Existence and regularity of solution for a stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Antonopoulou, Dimitra C.¹  Karali, Georgia^2,3  Millet, Annie^4,5,6 

[1] Univ Chester, Dept Math, Thornton Sci Pk, Chester CH2 4NU, Cheshire, England

[2] Univ Crete, Dept Appl Math, GR-71009 Iraklion, Crete, Greece

[3] FORTH, Inst Appl & Computat Math, GR-71110 Iraklion, Greece

[4] Univ Paris 01, SAMM EA 4543, 90 Rue Tolbiac, F-75634 Paris 13, France

[5] Univ Paris 06, CNRS, Lab Probabilites & Modeles Aleatoires, UMR 7599, Boite Courrier 188,4 Pl Jussieu, F-75252 Paris 05, France

[6] Univ Paris 07, CNRS, Lab Probabilites & Modeles Aleatoires, UMR 7599, Boite Courrier 188,4 Pl Jussieu, F-75252 Paris 05, France

关键词: Stochastic Cahn-Hilliard/Allen-Cahn equation;    Space-time white noise;    Convolution semigroup;    Galerkin approximations;    Unbounded diffusion;   

DOI  :  10.1016/j.jde.2015.10.004

来源: Elsevier

【 摘 要 】

The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diffusion coefficient of linear growth. Applying techniques from semigroup theory, we prove local existence and uniqueness in dimensions d = 1, 2, 3. Moreover, when the diffusion coefficient satisfies a sub-linear growth condition of order alpha bounded by 1/3, which is the inverse of the polynomial order of the nonlinearity used, we prove for d = 1 global existence of solution. Path regularity of stochastic solution, depending on that of the initial condition, is obtained a.s. up to the explosion time. The path regularity is identical to that proved for the stochastic Cahn-Hilliard equation in the case of bounded noise diffusion. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diffusion, for which previous results were established only in the framework of a bounded diffusion coefficient. As expected from the theory of parabolic operators in the sense of Petrovsk'ii, the bi-Laplacian operator seems to be dominant in the combined model. (C) 2015 Elsevier Inc. All rights reserved.

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