The Study Of Two Chemotaxis-Fluid Systems

The chemotaxis system is a partial differential system which describes the biased movement of cells or bacteria according to the concentration of chemical substances in the environment.The representative of the chemotaxis system is the standard Keller-Segel model.The standard Keller-Segel model mainly describes the interaction between cells and chemicals.However,in nature,many bacteria live in viscous liquids.Therefore,in addition to the effects of chemical substances or nutrient solutions on cells,the effects of the liquid environment in which the cells are located on the chemotactic movement of cells should also be considered.Bio-mathematicians use the coupling chemotaxis-fluid system to model the chemotactic behavior of bacteria in liquid environment.This paper mainly studies two chemotaxis-fluid systems.The first one is the following initial-boundary value problem:where ?(?)R3 is a bounded domain with smooth boundary.In contrast to the existing results,both the effect of gravity(potential force)on cells and the effect of the chemotactic force on fluid is considered here,which leads to the stronger coupling than the most-studied chemotaxis-fluid model.We shall establish the global existence of(1)through some suitable energy method and a series of a priori estimate.The second initial-boundary value problem we consider is where ?(?)R2 is a bounded domain with smooth boundary.It is known that when 0<?<1,there exists unique classical solution {n?,c?,u?)for any ??[0,1)We shall study the small convection limit problem.We will prove that[n?,c?,u?)will converges to(n0,c0,u0)as ??0+.The plan for this thesis is as follows:The introduction mainly introduces the research background of chemotaxis-fluid system and the related research progress.The first chapter introduces the main problems of this thesis and some premilaries.In the second chapter,we consider the global existence of the weak solution to(1).In the third chapter,the convergence of the solution to(2)is considered.