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 Lx norm, for any d d−1 ≤ q ≤ ∞. We prove for a general class of nonsymmetric initial data the finite time blowup of the corresponding solutions. 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.