Volume preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation

Abstract: "We study the asymptotic behaviour of radially symmetric solutions of the nonlocal equation [epsilon phi subscript t] - [epsilon delta phi] + 1/[epsilon]W(́[phi]) - [lambda subscript epsilon](t) = 0 in a bounded spherically symmetric domain [omega][contained within] R[superscript n], where [lambda subscript epsilon](t) = 1/[epsilon] f[subscript omega]W(́[phi])dx, with a Neumann boundary condition. The analysis is based on 'energy methods' combined with some a-priori estimates, the latter being used to approximate the solution by the first two terms of an asymptotic expansion. We only need to assume that the initial data as well as their energy are bounded. We show that, in the limit as [epsilon] -> 0, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a byproduct of our analysis, we obtain an L² estimate on the 'Lagrange multiplier' [lambda subscript epsilon](t). In addition we show rigorously that the nonlocal Ginzburg-Landau equation and the Cahn-Hilliard equation occur as special degenerate limits of a viscous Cahn-Hilliard equation."