Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models

Details Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models

2010-09-30

arxiv.org/abs/1009.6187v2

Mathematics

Analysis of PDEs

18 pages

Scientific paper

We examine the long-term asymptotic behavior of dissipating solutions to aggregation equations and Patlak-Keller-Segel models with degenerate power-law and linear diffusion. The purpose of this work is to identify when solutions decay to the self-similar spreading solutions of the homogeneous diffusion equations. Combined with strong decay estimates, entropy-entropy dissipation methods provide a natural solution to this question and make it possible to derive quantitative convergence rates in $L^1$. The estimated rate depends only on the nonlinearity of the diffusion and the strength of the interaction kernel at long range.

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