Volume-Preserving Mean Curvature Flow as a Limit of a Nonlocal Ginzburg-Landau Equation

We study the asymptotic behavior of radially symmetric solutions of the nonlocal equation $$ \varepsilon\phi_t- \varepsilon\Delta\phi +\frac{1}{\varepsilon}W'(\phi)-\lambda_\varepsilon (t) =0 $$ in a bounded spherically symmetric domain $\Omega\subset\RN$, where $\lambda_\varepsilon (t)=\frac{1}{\varepsilon} \int_{\Omega}{\!\!\!\!\!\!\!\!-} \ W'(\phi)\, dx$, 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 $\varepsilon\to 0$, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a by-product of our analysis, we obtain an $L^2$ estimate on the "Lagrange multiplier" $\lambda_\varepsilon (t)$, which holds in the nonradial case as well. In addition, we show rigorously (in general geometry) that the nonlocal Ginzburg--Landau equation and the Cahn--Hilliard equation occur as special degenerate limits of a viscous Cahn--Hilliard equation.