Refined blowup criteria and nonsymmetric blowup of an aggregation equation

Abstract

We consider an aggregation equation in R d , d ⩾ 2 , with fractional dissipation: u t + ∇ ⋅ ( u ∇ K ∗ u ) = − ν Λ γ u , where ν ⩾ 0 , 0 < γ < 1 , and K ( x ) = e − | x | . We prove a refined blowup criteria by which the global existence of solutions is controlled by its L x q norm, for any d d − 1 ⩽ q ⩽ ∞ . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν = 0 and the supercritical case ν > 0 and 0 < γ < 1 . Additionally, we present new proofs of blowup which does not use free energy arguments.