Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate

11. Stanislaus Maier-Paape, Thomas Wanner:
    Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate
    Communications in Mathematical Physics 195(2), pp. 435-464, 1998.

Abstract

This paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the Cahn-Hilliard equation $u_t = -\Delta(\epsilon^2 \Delta u + f(u))$ in $\Omega$, subject to $\frac{\partial u}{\partial \nu} = \frac{\partial \Delta u}{\partial \nu} = 0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$, $n = 1,2,3$, is a bounded domain with sufficiently smooth boundary, and $f$ is cubic-like, for example $f(u) = u - u^3$. We will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to Grant (1993). Furthermore, we derive certain probability and wavelength estimates. The probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium $u_0 \equiv \mu$ of the Cahn-Hilliard equation, with mass $\mu$ in the spinodal region, a strongly unstable manifold has dominating effects. This is demonstrated for the linearized equation, but will be essential for the nonlinear setting in the second paper as well. Moreover, we introduce the notion of a characteristic wavelength for the strongly unstable directions.

Links

The published version of the paper can be found at https://doi.org/10.1007/s002200050397.

Bibtex

@article{maier:wanner:98a,
   author = {Stanislaus Maier-Paape and Thomas Wanner},
   title = {Spinodal decomposition for the {C}ahn-{H}illiard equation in
            higher dimensions. {P}art {I}: {P}robability and wavelength
            estimate},
   journal = {Communications in Mathematical Physics},
   year = 1998,
   volume = 195,
   number = 2,
   pages = {435--464},
   doi = {10.1007/s002200050397}
   }