Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux

Feng Li; Yuxiang Li

doi:10.3934/dcdsb.2019064

Article Contents

2019, Volume 24, Issue 10: 5409-5436. Doi: 10.3934/dcdsb.2019064

This issue Previous Article A backscattering model based on corrector theory of homogenization for the random Helmholtz equation Next Article Effects of the noise level on nonlinear stochastic fractional heat equations

Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux

Feng Li^1, and
Yuxiang Li^2, ,

1.
School of Mathematics, Southeast University, Nanjing 210096, China
2.
Institute for Applied Mathematics, School of Mathematics, Southeast University, Nanjing 211189, China

^* Corresponding author
^* Corresponding author

Received: May 31, 2018

Revised: October 31, 2018

Early access: April 2019

Published: August 2019

The authors are supported in part by NSF of China (No. 11671079, No. 11701290, No. 11601127, No. 11871147 and No. 11871148), and NSF of Jiangsu Province (No. BK20170896).

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Abstract

In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by

$\begin{eqnarray*} \left\{\begin{array}{lll}n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\\[1mm]c_t+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,\ \ t>0,\\[1mm]u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\\[1mm]\nabla\cdot u=0,&x\in\Omega,\ \ t>0 \end{array}\right.\end{eqnarray*}$

in a bounded domain $\Omega\subset\mathbb{R}^3$, where $k\in\mathbb{R}$, $\phi\in W^{2,\infty}(\Omega)$ and the given tensor-valued function $S$: $\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$ satisfies

$|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$

Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming $m+\alpha>\frac{4}{3}$ and $m>\frac{1}{3}$.

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Mathematics Subject Classification: 35K65, 35Q35, 35Q51, 92C17.

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