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arXiv · 1909.12106

The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential

Luca Scarpa
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Abstract

We prove existence of martingale solutions for the stochastic Cahn-Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by C. M. Elliott and H. Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed.

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Luca Scarpa. 2019-09-26. The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential. https://doi.org/10.1088/1361-6544%2Fabf338

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