Refined blowup criteria and nonsymmetric blowup of an aggregation equation

We consider an aggregation equation in Rd, d⩾2, with fractional dissipation: ut+∇⋅(u∇K∗u)=−νΛγu, 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 norm, for any . 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.