On the stochastic Cahn-Hilliard equation. (English) Zbl 0729.60057

The Cahn-Hilliard equation is used in studying phase separation of binary alloys. The main result of this paper is the proof of the existence and properties of the strong solution \(u=u(t,x,\omega)\) for the equation \[ \partial u/\partial t+\gamma \Delta^ 2u-\Delta \phi (u)=\partial w/\partial t \] with boundary conditions of Neumann’s type and random initial condition \(u_ 0\), where \(\gamma >0\), \(\phi (u)=u^ 3-\beta u\), and w is a Wiener random field with given correlation function of the type min(t,s)k(x,y), x,y\(\in \Omega\), so that \(\partial w/\partial t\) is a process of white noise type.
According to the properties of \(u_ 0\), it is proven that, for example, in the case \(u_ 0\in H^ 1(\Omega)\), there exists a strong solution in the space \(L^ 2(0,T;V)\cap L^{\infty}(0,T;U)\cap C^ w([0,T];U'),\) where \(V=H^ 2_ E(\Omega)\), \(U=H^ 1(\Omega)\), and \(C^ w\) is the space of Hölder continuous functions connected with Lévy’s modulo of continuity for Brownian motion.
Reviewer: N.Elezović


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q35 PDEs in connection with fluid mechanics
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