A class of parabolic-elliptic Keller–Segel–Stokes systems generalizing the prototype

is considered under boundary conditions of homogeneous Neumann type for n (the density of the cell population) and c (the chemical concentration), and Dirichlet type for u (the velocity field), in a bounded domain \(\Omega \subseteq {\mathbb {R}}^2\) with smooth boundary, where \(C_S>0\) and \(\phi \) is a given sufficiently smooth function. The model is proposed to describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. Moreover, the chemical diffuses much faster than the cells move. It is shown that under the condition that

for any sufficiently smooth initial data \((n_0,u_0 )\) satisfying some compatibility conditions, the associated initial-boundary-value problem (KSF) possesses a global bounded classical solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on \(\alpha \) here is optimal. Building on this boundedness property, it can finally even be proved that the corresponding solution of the system decays to \(({\bar{n}}_0,{\bar{n}}_0,0)\) exponentially if \(C_S\) is smaller, where \({\bar{n}}_0=\frac{1}{|\Omega |}\int _{\Omega }n_0\). Our main tool is consideration of the energy functional

which is a new energy-like functional.

一类泛化原型的抛物线-椭圆 Keller-Segel-Stokes 系统

在n（细胞群密度）和c（化学浓度）的齐次 Neumann 型边界条件下以及u（速度场）的Dirichlet 型边界条件下，在有界域\(\Omega \subseteq {\ mathbb {R}}^2\)具有平滑边界，其中\(C_S>0\)和\(\phi \)是给定的足够平滑的函数。该模型旨在描述趋化性-流体相互作用，当趋化剂的进化主要由细胞产生时。此外，化学物质的扩散速度比细胞移动的速度要快得多。表明在条件

对于满足某些兼容性条件的任何足够平滑的初始数据\((n_0,u_0 )\)，相关的初始边界值问题 ( KSF ) 具有全局有界经典解。与相应的无流体系统的结果相比，很容易看出这里对\(\alpha \)的限制是最优的。基于这个有界性质，最终甚至可以证明系统的相应解以指数方式衰减到\(({\bar{n}}_0,{\bar{n}}_0,0)\)如果\( C_S\)较小，其中\({\bar{n}}_0=\frac{1}{|\Omega |}\int _{\Omega }n_0\)。我们的主要工具是考虑能量泛函

这是一种新能源般的功能。