Spinodal decomposition for the Cahn-Hilliard-Cook equation. (English) Zbl 0993.60061

A stochastic Cahn-Hilliard-Cook equation \[ \partial_{t}u = -\Delta(\varepsilon^2\Delta u + f(u)) + \sigma_0 \varepsilon ^\Sigma\xi \tag{1} \] in \(G\), \(\partial_\nu u = \partial_\nu\Delta u = 0\) on \(\partial G\) (that is, the Cahn-Hilliard equation driven by an additive Gaussian noise) is studied. It is assumed that \(G\subset \mathbb R^{d}\), \(d\leq 3\), is a bounded domain with a smooth boundary, \(-f\) is a derivative of a double-well potential, and \(\xi\) denotes a (generalized) derivative of a (possibly cylindrical) Wiener process on the state space \(X=\{v\in L^2(G)\); \(\int_{G} v dx =0\}\). S. Maier-Paape and T. Wanner [see e.g. Commun. Math. Phys. 195, No. 2, 435-464 (1998; Zbl 0931.35064) and Arch. Ration. Mech. Anal. 151, No. 3, 187-219 (2000; Zbl 0954.35089)] have given a condition under which solutions to the deterministic counterpart to (1) (i.e., to (1) with \(\sigma_0\equiv 0\)) exhibit with high probability spinodally decomposed patterns. In the paper under review, their results are extended to the stochastic equation (1).


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K55 Nonlinear parabolic equations
35B25 Singular perturbations in context of PDEs
82D35 Statistical mechanics of metals
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