Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels

Abstract

We consider a nonlocal evolution equation in R2: ℓtu +δ • (uK * u) = 0, where K(x) = μ x |x| , μ = ±1 and 1 > α > 2. We study wellposedness, continuation/blowup criteria and smoothness of solutions in Sobolev spaces. In the repulsive case (μ = 1), by using the sharp blowup criteria, we prove global wellposedness for any positive large initial data. In the attractive case (μ = -1), by using a novel free energy inequality together with a mass localization technique, we construct finite time blowups for a large class of smooth initial data.