scholarly journals Nonexistence of global weak solution with only one stable supersonic conic shock wave for the steady supersonic Euler flow past a perturbed cone

2012 ◽  
Vol 70 (2) ◽  
pp. 199-218 ◽  
Author(s):  
Xu Gang ◽  
Yin Huicheng
Keyword(s):  
Shock Wave ◽  
Weak Solution ◽  
Global Weak Solution ◽  
Euler Flow ◽  
Flow Past

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

2010 ◽  
Vol 199 ◽  
pp. 151-181 ◽  
Author(s):  
Gang Xu ◽  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Strong Shock ◽  
Weak Shock ◽  
Vertex Angle ◽  
Transonic Shock ◽  
Euler Flow ◽  
Flow Past ◽  
Curve Method ◽  
Sharp Cone ◽  
Supersonic Shock

AbstractIn 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.

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

2010 ◽  
Vol 199 ◽  
pp. 151-181 ◽  
Author(s):  
Gang Xu ◽  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Strong Shock ◽  
Weak Shock ◽  
Vertex Angle ◽  
Transonic Shock ◽  
Euler Flow ◽  
Flow Past ◽  
Curve Method ◽  
Sharp Cone ◽  
Supersonic Shock

AbstractIn 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 bookSupersonic Flow and Shock Wavesby 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.

scholarly journals A global weak solution to the Lorentzian harmonic map flow

2019 ◽  
Vol 63 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Xiaoli Han ◽  
Lei Liu ◽  
Liang Zhao
Keyword(s):  
Weak Solution ◽  
Harmonic Map ◽  
Global Weak Solution ◽  
Harmonic Map Flow

scholarly journals Existence and stability of fourth-order nonlinear plate problem

2019 ◽  
Vol 6 (1) ◽  
pp. 81-98
Author(s):  
Salim A. Messaoudi ◽  
Soh Edwin Mukiawa
Keyword(s):  
Weak Solution ◽  
Fourth Order ◽  
Stability Result ◽  
Suspension Bridge ◽  
Global Weak Solution ◽  
Restoring Force ◽  
Nonlinear Plate ◽  
Frictional Damping ◽  
Existence And Stability

AbstractIn this paper, we study a fourth-order plate problem as a model for a suspension bridge in the presence of a nonlinear frictional damping and a hanger restoring force. We establish the existence of a global weak solution and prove a stability result.

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