Unexpectedly Linear Behavior for the Cahn-Hilliard Equation

This paper gives theoretical results on spinodal decomposition for the CahnHillard equation. We prove a mechanism which explains why most solutions for the Cahn-Hilliard equation which start near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected. The Cahn-Hilliard equation depends on a small parameter ", modeling the (atomic scale) interaction length; we quantify the behavior of solutions as " ! 0. Speci cally, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance O("2 n=2) up to a ball of radius R, where R is proportional to " 1+%+n=4 as " ! 0. Here, n 2 f1; 2; 3g denotes the dimension of the considered domain, and % > 0 can be chosen arbitrarily small. Not only does this approach signi cantly increase the radius of explanation for spinodal decomposition, but it also gives a clear picture of why the phenomenon occurs. While these results hold for the standard cubic nonlinearity, we also show that considerably better results can be obtained for similar higher order nonlinearities. In particular, we obtain R " 2+%+n=2 for every % > 0 by choosing a suitable nonlinearity. AMS subject classi cations: 35K35, 35B05, 35P10.