Study On A Coupled Chemotaxis-Fluid System

The chemotaxis system is a mathematical model that characterizes the behavior of cell self-organization and chemotaxis.The classical research model of this kind of problem is the Keller-Segel model.The classical Keller-Segel model mainly describes the interaction between the cells and the signal substance.However,in the actual biological background,the fluid environment in which cells are located also has a non-negligible effect on the chemotaxis behavior of cells.This biological phenomenon is characterized by the chemotaxis-fluid systems,we shall do some qualitative study on the chemotaxis-fluid model in this paper.In particular,this paper mainly studies the following initial-boundary value problemin a bounded(convex)domain3∈ΩR with smooth boundary,where =(txnn,)denotes the density of bacteria and =(txcc,)denotes the oxygen concentration.S =(cnx S,,)is the sensitivity function,while =(txuu,)and P represents the fluid velocity and the associated pressure,respectively.The parameterκ∈R is related to the strength of fluid convection and φ is a given function.The paper is structured as followsThe introduction reviews the background and development of chemotaxis-fluid systems,and makes a clarified exposition on the research progress and important results of the corresponding derivative models.The first chapter clarifies the research content and main results of this paper,and give the necessary preparation knowledge.The second chapter considers the energy inequality on the condition of(cnxS)1,,≡ in the model(*).The third chapter considers the case where the chemotactic sensitivity function is a tensor value function,that isand S=(x,n,c) satisfies(?),for constants.(?) We shall establish the uniform boundedness of solutions for any α>0.