Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model
Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model
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DOI：
10.1016/j.jfa.2011.12.012
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发表时间：
2012-03-01
影响因子：
1.7
通讯作者：
Carrillo, Jose A.
中科院分区：
文献类型：
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作者：
Blanchet, Adrien;Carlen, Eric A.;Carrillo, Jose A.
研究了临界质量Patlak-Keller-Segel方程的长时间行为.该方程有一个单参数稳态解族Q（lambda），lambda> 0，具有厚尾，其二阶矩无界。我们表明，这些稳态的解决方案是稳定的，并找到盆地的吸引力，他们使用的熵泛函H-λ，来自R-2中的临界快速扩散方程。我们构造了满足H-λ的熵-熵耗散不等式的Patlak-Keller-Segel方程的解.虽然H-λ的熵耗散是严格正的，但它是两项之差，当耗散很小时，这两项都不需要很小。我们引入了一种控制浓度的策略来处理这个问题，然后使用从熵-熵耗散不等式得到的规律性来证明由一定的初始数据组成的每个定态的吸引盆的存在性收敛到Q（λ）。(C)2011 Elsevier Inc. All rights reserved.
We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions Q(lambda), lambda > 0, with thick tails whose second moment is unbounded. We show that these steady-state solutions are stable, and find basins of attraction for them using an entropy functional H-lambda, coming from the critical fast diffusion equation in R-2. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy entropy dissipation inequality for H-lambda. While the entropy dissipation for H-lambda is strictly positive, it turns out to be a difference of two terms, neither of which needs to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards Q(lambda). (C) 2011 Elsevier Inc. All rights reserved.