Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian   Operators
  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". 
Sylvia Serfaty and Juan Luis Vazquez
math.AP