Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation

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• 作者：Yulan Wang^a ; ^{wangyulan-math@163.com" class="auth_mail" title="E-mail the corresponding author} ; Zhaoyin Xiang^b ; ^{zxiang@uestc.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K55 ; 35Q92 ; 35Q35 ; 92C17
• 刊名：Journal of Differential Equations
• 出版年：2015
• 出版时间：15 December 2015
• 年：2015
• 卷：259
• 期：12
• 页码：7578-7609
• 全文大小：445 K

文摘

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller–Segel–Stokes system Here, one of the novelties is that the chemotactic sensitivity S$\mathcal{S}$ is not a scalar function but rather attains values in R^2×2${\mathbb{R}}^{2\times 2}$, and satisfies |S(x,n,c)|≤C_S(1+n)^−伪$|\mathcal{S}(x,n,c)|\le C_{\mathcal{S}}{(1+n)}^{-伪}$ with some C_S>0$C_{\mathcal{S}}>0$ and 伪>0$伪>0$. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued S$\mathcal{S}$. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality.

To the best of our knowledge, this is the first result on global existence and boundedness in a Keller–Segel–Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved.