Mathematical Theory Of Chemotaxis-fluid Systems With Nonlinear Cell Diffusion

Diffusion is one of the most omnipresent natural phenomena.Generally speaking,bacterial diffusion in viscous fluids can be regarded as porous medium diffusion and its movement is also affected by other chemicals(such as bacterial chemotaxis,in which the direction of bacterial movement is related to the increase or decrease of chemical signal concentration).To describe the dynamic behavior of bacteria in viscous fluids,Francesco et al.came up with a chemotaxis-fluid system with porous media cell diffusion Δnm,where n denotes the density of bacteria.In view of the difference between the observation in experiments and the setting in analysis,the dissertation aims at investigating a three-dimensional chemotaxis-Stokes system with porous medium diffusion and general sensitivity under different signal boundary conditions.The research contents are as follows:First,the work investigates the global bounded weak solutions and the large-time behavior of a three-dimensional chemotaxis-Stokes system with general sensitivity,porous medium cell diffusion,and Robin signal boundary for each m>7/6.A Lions-Magenes type transformation and an iterative method guarantee the global existence of weak bounded solutions.Here,the key is to deal with the extra terms coming from the transformation.In the case of homogeneous Robin boundary value,the large-time behavior and the eventual smoothness of the preceding weak solutions are shown by adopting Winkler’s method(Calc.Var.,2015).Then,the work considers the global bounded weak solutions of a three-dimensional chemotaxis-Stokes system with general sensitivity,porous medium diffusion,and Neumann signal boundary for each m≥65/63.In particular,this extends the precedent results which asserted global solvability within the larger range(m>7/6)for general sensitivity,Winkler,Calc.Var.,2015 or m>9/8 for scalar sensitivity,Winkler,J.Differ.Equ.,2018).The proof is based on a new observation of a quasi-energy type functional and an induction argument.Finally,the work explores the global bounded weak solutions of a three-dimensional chemotaxis-Stokes system with general sensitivity,porous medium diffusion,and inhomogeneous Dirichlet signal boundary for each m>13/12.Compared with the quite welldeveloped solvability for the no-flux signal boundary value with m>7/6(Winkler,Calc.Var.,2015),to our best knowledge,this seems to be the first result on a chemotaxis-fluid system with general matrix-valued sensitivity for such a Dirichlet signal boundary condition,under which even for the scalar sensitivity,this result also extends the recent range m>7/6(Wu,Xiang,J Differ.Equ.,2022).This proof will depend on a new supervision of the boundary estimate and on an iterative method.The same technique can be applied to the two-dimensional chemotaxis-Navier-Stokes setting to confirm a similar conclusion for any m>1.