Global conic shock wave for the steady supersonic flow past a cone: isothermal case

Abstract

We establish the global existence and stability of a steady symmetric conic shock wave for the perturbed supersonic isothermal flow past an infinitely long circular cone with an arbitrary vertex angle. The flow is assumed to be described by a steady potential equation. By establishing the uniform weighted energy estimate on the linearized problem, we show that the symmetric conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large.

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 we derive some useful estimates on the coefficients of the reformulated nonlinear systems and its boundary conditions. From these, as in [4,13], a kind of "dissipative" property on the solution is established in Lemma 3.5. In Sect.…”

“…Based on such estimates we can use the continuity method for hyperbolic system to obtain the global existence of a shock solution. In [3,4,12,13], the key ingredients are to look for the suitable multipliers so that the weighted energy estimates on the solution and shock can be derived. Finding such suitable multipliers is rather involved and complicated.…”

“…Furthermore, the smallness of b 0 and the decay property of b(x 3 ) − b 0 x 3 for large x 3 play crucial roles in finding such multipliers. Thus, for the appropriately large b 0 and large curved cone surface, it seems very difficult for us to use the method in [3,4,12,13]. In this paper, as in [9], we intend to use the reflected characteristics method to treat the problem (1.4) with (1.6)-(1.9).…”

The Global Stability of 2-D Subsonic Circulatory Flows for the Steady Isothermal Gas

“…In the present paper, our goal is to establish the global existence and stability of subsonic circulatory flows solution for the subsonic isothermal gas around a perturbed circular body. The so-called isothermal gas means that the pressure P and the density ρ of gas are described by the state equation P = Aρ for some constant A > 0 (see Cui & Yin, 2007 and the references therein). In this case, the sound speed is a constant independent of the density ρ.…”