The rates of convergence for the chemotaxis-Navier–Stokes equations in a strip domain

ABSTRACT

In this paper, we study the long-time behavior of the chemotaxis-Navier–Stokes system $\begin{array}{rl}& {\mathrm{\partial }}_{t}n+\mathit{u}\cdot \mathrm{\nabla }n=\lambda \mathrm{\Delta }n-\mathrm{\nabla }\cdot \left(\chi \right(c\left)n\mathrm{\nabla }c\right),\\ & {\mathrm{\partial }}_{t}c+\mathit{u}\cdot \mathrm{\nabla }c=\mu \mathrm{\Delta }c-f\left(c\right)n,\\ & {\mathrm{\partial }}_t\mathit{u}+\mathit{u}\cdot \mathrm{\nabla }\mathit{u}+\mathrm{\nabla }P=\zeta \mathrm{\Delta }\mathit{u}-n\mathrm{\nabla }\phi ,\\ & \mathrm{\nabla }\cdot \mathit{u}=0,t>0,x\in \mathrm{\Omega }\end{array}$ posed in a strip domain $\mathrm{\Omega }:={\mathbb{R}}^2\times [0,1]\subset {\mathbb{R}}^3$. In Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920), the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for n and c and non-slip boundary conditions for $\mathit{u}$. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic $L^p$ interpolation and the iterative techniques.

Keywords:

• Chemotaxis-fluid interaction
• strip domain
• rates of convergence

AMS (2000) Subject Classifications:

• 35K55
• 35Q92
• 35Q35
• 92C17

Acknowledgements

The authors are very grateful to the reviewers for their detailed comments and valuable suggestions, which greatly improved the manuscript. Hongxia Lin acknowledges support by the NSF of China under Grant 11701049, the China Postdoctoral Science Foundation under Grant 2017M622989, and the Scientific fund of the Education Department of Sichuan Province under Grant 18ZB0069.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Hongxia Lin acknowledges support by the NSF of China [grant number 11701049], the China Postdoctoral Science Foundation [grant number 2017M622989], and Scientific Research Foundation of the Education Department of Sichuan Province [grant number 18ZB0069].