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

We study the asymptotic behaviour of radially symmetric solutions of the nonlocal equation In a bounded spherically symmetric domain ft C R"f where A,(<) m 1 fftW(<p) «fcr, 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 sohtion by the first two terms of an asymptotic expansionWe only need to assume that the initial data as well as their energy are bounded. We show that, in the limit as e -+ 0, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a byproduct of our analysis, we obtain an I? estimate on the "Lagrange multiplier A*(<). 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. Section 1: Introduction. We consider the nonlocal reaction-diffusion equation introduced recently by Rubinstein and Sternberg [RS]