Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

Abstract

In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the "dissipative" property. From this property, by use of the reflected characteristics method and the special form of the shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].

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

“…As illustrated in [6], if a uni-form supersonic flow (ρ 0 , 0, 0, q 0 , P 0 ) comes from minus infinity, and the flow hits the sharp circular cone x 2 1 + x 2 2 = b 0 x 3 along the axis x 3 -direction, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak or a strong self-similar shock attached at the vertex of the cone. With respect to the weak shocks, under some different assumptions, the authors in [4], [5], [7], [8], and [15] have established the local or global existence and stability for the perturbed supersonic incoming flow past a sharp cone when the pressure of downstream region at infinity is appropriately smaller than that of the incoming flow. With respect to transonic shocks, for the symmetrically or multidimensionally perturbed supersonic incoming flow and the potential equation, we have shown the global existence and stability of a steady transonic shock wave solution in [16] and [17], respectively.…”

Global Conic Shock Wave for the Steady Supersonic Flow Past a Large Curved Cone at Infinity

“…2 in [5], when the uniform supersonic coming flow (0, 0, q 0 ; ρ 0 ) hits the cone r = b 0 x 3 , a conic self-similar shock r = s 0 x 3 appears if only if b 0 < b * holds for large q 0 .…”

“…In this procedure, more delicate asymptotic expansions on the background solutions and involved computation on the coefficients of (1.4) with (1.6)-(1.8) are required since these coefficients are closely related to the supersonic coming flow and the cone vertex angle. Additionally, compared with the case in [5], which the perturbed conic body r = b(x 3 ) tends to r = b 0 x 3 when x 3 tends to infinity, the special attentions should be paid since we have to treat the troubles induced by the large perturbation of r = b(x 3 ) and give more detailed computation.…”

“…2, we will list some elementary estimates on the background solution for the polytropic gas when the speed of the supersonic coming flow is appropriately large, most of these estimates have been given in [3,5]. In Sect.…”

“…In this section we will mainly recall some properties on the self-similar background solution which are given in [3,5] respectively. In the book [2] of R. Courant and K.O.…”