Well-posedness Of Chemotaxis Systems Related To Fluid Motion

The classical Keller-Segel model and its variants mainly describe the interaction be-tween cells and the signaling substance mainly produced by cells themselves.Obviously,the alternation of the surrounding environment,such as fluids,may have a considerable impact on the migration of cells and vice versa.Consequently,investigating the interac-tion mechanism between cells,chemical signal and fluids is an interesting research topic in applied mathematics and biology.From the viewpoint of mathematical analysis,the chemotaxis-fluid model couples the typical difficulties in the study of fluid equations and the one of chemotaxis equations.This dissertation mainly focuses on the global well-posedness and stabilization of solutions to the initial boundary value problems for three types of the fluid-related chemotaxis models.The specific research contents are as fol-lows:1.This work is devoted to investigating the critical asymptotic behavior of the chemotactic sensitivity functions S₁and S₂for a class of double chemotaxis-Stokes system in a three-dimensional bounded domain.Firstly,the applications of the maximal Sobolev regularity of the heat and the Stokes evolution equations guarantee the spatio-temporal es-timates of solutions,so that the uniform a priori estimates of solutions may be established by introducing an appropriate weight function.Finally,with the help of the variation-of-constants representation of solutions,the global existence and uniform boundedness of classical solutions to the initial boundary value problem for the double chemotaxis-Stokes system under the natural assumption of saturation effect are established.From the analy-sis of the relevant findings,the proved critical exponents with respect to the chemotactic sensitivity functions S₁and S₂are optimal.2.This work investigates the relevant properities of solutions for a class of chemo-taxis models with cell proliferation in a given fluid environment.On the basis of energy es-timation and the bounedness of fluid,the global well-posedness of classical solutions to the initial boundary value problem for the chemotaxis model are proved in a two-dimensional bounded convex domain by introducing an appropriate weight function and using the rele-vant properties of the Neumann heat semigroup.When appropriate assumptions are made on the chemotactic sensitivity functionχ,the corresponding conclusion still holds for the three-dimensional bounded convex domain.3.This work studies the relevant properties of solutions for the chemotaxis model with cell proliferation coupled to the（Navier-）Stokes equations.The global well-posedness of classical solutions to the initial boundary value problem for a two-dimensional chemotaxis-Navier-Stokes system with cell proliferation and the global ex-istence of weak solutions to the initial boundary value problem for a three-dimensional chemotaxis-Stokes system are established by making use of a coupled entropy estimate together with a series of optimal estimates involving the Gagliardo-Nierenberg inequal-ities,and further,it is shown that the global classical solutions of the two-dimensional chemotaxis-Navier-Stokes system will stabilize to a constant spatial equilibrium state as t→∞by the use of the H（?）lder continuity and basic integrability of the solutions.