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# 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: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a5

#### Authors

#### 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