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2016, 15(6): 2007-2021. doi: 10.3934/cpaa.2016025

Uniform global existence and convergence of Euler-Maxwell systems with small parameters

1. 

20 Rue de Vialle, Lamothe, 43100, France

Received  April 2015 Revised  April 2016 Published  September 2016

The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.
Citation: Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model,, \emph{SIAM J. Appl. Math.}, 35 (2003), 389. 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,, \emph{Nonlinear Analysis TMA}, 72 (2010), 4410. 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,, \emph{Math. Models Methods Appl. Sci.}, 14 (2004), 393. 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,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1559. 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,, \emph{Comm. Math. Sci.}, 1 (2003), 437.

[6]

G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation,, \emph{J. Differential Equations}, 246 (2009), 291. doi: 10.1016/j.jde.2008.05.015.

[7]

J. Y. Chemin, Fluides Parfaits Incompressibles,, Ast\'erisque No. 230, (1995).

[8]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Vol. 1, (1984).

[9]

G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations,, \emph{Transport theory and statistical physics}, 29 (2000), 311. doi: 10.1080/00411450008205877.

[10]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations,, \emph{Transactions Amer. Math. Soc.}, 359 (2007), 637. 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,, \emph{Journal of computational physics}, 231 (2012), 1917. doi: 10.1016/j.jcp.2011.11.011.

[12]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case,, \emph{J. Hyper. Diff. Equations}, 8 (2011), 375. doi: 10.1142/S0219891611002421.

[13]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations,, \emph{J. Differential Equations}, 123 (1995), 523. doi: 10.1006/jdeq.1995.1172.

[14]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system,, \emph{Annales Scientifiques de l'ENS}, 47 (2014), 469.

[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,, \emph{Arch. Ration. Mech. Anal.}, 169 (2003), 89. 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,, \emph{J. Differential Equations}, 192 (2003), 111. 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,, \emph{J. Eur. Math. Soc.}, 16 (2014), 2355. 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,, \emph{Math. Models Methods Appl. Sci.}, 4 (1994), 677. doi: 10.1142/S0218202594000388.

[20]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1007. doi: 10.1080/03605309908821456.

[21]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 58 (1975), 181.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, \emph{Comm. Pure Math. Appl.}, 34 (1981), 481. 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,, \emph{Math. Models Methods Appl. Sci.}, 10 (2000), 351. 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,, \emph{Discrete Contin. Dyn. Syst.}, 5 (1999), 449. doi: 10.3934/dcds.1999.5.449.

[25]

C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 20 (2013), 447. doi: 10.1007/s00030-012-0159-0.

[26]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (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,, \emph{Arch. Ration. Mech. Anal.}, 129 (1995), 129. doi: 10.1007/BF00379918.

[28]

P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations,, Springer-Verlag, (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,, \emph{Chinese Annals of Mathematics}, 28-B (2007), 583. doi: 10.1007/s11401-005-0556-3.

[30]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations,, \emph{Communications in Partial Differential Equations}, 33 (2008), 349. doi: 10.1080/03605300701318989.

[31]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations,, \emph{SIAM J. Math. Anal.}, 40 (2008), 349. 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,, \emph{SIAM J. Math. Anal.}, 43 (2011), 944. doi: 10.1137/100786927.

[33]

Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations,, \emph{J. Math. Pure Appl.}, 103 (2015), 39. 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,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1355. doi: 10.1137/140983276.

[35]

Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, AN (2015).

[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,, \emph{Hokkaido Math. J.}, 14 (1985), 249. doi: 10.14492/hokmj/1381757663.

[38]

J. Simon, Compact sets in the space $L^p(0, T; B)$,, \emph{Ann. Mat. Pura Appl.}, 146 (1987), 65. doi: 10.1007/BF01762360.

[39]

B. Texier, WKB asymptotics for the Euler-Maxwell equations,, \emph{Asymptot. Anal.}, 42 (2005), 211.

[40]

B. Texier, Derivation of the Zakharov equations,, \emph{Arch. Ration. Mech. Anal.}, 184 (2007), 121. doi: 10.1007/s00205-006-0034-4.

[41]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system,, \emph{Methods Appl. Anal.}, 18 (2011), 245. doi: 10.4310/MAA.2011.v18.n3.a1.

[42]

W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors,, \emph{SIAM J. Appl. Math.}, 64 (2004), 1737. doi: 10.1137/S0036139903427404.

[43]

W. A. Yong, Entropy and global existence for hyperbolic balance laws,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 247. doi: 10.1007/s00205-003-0304-3.

