Communications in Mathematical Physics

, Volume 223, Issue 3, pp 553–582

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

  • Dirk Blömker
  • Stanislaus Maier-Paape
  • Thomas Wanner

DOI: 10.1007/PL00005585

Cite this article as:
Blömker, D., Maier-Paape, S. & Wanner, T. Commun. Math. Phys. (2001) 223: 553. doi:10.1007/PL00005585
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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 start 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 ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?ε is high as long as the solution stays within distance rε=OR) of the homogeneous state. The subspace ?ε 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Dirk Blömker
    • 1
  • Stanislaus Maier-Paape
    • 1
  • Thomas Wanner
    • 2
  1. 1.Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany.¶E-mail: bloemker@math.uni-augsburg.de; maier@math.uni-augsburg.deDE
  2. 2.Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore,¶MD 21250, USA. E-mail: wanner@math.umbc.eduUS

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