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On a 1D transport equation with nonlocal velocity and supercritical dissipation. (English) Zbl 1286.35058
Summary: We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35R09 Integral partial differential equations 35R11 Fractional partial differential equations 35B50 Maximum principles in context of PDEs
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##### References:
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