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.

摘要

在本文中，我们研究了趋化性-Navier-Stokes 系统的长期行为$\begin{array}{rl}& {\mathrm{\partial }}_{吨}n+\mathit{你}\cdot \mathrm{\nabla }n=\lambda \mathrm{\Delta }n-\mathrm{\nabla }\cdot \left(\chi \right(C\left)n\mathrm{\nabla }C\right),\\ & {\mathrm{\partial }}_{吨}C+\mathit{你}\cdot \mathrm{\nabla }C=\mu \mathrm{\Delta }C-F\left(C\right)n,\\ & {\mathrm{\partial }}_{吨}\mathit{你}+\mathit{你}\cdot \mathrm{\nabla }\mathit{你}+\mathrm{\nabla }磷=\zeta \mathrm{\Delta }\mathit{你}-n\mathrm{\nabla }\phi ,\\ & \mathrm{\nabla }\cdot \mathit{你}=0,吨>0,X\in \mathrm{\Omega }\end{array}$在带状域中构成$\mathrm{\Omega }:={\mathbb{R}}^2\times [0,1]\subset {\mathbb{R}}^3$. 在 Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920) 中，作者已经建立了该系统的强解的全局存在性，其中n和c的非通量边界条件和非通量边界条件滑动边界条件$\mathit{你}$. 我们的主要目的是建立此类解决方案的时间衰减率。这将通过使用各向异性来完成${\mathrm{大号}}^p$插值和迭代技术。