随机Couette流对二维Keller--Segel系统爆破解的抑制

中文摘要 AI 辅助

我们研究由随机Couette流平流的$\mathbb R^2$上的Keller--Segel系统。我们证明足够强的混合几乎必然抑制趋化爆破解，从而对任意质量的初始数据产生唯一的全局温和解。为证明此结果，我们利用被动标量输运方程的随机解算子的先验估计和不动点论证构造极大局部温和解。全局存在性则通过将时间分解为块、自举论证和第一个Borel--Cantelli引理得出。

英文摘要

We study Keller--Segel systems on $ \mathbb{T}\times\mathbb{R}$ subject to a stochastic Couette transport flow. We prove that sufficiently strong mixing suppresses chemotactic finite-time blow-up with high probability; thus yielding a unique global-in-time mild solution for initial data of arbitrary mass. Moreover, we obtain a quantitative estimate for the probability of blow-up, showing that it decays exponentially with increasing strength of the stochastic shear. Our analysis is based on the enhanced dissipation generated by the stochastic shear flow. To prove our main result, we first construct a maximal local mild solution through pathwise estimates for the random solution operator of passive scalar transport equations and a fixed-point argument. To obtain global control, we decompose the evolution into time blocks and exploit mixing-induced decay of the nonzero spatial Fourier modes. A nonlinear bootstrap argument then yields global existence above an almost-sure finite random threshold for the shear strength which is obtained by an application of the Borel-Cantelli theorem.

发表机构

• Department of Applied Mathematics, University of Washington（华盛顿大学应用数学系）

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