VOLUME-PRESERVING MEAN-CURVATURE FLOW AS A LIMIT OF A NONLOCAL GINZBURG-LANDAU EQUATION

We study the asymptotic behavior of radially symmetric solutions of th e nonlocal equation epsilon phi(t) - epsilon Delta phi + 1/epsilon W'( phi) - lambda(epsilon)(t) = 0 in a bounded spherically symmetric domai n Omega subset of R-n, where lambda(epsilon)(t) = 1/epsilon f(Omega)W' (phi) dx, with a Neumann boundary condition. The analysis is based on ''energy methods'' combined with some a priori estimates, the latter b eing used to approximate the solution by the first two terms of an asy mptotic expansion. We only need to assume that the initial data as wel l as their energy are bounded. We show that, in the limit as epsilon - -> 0, the interfaces move by a nonlocal mean curvature flow, which pre serves mass. As a by-product of our analysis, we obtain an L-2 estimat e on the ''Lagrange multiplier'' lambda(epsilon)(t), which holds in th e nonradial case as well. In addition, we show rigorously (in general geometry) that the nonlocal Ginzburg-Landau equation and the Cahn-Hill iard equation occur as special degenerate limits of a viscous Cahn-Hil liard equation.