On Dirichlet’s Problem for Elliptic Equations in Sectionally Smooth <i>n</i>-Dimensional Domains

Azzam, A.

doi:10.1137/0511024

Cited by 28 publications

(18 citation statements)

References 3 publications

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“…The complete steady Euler systems in two-dimensional spaces and three-dimensional spaces are ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 1 (ρu 1 ) + ∂ 2 (ρu 2 ) = 0, ∂ 1 P + ρu 2 1 + ∂ 2 (ρu 1 u 2 ) = 0, ∂ 1 (ρu 1 u 2 ) + ∂ 2 P + ρu 2 2 = 0, ∂ 1 ρe + 1 2 ρ|u| 2 + P u 1 + ∂ 2 ρe + 1 2 ρ|u| 2 + P u 2 = 0 (1.1) and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 1 (ρu 1 ) + ∂ 2 (ρu 2 ) + ∂ 3 (ρu 3 ) = 0, ∂ 1 ρu 2 1 + P + ∂ 2 (ρu 1 u 2 ) + ∂ 3 (ρu 1 u 3 ) = 0, ∂ 1 (ρu 1 u 2 ) + ∂ 2 ρu 2 2 + P + ∂ 3 (ρu 2 u 3 ) = 0, ∂ 1 (ρu 1 u 3 ) + ∂ 2 (ρu 2 u 3 ) + ∂ 3 ρu 2 3 + P = 0,…”

Section: Introduction and The Main Resultsmentioning

confidence: 98%

“…More precisely, we assume that the equation of Γ 2 i is represented by x 2 = (−1) i x 1 tg θ 0 for x 1 > 0 and X 0 + 1 4 < r < X 0 + 1, here 0 < θ 0 < π 2 . Besides, we suppose that the C 3 -smooth supersonic incoming flow 2), and S − 0 is a constant (this assumption can be easily realized by the hyperbolicity of the supersonic incoming flow and the symmetric property of the nozzle for X 0 + 1 4 < r < X 0 + 1).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

Xin

Yin

2008

Journal of Differential Equations

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In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P 1 and P 2 with P 1 < P 2 such that for given constant exit pressure P e ∈ (P 1 , P 2 ), a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by P e . Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure P e when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.

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Section: Introduction and The Main Resultsmentioning

confidence: 98%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

Xin

Yin

2008

Journal of Differential Equations

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“…Now, by the classical Schauder estimates (see [19, lemma 1] and [1,3]) applied to (5.14) on domains D 0 andD 0 , one gets…”

Section: Proof: As Inmentioning

confidence: 99%

“…Motivated by the analysis in [3], we will consider the mixed boundary value problem (5.3) in the following domains in G 1/2 (r 0 ):…”

Section: Hencementioning

confidence: 99%

Transonic shock in a nozzle I: 2D case

Xin

Yin

2004

Comm Pure Appl Math

141

117

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In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two-dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation and is supersonic upstream, has no-flow boundary conditions on the nozzle walls, and an appropriate boundary condition at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C 1,1−δ 0 flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This problem is motivated by the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities.

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“…(7.26) It is noted that 2 i, j=1 ∂ 1 (ȧ ij )∂ i ϕ C 0,δ 0 (D 1 2 (z)) C ε ∇(∂ 1 ϕ) C 0,δ 0 (D 1 2 (z)) C ε ∂ 1 ϕ C 1,δ 0 (D 1 2 (z)) , (7.27) then substituting (7.27) into (7.26) yields ∂ 1 ϕ C 1,δ 0 (D 1 2 (z)) C ∂ 1 ϕ C (D 1 2 (z)) + ∂ 1 ϕ C (B(z, 1 2 )∩Γ 0 ) + ∂ 1 ϕ C (B (z, 1 2 )∩Γ 1 ) .…”

Section: Asymptotic Behavior Of Subsonic Solution At Infinitymentioning

confidence: 98%

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

Yin¹,

Zhou²

2009

Journal of Differential Equations

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In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317-318): when a supersonic flow hits a sharp wedge, it follows from the RankineHugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.

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On Dirichlet’s Problem for Elliptic Equations in Sectionally Smooth n-Dimensional Domains

Cited by 28 publications

References 3 publications

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

Transonic shock in a nozzle I: 2D case

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

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