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On a quasilinear hyperbolic system in blood flow modeling
Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system
1. | Department of Mathematics, Shanghai Normal University, Shanghai 200234 |
References:
[1] |
P. Amster and M. P. Beccar Varela, Subsonic solutions to a one-dimensional nonisentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 258 (2001), 52-62.
doi: 10.1006/jmaa.2000.7359. |
[2] |
A. Ambrose, F. Méhats and P. Raviart, On singular perturbation problems for the nonlinear Poisson equation, Asymptot. Anal., 25 (2001), 39-91. |
[3] |
U. Ascher, P. A. Markowich and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Meth. Appl. Sci., 11 (1991), 347-376.
doi: 10.1142/S0218202591000174. |
[4] |
H. Brézis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas, C. R. Acad. Sci. Paris, 321 (1995), 953-959. |
[5] |
S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.
doi: 10.1080/03605300008821542. |
[6] |
P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Letters, 3 (1990), 25-29.
doi: 10.1016/0893-9659(90)90130-4. |
[7] |
P. Degond and P. A. Markowich, A steady state potential flow model for semiconductors, Ann. Math. Pura Appl.(4), 165 (1993), 87-98. |
[8] |
I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations, 17 (1992), 553-577. |
[9] |
T. Goudon, A. Jüngel and Y.-J. Peng, Zero-mass-electrons limits in hydrodynamic models for plasmas, Appl. Math. Letters, 12 (1999), 75-79.
doi: 10.1016/S0893-9659(99)00038-5. |
[10] |
D. Gilbarg and N. S. Trüdinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1998. |
[11] |
L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift diffusion equations, J. Differential Equations, 165 (2000), 315-354.
doi: 10.1006/jdeq.2000.3780. |
[12] |
A. Jüngel and Y.-J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisted, Z. Angew. Math. Phys., 51 (2000), 385-396.
doi: 10.1007/s000330050004. |
[13] |
A. Jüngel, "Quasi-Hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations, Birkhäuser, 2001. |
[14] |
C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dyn. Syst., 5 (1999), 449-455.
doi: 10.3934/dcds.1999.5.449. |
[15] |
Y.-P. Li and J.-Z. Zhang, Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors, Math. Comput. Modeling, 49 (2009), 163-177.
doi: 10.1016/j.mcm.2008.05.006. |
[16] |
Y.-P. Li, Stationary solutions for a one-dimensional nonisentropic hydrodynamic model for semiconductors, Acta Math. Sci., B28 (2008), 479-488. |
[17] |
Y.-P. Li, Asymptotic profile in a one-dimensional nonisentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 325 (2007), 949-967.
doi: 10.1016/j.jmaa.2006.02.018. |
[18] |
O. A. Ladyzhenskaya and N. U. Uraltsera, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968. |
[19] |
Y.-P. Li, Asymptotic profile in a multi-dimensional nonisentropic hydrodynamic model for semiconductors, Nonlinear Anal. Real World Appl., 8 (2007), 1235-1251.
doi: 10.1016/j.nonrwa.2006.06.011. |
[20] |
P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew Math. Phys., 42 (1991), 387-407.
doi: 10.1007/BF00945711. |
[21] |
P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145.
doi: 10.1007/BF00379918. |
[22] |
P. A. Markowich, C. A. Ringhofev and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Wien, New York, 1990. |
[23] |
Y.-J. Peng, Some analysis in steady-state Euler-Poisson equations for potential flow, Asymp. Anal., 36 (2003), 75-92. |
[24] |
Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41 (2005), 141-160. |
[25] |
Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity, 17 (2004), 835-849.
doi: 10.1088/0951-7715/17/3/006. |
[26] |
Y.-J. Peng and I. Violet, Asymptotic expansions in a steady state Euler-Poisson equations to incompressible Euler equations, Math. Models Meth. Appl. Sci., 15 (2005), 717-736.
doi: 10.1142/S0218202505000546. |
[27] |
Y.-J. Peng and Y.-F. Yang, Junction layer analysis in one-dimensional steady-state Euler-Poisson equations, J. Math. Anal. Appl., 344 (2008), 440-448.
doi: 10.1016/j.jmaa.2008.02.062. |
[28] |
M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model, Quart. Appl. Math., 2003. |
[29] |
M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system, J. Nonlinear Sci., 11 (2001), 193-209.
doi: 10.1007/s00332-001-0004-9. |
[30] |
I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow, Proc. Roy. Soc. Edinburgh, 137 (2007), 1101-1118.
doi: 10.1017/S0308210505001216. |
[31] |
S. Wang, Quasineutral limit of Euler-Poinsson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456.
doi: 10.1081/PDE-120030403. |
[32] |
L.-M. Yeh, On a steady state Euler-Poisson model for semiconductors, Comm. Partial Differential Equations, 21 (1996), 1007-1034.
doi: 10.1080/03605309608821216. |
[33] |
F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351 (2009), 480-490.
