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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

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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

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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

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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($E_{\pm}>0, u_{\pm}>0$)
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Figure 2.  Transonic shocks of Euler system(left: well-posed, right: ill-posed)
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Figure 3.  Transonic shock of E-P system in a flat nozzle (conditionally well-posed)
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