Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

Abstract

Abstract. In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed. §1. Introduction and main resultsIf there is a uniform supersonic flow (ρ 0 , 0, 0, q 0 , P 0 ) which comes from minus infinity, and the flow hits the circular cone x 2 1 + x 2 2 = b 0 x 3 along the axis x 3 -direction, when the vertex angle 2 arctan b 0 of the cone is less than a critical value θ * , the following conic shock problems are illustrated in [6, pages 313-314, 414]: it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there will appear a weak or a strong self-similar shock attached at the vertex of the cone in terms of the different

On the instability problem of a 3-D transonic oblique shock wave

“…For the potential equation, under various assumptions on the supersonic incoming flows and the sharp vertex angles of the conic bodies, the authors have established the local or global existence and stability of the weak conic shocks or strong conic shocks, one can see [5][6][7][8][9]15,16,19,[23][24][25]29] and the references therein. For the full Euler system, because of the essential influences of the rotations, the authors in [28] and [26] have shown the nonexistence of the global weak solution with only one stable weak conic shock and the instability of a global transonic conic shock for the steady supersonic flow past a sharp conic body, respectively. Therefore, these results have given a basic answer for the global stability or instability of weak and strong conic shocks.…”

On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave

“…It was frequently stated that the strong shock is unstable and that, therefore, only the weak shock is present in real situations. In [32][33][34], the global instability of an attached strong conic shock in the whole space was systematically studied (in this case, the corresponding subsonic potential equation is nonlinear elliptic and the steady Euler system becomes elliptic-hyperbolic) which especially showed that a global strong conic shock is actually unstable as long as the perturbation of the sharp circular cone satisfies some suitable assumptions. On the other hand, from the result in this paper, one infers the global stability of a supersonic conic shock.…”