Some Problems For Models Of Keller-Segel-Navier-Stokes

In this paper,we consider the well-posedness of Keller-Segel-Navier-Stokes equations.The model is as follows:Here,the unknows,(?)and are the cell density,the chemical concentration,the fluid velocity and the pressure of the fluid respectively.Using the Fourier localization technique,we first estabilish the blow up criterion of smooth solutions for the incompressible Keller-Segel-Euler equations.Then,we consider the inviscid limit of the three dimensional Keller-Segel-Navier-Stokes equations and establish the conver-gence rate of the inviscid limit.Next,by making use of weighted function generated by heat kernel and fourier localization technique,we obtain the existence and uniqueness of smooth solutions in Besov spaces.What is more,we gain a Beal-Kato-Madja type blow-up criterion with the help of a logarithmic inequality.Finally,it is proved that we obtain the existence and uniqueness of weak solutions for the two dimensional equations.We also obtain the global well-posedness for the three dimensional incompressible equations with large initial vertical component.