Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Upper bounds for coarsening for the degenerate Cahn-Hilliard equation

Pages: 251 - 272, Volume 25, Issue 1, September 2009      doi:10.3934/dcds.2009.25.251

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Amy Novick-Cohen - Department of Mathematics, Technion-IIT, Haifa 32000, Israel (email)
Andrey Shishkov - Institute of Applied Mathematics and Mechanics, 83114 Donetsk, Ukraine (email)

Abstract: The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 9],

$u_t=\nabla \cdot (1-u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)-\ln(1-u)\} - \alpha u -$ Δu$],$

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 a $W^{1, \infty}$ predual 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 [14]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.

Keywords:  phase separation, coarsening, the Cahn-Hilliard equation, higher order degenerate parabolic equations.
Mathematics Subject Classification:  Primary: 35K65, 35K35, 74N20 ; Secondary: 35K55.

Received: December 2007;      Revised: October 2008;      Available Online: June 2009.