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July & August  2021, 20(7&8): 2421-2440. doi: 10.3934/cpaa.2021079

Three-dimensional supersonic flows of Euler-Poisson system for potential flow

Myoungjean Bae 1,2,, and  Hyangdong Park 3,

1. 

Department of Mathematical Sciences, KAIST, 291 Daehak-Ro, Yuseong-Gu, Daejeon 34141, Republic of Korea
2. 

Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Republic of Korea
3. 

Center for Mathematical Analysis and Computation (CMAC), Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea

* Corresponding author

Received  February 2021 Revised  March 2021 Published  July & August 2021 Early access  May 2021

Fund Project: The first author is supported in part by Samsung Science and Technology Foundation under Project Number SSTF-BA1502-51. The second author is supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT and MOE) (No. 2015R1A5A1009350 and No. 2020R1I1A1A01058480)

We prove the unique existence of supersonic solutions of the Euler-Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [1], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.

Keywords: Euler-Poisson system, potential flow, three-dimensional, hyperbolic-elliptic coupled system, supersonic flow.
Mathematics Subject Classification: Primary: 35G60, 35J66, 35L72, 35M32, 76J20, 76N10.
Citation: Myoungjean Bae, Hyangdong Park. Three-dimensional supersonic flows of Euler-Poisson system for potential flow. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2421-2440. doi: 10.3934/cpaa.2021079
References:
[1]

M. Bae, B. Duan, J. Xiao and C. Xie, Structural Stability of Supersonic Solutions to the Euler-Poisson System, Arch. Rational Mech. Anal., 239 (2021), 679-731.  doi: 10.1007/s00205-020-01583-7.

[2]

M. Bae, B. Duan and C. Xie, Subsonic Flow for the Multidimensional Euler-Poisson System, Arch. Rational Mech. Anal., 220 (2016), 155–191. doi: 10.1007/s00205-015-0930-6.

[3]

M. Bae, B. Duan and C. Xie, Subsonic solutions for steady Euler-Poisson system in two-dimensional nozzles, SIAM J. Math. Anal., 46 (2014), 3455–3480. doi: 10.1137/13094222X.

[4]

M. Bae, B. Duan and C. Xie, Two dimensional subsonic flows with self-gravitation in bounded domain, Math. Models Methods Appl. Sci., 25 (2015), 2721–2747. doi: 10.1142/S0218202515500591.

[5]

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.

[6]

P. Degond and P. A. Markowich, A steady state potential flow model for semiconductors, Annali di Matematica pura ed applicata, 165 (1993), 87-98.  doi: 10.1007/BF01765842.

[7]

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.

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin.

[9]

F. M. Huang, R. H. Pan and H. M. Yu, Large time behavior of Euler-Poisson system for semiconductor, Sci. China Ser. A, 51 (2008), 965-972.  doi: 10.1007/s11425-008-0049-4.

[10]

T. Luo, R. Natalini and Z. P. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1999), 810-830.  doi: 10.1137/S0036139996312168.

[11]

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

[12]

T. Luo and Z. P. 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.

[13]

P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407.  doi: 10.1007/BF00945711.

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[15]

Y. J. Peng and I. Violet, Example of supersonic solutions to a steady state Euler-Poisson system, Appl. Math. Lett., 19 (2006), 1335-1340.  doi: 10.1016/j.aml.2006.01.015.

[16]

M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model, Quart. Appl. Math., 63 (2005), 251-268.  doi: 10.1090/S0033-569X-05-00942-1.

[17]

L. M. Yeh, On a steady state Euler-Poisson model for semiconductors, Commun. Partial Differ. Equ., 21 (1996), 1007-1034.  doi: 10.1080/03605309608821216.

show all references

References:
[1]

M. Bae, B. Duan, J. Xiao and C. Xie, Structural Stability of Supersonic Solutions to the Euler-Poisson System, Arch. Rational Mech. Anal., 239 (2021), 679-731.  doi: 10.1007/s00205-020-01583-7.

[2]

M. Bae, B. Duan and C. Xie, Subsonic Flow for the Multidimensional Euler-Poisson System, Arch. Rational Mech. Anal., 220 (2016), 155–191. doi: 10.1007/s00205-015-0930-6.

[3]

M. Bae, B. Duan and C. Xie, Subsonic solutions for steady Euler-Poisson system in two-dimensional nozzles, SIAM J. Math. Anal., 46 (2014), 3455–3480. doi: 10.1137/13094222X.

[4]

M. Bae, B. Duan and C. Xie, Two dimensional subsonic flows with self-gravitation in bounded domain, Math. Models Methods Appl. Sci., 25 (2015), 2721–2747. doi: 10.1142/S0218202515500591.

[5]

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.

[6]

P. Degond and P. A. Markowich, A steady state potential flow model for semiconductors, Annali di Matematica pura ed applicata, 165 (1993), 87-98.  doi: 10.1007/BF01765842.

[7]

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.

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin.

[9]

F. M. Huang, R. H. Pan and H. M. Yu, Large time behavior of Euler-Poisson system for semiconductor, Sci. China Ser. A, 51 (2008), 965-972.  doi: 10.1007/s11425-008-0049-4.

[10]

T. Luo, R. Natalini and Z. P. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1999), 810-830.  doi: 10.1137/S0036139996312168.

[11]

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

[12]

T. Luo and Z. P. 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.

[13]

P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407.  doi: 10.1007/BF00945711.

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[15]

Y. J. Peng and I. Violet, Example of supersonic solutions to a steady state Euler-Poisson system, Appl. Math. Lett., 19 (2006), 1335-1340.  doi: 10.1016/j.aml.2006.01.015.

[16]

M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model, Quart. Appl. Math., 63 (2005), 251-268.  doi: 10.1090/S0033-569X-05-00942-1.

[17]

L. M. Yeh, On a steady state Euler-Poisson model for semiconductors, Commun. Partial Differ. Equ., 21 (1996), 1007-1034.  doi: 10.1080/03605309608821216.

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