## 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

### Keywords:

Cahn-Hilliard-Cook equation; spinodal decomposition

### Citations:

Zbl 0931.35064; Zbl 0954.35089
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