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A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators. (English) Zbl 1290.35316

Summary: 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”.

MSC:

35R11 Fractional partial differential equations
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
76S05 Flows in porous media; filtration; seepage
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