Global Solutions To The Coupled Chemotaxis-fluids System In Horizontal Periodic Domain

Chemotaxis is a common biological phenomenon,which describes the tendency move-ment of cells or other living organisms moving towards a chemically more favorable en-vironment.For example,Bacillus subtilis often swim towards higher concentration of oxygen to survive.In nature,cells often live in a viscous fluid so that cells and chem-ical substrates are also transported with the fluid.On the other hand,the movement of the fluid is under the influence of the gravitational forcing generated by aggregation of cells.The cell-fluid interaction can be modeled by the coupled chemotaxis-fluid model.Therefore,the study of the solution of the coupled chemotaxis-fluid model with biological background has great significance in biomathematics.In this paper,we focus on the initial boundary value problem in the three-dimensional horizontal periodic domain of a system coupled with the oxygen consumption chemotaxis model and the viscous incompressible fluid equation,and the corresponding model is (?) where?(?)R~3is a horizontal periodic domain with finite depth.Here,the unknown functions are n,c and u,denoting the cell density,the chemical substance concentration and the fluid velocity field,respectively.In particular,we assume that n and c satisfy the no-flux boundary conditions on the upper and lower boundary of?,and u satisfies the no-slip boundary condition on the upper and lower boundary of?.Under some basic assumptions on the parameter function?and the initial data(n(x,0),c(x,0),u(x,0)),then we establish the global existence and uniqueness of strong solutions near constant states for the chemotaxis-Navier-Stokes system.Firstly,we shall chain?to?_tto validate the local existence and uniqueness of the classical solution.Secondly,using the correspond energy functional and the appropriate inequality,we show that the uniform prior estimates of solution for the system.Finally,we obtain the global existence and boundedness of the classical solution by using the local existence,uniform a prior estimates and the standary continuity theory.