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation,, \emph{Arch. Ration. Mech. Anal.}, 150 (1999), 225. doi: 10.1007/s002050050188.

show all references

References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model,, \emph{SIAM J. Appl. Math.}, 35 (2003), 389. 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,, \emph{Nonlinear Analysis TMA}, 72 (2010), 4410. 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,, \emph{Math. Models Methods Appl. Sci.}, 14 (2004), 393. 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,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1559. 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,, \emph{Comm. Math. Sci.}, 1 (2003), 437.

[6]

G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation,, \emph{J. Differential Equations}, 246 (2009), 291. doi: 10.1016/j.jde.2008.05.015.

[7]

J. Y. Chemin, Fluides Parfaits Incompressibles,, Ast\'erisque No. 230, (1995).

[8]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Vol. 1, (1984).

[9]

G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations,, \emph{Transport theory and statistical physics}, 29 (2000), 311. doi: 10.1080/00411450008205877.

[10]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations,, \emph{Transactions Amer. Math. Soc.}, 359 (2007), 637. 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,, \emph{Journal of computational physics}, 231 (2012), 1917. doi: 10.1016/j.jcp.2011.11.011.

[12]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case,, \emph{J. Hyper. Diff. Equations}, 8 (2011), 375. doi: 10.1142/S0219891611002421.

[13]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations,, \emph{J. Differential Equations}, 123 (1995), 523. doi: 10.1006/jdeq.1995.1172.

[14]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system,, \emph{Annales Scientifiques de l'ENS}, 47 (2014), 469.

[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,, \emph{Arch. Ration. Mech. Anal.}, 169 (2003), 89. 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,, \emph{J. Differential Equations}, 192 (2003), 111. 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,, \emph{J. Eur. Math. Soc.}, 16 (2014), 2355. 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,, \emph{Math. Models Methods Appl. Sci.}, 4 (1994), 677. doi: 10.1142/S0218202594000388.

[20]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1007. doi: 10.1080/03605309908821456.

[21]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 58 (1975), 181.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, \emph{Comm. Pure Math. Appl.}, 34 (1981), 481. 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,, \emph{Math. Models Methods Appl. Sci.}, 10 (2000), 351. 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,, \emph{Discrete Contin. Dyn. Syst.}, 5 (1999), 449. doi: 10.3934/dcds.1999.5.449.

[25]

C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 20 (2013), 447. doi: 10.1007/s00030-012-0159-0.

[26]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (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,, \emph{Arch. Ration. Mech. Anal.}, 129 (1995), 129. doi: 10.1007/BF00379918.

[28]

P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations,, Springer-Verlag, (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,, \emph{Chinese Annals of Mathematics}, 28-B (2007), 583. doi: 10.1007/s11401-005-0556-3.

[30]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations,, \emph{Communications in Partial Differential Equations}, 33 (2008), 349. doi: 10.1080/03605300701318989.

[31]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations,, \emph{SIAM J. Math. Anal.}, 40 (2008), 349. 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,, \emph{SIAM J. Math. Anal.}, 43 (2011), 944. doi: 10.1137/100786927.

[33]

Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations,, \emph{J. Math. Pure Appl.}, 103 (2015), 39. 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,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1355. doi: 10.1137/140983276.

[35]

Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, AN (2015).

[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,, \emph{Hokkaido Math. J.}, 14 (1985), 249. doi: 10.14492/hokmj/1381757663.

[38]

J. Simon, Compact sets in the space $L^p(0, T; B)$,, \emph{Ann. Mat. Pura Appl.}, 146 (1987), 65. doi: 10.1007/BF01762360.

[39]

B. Texier, WKB asymptotics for the Euler-Maxwell equations,, \emph{Asymptot. Anal.}, 42 (2005), 211.

[40]

B. Texier, Derivation of the Zakharov equations,, \emph{Arch. Ration. Mech. Anal.}, 184 (2007), 121. doi: 10.1007/s00205-006-0034-4.

[41]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system,, \emph{Methods Appl. Anal.}, 18 (2011), 245. doi: 10.4310/MAA.2011.v18.n3.a1.

[42]

W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors,, \emph{SIAM J. Appl. Math.}, 64 (2004), 1737. doi: 10.1137/S0036139903427404.

[43]

W. A. Yong, Entropy and global existence for hyperbolic balance laws,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 247. doi: 10.1007/s00205-003-0304-3.

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation,, \emph{Arch. Ration. Mech. Anal.}, 150 (1999), 225. doi: 10.1007/s002050050188.

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