Study On The Small Parameter Limit Problems Of The Keller-segel Model

In this dissertation,we mainly study the small parameter limit problems of several biological chemotaxis models.The thesis consists of six chapters:In Chapter 1,we introduce the background and the development history of chemotaxis and the main results of the thesis.In Chapter 2,we mainly study the small parameter limit problem for a system arising from the Keller-Segel equations in one space dimension under Dirichlet boundary conditions.First,we construct an accurate approximate solution which incorporates the effects of boundary layers.Then we prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.Our approach is based on the method of matched asymptotic expansions of singular perturbation theory and the classical energy estimates.In Chapter 3,the same Keller-Segel model in 3-D(or 2-D)under Dirichlet boundary conditions is considered.It is more difficult to construct the approximate solutions because of the change of dimension.Moreover,the initial values also need to satisfy higher regularization conditions.Classical energy estimation is also used to study the structural stability of approximate solutions.In Chapter 4,the small parameter limit problem of a two-dimensional KellerSegel model with linear chemotaxis under Neumann boundary condition is studied.The approximate solution is constructed by the asymptotic expansion method of singularly perturbed theory,and the influence of boundary layer is taken into account.Finally,the energy method is used to prove the structural stability of the approximate solution as the chemical diffusion coefficient approaches zero.In Chapter 5,we consider the small parameter limit problem of 2-D incompressible Keller-Segel-Stokes model with zero-flux boundary for the microbial concentration,chemical concentration and no-slip boundary for the fluid velocity.The construction of fluid velocity induces much more difficulty due to the incompressibility condition.In this chapter,only an approximate solution is constructed,and further analysis on this approximate solution will be carried out in future work.In Chapter 6,we summarize the main results of the thesis,and proposes some interesting problems for further investigation.