Spinodal decomposition for multicomponent Cahn-Hilliard systems
In the papers of \textit{S. Maier-Paape} and \textit{T. Wanner} [Comm. Math. Phys. 195, 435-464 (1998; Zbl 0931.35034) and Arch. Ration. Mech. Anal. 151, No. 3, 187-219 (2000; Zbl 0931.35034)] a framework for the analysis of the early stages of spinodal decomposition in the Cahn-Hilliard equation is presented. In this paper the authors extend to this framework to the case of systems. It must be noted that the paper is not self-contained and relies heavily on the above two references. Consider a system of Cahn-Hilliard equations, \[ u_t = -\Delta(\epsilon^2 \Delta u +f(u)), \quad x \in \Omega, \tag\text{CH} \] with the boundary conditions \[ \partial u/\partial \nu = \partial \Delta u/\partial \nu =0 \text{ on } \partial \Omega. \] Here \(u \in \mathbb R^N\), \(N > 1\) and the nonlinearity \(f(\cdot)\) is derived from the Gibbs free energy (see (5) in the paper). Suppose the system (CH) has an unstable constant equilibrium state \(\overline u\). Spinodal decomposition is the process of evolution of infinitesimal perturbations from the unstable steady state. The heart of the paper is Theorem 2.9, which, roughly, says that there is a finite-dimensional subspace (the strongly unstable subspace) in the phase space such that for sufficiently small \(\epsilon\) and \(0<p \leq 1\) which dominates the evolution of proportion \(p\) of initial conditions lying in a ball of radius \(r\) (depending on \(p\)) around the unstable state \(\overline u\). The proof of this theorem is based on a detailed analysis of the behaviour of the semigroup generated by the linearised Cahn-Hilliard operator (see Lemma 2.6). In addition, in Theorems 1.2 and 1.3 the authors describe the structure of this subspace, which depends on the spectrum of the linearization of the mapping \(f(u)\) at \(\overline u\). Finally, in Section 3 these abstract results are applied to the case of ternary alloys and related to numerical work on this system.
scientific article; zbMATH DE number 1441254