The Multidimensional Supersonic Conic Shock Of Isothermal Gas

When a supersonic flow hits a sharp cone, there will appear a supersonic shock naturally. Understanding the formation mechanisms of supersonic shock and transonic shock are fundamental problems in compressible fluid dynamics(see the classical book 《Supersonic flow and shock wave》 of Courant and Friedrichs), which have been exten-sively studied in physical experiments and numerical computations so far. The related physical descriptions are:if there is a uniform supersonic incoming flow which comes from minus infinity, and the flow hits the circular cone along the axis direction, when the vertex angle of the cone is less than a critical value, in theory, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak or a strong shock attached at the vertex of the cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively.This thesis will be mainly focused on some problems of these phenomena.We now mention some recent progresses on this topic which are related to our works. It is noted that there are several interesting and significant results regarding the global existence of the supersonic shock solution for the uniform supersonic in-coming flow past a sharp pointed body by Taiping Liu, Zhouping Xin, and Huicheng Yin, Dacheng Cui, Jun Li etc. See the references [5],[8],[10],[11],[24],[37]. In the refer-ence [11], under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, Dacheng Cui and Huicheng Yin have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic potential flow past a circular cone. Es-pecially, Jun Li, Ingo Witt and Huicheng Yin [22] has established the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic potential flow past an infinitely long circular cone with a sharp angle. For the results of nonexistence of a globally stable supersonic conic shock wave, we should mention that, when the surface of the cone is perturbed, Gang Xu and Huicheng Yin [38] have shown that there cannot exit a globally stable multidimensional supersonic conic shock for a steady supersonic Euler flow in the poly tropic case.This thesis is arranged as follows:First, we will remove all the symmetric assumptions in the previous paper [11] and study both the local and global existence problems on a really multidimensional conic shock wave. More concretely, we establish both the local and global existence and stability of a three dimensional supersonic conic shock wave for a uniform steady supersonic isothermal potential flow past an infinitely long and curved conic body.Second, contrasted with potential flow, for the full Euler system of isothermal gas, we show that a globally stable supersonic conic shock wave solution does not exist when a uniform supersonic incoming Euler flow hits an infinitely long and curved sharp conic body. This results can be achieved by the influence of entropy and rotation on the flow movements.The whole thesis is organized as follows:In Chapter1, we recall some related physical backgrounds on supersonic conical shocks. In addition, some recent results and progresses on these problems are presented. Meanwhile, some main results and their significance in this thesis are illustrated and commented.Chapter2is devoted to proving the local and global existence of a three dimen-sional supersonic conic shock wave for a uniform steady supersonic isothermal potential flow past an infinitely long and curved conic bodyChapter3is devoted to showing that there does not exist a globally stable su-personic conic shock wave solution when a uniform steady supersonic isothermal Euler flow hits an infinitely long and curved sharp conic body.