Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Global existence and boundedness in a 2D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux

Pages: 1889 – 1910

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n7.a5

Authors

Xie Li (School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China; and College of Mathematic and Information, China West Normal University, Nanchong, China)

Yulan Wang (School of Science, Xihua University, Chengdu, China)

Zhaoyin Xiang (School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China)

Abstract

In this paper, we investigate the degenerate Keller–Segel–Stokes system (KSS) in a bounded convex domain $\Omega \subset \mathbb{R}^2$ with smooth boundary. A particular feature is that the chemotactic sensitivity $S$ is a given parameter matrix on $\Omega \times [0,\infty)^2$ whose Frobenius norm satisfies $\lvert S(x,n,c)\rvert \leq C_S$ with some $C_S \gt 0$. It is shown that for any porous medium diffusion $m \gt 1$, the system (KSS) with nonnegative and smooth initial data possesses at least a global-in-time weak solution, which is uniformly bounded.

Keywords

global existence, boundedness, Keller–Segel–Stokes system, tensor-valued sensitivity

2010 Mathematics Subject Classification

35K55, 35Q35, 35Q92, 92C17

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Published 14 September 2016