## Hydrodynamic Limit for the Ginzburg-Landau $\nablaÏ$ Interface Model with a Conservation Law

J. Math. Sci. Univ. Tokyo
Vol. 9 (2002), No. 3, Page 481--519.

Nishikawa, Takao
Hydrodynamic Limit for the Ginzburg-Landau $\nablaÏ$ Interface Model with a Conservation Law
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

Abstract:
Hydrodynamic limit for the Ginzburg-Landau $\nablaÏ$ interface model was established in [5] under the periodic boundary conditions. This paper extends their results to the modified dynamics which preserve the total volume of each microscopic phase. Nonlinear partial differential equation of fourth order % $\frac{\partial h}{\partial t} =-Î\left[\dive\left\{(\nablaÏ)(\nabla h(t,Î¸))\right\}\right], Î¸\in{\mathbb T}^d\equiv[0,1)^d,\,t>0$ % is derived as the macroscopic equation, where $Ï=Ï(u)$ is the surface tension of the surface with tilt $u\in\real^d$. The main tool is $H^{-2}$-method, which is a modification of $H^{-1}$-method used in [5]. The Gibbs measures associated with the dynamics are characterized.

Keywords: Hydrodynamic Limit, Ginzburg-Landau Model, Effective Interfaces, Massless Fields, Surface Diffusion

Mathematics Subject Classification (2000): 60K35, 82C24, 35K55
Mathematical Reviews Number: MR1930416