The Well Posedness Of Two Types Of Boundary Layer Equations For Compressible Isentropic Flow

Boundary layer equation is a kind of equation which is used to describe the motion state of gas or liquid near the boundary and it is widely applied in gas or liquid dynamics.In this paper,we study two kinds of boundary layer equations:Navier-Stokes boundary layer equations and two phase flow boundary layer equations.We use energy method to obtain the uniform estimate of the solutions.Then through the energy estimate,the local well-posedness of the equations can be obtained.There are many conclusions about the Navier-Stokes boundary layer equations,but there is no relevant work about the boundary layer equations of two phase flow at present.This paper is the first exploration of the research on the well-posdness of two phase flow boundary layer equations.Compared with the previous works,we mainly do three different works in this paper:1.we directly use the energy method to get the well-posedness of equations;2.most of the literature researches focus on the form of incompressible equations.These two kinds of equations we study are all compressible,so it is much more difficult to apply energy estimate than the equations with incompressible condition;3.The function of density in the two phase boundary layer equations is unknown,the uniform energy estimate of density need to be estimated,so the progress of energy method is more complicated.In the first chapter,we mainly introduces two kinds of boundary layer equations and related progress of the boundary layer equations.In the second chapter,the basic conclusions used frequently in this paper are intro-duced,such as Hardy type inequality,Sobolev type inequality,comparision principle,etc.,which are prepared for the proof of the main results.In the third chapter,the well-posedness of Navier-Stokes boundary layer equations is obtained by using energy estimate.In the first section,we apply energy estimate to the regularized equations.The proof is divided into two parts:1.When ?<s,?1?s-1,we directly calculate uniform weighted L² estimate of D^??;2.When ?₁=s,because of the derivative loss caused by compressible condition,the new variable g_s is introduced to avoiding the difficulty,then the uniform weighted L² estimate is calculated.In the second section,the well-posedness of Navier-Stokes boundary layer equations can be obtained by the uniform weighted H^s estimate which is obtained in the first section.In the forth chapter,the well-posedness of two phase flow boundary layer equations is proved by energy estimate.In the first section,we apply energy estimate to the regularized equations.The proof is divided into four parts.In the first part,the uniform weighted L² estimate of D^?? is estimated when |?|?s:?1?s-1.In the second part,in order to avoid the difficulty of derivative loss caused by compressible condition,we introduce a new variable g_s,then the uniform weighted L² estimation of g_s is estimated.In third part,the uniform weighted L² estimate of Dap is estimated when |?|?s,?1?s-1.In the forth part,the uniform weighted L² estimate of hs which is introduced to avoiding the derivative loss is estimated.Then in the second section,the well-posedness of two phase flow boundary layer equations is obtained by using the uniform estimate obtained in the first section.