# Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

Source: [Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics](<https://www.sota2.com/research/paper/68fa54031e209131ce98803e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:42.952Z

Authors: Xinru Cao, Shunsuke Kurima, Masaaki Mizukami

Published: 2017-06-24T05:51:52.000Z

## Abstract

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n\_1)\_t + u\\cdot\\nabla n\_1 =\\Delta n\_1 - \\chi\_1\\nabla\\cdot(n\_1\\nabla c)+ \\mu\_1n\_1(1- n\_1 - a\_1n\_2)$, $(n\_2)\_t + u\\cdot\\nabla n\_2 =\\Delta n\_2 - \\chi\_2\\nabla\\cdot(n\_2\\nabla c) + \\mu\_2n\_2(1- a\_2n\_1 - n\_2), c\_t + u\\cdot\\nabla c =\\Delta c - c + \\alpha n\_1 +\\beta n\_2$, $u\_t= \\Delta u + \\nabla P+ (\\gamma n\_1 + \\delta n\_2)\\nabla\\phi$, $ \\nabla\\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $\\Omega \\subset \\mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+\\alpha n\_1+\\beta n\_2$ is replaced with $-(\\alpha n\_1+\\beta n\_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $\\mu\_1,\\mu\_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $\\mu\_1,\\mu\_2$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1706.07910>)

[PDF](<https://arxiv.org/pdf/1706.07910.pdf>)