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

Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators