

Burger, M.; Chu, S.-Y.; Markowich, P. A.; Schonlieb, C. -B. The Willmore functional and instabilities in the Cahn-Hilliard equation. Commun. Math. Sci. 6 (2008), no. 2, 309--329. https://projecteuclid.org/euclid.cms/1214949925


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