2008
DOI: 10.4310/cms.2008.v6.n2.a3
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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 solution… Show more

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Cited by 8 publications
(7 citation statements)
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“…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…”
Section: Introductionmentioning
confidence: 99%
See 2 more Smart Citations
“…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…”
Section: Introductionmentioning
confidence: 99%
“…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…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…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 ( ):…”
Section: A Residual Upper Boundmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%