On The Instability Of 3-D Transonic Oblique Shock Waves

The present Ph.D dissertation is devoted to studying the instability problem of a transonic oblique shock for the steady supersonic flow past an infinitely long sharp wedge.It is well known that when a supersonic flow attacks a wedge, there will appear shocks naturally. The formation mechanisms of shocks are fundamental problems in compressible fluid dynamics which is a very active research field and has been exten-sively studied in experiments. On the other hand, since the instable shock dose not actually occur, then the study on the stability or instability of shocks becomes ex-tremely important. As R. Courant and K.O. Friedrichs described in the classical book 《Supersonic flow and shock waves》:if a supersonic steady flow comes from minus in-finity and hits a sharp symmetric wedge (vertex angle less than a critical value), then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock (with supersonic flow behind the shock fronts) or a transonic shock (with subsonic flow behind the shock fronts), respectively. The question arises which of the two shocks actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this thesis is to understand such a longstanding open question, modeled by 3-D potential flow equations. We will show that the attached 3-D transonic oblique shock problem is overdetermined, which implies that the 3-D transonic shock is insta-ble in general and further gives a rather positive illustration on the instability of a 3-D transonic oblique shock.The dissertation is organized as follows:Chapter 1 is devoted to introducing physical background and previous mathemat-ical research works on oblique shocks. The main problem, main results and method in this dissertation are also illustrated.In Chapter 2, we focus on the solvability for an Neumann boundary problem of a second order elliptic equation in an unbounded angular domain with the value only at one point. At first, we give some preliminaries include weighted Holder spaces and properties of the modified Bessel functions so that one can use the separation method to study the problems. Secondly, a cut-off problem with a suitable Neumann boundary condition on the cut-off surface is studied in details, where its solvability and the rough regularity of the solution vL in related weighted Holder space are shown. Thirdly, the higher regularities of VL are obtained by the classical Schauder estimate and the regularity theory of solutions to the second order elliptic equations in a 3-D bounded angular region. Moreover, the global solvability and estimates of the solution to the problem in the unbounded angular domain are established. Finally, the uniqueness of solution to the problem is proved by the separation variable method other than by the usual maximum principle for the second order elliptic equations since it seems that there is no maximum principle for the problem due to the 3-D unbounded angular region and the Neumann boundary conditions.Chapter 3 studies the instability of 3-D transonic oblique shocks by taking poten-tial equation as the model. Such a problem can be formulated into a free boundary value problem of nonlinear elliptic-hyperbolic mixed equation of second order in an unbounded domain with the value on the edge. Employing the uniform estimates ob-tained in Chapter Two, we obtain the solvability for the free boundary value problem with the value only at one point via linearized technique, partial hodograph transfor-mation and a suitable nonlinear iteration scheme. This resolves the instability of a 3-D transonic oblique shockIn addition, some complicated and useful computations are carried out in the Appendix which used in Chapter Two.