The Well-Posedness Of Several Types Of Mathematical Models In Relation To Chemotaxis

In this master’s dissertation,we focus on the initial boundary problem for the two-types of the two-species of Chemotaxis-Navier-Stokes（CNS）equations in bounded domains,and mainly study the global existence and asymptotic stability of the classical solutions of the consumptive and signal-producting CNS systems with a large class of dynamic functions including sub-logistic source.This thesis consists of four chapters:In Chapter 1,we mainly introduce the research background of Chemotaxi-Navier-Stokes equations,and give the recent research status and relevant progress.In particular,the influence of the existence of logistic source on the well-posedness of the solution is introduced.In Chapter 2,we present the basic knowledge of differential equations in this paper,and list some basic results and lemmas.In Chapter 3,we study the global existence and convergence of classical solution of the consumptive two-species Chemotaxis-Navier-Stokes model in a two-dimensional bounded smooth domain under homogeneous Neumann boundary conditions.By making full use of the characteristics of c-equation,such as the boundedness of c,the energy method is used to obtain the global priori estimates,so as to break through the blow-up criterion and get the global existence of the classical solution.In the study of large time behavior,we use the H?lder regularity to get the convergence result of the global solution.Compared with standard Lotka-Volterra competitive dynamic functions,our conclusions extend existing results for two-species chemotaxis-fluid models.In Chapter 4,we consider the Neumann initial boundary value problem of signal-producting two-species CNS model in a two-dimensional bounded domain.Compared with the consumptive model in Chapter 3,we cannot directly obtain‖c‖_L?（?）of the model.Hence,we need to construct different energy functional to overcome this difficulty.We prove the global existence of the classical solution by using the energy method and the heat semigroup estimates.Based on the results of the existence of global solution,the asymptotic stability of classical solution is further studied,and the specific decay form is given.