Figure 1.
Transonic shock of E-P system in a convergent domain($E_{\pm}>0, u_{\pm}>0$ )
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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
| 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 |
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
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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. |
| [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. |
| [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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
M. W. Hirsch, S. Smale and R. L. Devney, Differential Equations, Dynamical Systems and An introduction to Chaos, 2nd edn., Elsevier, USA, 2004. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
Transonic shocks of Euler system(left: well-posed, right: ill-posed)
Figure 3.
Transonic shock of E-P system in a flat nozzle (conditionally well-posed)
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