Study On The Zero Dissipation Limit Of Two Types Of Compressible Fluid Dynamic Equations

This thesis is mainly concerned with the zero dissipation limit of solutions toward the contact discontinuity for two types of compressible fluid dynamic equations.Firstly,we study the zero dissipation limit of solutions to contact discontinuity for the following Cauchy problem of the one-dimensional com-pressible Navier-Stokes system:Here v>0,u,?>0,p and e denote the specific volume,the velocity,the abso-lute temperature,the pressure and the internal energy of the fluid,respectively;? and are the viscosity coeeffcient and the heat conductivity coeeffcient,re-spectively.v±>0,u±?±>0 are given constant.We assume the pressure p nd the internal energy e are given by p=R?/v,e=R?/?-1,heree R>0 is a constant,and ?>1 is the adiabatic exponent.y using a new a priori assumption and some refined energy estimates,e show that when the Riemann problem of the compressible Euler equations dmits a contact discontinuity solution,the Cauchy problem(1)of the corre-ponding Navier-Stokes equations has a unique global smooth solution,which onverges to the contact discontinuity at a rate ?7/8 when the viscosity coeeffi-cient ? and the heat conductivity coeeffcient ? satisfy ?=(?)or ?=O(?),and ??0.Here the strength of the contact discontinuity has no need to be mall.Secondly,we consider the zero dissipation limit of solutions to contact discontinuity for the following Cauchy problem of the one-dimensional com-pressible micropolar fluid model:(v,u,?,?)(x,0)=(v0,u0,?0,?0)(x)?(v±,u±,?±,0),x?±?(3)Here v>0,u,?>0,?,p and edenote the specific volume,the velocity,the absolute temperature,the microrotation velocity,the pressure and the internal energy of the fluid,respectively;?,?,and A are the viscosity coefficient,the heat conductivity coefficient and the microviscosity coefficient,respectively.v±>0,u±,?±>0 are given constant.We assume the pressure p and the internal energy e are given by p=R?/v,e=R?/?-1,where R>0 is a constant,and ?>1 is the adiabatic exponent.By using a priori assumption and energy estimates,we prove that when the Riemann problem of the compressible Euler equations admits a contact discontinuity solution,the Cauchy problem(2)-(3)of the corresponding com-pressible micropolar fluid model has a unique global smooth solution,which converges to the contact discontinuity at a rate ?7/8 when the viscosity coeffi-cient ? and the heat conductivity coefficient ? satisfy ?=O(?)and ??0.Moreover,we also obtain the large-time behavior toward the contact disconti-nuity for such a smooth solution.Here the strength of the contact discontinuity has also no need to be small.This thesis is divided into four chapters.In Chapter 1,we introduce our problem and some related background,and then state the two main theorems in the thesis.In Chapter 2,we shall prove Theorem 1.1,i.e,the zero dissipa-tion limit of smooth solutions of the Cauchy problem(1)toward the contact discontinuity.To do so,we first make an a priori assumption(2.6)which de-pends on the heat conduct coefficient ?.Under some smallness assumptions on the viscosity coefficient ? and the heat conductivity coefficient ?,we ob-tain the zero dissipation limit for the smooth solution of the problem(1),and a faster convergence rate compared with the formal results by some refined energy estimates.In Chapter 3,we shall prove Theorem 1.2,i.e,the zero dis-sipation limit of solutions of the Cauchy problem(2)-(3)toward the contact discontinuity.To this end,we first make an a priori assumption(3.6)which depends on the heat conduct coefficient n.Under some smallness assumptions on the viscosity coefficient ? and the heat conductivity coefficient ?,we obtain the zero dissipation limit and large-time behavior of smooth solutions of the Cauchy problem of(2)-(3)by utilizing the estimate of the heat kernel and some refined energy estimates.In Chapter 4,we summarize the whole thesis,and then present some problems which deserve further investigation in the future.