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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).

MSC:
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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