Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs
Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs
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DOI：
10.1090/tran/8195
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发表时间：
2019-08
影响因子：
1.3
通讯作者：
Gautam Iyer;Xiaoqian Xu;Andrej Zlatoš
中科院分区：
文献类型：
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作者：
Gautam Iyer;Xiaoqian Xu;Andrej Zlatoš
本文研究了一类一般的非线性抛物型偏微分方程解的增长受对流项的影响，以及由此引起的耗散率的增加。特别地，我们证明了如果附加漂移具有足够小的耗散时间，这些模型中的爆破总是可以被防止的。我们还证明了定常胞状流的耗散时间和有效扩散率之间的一个一般结果，这使得我们可以得到具有任意小的耗散时间的简单不可压缩流动的例子。作为应用，我们证明了当周围流体的速度场具有足够小的耗散时间时，趋化性的Keller-Segel模型中的爆破总是可以防止的。我们还研究了具有着火型非线性的反应扩散方程，证明了只要初始平均温度低于点火阈值，只要增加一个具有足够小的耗散时间的对流项，反应总是可以熄灭的。
In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times. As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.