doi: 10.1016/j.jmaa.2008.10.032. |
show all references
References:
[1] |
P. Amster and M. P. Beccar Varela, Subsonic solutions to a one-dimensional nonisentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 258 (2001), 52-62.
doi: 10.1006/jmaa.2000.7359. |
[2] |
A. Ambrose, F. Méhats and P. Raviart, On singular perturbation problems for the nonlinear Poisson equation, Asymptot. Anal., 25 (2001), 39-91. |
[3] |
U. Ascher, P. A. Markowich and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Meth. Appl. Sci., 11 (1991), 347-376.
doi: 10.1142/S0218202591000174. |
[4] |
H. Brézis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas, C. R. Acad. Sci. Paris, 321 (1995), 953-959. |
[5] |
S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.
doi: 10.1080/03605300008821542. |
[6] |
P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Letters, 3 (1990), 25-29.
doi: 10.1016/0893-9659(90)90130-4. |
[7] |
P. Degond and P. A. Markowich, A steady state potential flow model for semiconductors, Ann. Math. Pura Appl.(4), 165 (1993), 87-98. |
[8] |
I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations, 17 (1992), 553-577. |
[9] |
T. Goudon, A. Jüngel and Y.-J. Peng, Zero-mass-electrons limits in hydrodynamic models for plasmas, Appl. Math. Letters, 12 (1999), 75-79.
doi: 10.1016/S0893-9659(99)00038-5. |
[10] |
D. Gilbarg and N. S. Trüdinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1998. |
[11] |
L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift diffusion equations, J. Differential Equations, 165 (2000), 315-354.
doi: 10.1006/jdeq.2000.3780. |
[12] |
A. Jüngel and Y.-J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisted, Z. Angew. Math. Phys., 51 (2000), 385-396.
doi: 10.1007/s000330050004. |
[13] |
A. Jüngel, "Quasi-Hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations, Birkhäuser, 2001. |
[14] |
C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dyn. Syst., 5 (1999), 449-455.
doi: 10.3934/dcds.1999.5.449. |
[15] |
Y.-P. Li and J.-Z. Zhang, Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors, Math. Comput. Modeling, 49 (2009), 163-177.
doi: 10.1016/j.mcm.2008.05.006. |
[16] |
Y.-P. Li, Stationary solutions for a one-dimensional nonisentropic hydrodynamic model for semiconductors, Acta Math. Sci., B28 (2008), 479-488. |
[17] |
Y.-P. Li, Asymptotic profile in a one-dimensional nonisentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 325 (2007), 949-967.
doi: 10.1016/j.jmaa.2006.02.018. |
[18] |
O. A. Ladyzhenskaya and N. U. Uraltsera, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968. |
[19] |
Y.-P. Li, Asymptotic profile in a multi-dimensional nonisentropic hydrodynamic model for semiconductors, Nonlinear Anal. Real World Appl., 8 (2007), 1235-1251.
doi: 10.1016/j.nonrwa.2006.06.011. |
[20] |
P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew Math. Phys., 42 (1991), 387-407.
doi: 10.1007/BF00945711. |
[21] |
P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145.
doi: 10.1007/BF00379918. |
[22] |
P. A. Markowich, C. A. Ringhofev and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Wien, New York, 1990. |
[23] |
Y.-J. Peng, Some analysis in steady-state Euler-Poisson equations for potential flow, Asymp. Anal., 36 (2003), 75-92. |
[24] |
Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41 (2005), 141-160. |
[25] |
Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity, 17 (2004), 835-849.
doi: 10.1088/0951-7715/17/3/006. |
[26] |
Y.-J. Peng and I. Violet, Asymptotic expansions in a steady state Euler-Poisson equations to incompressible Euler equations, Math. Models Meth. Appl. Sci., 15 (2005), 717-736.
doi: 10.1142/S0218202505000546. |
[27] |
Y.-J. Peng and Y.-F. Yang, Junction layer analysis in one-dimensional steady-state Euler-Poisson equations, J. Math. Anal. Appl., 344 (2008), 440-448.
doi: 10.1016/j.jmaa.2008.02.062. |
[28] |
M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model, Quart. Appl. Math., 2003. |
[29] |
M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system, J. Nonlinear Sci., 11 (2001), 193-209.
doi: 10.1007/s00332-001-0004-9. |
[30] |
I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow, Proc. Roy. Soc. Edinburgh, 137 (2007), 1101-1118.
doi: 10.1017/S0308210505001216. |
[31] |
S. Wang, Quasineutral limit of Euler-Poinsson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456.
doi: 10.1081/PDE-120030403. |
[32] |
L.-M. Yeh, On a steady state Euler-Poisson model for semiconductors, Comm. Partial Differential Equations, 21 (1996), 1007-1034.
doi: 10.1080/03605309608821216. |
[33] |
F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351 (2009), 480-490.
doi: 10.1016/j.jmaa.2008.10.032. |
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