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

Li, Liang; Xu, Gang; Yin, Huicheng

doi:10.1016/j.aim.2015.06.018

Cited by 14 publications

(13 citation statements)

References 26 publications

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“…A different approach may be required to handle this case, which is currently under investigation. In this regard, we notice that an instability result has been observed recently in Li-Xu-Yin [25].…”

Section: Introductionsupporting

confidence: 69%

“…Then there exist constants δ 0 > 0 and σ s > 0 depending on the background solution such that, for any −1 < σ ∞ ≤ 0 < σ 0 < σ s , if the wedge edge is not perturbed, that is, where K > 0 depends on the background solution, but is independent of δ 0 > 0. 4) is over-determined, which is exactly the instability mechanism for strong transonic solutions shown in [25]. In Theorem 5.2, since σ 0 > 0, estimate (5.6) yields that Du − Du 0 ≡ 0 on edge y 1 = 0, y 2 = 0, y ′ ∈ R n−2 , which indicates that, as the wedge edge is not perturbed such that (3.10) holds, conditions (5.4) hold automatically, and solution u(y) to problem (5.1)-(5.3) is indeed a solution to problem (5.1)-(5.4).…”

Section: The Partial Hodograph Transformationmentioning

confidence: 86%

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Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Fang

2017

Advances in Mathematics

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Abstract. We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.

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Section: Introductionsupporting

confidence: 69%

Section: The Partial Hodograph Transformationmentioning

confidence: 86%

Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Fang

2017

Advances in Mathematics

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“…It would be interesting to investigate further problems for the stability of M-D shocks in steady supersonic flow past M-D wedges. In this regard, we notice that an instability result has been observed in Liang-Xu-Yin [46].…”

Section: Further Problems and Remarksmentioning

confidence: 86%

“…Remark 3.1. For the global stability of weak transonic shocks for the 3-D wedge problem, see [26,28]; also see the instability phenomenon for strong transonic shocks for the 3-D wedge problem in [75]. For the global stability of conical shocks for the M-D conic problem, see [27] for the transonic shock case, and [38,49,80] for the supersonic shock case.…”

Section: Approach II For Problem 32 (St) and (Wt)mentioning

confidence: 99%

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

Chen

2017

Sci. China Math.

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When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue -whether one or both of the steady weak and strong shocks are physically admissible solutions -has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

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“…In the following, by the separation variable method as in (Cui & Li, 2011;Li, Xu & Yin, 2015), the estimate of the infinity state of the solution to (19) is established. This together with the weighted Hölder estimate in (Gilbarg & Tudinger, 1998), we can get the uniform weighted Hölder estimate.…”

Section: The Proof Of the Theorem 21mentioning

confidence: 99%

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

Cui¹

2017

JMR

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This paper is a complement of our work in (Cui & Li, 2011) where we have established the global subsonic circulatory solution for the polytropic gas. In this paper, we are concerned with the global stability of the 2-D subsonic circulatory flow around a perturbed circular body for the isothermal gas. The flow is assumed to be isothermal, isentropic, irrotational and described by a steady Euler equations, which can be reduced into a second order quasilinear elliptic equation in a exterior domain with suitable physical conditions. The unique existence and the state of the flow at infinity are obtained under nature physical assumption.

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

Cited by 14 publications

References 26 publications

Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

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

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