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

Dong Li; Xiaoyi Zhang

doi:10.3934/cpaa.2010.9.1591

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2010, Volume 9, Issue 6: 1591-1606. Doi: 10.3934/cpaa.2010.9.1591

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

Dong Li^1, and
Xiaoyi Zhang^1,

1.
Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242

Received: September 30, 2009

Revised: January 31, 2010

Published: October 2010

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Abstract

We consider a nonlocal evolution equation in $R^2$: $\partial_t u + \nabla \cdot (u K*u )= 0$, where $K(x) = \mu \frac x {|x|^\alpha}$, $\mu=\pm 1$ and $1 < \alpha < 2 $. We study wellposedness, continuation/blowup criteria and smoothness of solutions in Sobolev spaces. In the repulsive case ($\mu=1$), by using the sharp blowup criteria, we prove global wellposedness for any positive large initial data. In the attractive case ($\mu=-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.

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Mathematics Subject Classification: Primary: 35Q92; Secondary: 35Q35.

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