Small Parameter Limits In Chemotaxis Systems And Related Problems

Chemotaxis is a common phenomenon in biology and has received widespread attention.From a mathematical perspective,in order to describe the chemotactic phenomenon of bacteria aggregation due to the attraction of chemical substances,Keller and Segel proposed the classical Keller-Segel system in 1970.Based on the experimental facts and relevant chemotaxis background,in recent years,many scholars have made in-depth research on the Keller-Segel-fluid coupling system,the Keller-Segel system with indirect signal production mechanism and other variants of chemotaxis systems.This dissertation mainly concentrates on the small parameter limit problems of the Keller-Segel-(Navier-)Stokes system and the Keller-Segel-ODE system with indirect signal production mechanism.The specific research contents are as follows:1.We study the fast signal diffusion limit of solutions of the parabolic-parabolic-fluid system of the Keller-Segel-Stokes type to the solution of the corresponding parabolicelliptic-fluid system in 2D and 3D bounded domains.Under the assumption of natural volume filling effect,we establish an algebraic convergence rate of the fast signal diffusion limit with large initial data by utilizing a series of very subtle bootstrap arguments for combinational functionals,an induction argument and some maximal regularities.In particular,we eliminate the limitation of asserting convergence only along some subsequence in Wang-Winkler-Xiang(Calc.Var.,2019).2.We explore the stability of a parabolic-parabolic-fluid system of the Keller-SegelNavier-Stokes type in fast signal diffusion on a bounded planar domain.Under the assumption of natural volume filling effect,we first prove that the global classical solutions of the parabolic-parabolic-fluid system will converge to the solution of the parabolicelliptic-fluid counterpart by using space-time estimates and bootstrap arguments.Meanwhile,we also gain the global well-posedness of the parabolic-elliptic-fluid system for large initial data.Secondly,for appropriately small initial cell mass,we establish some new exponential time decay estimates,which especially guarantee an improvement of convergence rate in time.Finally,in order to further explore the stability,we carry out three different types of numerical simulations: the nontrivial equilibrium,the trivial equilibrium and the rotating aggregation.The simulation results demonstrate the possibility to achieve the optimal convergence and illustrate the disappearance of equilibrium deviations between the parabolic-parabolic-fluid system and the parabolic-elliptic-fluid system,as well as the sharp fluctuation of errors for the rotating solution.3.We investigate the small parameter limit problem of the Keller-Segel-ODE system with indirect signal production mechanism in a bounded planar domain,and conduct the rigorous convergence analysis.Precisely,we reveal that the system has a uniformly global-in-time convergence under the appropriate assumption of the initial population mass as some parameter approaches zero by using Lyapunov functionals and a series of combinational functionals.In particular,we confirm that the solution of the system will converge to the solution of the corresponding Keller-Segel system with direct signal production mechanism under this assumption,and also show that its solution will blow up in infinite time,which improves the existing blow-up results of Lauren(?)ot(Discrete Cont.Dyn.Syst.-B,2019).