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

Myoungjean Bae; Hyangdong Park

doi:10.3934/cpaa.2021079

Article Contents

2021, Volume 20, Issue 7&8: 2421-2440. Doi: 10.3934/cpaa.2021079

This issue Previous Article Preface: Special issue in honor of Professor Shuxing Chen on the occasion of his 80th birthday Next Article Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

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
^* Corresponding author

Received: January 31, 2021

Revised: February 28, 2021

Early access: May 2021

Published: August 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35G60, 35J66, 35L72, 35M32, 76J20, 76N10.

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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.