Study On A Quasi-linear Chemotaxis Model Coupling With Fluids

In this paper,we study a two-dimensional homogeneous initial boundary value problem of the following quasi-linear chemotaxis model coupling with fluids:where ?(?)R2 is a bounded domain with smooth boundary.This model describes the broadcast fertilization phenomenon of corals,sea urchins,sea anemones and ect,in which n,?,c denote respectively the density of sperm cells,the density of egg cells and the concentration of enzymes that secreted by egg cells;the chemotactic sensitivity function S(x,n,c)is a tensor-valued function;u denotes the velocity of fluid,n,?,c satisfy the no-flux boundary condition and u satisfies the homogeneous Dirichlet boundary condition.Since this model is a degenerate parabolic equations,the corresponding initial boundary value problem has no classical solutions.In order to obtain the global existence of the weak solutions to the initial boundary value problem which is first regularized,and under the condition of exponential m>1 and l>0,by using the Gagliardo-Nerenberg inequality,Gronwall inequality,energy estimation and other methods,a series of uniform prior estimates for local classical solutions of the regularized problem is established.Then,the global bounded classical solutions of the regularized problem is further obtained by using the Moser iterative and the solution of continuation criterion.Finally,by using the regularization approximation method,the global bounded weak solutions to the quasi-linear initial boundary value problem are obtained with the the condition that m>1 and l>0.