Wellposedness and regularity of solutions of an aggregation equation

Abstract

We consider an aggregation equation in ℝd, d ≥ 2 with fractional dissipation: ut + ∇· (u∇K*u) = -vΛΥu, where v≥ 0, 0<Υ≤2 and K(x) = e-|x| In the supercritical case, 0 < Υ < 1, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, Υ = 1, we prove the global wellposedness for initial data having a small L1x norm. In the subcritical case, Υ > 1, we prove global wellposedness and smoothing of solutions with general L 1x initial data.