Global Existence And Asymptotic Stability Of The Fractional Chemotaxis-Fluid System

The study of the anomalous duffusion has become one of the most popular topics in partial differential equations.It has also attracted the attention of several famous scholars,such as the Wolf prize winner Caffarelli,the famous German math-ematician Valdinoci,the famous Italy mathematician Terracini and so on.In this paper,we focus on the fractional chemotaxis-fluid system.Chemotaxis is a biologi-cal process in which cells(e.g.,bacteria)move towards a chemically more favorable environment.For example,bacteria often swim towards higher concentration of oxygen to survive.In 1970s,Keller and Segel[1]first proposed Keller-Segel system.This system is one of the best-studied model for chemotaxis.Recently,a strong theoretical and empirical evidence has appeared for replacing the classical diffusion with a fractional one in Keller-Segel system in order to model feeding strategies of a wide range of organisms.In nature,cells often live in a viscous fluid so that cells and chemical substrates are also transported with fluid,and meanwhile the motion of the fluid is under the influence of gravitational forcing generated by aggregation of cells.Generally,the motion of the fluid is determined by the well-known incompressible Navier-Stokes equations or Stokes equations.This kind of cell-fluid interaction becomes more complicated since it not consists of chemotaxis and diffusion,but also includes transport and fluid dynamics.To describe the coupled biological phenomena men-tioned above,Tuval et al.[2]proposed the classical chemotaxis-fluid system which has received much attention from the mathematical community over the past few years.In this paper,we mainly concern the global existence and asymptotic stability of fractional chemotaxis-fluid system.The outline of the paper is as follows.In Chapter 1,we introduce the research background and some progress of fractional chemotaxis-fluid system.In Chapter 2,we investigate the generalized the Keller-Segel sysytem with two fractional parabolic equations and a classical elliptic equation in Rn with n?2.We develop a framework for a unified treatment of the existence,uniqueness and decay estimates.With the help of selecting appropriate functional space,the existence,uniqueness and decay estimates of the global classical solution are shown at the same time under the assumption that the initial data are small enough.In Chapter 3,we consider a model arising from biology,consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in R3.We establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument.Moreover,the decay rates of the solutions and their higher-order spa-tial derivatives towards the equilibrium by introducing the negative Sobolev space H-s(0?s<3/2).