Second Grade Incompressible Fluid Flow With Slip Boundary Conditions

In this thesis,we study the equation governing the motion of second grade incompressible fluids in a bounded domain of R~3, with slip boundary condition by Galerkin method and Hodge Decomposition theory. We proof the nonlinearity in the second grade fluid equation(1.1) matches smoothly with the slip boundary condition(1.2). In dimension three, we obtain global existence for small initial data and positive viscosity and local existence of H~3 solution for arbitrary initial data. In the light of contents, this thesis is divided into four chapters.The first chapter is to introduce the main problems that we are concerned and the development of the problem in the internal and external.In the second Chapter,we introduce notations,some important theorems and several classical results that will be used in the following proofs.In the third chapter,we consider the equations governing the motion of second grade fluid and averaged equations. As we know, the existence of solution has many classical results for the problem,but seldom about the equation with slip boundary condition.In §3.1we show the nonlinearity in the second grade fluid equation(1.1) matches smoothly with the slip boundary condition(1.2) with Hodge decomposition theory.In §3.2 we prove some priori estimates;In §3.3,we discuss the global existence for small initial data and positive viscosity;In §3.4,we discuss the local existence of H~3 solution for arbitrary initial data;In §3.5, we use Galerkin method, and priori estimates to get the existence of the solution.In §3.6 we present the proof of the uniqueness.At the last chapter, we sum up the work we have done and point out the issue that we can continue to study.