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Communications in Mathematical Physics

, Volume 223, Issue 3, pp 553–582 | Cite as

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

  • Dirk Blömker
  • Stanislaus Maier-Paape
  • Thomas Wanner

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 r ε=O R ) 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.

Keywords

Phase Separation Homogeneous State Positive Parameter Atomic Scale Spinodal Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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