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 Elliott and 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.
Highlights
The Cahn–Hilliard equation was firstly proposed in [14] in order to describe the spinodal decomposition occurring in binary metallic alloys
The field of applications of diffuse-interface modelling is enormous. In physics it is used in the context of evolution of separating materials, phase-transition phenomena, and dynamics of mixtures of fluids; in biology phase-field modelling is crucial in the description of evolution of interacting cells, tumour growths, and dynamics of interacting populations; in engineering it plays a central role in modelling of damage and deterioration in continuous media
Our results have important consequences to all fields, in particular physics and engineering, where phase-separation is usually studied under random forcing, as we provide the first mathematical validation of the stochastic model in its more relevant form, i.e. with degenerate mobility mpol and logarithmic potential Flog
Summary
The Cahn–Hilliard equation was firstly proposed in [14] in order to describe the spinodal decomposition occurring in binary metallic alloys. Our results have important consequences to all fields, in particular physics and engineering, where phase-separation is usually studied under random forcing, as we provide the first mathematical validation of the stochastic model in its more relevant form, i.e. with degenerate mobility mpol and logarithmic potential Flog. In this paper we are interested in studying the stochastic Cahn–Hilliard equation (1.10)–(1.13) from a variational approach, including the cases of degenerate mobility mpol, the logarithmic double-well potential Flog, and the double-obstacle potential Fob. As we have pointed out, these choices are the most relevant in terms of the thermodynamical coherency of the model.
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