Mathematical Analysis On The Stability Of Subsonic Flows And Transonic Shocks In Two-dimensional Ducts

In this paper,we focus on the stability mechanisms of subsonic flows and transonic shocks.We consider the effects of heat exchange on the stability of transonic shocks and mass of change on the stability of subsonic flows in two-dimensional straight ducts.This paper first studies the effect of heat exchange in two-dimensional pipe on the stability of transonic shocks.Transonic shocks play a pivotal role in design of supersonic nozzles in gas dynamics.Previous studies have shown that for sta-tionary compressible Euler flows in rectilinear ducts with constant cross-sections,they are not stable under perturbations of upcoming supersonic flows and back pressures at the exits of the ducts,while expanding of the nozzle and frictions have stabilization effects.In this paper we study the Rayleigh flows,namely the effects of heat-exchange,on stabilization of transonic shocks,in two-dimensional rectilinear ducts.We proved that for given heat-exchange per unit mass of gas,almost all the associated unidimensional transonic shocks are stable,provided that the perturbations of the upstream supersonic flows and downstream back pressures satisfy some symmetry conditions;while for given heat-exchange per unit volume of gas,the resultant unidimensional transonic shocks are not stable.Mathematically we study a nonlinear free-boundary problem of a system of con-servation laws of hyperbolic-elliptic composite-mixed type.The proof depends on decoupling the elliptic and hyperbolic part of the subsonic Euler system in Lagrangian coordinates by characteristic decomposition.Since heat-exchange in-troduces complicate interactions in the Euler system,we need to study a general linear variable-coefficient first-order elliptic-hyperbolic strongly-coupled system with nonlocal boundary conditions,by Fourier analysis and careful investigation of reduced boundary-value problems of ordinary differential equations.The second problem of this paper is to study the effect of mass of change for steady subsonic flows in two-dimensional ducts.The purpose of this study is to further explore whether the mass addition effect has the stability of tran-sonic shocks under the disturbance of supersonic flows at the inlet and pressure at the outlet.We construct a class of subsonic solutions in two-dimensional straight ducts with constant cross-section,which only depend on the normal di-rection of the ducts.By proving the well posedness of the subsonic solutions of this special subsonic flows with respect to the two-dimensional perturbation of the inlet and outlet boundary conditions,the rationality of the formulation of the boundary value problem is proved.Since the subsonic Euler equations are quasilinear elliptic-hyperbolic mixed type,the general method to deal with this kind of problem is to separate the ellipse from the hyperbolic model and elliptic model.Therefore,we construct a new decomposition method which separates the principal part of elliptic mode and hyperbolic mode of Euler equations and couples the lower order terms.Due to the mass addition effect,the flow field has a stronger interaction,which induces a class of second-order elliptic equation mixed boundary value problems with multiple integral nonlocal terms.By using Fourier analysis,linear algebra,analytic function theory and regularity theory of second order elliptic equation,we obtain the well posedness of the problem.In particular,we study the regularities of transport equations and second-order elliptic equations in a class of x-direction anisotropic Holder spaces and ordinary Holder spaces.Based on this,a nonlinear iterative scheme is designed,and all the physical quantities obtained have the same regularity.The following is a brief introduction to the structure of this article.The first chapter is the introduction of this paper,which introduces the research background of this paper,and puts forward the problems and main results of this paper.In Chapter 2,we give some basic knowledge needed for this article.In Chapter 3,we use the implicit function theorem to construct the subsonic,supersonic and transonic shocks solutions for one-dimensional Rayleigh flow and the subsonic shock wave solutions for the mass addition problem.In Chapter 4,we reformulated the original problem in Lagrange coordinates in Section 4.1,and transformed it into a first-order linear hyperbolic-elliptic com-posite system with non-local boundary conditions and a fixed boundary problem by linearization.Update the Cauchy problem of ordinary differential equations for shock shape.In Section 4.2,we study the well posedness of a class of first order linear hyperbolic elliptic systems with nonlocal boundary conditions.In Section 4.3,we construct a nonlinear mapping and prove the first main result of this paper by mapping compressibility.In Chapter 5,in Section 5.1,we give a new equivalent decomposition method of Euler equations with mass addition effect in two-dimensional ducts,including Cauchy problem of transport equations with entropy and total enthalpy,mixed boundary value problem of pressure satisfying second-order elliptic equation in ducts,two-point boundary value problem of ordinary differential equation with tangential velocity along y-axis on arbitrary section.In Section 5.2,the equa-tions and boundary conditions obtained by the new decomposition method are linearized at the background solution respectively,and the corresponding lin-earization problem is obtained.In section 5.3,the well posedness and regularity theorems of solutions for three typical problems-Cauchy problem of transport equations with variable coefficients along the x-axis,mixed boundary value prob-lem of second order elliptic Equations with multiple integral nonlocal terms,and two-point boundary value problem of ordinary equation along the y-axis on ar-bitrary cross section are given.In section 5.4,the stability of the subsonic flow with mass addition effect is proved,and the proof of the second main result of this paper is completed.Chapter 6 contains the details of the mathematical tools used in this arti-cle.In Section 6.1,the regularity of the solutions of linear ordinary differential equations in Holder space is proved.In section 6.2,we give some properties of x-direction anisotropy Holder space.In section 6.3,we prove the well posedness theorem of the solutions of the transport equations in the x-direction anisotropy Holder space.The seventh chapter is the assumption of the follow-up work.