The Stability Of The 2-D E-H Type Reflection And Refraction Of Shocks

In this paper, we are concerned with the stability of the 2-D E-H type Reflection and Re-fraction of shocks on the interface between two different media. In general, when a shock front attacks an interface between two media, a complicated wave structure will form. If the incident angle less than some critical value, a reflected wave and a refracted shock will appear with the interface being deflected. Such a reflection-refraction wave struc-ture is called regular refraction. According to different parameters of the incident shock and the media, both the reflected wave and the refracted shock can be a supersonic-supersonic shock(supersonic shock for short) or a supersonic-subsonic shock(transonic shock for short). Correspondingly, these cases of regular refraction can be classified as H-H type, E-E type and E-H type. In this paper, we concentrate on the E-H type regular refraction. Our result indicates that the steady flat E-H type regular refraction is globally stable. We use the 2-D steady potential equations as the mathematical model to describe the motion of the fluid. The system of potential equations is hyperbolic for supersonic flows, while it is elliptic for subsonic ones. Hence, the stability problem of the flat E-H type regular refraction can be reduced to a free boundary problem of nonlinear mixed type equations in an unbounded domain for the unique existence of the solution near an unper-turbed piecewise constant solution. The unique existence of the solution to the nonlinear free boundary problem implies the global stability of the flat E-H type regular refraction.The whole dissertation is organized as follows:Chapter One is Introduction. This chapter is devoted to the physical and mathe-matical background. The main results and difficult of this Ph.D. dissertation are also illustrated.Chapter Two studies the boundary value problem for the linear mixed type system. We introduce weighted Holder spaces. Then we prove unique existence of the solution to a special boundary value problem for the linear mixed type system. Some estimates have also been obtained. Chapter Three studies the stability of 2-D E-H type Reflection and Refraction of shocks. First, the Lagrange transformation is introduced to fix the free boundary of the interfaces. Then a boundary iteration scheme is employed to fix the free boundary of the shocks to reduce the problem to a fixed boundary value problem for mixed type system. Finally, we use iteration scheme and linearize the fixed boundary problem, the results in Chapter Two will be applied to this boundary value problem for this linear mixed type system to prove the main theorem.