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November  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
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show all references

References:
[1]

SIAM J. Appl. Math., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.  Google Scholar

[2]

Nonlinear Analysis TMA, 72 (2010), 4410-4427. doi: 10.1016/j.na.2010.02.016.  Google Scholar

[3]

Math. Models Methods Appl. Sci., 14 (2004), 393-415. doi: 10.1142/S0218202504003283.  Google Scholar

[4]

Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195.  Google Scholar

[5]

Comm. Math. Sci., 1 (2003), 437-447.  Google Scholar

[6]

J. Differential Equations, 246 (2009), 291-319. doi: 10.1016/j.jde.2008.05.015.  Google Scholar

[7]

Astérisque No. 230, 1995.  Google Scholar

[8]

Vol. 1, Plenum Press, New York, 1984. Google Scholar

[9]

Transport theory and statistical physics, 29 (2000), 311-331. doi: 10.1080/00411450008205877.  Google Scholar

[10]

Transactions Amer. Math. Soc., 359 (2007), 637-648. doi: 10.1090/S0002-9947-06-04028-1.  Google Scholar

[11]

Journal of computational physics, 231 (2012), 1917-1946. doi: 10.1016/j.jcp.2011.11.011.  Google Scholar

[12]

J. Hyper. Diff. Equations, 8 (2011), 375-413. doi: 10.1142/S0219891611002421.  Google Scholar

[13]

J. Differential Equations, 123 (1995), 523-566. doi: 10.1006/jdeq.1995.1172.  Google Scholar

[14]

Annales Scientifiques de l'ENS, 47 (2014), 469-503.  Google Scholar

[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.  Google Scholar

[16]

Arch. Ration. Mech. Anal., 169 (2003), 89-117. doi: 10.1007/s00205-003-0257-6.  Google Scholar

[17]

J. Differential Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.  Google Scholar

[18]

J. Eur. Math. Soc., 16 (2014), 2355-2431. doi: 10.4171/JEMS/489.  Google Scholar

[19]

Math. Models Methods Appl. Sci., 4 (1994), 677-703. doi: 10.1142/S0218202594000388.  Google Scholar

[20]

Comm. Partial Differential Equations, 24 (1999), 1007-1033. doi: 10.1080/03605309908821456.  Google Scholar

[21]

Arch. Ration. Mech. Anal., 58 (1975), 181-205.  Google Scholar

[22]

Comm. Pure Math. Appl., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.  Google Scholar

[23]

Math. Models Methods Appl. Sci., 10 (2000), 351-360. doi: 10.1142/S0218202500000215.  Google Scholar

[24]

Discrete Contin. Dyn. Syst., 5 (1999), 449-455. doi: 10.3934/dcds.1999.5.449.  Google Scholar

[25]

NoDEA Nonlinear Differential Equations Appl., 20 (2013), 447-461. doi: 10.1007/s00030-012-0159-0.  Google Scholar

[26]

Springer-Verlag, New-York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.  Google Scholar

[28]

Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[29]

Chinese Annals of Mathematics, 28-B (2007), 583-602. doi: 10.1007/s11401-005-0556-3.  Google Scholar

[30]

Communications in Partial Differential Equations, 33 (2008), 349-376. doi: 10.1080/03605300701318989.  Google Scholar

[31]

SIAM J. Math. Anal., 40 (2008), 349-376. doi: 10.1137/070686056.  Google Scholar

[32]

SIAM J. Math. Anal., 43 (2011), 944-970. doi: 10.1137/100786927.  Google Scholar

[33]

J. Math. Pure Appl., 103 (2015), 39-67. doi: 10.1016/j.matpur.2014.03.007.  Google Scholar

[34]

SIAM J. Math. Anal., 47 (2015), 1355-1376. doi: 10.1137/140983276.  Google Scholar

[35]

Ann. Inst. H. Poincaré Anal. Non Linéaire, AN (2015), http://dx.doi.org/10.2016/j.anihpc.2015.03.006 Google Scholar

[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.  Google Scholar

[37]

Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.  Google Scholar

[38]

Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[39]

Asymptot. Anal., 42 (2005), 211-250.  Google Scholar

[40]

Arch. Ration. Mech. Anal., 184 (2007), 121-183. doi: 10.1007/s00205-006-0034-4.  Google Scholar

[41]

Methods Appl. Anal., 18 (2011), 245-267. doi: 10.4310/MAA.2011.v18.n3.a1.  Google Scholar

[42]

SIAM J. Appl. Math., 64 (2004), 1737-1748. doi: 10.1137/S0036139903427404.  Google Scholar

[43]

Arch. Ration. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.  Google Scholar

[44]

Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188.  Google Scholar

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