Abstract
Well-posedness is proved for the stochastic viscous Cahn–Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn–Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case.
Highlights
The deterministic Cahn–Hilliard equation was first proposed in [6] to describe the spinodal decomposition phenomenon occurring during the phase separation in a binary metallic alloy
To the best of our knowledge, the only available results dealing with the stochastic viscous Cahn–Hilliard equation seem to be [41,45]: here, the authors prove existence of mild solution and attractors under the classical case of a polynomial double-well potential
As an direct application of the vanishing viscosity limit, we prove some refined regularity results for the both the viscous and the pure case
Summary
The deterministic Cahn–Hilliard equation was first proposed in [6] to describe the spinodal decomposition phenomenon occurring during the phase separation in a binary metallic alloy. To the best of our knowledge, the only available results dealing with the stochastic viscous Cahn–Hilliard equation seem to be [41,45]: here, the authors prove existence of mild solution and attractors under the classical case of a polynomial double-well potential. The aim and novelty of this paper is to carry out a unifying and self-contained mathematical analysis of the stochastic viscous Cahn–Hilliard system, under no growth nor smoothness assumptions on the double-well potential This is motivated by the fact that, in applications to phase-transitions, degenerate potentials (possibly defined on bounded domains) play an important role (see again [14]): from the mathematical point of view, it is worth trying to give sense to the equation with as less constraints as possible on the growth of β. We are able to give sufficient conditions yielding H 3-regularity in space for the solution
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