The zero-energy limit and quasi-neutral limit of scaled Euler-Maxwell system and its corresponding limiting models

Tariq Mahmood; Tariq Mahmood

doi:10.3934/math.2019.3.910

AIMS Mathematics

2019, Volume 4, Issue 3: 910-927. doi: 10.3934/math.2019.3.910

Previous Article Next Article

Research article

The zero-energy limit and quasi-neutral limit of scaled Euler-Maxwell system and its corresponding limiting models

Tariq Mahmood ^,

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

Received: 08 May 2019 Accepted: 10 July 2019 Published: 26 July 2019
MSC : Primary: 34E05, 35Q35, 35L60; Secondary: 35Q61

In this paper, we apply a new scaling for Euler-Maxwell system. As a result, zero-energy limit is introduced, combining it with quasi-neutral limit, we obtain a complete system of the limiting models for Euler-Maxwell system. The asymptotic analysis and a weighted energy method are used to rigorously justify the zero-energy limit for e-MHD, which is the limiting model of Euler-Maxwell system as the Debye length tends to zero. For other limits, we provide a formal derivation and obtain an approximate asymptotic expansion.
- Euler-Maxwell system,
- zero-energy limit,
- magnetic curl
Citation: Tariq Mahmood. The zero-energy limit and quasi-neutral limit of scaled Euler-Maxwell system and its corresponding limiting models[J]. AIMS Mathematics, 2019, 4(3): 910-927. doi: 10.3934/math.2019.3.910

Related Papers:

Abstract

In this paper, we apply a new scaling for Euler-Maxwell system. As a result, zero-energy limit is introduced, combining it with quasi-neutral limit, we obtain a complete system of the limiting models for Euler-Maxwell system. The asymptotic analysis and a weighted energy method are used to rigorously justify the zero-energy limit for e-MHD, which is the limiting model of Euler-Maxwell system as the Debye length tends to zero. For other limits, we provide a formal derivation and obtain an approximate asymptotic expansion.

References

[1]	S. I. Braginskii, Transport processes in a plasma, Reviews of Plasma Physics, Vol. 1, New York: Consultants Bureau, 1965.
[2]	Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Commun. Math. Sci., 1 (2003), 437-447. doi: 10.4310/CMS.2003.v1.n3.a4
[3]	S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Commun. Part. Diff. Eq., 25 (2000), 1099-1113. doi: 10.1080/03605300008821542
[4]	F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, New York: Plenum Press, 1984.
[5]	P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, J. Comput. Phys., 231 (2012), 1917-1946. doi: 10.1016/j.jcp.2011.11.011
[6]	R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbol. Differ. Eq., 8 (2011), 375-413. doi: 10.1142/S0219891611002421
[7]	L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differ. Eq., 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. An., 58 (1975), 181-205. doi: 10.1007/BF00280740
[9]	S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405
[10]	P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford Lecture, Ser. Math. Appl., New York: Oxford University Press, 1996.
[11]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Math. Sci., Vol. 53, New York: Springer-Verlag, 1984.
[12]	Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41 (2005), 141-160.
[13]	Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Ann. Math. B, 28 (2007), 583-602. doi: 10.1007/s11401-005-0556-3
[14]	Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 540-565. doi: 10.1137/070686056
[15]	Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Commun. Part. Diff. Eq., 33 (2008), 349-376. doi: 10.1080/03605300701318989
[16]	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
[17]	H. Rishbeth and O. K. Garriott, Introduction to Ionospheric Physics, International geophysics series, Academic Press, 1969.
[18]	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
[19]	B. Texier , WKB asymptotics for the Euler-Maxwell equations, Asymptotic Anal., 42 (2005), 211-250.
[20]	B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. An., 184 (2007), 121-183. doi: 10.1007/s00205-006-0034-4
[21]	J. Xu, Global classical solutions to the compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 2688-2718. doi: 10.1137/100812768

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)