Refined blowup criteria and nonsymmetric blowup of an aggregation equation

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Abstract

We consider an aggregation equation in R-d, d >= 2, with fractional dissipation: u(t) +del . (u del K * u) = -nu Lambda(gamma)u, where nu >= 0, 0 < gamma < 1, and K(x) = e(-vertical bar x vertical bar). 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 <= infinity. . 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 nu = 0 and the supercritical case nu > 0 and 0 < gamma < 1. Additionally, we present new proofs of blowup which does not use free energy arguments. (C) 2008 Elsevier Inc. All rights reserved.

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