| Bacterial chemotaxis has intrigued many investigators since it was first observed by Engelmann in 1881.In order to use the mathematical language to describe the phenomenon where bacterial chemotaxis leads to the aggregation of bacterial population,Keller and Segel in 1970 proposed the famous Keller-Segel model,which consists of a coupled system of two parabolic partial differential equations.Since then,partial differential equations describing the phenomenon of bacterial chemotaxis have attracted the extensive attention of biologists and mathematicians.Considering that bacteria or microorganisms usually live in a viscous fluid,further,Tuval et al.introduced a chemotaxis-fluid coupling model with signal consumption in 2005.This dissertation mainly focuses on the global existence,boundedness,large time behavior and convergence of solutions to the initial-boundary value problems of chemotaxisfluid coupling models.The specific research contents are as follows:1.This work investigates the small-convection limit of the classical solutions of the initial-boundary value problems for Keller-Segel-Navier-Stokes equations with subcritical sensitivity and no-flux-no-flux-no-slip boundary conditions in a two-dimensional bounded convex domain.It is shown that the classical solutions for the initial-boundary value problems of the Keller-Segel-Navier-Stokes equations will stabilize to the classical solution for the initial-boundary value problem of the corresponding Keller-Segel-Stokes equations as the strength coefficient of nonlinear fluid convection κ→0+.For the small initial data case,it is established that these solutions have uniform decay estimates with respect to time in the limit α→ 0+.2.This work is devoted to studying the global existence and boundedness of weak solutions of the two-dimensional chemotaxis-Navier-Stokes equations with porous-media type diffusion Δnm(m>1)and no-flux-inhomogeneous Robin-no-slip boundary conditions in a bounded domain.The main methods to the proof are to introduce a LionsMagenes type boundary transformation to homogenize the inhomogeneous boundary condition for signal concentration and then make use of energy estimation technique.Further,in the incoming oxygen-free case,it is proved that the constructed solutions will stabilize to a constant spatial equilibrium state as t→∞.3.This work researches the global existence and boundedness of weak solutions of the initial-boundary value problem for the chemotaxis-Stokes equations with porousmedia type diffusion Δnm(m>3N-2/2N)and no-flux-inhomogeneous Dirichlet-no-slip boundary conditions in N-dimensional bounded domains(N=2,3).The main methods to the proof are to operate a series of uniform a priori estimates of solutions for the approximated system by means of the bootstrap procedure and decay estimates of homogeneous Dirichlet heat semigroups. |
