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 …