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

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3-dimensional bounded domain < subset of>R3 with smooth boundary. Both chemotaxis-fluid systems and 2-species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2-species-fluid systems. Recently, global existence and asymptotic stability in the above problem with (u delta)u in the fluid equation were established in the 2-dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3-dimensional case when <mml:mfrac>max{1,2}min{1,2}</mml:mfrac>c0L() is sufficiently small.