TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES

doi:10.1007/s10473-022-0414-3

Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (4): 1536-1546.doi: 10.1007/s10473-022-0414-3

• Articles • Previous Articles Next Articles

TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES

Ben DUAN¹, Ao LAN², Zhen LUO³

1. School of Mathematical Sciences, Jilin University, Changchun, 130012, China;
2. School of Mathematics, Dalian University of Technology, Dalian, 116024, China;
3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

Received:2021-02-20 Revised:2021-05-24 Online:2022-08-25 Published:2022-08-23
Contact: Zhen LUO,E-mail:zluo@xmu.edu.cn E-mail:zluo@xmu.edu.cn
Supported by:
The research of Ben Duan is partially supported by NSFC (11871133, 12171498). The research of Zhen Luo is partially supported by NSFC (11971402, 12171401) and the NSF of Fujian province, China (2020J01029).

Abstract

Abstract: In this paper, we study the transonic shock solutions to the steady Euler system in a quasi-one-dimensional divergent-convergent nozzle. For a given physical supersonic inflow at the entrance, we obtain exactly two non-isentropic transonic shock solutions for the exit pressure lying in a suitable range. In addition, we establish the monotonicity between the location of the transonic shock and the pressure downstream.

Key words: Euler system, transonic shocks, steady solutions, nozzle

CLC Number:

35Q31

Ben DUAN, Ao LAN, Zhen LUO. TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES[J].Acta mathematica scientia,Series B, 2022, 42(4): 1536-1546.

Trendmd

References

[1] Bethe H A. On the theory of shock waves for an arbitrary equation of state//Classic papers in shock compression science. New York:Springer, 1998:421-492
[2] Liu T P. Nonlinear satbility and instability of transonic gas flow through a nozzle. Comm Math Phys, 1982, 83:243-260
[3] Chen C, Xie C J. Three dimensional steady subsonic Euler flows in bounded nozzles. J Differential Equations, 2014, 256:3684-3708
[4] Chen G Q, Huang F M, Wang T Y, et al. Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles. Adv Math, 2019, 346:946-1008
[5] Chen S X, Xin Z P, Yin H C. Global shock waves for the supersonic flow past a perturbed cone. Comm Math Phys, 2002, 228:47-84
[6] Du L L, Xie C J, Xin Z P. Steay subsonic ideal flows through an infinitely long nozzle with large vorticity. Comm Math Phys, 2014, 28:327-354
[7] Wang C P, Xin Z P. Regular subsonic-sonic flows in general nozzles. Adv Math, 2021, 380:107578
[8] Xie C J,Xin Z P. Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana Univ Math J, 2007, 56:2991-3023
[9] Courant R, Friedrichs K O. Supersonic flow and shock waves. New York:Interscience Publishers Inc, 1948
[10] Chen S X. Compressible flow and transonic shock in a diverging nozzle. Comm Math Phys, 2009, 289:75-106
[11] Li J, Xin Z P, Yin H C. On transonic shocks in a nozzle with variable end pressures. Comm Math Phys, 2009, 291:111-150
[12] Xin Z P, Yin H C. Transonic shock in a nozzle I:two dimensional case. Comm Pure Appl Math, 2005, 58:999-1050
[13] Xin Z P, Yin H C. The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. J Differential Equations, 2008, 245:1014-1085
[14] Bae M, Feldman M. Transonic shocks in multidimensional divergent nozzles. Arch Ration Mech Anal, 2011, 201:777-840
[15] Chen G Q, Feldman M. Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections. Arch Ration Mech Anal, 2007, 184:185-242
[16] Xie F, Wang C P. Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder. J Differential Equations, 2007, 242:86-120
[17] Chen S X. Stability of transonic shock fronts in two-dimensional Euler systems. Trans Am Math Soc, 2005, 357:287-308
[18] Chen G Q, Fang B X. Stability of transonic shocks in steady supersonic flow past multidimensional wedges. Adv Math, 2017, 314:493-539
[19] Liu T P. Nonlinear resonance for quasilinear hyperbolic equation. J Math Phys, 1987, 28:2593-2602
[20] Rauch J, Xie C J, Xin Z P. Global stability of transonic shock solutions in quasi-onedimensional nozzles. J Math Pures Appl, 2010, 99:395-408
[21] Chen S X. Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems. Trans Amer Math Soc, 2008, 360:5265-5289
[22] Chen S X, Yuan H R. Transonic shocks in compressible flow passing a duct for three-dimensional Euler system. Arch Ration Mech Anal, 2008, 187:523-556
[23] Xin Z P, Yan W, Yin H C. Transonic shock problem for the Euler system in a nozzle. Arch Ration Mech Anal, 2009, 194:1-47
[24] Yuan H R. On transonic shocks in two-dimensional variable-area ducts for steady Euler system. SIAM J Math Anal, 2006, 38:1343-1370
[25] Morawetz C S. On the non-existence of continuous transonic flows past profiles III. Commun Pure Appl Math, 1958, 11:129-144
[26] Wang C P, Xin Z P. Smooth transonic flows of meyer type in de Laval nozzles. Arch Ration Mech Anal, 2019, 232:1597-1647
[27] Luo T, Xin Z P. Transonic shock solutions for a system of Euler-Poisson equations. Comm Math Sci, 2012, 10:419-462
[28] Luo T, Rauch J, Xie C J, Xin Z P. Stability of transonic shock solutions for onedimensional Euler-Poisson equations. Arch Rat Mech Anal, 2011, 202:787-827
[29] Duan B, Luo Z, Xiao J J. Transonic shock solutions to the Euler-Poisson system in quasi-one-dimensional nozzles. Commun Math Sci, 2016, 14:1023-1047

TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 3

Metrics

Comments

Recommended 10

[1]	Fei WU, Zejun WANG. EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM [J]. Acta mathematica scientia,Series B, 2022, 42(1): 155-171.
[2]	Hairong YUAN, Qin ZHAO. STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS [J]. Acta mathematica scientia,Series B, 2020, 40(2): 470-502.
[3]	Chunyin JIN. EXISTENCE AND UNIQUENESS OF ENTROPY SOLUTION TO PRESSURELESS EULER SYSTEM WITH A FLOCKING DISSIPATION [J]. Acta mathematica scientia,Series B, 2016, 36(5): 1262-1284.