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Preface: Special issue in honor of Professor Shuxing Chen on the occasion of his 80th birthday
Three-dimensional supersonic flows of Euler-Poisson system for potential flow
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 |
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 [
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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