An Aggregation Equation with Degenerate Diffusion: Qualitative Property of Solutions

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 foc...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2013-01, Vol.45 (5), p.2995-3018
Main Authors: Chayes, Lincoln, Kim, Inwon, Yao, Yao
Format: Article
Language:English
Subjects:
Applied mathematics
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Summary: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. [PUBLICATION ABSTRACT]
ISSN:0036-1410
1095-7154
DOI:10.1137/120874965