Cauchy Problem For The Incompressible Navier-stokes Equation With An External Force And Gevrey Smoothing Effect For The Prandtl Equation

This thesis deals with equations of fluid dynamics.We consider the following two models:one is the Navier-Stokes equation in R3 with an external force,the other one is the Prandtl equation on the half plane R2+.For the Navier-Stokes system,we focus on the local in time existence,uniqueness,long-time behavior and blow-up criterion.For the Prandtl equation on the half-plane,we consider the Gevrey regularity.This thesis consists in four chapters.In the first chapter,we introduce some back-ground on equations of fluid dynamics and recall the physical meaning of the above two models as well as some well-known mathematical results.Next,we state our main results and motivations briefly.At last we mention some open problems.The second chapter is devoted to the Cauchy problem for the Navier-Stokes e-quation equipped with a small rough external force in R3.We show the local in time existence for this system for any initial data belonging to a critical Besov space with negative regularity.Moreover we obtain three kinds of uniqueness results for the above solutions.Finally,we study the long-time behavior and stability of priori global solu-tions.The third chapter deals with a blow-up criterion for the Navier-Stokes equation with a time independent external force.We develop a profile decomposition for the forced Navier-Stokes equation.The decomposition enables us to connect the forced and the unforced equations,which provides the blow-up information from the unforced solution to the forced solution.In Chapter 4,we study the Gevrey smoothing effect of the local in time solution to the Prandtl equation in the half plane.It is well-known that the Prandtl boundary layer equation is unstable for general initial data,and is well-posed in Sobolev spaces for monotonic initial data.Under a monotonicity assumption on the tangential velocity of the outflow,we prove Gevrey regularity for the solution to Prandtl equation in the half plane with initial data belonging to some Sobolev space.