Global Boundedness To A Chemotaxis-Navier-Stokes System With Porous Medium Cell Diffusion

Chemotaxis describes the directional movement of organisms induced by some chemical signal substances.In nature,most cells or cell populations live in viscous fluid,so it is of obvious practical significance to study chemotaxis-fluid models.The thesis mainly considers the following chemotaxis-Navier-Stokes model with porous medium cell diffusion and matrix-valued sensitivity:in a bounded domain Ω (?) R2 with smooth boundary,where the parameter m>1,the unknown functions n,c,u and P represent cell density,chemical signal concentration,fluid velocity and the associated pressure,respectively,the matrix-valued function measures the chemotactic sensitivity S,while φ is a given potential function.The main result of the thesis is as follows:If n and c satisfy the no-flux boundary conditions and u satisfies the homogeneous Dirichlet boundary condition,the corresponding initial-boundary value problem admits a global weak solution for any smooth initial data and bounded matrix-valued function S.The basic idea of our proof is as follows:firstly,the regularized system is constructed for the initial-boundary value problem,and the global classical solvability of the regularized system is established;then the boundedness of the regularized solution is obtained with the aid of energy estimates,Moser iteration and Neumann heat semigroup;finally,the global existence of the weak solution of the original problem will be obtained by taking the limit in the regularizaed system.