Title: Phase separation of the slightly viscous Cahn-Hilliard equation in the singular perturbation limit

Authors: Z. Artstein and M. Slemrod

Issue: Volume 47 (1998), Issue 3, 1147-1166

Abstract: We prove the existence of cluster points in $L^1$ as $\varepsilon \to 0$, say $\bar u$, of solutions $\{ u^{\varepsilon} $ to a Cahn-Hilliard equation on a domain $Q_T = \Omega \times (0,T)$, $\Omega \in \mathbb{R}^N$, with $O(\varepsilon)$ viscous damping and finite energy initial data. The function $\bar u$ is then in $BV(Q_T)$ and has values in $\{-1, +1\}$ for almost all $x$, $t \in Q_T$. Furthermore the two separated phases $Q_{+}(t) = \{x \in \Omega: \bar u(x,t) = +1\} and $Q_{-}(t) = \{x \in \Omega: \bar u(x,t) = -1\}$ are well defined and the perimeter of the interface $\partial Q_{+}(t) \cap \partial Q_{-}(t)$ is bounded. We examine also the limit behavior as $t \to \infty$ of the separated phases.