Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators

Sylvia Serfaty, Juan Luis Vazquez

Published 2012-05-29, updated 2012-06-28Version 2

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".