Boundedness Of Weak Solutions In A 3D Chemotaxis-Stokes System With Doubly Degenerate Diffusion

In biology,chemotaxis describes the directional movement of cells under the influ-ence of some chemical signal substances in the environment.Because cells live in viscous fluid,and the movement of fluid is usually expressed by the famous(Navier-)Stokes equa-tions,chemotaxis-Stokes model has become one of the hot topics in the frontier research of biomathematics.In particular,the existence,uniqueness,stability,blow-up,asymp-totic behavior and viscosity vanishing limit of the solutions of the relevant models have attracted the interest of many scholars.In the thesis,we study the initial boundary value problem of a chemotaxis-Stokes model with nonlinear doubly degenerate diffusion in the three dimensional space under the proper definition of weak solutions.The boundary value satisfies Neumann-Neumann-Dirichlet boundary conditions.If the unknown function n denotes the bacterial density,the typical feature of this model is the doubly degenerate diffusion term?·(|?n^m|^p-2?n^m)by coupling p-Laplace with porous medium with p?2 and m?1.When m and p further satisfy 8mp-8m+3p>15,the thesis establishes the existence of globally bounded weak solutions for this initial boundary value problem.The main methods of this thesis are as follows:we first prove the global existence of the solutions of the regularized sys-tem by using parabolic equation theory.Then,we establish priori estimates of solutions of the regularized system by using the delicate technique of energy estimation,and ob-tain the boundedness of solutions of the regularized system by using the Moser iteration method.Finally,we obtain the global existence of bounded weak solutions for the original boundary value problems by taking the limit of regularized systems.