Spinodal decomposition for the Cahn-Hilliard-Cook equation

22. Dirk Blömker, Stanislaus Maier-Paape, Thomas Wanner:
    Spinodal decomposition for the Cahn-Hilliard-Cook equation
    Communications in Mathematical Physics 223(3), pp. 553-582, 2001.

Abstract

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn-Hilliard-Cook equation, which is a Cahn-Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which starts at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn-Hilliard-Cook equation depends on a small positive parameter $\epsilon$ which models atomic scale interaction length. We quantify the behavior of solutions as $\epsilon \rightarrow 0$. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace $Y_\epsilon$ is high as long as the solution stays within distance $r_\epsilon = O(\epsilon^R)$ of the homogeneous state. The subspace $Y_\epsilon$ is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent $R$ depends on both the strength and the regularity of the noise.

Links

The published version of the paper can be found at https://doi.org/10.1007/PL00005585.

Bibtex

@article{bloemker:etal:01b,
   author = {Dirk Bl\"omker and Stanislaus Maier-Paape
             and Thomas Wanner},
   title = {Spinodal decomposition for the
            {C}ahn-{H}illiard-{C}ook equation},
   journal = {Communications in Mathematical Physics},
   year = 2001,
   volume = 223,
   number = 3,
   pages = {553--582},
   doi = {10.1007/PL00005585}
   }