Three Dimensional Supersonic Shock Wave
DescriptionThe aim of this project is to develop the mathematical theory of the three dimensional supersonic shock waves, particularly focusing on the rigorous mathematical analysis of the attached supersonic shock over a delta wing and the global stability of relativistic conic shocks and the non-relativistic limit.Shock waves can be frequently observed by gas against a wedge, a sharp corner, or the head of an airfoil. More description about the steady, isentropic and irrotational shock flow can be found in Courant-Friedrichs’ classic book (cf. [24, p294–297]). In this project, we will study the global existence and stability of three dimensional supersonic shock waves of specific structures. Three dimensional shock is different from the two dimensional one because of the additional x3-direction. Actually, the unstability of attached three dimensional strong planar shock governed by the potential flow equation is obtained in  recently. It is completely different from the known results for the two dimensional case for which the stability of attached transonic shock is obtained in [4, 21, 44]. Moreover, as far as the P.I. know, there are very few results on the global existence of three dimensional supersonic shock waves. Therefore, as a typical prototype, the study of the proposed project contributes a lot to the mathematical theory of the fluids dynamics.It is a long history to study the steady shock waves since Von Neumann [40, 41] and Courant-Friedriches  introduce the open problem on the uniqueness of the configuration,i.e., which one of the strong shock and the weak shock is physical? Up to now, satisfactory results on the two-dimensional steady attached shock waves have been obtained in many literatures for the uniqueness and the global stability of the configuration of both the strong shock and weak shock. However, for the three dimensional case, all the study is on the solutions with additional assumptions, for example, for the stability of conic shocks in [7, 19, 22, 32, 33, 34, 43]. Recently, the unstability of the strong shock with sufficiently small wedge angle and the stability of the transonic weak shock have been established in [8, 31]. On the other hand, for the global existence and stability of the supersonic shock without conic-like structure, very few results are available, for example, (see [18, 23]) for the supersonic flow over a delta wing with sufficiently large vertex angle. Because the three dimensional supersonic shock is fundamental and important even though there are few results, it is valuable for us to work on the objectives proposed in this project.We have been working on the related problems for several years. Recently, we have established the uniqueness of the two-dimensional attached shock in ; the convexity of the shock for the regular shock reflection-diffraction problem governed by the potential flow equation in ; the Lighthill problem governed by the nonlinear wave equation in  and by the potential flow equation in . In this project, we will mainly focus on a further development of the technics of proving the obtained results, to study the mentioned objectives.
|Effective start/end date||1/01/19 → …|