On The Navier-Stokes Equation Of Vorticity Type With A Slip Boundary Conditions

In this thesis,we study the Navier-Stokes equation of vorticity type in a bounded domain of R~3, with slip boundary condition by Galerkin method and Hodge Decomposition theory. We obtain the local existence and uniqueness of H~1 solution for arbitrary initial data.and the regularity of the weak solution .we close by find the relation between N-S equation and the form for vorticity of N-S equation. 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 in domestic and foreign.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 form for vorticity of N-S equations. In§3.1we show the boundary effect of the nonlinearity.In§3.2 we prove some priori estimates;In§3.3,we discuss the existence and uniqueness of H~1 weak solution for arbitrary initial data;In§3.4,we discuss the regularity of the above weak solution;In§3.5, we discuss some problems about the equation itself.