American Institute of Mathematical Sciences

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

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

 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.

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