American Institute of Mathematical Sciences

Advanced
October  2018, 11(5): 773-791. doi: 10.3934/dcdss.2018049

Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

Myoungjean Bae 1,2,, and  Yong Park 1,

1. 

Department of Mathematics, POSTECH, Pohang Gyungbuk, 37673, Republic of Korea
2. 

Korea Institute for Advanced Study 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea

* Corresponding author: Myoungjean Bae

Received  February 2017 Revised  May 2017 Published  June 2018

Fund Project: The first author is supported in part by Samsung Science and Technology Foundation under Project Number SSTF-BA1502-02. The second author supported in part under the framework of international cooperation program managed by National Research Foundation of Korea(NRF- 2015K2A2A2002145).

Given constant data of density $ρ_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(ρ_0, u_0, p_0)$ and the length of the nozzle.

Keywords: Euler-Poisson system, compressible, supersonic, subsonic, transonic shock, radial flow, convergent nozzle, exit pressure, electric field, monotonicity.
Mathematics Subject Classification: 34A12, 34A34, 76H05, 76L05, 76N10, 76X05.
Citation: Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049
References:
[1]

M. Bae and M. Feldman, Transonic shocks in multidimensional divergent nozzles, Arch. Ration. Mech. Anal., 201 (2011), 777-840.  doi: 10.1007/s00205-011-0424-0.  Google Scholar

[2]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3 (1990), 25-29.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[3]

I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Commun. Partial Differ. Equ., 17 (1992), 553-577.  doi: 10.1080/03605309208820853.  Google Scholar

[4]

M. W. Hirsch, S. Smale and R. L. Devney, Differential Equations, Dynamical Systems and An introduction to Chaos, 2nd edn., Elsevier, USA, 2004.  Google Scholar

[5]

T.-P. Liu, Nonlinear stability and instability of tranosnic gas flow through a nozzle, Commun. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[6]

T. Luo, J. Rauch, C. Xie and Z. Xin, Stability of transonic shock solutions for one-dimensional euler-poisson equations, Arch. Ration. Mech. Anal., 202 (2011), 787-827.  doi: 10.1007/s00205-011-0433-z.  Google Scholar

[7]

T. Luo and Z. Xin, Transonic shock solutions for a system of Euler-Poisson equations, Commun. Math. Sci., 10 (2012), 419-462.  doi: 10.4310/CMS.2012.v10.n2.a1.  Google Scholar

[8]

H. R. Yuan, A remark on determination of transonic shocks in divergent nozzle for steady compressible Euler flows, Nonlinear Anal. Real World Appl., 9 (2008), 316-325.  doi: 10.1016/j.nonrwa.2006.10.006.  Google Scholar

show all references

References:
[1]

M. Bae and M. Feldman, Transonic shocks in multidimensional divergent nozzles, Arch. Ration. Mech. Anal., 201 (2011), 777-840.  doi: 10.1007/s00205-011-0424-0.  Google Scholar

[2]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3 (1990), 25-29.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[3]

I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Commun. Partial Differ. Equ., 17 (1992), 553-577.  doi: 10.1080/03605309208820853.  Google Scholar

[4]

M. W. Hirsch, S. Smale and R. L. Devney, Differential Equations, Dynamical Systems and An introduction to Chaos, 2nd edn., Elsevier, USA, 2004.  Google Scholar

[5]

T.-P. Liu, Nonlinear stability and instability of tranosnic gas flow through a nozzle, Commun. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[6]

T. Luo, J. Rauch, C. Xie and Z. Xin, Stability of transonic shock solutions for one-dimensional euler-poisson equations, Arch. Ration. Mech. Anal., 202 (2011), 787-827.  doi: 10.1007/s00205-011-0433-z.  Google Scholar

[7]

T. Luo and Z. Xin, Transonic shock solutions for a system of Euler-Poisson equations, Commun. Math. Sci., 10 (2012), 419-462.  doi: 10.4310/CMS.2012.v10.n2.a1.  Google Scholar

[8]

H. R. Yuan, A remark on determination of transonic shocks in divergent nozzle for steady compressible Euler flows, Nonlinear Anal. Real World Appl., 9 (2008), 316-325.  doi: 10.1016/j.nonrwa.2006.10.006.  Google Scholar

Figure 1.  Transonic shock of E-P system in a convergent domain($E_{\pm}>0, u_{\pm}>0$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Transonic shocks of Euler system(left: well-posed, right: ill-posed)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Transonic shock of E-P system in a flat nozzle (conditionally well-posed)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108
[2]

Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583
[3]

Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201
[4]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085
[5]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573
[6]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086
[7]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[8]

Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125
[9]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[10]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577
[11]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345
[12]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775
[13]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449
[14]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549
[15]

Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515
[16]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981
[17]

Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448
[18]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743
[19]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011
[20]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (65)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences