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Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities
Uniform global existence and convergence of Euler-Maxwell systems with small parameters
1. | 20 Rue de Vialle, Lamothe, 43100, France |
References:
[1] |
G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model, SIAM J. Appl. Math., 35 (2003), 389-422.
doi: 10.1137/S0036141001393225. |
[2] |
G. Alì, L. Chen, A. Jungel and Y.-J. Peng, The zero-electron-mass limit in the hydrodynamic models for plasmas, Nonlinear Analysis TMA, 72 (2010), 4410-4427.
doi: 10.1016/j.na.2010.02.016. |
[3] |
C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci., 14 (2004), 393-415.
doi: 10.1142/S0218202504003283. |
[4] |
S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.
doi: 10.1002/cpa.20195. |
[5] |
Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Comm. Math. Sci., 1 (2003), 437-447. |
[6] |
G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319.
doi: 10.1016/j.jde.2008.05.015. |
[7] |
J. Y. Chemin, Fluides Parfaits Incompressibles, Astérisque No. 230, 1995. |
[8] |
F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. |
[9] |
G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport theory and statistical physics, 29 (2000), 311-331.
doi: 10.1080/00411450008205877. |
[10] |
J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Transactions Amer. Math. Soc., 359 (2007), 637-648.
doi: 10.1090/S0002-9947-06-04028-1. |
[11] |
P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of computational physics, 231 (2012), 1917-1946.
doi: 10.1016/j.jcp.2011.11.011. |
[12] |
R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case, J. Hyper. Diff. Equations, 8 (2011), 375-413.
doi: 10.1142/S0219891611002421. |
[13] |
W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566.
doi: 10.1006/jdeq.1995.1172. |
[14] |
P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Annales Scientifiques de l'ENS, 47 (2014), 469-503. |
[15] |
Y. Guo, A. D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D,, preprint, ().
doi: 10.4007/annals.2016.183.2.1. |
[16] |
B. Hanouzet and R. Natalini, Global existence of smooth solutions for partial dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 169 (2003), 89-117.
doi: 10.1007/s00205-003-0257-6. |
[17] |
L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133.
doi: 10.1016/S0022-0396(03)00063-9. |
[18] |
A. D. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein-Gordon equations in 3D, J. Eur. Math. Soc., 16 (2014), 2355-2431.
doi: 10.4171/JEMS/489. |
[19] |
A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703.
doi: 10.1142/S0218202594000388. |
[20] |
A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033.
doi: 10.1080/03605309908821456. |
[21] |
T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. |
[22] |
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Math. Appl., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[23] |
C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semi-conductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360.
doi: 10.1142/S0218202500000215. |
[24] |
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. |
[25] |
C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 447-461.
doi: 10.1007/s00030-012-0159-0. |
[26] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New-York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[27] |
P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equations, Arch. Ration. Mech. Anal., 129 (1995), 129-145.
doi: 10.1007/BF00379918. |
[28] |
P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[29] |
Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Annals of Mathematics, 28-B (2007), 583-602.
doi: 10.1007/s11401-005-0556-3. |
[30] |
Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Communications in Partial Differential Equations, 33 (2008), 349-376.
doi: 10.1080/03605300701318989. |
[31] |
Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 349-376.
doi: 10.1137/070686056. |
[32] |
Y. J. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970.
doi: 10.1137/100786927. |
[33] |
Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pure Appl., 103 (2015), 39-67.
doi: 10.1016/j.matpur.2014.03.007. |
[34] |
Y. J. Peng, Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters, SIAM J. Math. Anal., 47 (2015), 1355-1376.
doi: 10.1137/140983276. |
[35] |
Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, AN (2015), http://dx.doi.org/10.2016/j.anihpc.2015.03.006 |
[36] |
Y. J. Peng and V. Wasiolek, Uniform global existence and parabolic limit for partially dissipative hyperbolic Systems,, preprint., ().
doi: 10.1016/j.jde.2016.01.019. |
[37] |
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[38] |
J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[39] |
B. Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal., 42 (2005), 211-250. |
[40] |
B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal., 184 (2007), 121-183.
doi: 10.1007/s00205-006-0034-4. |
[41] |
Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-267.
doi: 10.4310/MAA.2011.v18.n3.a1. |
[42] |
W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors, SIAM J. Appl. Math., 64 (2004), 1737-1748.
doi: 10.1137/S0036139903427404. |
[43] |
W. A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172 (2004), 247-266.
doi: 10.1007/s00205-003-0304-3. |
[44] |
Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.
doi: 10.1007/s002050050188. |
show all references
References:
[1] |
G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model, SIAM J. Appl. Math., 35 (2003), 389-422.
doi: 10.1137/S0036141001393225. |
[2] |
G. Alì, L. Chen, A. Jungel and Y.-J. Peng, The zero-electron-mass limit in the hydrodynamic models for plasmas, Nonlinear Analysis TMA, 72 (2010), 4410-4427.
doi: 10.1016/j.na.2010.02.016. |
[3] |
C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci., 14 (2004), 393-415.
doi: 10.1142/S0218202504003283. |
[4] |
S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.
doi: 10.1002/cpa.20195. |
[5] |
Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Comm. Math. Sci., 1 (2003), 437-447. |
[6] |
G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319.
doi: 10.1016/j.jde.2008.05.015. |
[7] |
J. Y. Chemin, Fluides Parfaits Incompressibles, Astérisque No. 230, 1995. |
[8] |
F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. |
[9] |
G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport theory and statistical physics, 29 (2000), 311-331.
doi: 10.1080/00411450008205877. |
[10] |
J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Transactions Amer. Math. Soc., 359 (2007), 637-648.
doi: 10.1090/S0002-9947-06-04028-1. |
[11] |
P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of computational physics, 231 (2012), 1917-1946.
doi: 10.1016/j.jcp.2011.11.011. |
[12] |
R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case, J. Hyper. Diff. Equations, 8 (2011), 375-413.
doi: 10.1142/S0219891611002421. |
[13] |
W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566.
doi: 10.1006/jdeq.1995.1172. |
[14] |
P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Annales Scientifiques de l'ENS, 47 (2014), 469-503. |
[15] |
Y. Guo, A. D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D,, preprint, ().
doi: 10.4007/annals.2016.183.2.1. |
[16] |
B. Hanouzet and R. Natalini, Global existence of smooth solutions for partial dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 169 (2003), 89-117.
doi: 10.1007/s00205-003-0257-6. |
[17] |
L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133.
doi: 10.1016/S0022-0396(03)00063-9. |
[18] |
A. D. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein-Gordon equations in 3D, J. Eur. Math. Soc., 16 (2014), 2355-2431.
doi: 10.4171/JEMS/489. |
[19] |
A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703.
doi: 10.1142/S0218202594000388. |
[20] |
A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033.
doi: 10.1080/03605309908821456. |
[21] |
T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. |
[22] |
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Math. Appl., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[23] |
C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semi-conductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360.
doi: 10.1142/S0218202500000215. |
[24] |
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. |
[25] |
C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 447-461.
doi: 10.1007/s00030-012-0159-0. |
[26] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New-York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[27] |
P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equations, Arch. Ration. Mech. Anal., 129 (1995), 129-145.
doi: 10.1007/BF00379918. |
[28] |
P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[29] |
Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Annals of Mathematics, 28-B (2007), 583-602.
doi: 10.1007/s11401-005-0556-3. |
[30] |
Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Communications in Partial Differential Equations, 33 (2008), 349-376.
doi: 10.1080/03605300701318989. |
[31] |
Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 349-376.
doi: 10.1137/070686056. |
[32] |
Y. J. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970.
doi: 10.1137/100786927. |
[33] |
Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pure Appl., 103 (2015), 39-67.
doi: 10.1016/j.matpur.2014.03.007. |
[34] |
Y. J. Peng, Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters, SIAM J. Math. Anal., 47 (2015), 1355-1376.
doi: 10.1137/140983276. |
[35] |
Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, AN (2015), http://dx.doi.org/10.2016/j.anihpc.2015.03.006 |
[36] |
Y. J. Peng and V. Wasiolek, Uniform global existence and parabolic limit for partially dissipative hyperbolic Systems,, preprint., ().
doi: 10.1016/j.jde.2016.01.019. |
[37] |
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[38] |
J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[39] |
B. Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal., 42 (2005), 211-250. |
[40] |
B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal., 184 (2007), 121-183.
doi: 10.1007/s00205-006-0034-4. |
[41] |
Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-267.
doi: 10.4310/MAA.2011.v18.n3.a1. |
[42] |
W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors, SIAM J. Appl. Math., 64 (2004), 1737-1748.
doi: 10.1137/S0036139903427404. |
[43] |
W. A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172 (2004), 247-266.
doi: 10.1007/s00205-003-0304-3. |
[44] |
Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.
doi: 10.1007/s002050050188. |
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