Spinodal Decomposition for the Cahn–Hilliard Equation in Higher Dimensions.¶Part I: Probability and Wavelength Estimate

Abstract:

This paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the Cahn–Hilliard equation

where , 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 [26]. 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≡μ of the Cahn–Hilliard equation, with mass μ 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 [37] as well. Moreover, we introduce the notion of a characteristic wavelength for the strongly unstable directions.