Upper Bounds for Coarsening for the
Degenerate Cahn-Hilliard Equation



Amy Novick-Cohen
Department of Mathematics
Technion-IIT
Haifa, 32000, Israel



Andrey Shishkov
Institute of Applied Mathematicisi and Mechanics
83114 Donetsk, Ukraine

Abstract. The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 10],

\begin{displaymath}u_t = \nabla \cdot (1 - u^2)\nabla \left[ \frac{\Theta}{2} \l... ...u) - ln(1-u) \right\} - \alpha u - \epsilon^2 \Delta u\right], \end{displaymath}

is characterized by the growth of domains in which $u(x, t) \approx u_{\pm}$, where $u_{\pm}$ denote the ``equilibrium phases;'' this process is known as coarsening. The degree of coarsening can be quantified in terms of a characteristic length scale, $l(t)$, where $l(t)$ is prescribed via a Liapunov functional and the $(W^{1,\infty})^*$ norm of $u(x, t)$. In this paper, we prove upper bounds on $l(t)$ for all temperatures $\Theta \in (0, \Theta_c)$, where $\Theta_c$ denotes the ``critical temperature,'' and for arbitrary mean concentrations, $\bar{u} \in (u_-, u_+)$. Our results generalize the upper bounds obtained by Kohn & Otto [15]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.

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