Quelques résultats sur l'équation de Cahn-Hilliard stochastique et déterministe

Abstract : In a first part, we are concerned with the stochastic Cahn-Hilliard partial differential equation in dimension 1 with one singularity. This is an equation of order 4 driven by the derivative of a space-time white noise. There is a logarithmic nonlinearity or a negative power $x^{-\alpha}$ nonlinearity. Thanks to Lipschitz approximated equations, we show existence and uniqueness of a solution. It is necessary to add a reflection measure to ensure existence. We study it with the associated Revuz measure. Thanks to the associated integration by parts formula, we can show that the reflection measure vanishes for alpha larger than 3. In a second part, we consider the same equation but with two logarithmic singularities in +1 and -1. This is the full Cahn-Hilliard's model. With polynomial approximated equations, we show the existence and uniqueness. We should add two reflection measures to ensure the existence. Moreover, we establish that the invariant measure is ergodic. Finally, we are concerned with the deterministic equation. Some numerical simulations based on a high order finite elements method have been computed. We study the bifurcations around the first eigenvalue of the Laplace operator on general domains, the interfaces and the stationary states. Furthermore, some stochastic simulations have been computed in order to show the contacts with the singular values. The long time evolutions and the jump between stationary states are also treated.
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Submitted on : Saturday, December 5, 2009 - 7:01:17 PM
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Ludovic Goudenège. Quelques résultats sur l'équation de Cahn-Hilliard stochastique et déterministe. Mathématiques [math]. École normale supérieure de Cachan - ENS Cachan, 2009. Français. ⟨tel-00439022⟩

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