Upper bounds for coarsening for the degenerate CahnHilliard equation
Amy NovickCohen  Department of Mathematics, TechnionIIT, Haifa 32000, Israel (email) Abstract: The long time behavior for the degenerate CahnHilliard equation [4, 5, 9], $u_t=\nabla \cdot (1u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)\ln(1u)\}  \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 CahnHilliard equation, higher order degenerate parabolic equations.
Received: December 2007; Revised: October 2008; Available Online: June 2009. 
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