Mathematical Research On The Chemotaxis-Navier-Stokes Equations And The Related Problems

Many problems in nature can be simulated and characterized by partial differential equations?PDEs?.In particular,the use of PDEs to describe chemotaxis in biology orig-inated from the pioneering works of Patlak in the 1950s and Keller-Segel in the 1970s.After more than half a century of development,the mathematical research of chemotaxis model based on PDEs has made great progress.Considering that bacteria usually live in viscous fluid environments,the chemotaxis-fluid coupled PDEs has become one of the mathematical models widely concerned by biologists and applied mathematicans in recent ten years.In this dissertation,the mathematical research on the chemotaxis-Navier-Stokes models and the related problems.The specific research contents are as follows:1.The global existence of weak solutions and classical solutions for Cauchy problem and initial-boundary value problem of chemotaxis-Navier-Stokes system in two-dimensional whole space and bounded domian is studied respectively under appropriate conditions.In particular,we improved the results for global existence of Duan-Li-Xiang?J.Differential Equations,2017?.2.We study the decay estimates of the strong solutions of the three-dimensional chemotaxis-Navier-Stokes system in a strip domain.Firstly,several very important in-equalities are obtained by using the anisotropic L^pinterpolation inequality.Then,the decay rate of chemical concentration c is obtained from these inequalities,and then the cell density n is finally stable to the equilibrium state n_?.Thirdly,the decay rate of fluid velocity u is obtained by using the decay rate of n,the strong coupling of the system and iterative techniques.Finally,the De Giorgi truncation technique is used to improve the decay estimates to c _L?_???.3.The boundedness and asymptotic behavior of global solution on the chemotaxis-fluid model with general logistic source and logarithmic sensitive function are studied.In particular,the global boundedness and exponential decay rate estimates of the classical solutions of the chemotaxis-Navier-Stokes system initial-boundary value problem with general logistic source and logarithmic sensitive function are established under appropriate conditions.4.The global solution and asymptotic behavior of coral fertilization model are stud-ied.The technique used is the smooth property of Neumann heat semigroup.