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arXiv · 1204.3938

An aggregation equation with degenerate diffusion: qualitative property of solutions

Lincoln Chayes·Inwon Kim·Yao Yao
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Abstract

We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter.

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Lincoln Chayes, Inwon Kim, Yao Yao. 2012-04-17. An aggregation equation with degenerate diffusion: qualitative property of solutions. https://arxiv.org/abs/1204.3938

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