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The sharp interface limit for the stochastic Cahn-Hilliard equation. (English. French summary) Zbl 1391.35192

Summary: We study the \(\varepsilon\)-dependent two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit \(\varepsilon\to0\). The parameter \(\varepsilon\) is positive and measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling, our results indicate that the stochastic Cahn-Hilliard equation converge to a Hele-Shaw problem with stochastic forcing on the curvature equation. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.

MSC:

35K55 Nonlinear parabolic equations
35K40 Second-order parabolic systems
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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