Abstract
This paper is concerned with the periodic problem to the two-fluid non-isentropic Euler–Maxwell (N-E-M) equations. The equations arises in the modeling of magnetic plasma, in which appear two physical parameters, the mass of an electron \(m_\mathrm{e}\) and the mass of an ion \(m_{\mathrm{i}}\). With the help of methods of asymptotic expansions, we prove the local-in-time convergence of smooth solutions to this problem by setting \(m_\mathrm{e} = 1\) and letting \(m_{\mathrm{i}} \rightarrow +\infty \). Moreover, when the initial data are near constant equilibrium states, by means of uniform energy estimates and compactness arguments, we rigorously prove the infinity-ion-mass convergence of the system for all time. The limit system is the one-fluid N-E-M system.
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Alì, G., Chen, L., Jüngel, A., Peng, Y.J.: The zero-electron-mass limit in the hydrodynamic model for plasmas. Nonlinear Anal. 72, 4415–4427 (2010)
Alì, G., Chen, L.: The zero-electron-mass limit in the Euler–Poisson system for both well- and ill-prepared initial data. Nonlinearity 24, 2745–2761 (2011)
Chen, F.: Introduction to Plasma Physics and Controlled Fusion, vol. 1. Plenum Press, New York (1984)
Chen, G.Q., Jerome, J.W., Wang, D.H.: Compressible Euler–Maxwell equations. Transp. Theory Stat. Phys. 29, 311–331 (2000)
Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)
Deng, Y., Ionescu, A.D., Pausade, B.: The Euler-Maxwell system for electrons: global solutions in 2D. Arch. Ration. Mech. Anal. (2) 225, 771–871 (2017)
Duan, R.J.: Global smooth flows for the compressible Euler–Maxwell system: the relaxation case. J. Hyper. Diff. Equ. 8, 375–413 (2011)
Duan, R.J., Liu, Q.Q., Zhu, C.J.: The Cauchy problem on the compressible two-fluids Euler-Maxwell equations. SIAM J. Math. Anal. 44, 102–133 (2012)
Feng, Y.H., Wang, S., Kawashima, S.: Global existence and asymptotic decay of solutions to the non-isentropic Euler–Maxwell system. Math. Models Methods Appl. Sci. 24, 2851–2884 (2014)
Germain, P., Masmoudi, N.: Global existence for the Euler–Maxwell system. Ann. Sci. éc. Norm. Supér. (4) 47, 469–503 (2014)
Goudon, T., Jüngel, A., Peng, Y.J.: Zero-mass-electrons limits in hydrodynamic models for plasmas. Appl. Math. Lett. 12, 75–79 (1999)
Guo, Y., Ionescu, A.D., Pausade, B.: Global solutions of the Euler–Maxwell two-fluid system in 3D. Ann. Math. 183, 377–498 (2016)
Jüngel, A., Peng, Y.J.: A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 83–118 (2000)
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205 (1975)
Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Markowich, P., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Berlin (1990)
Peng, Y.J.: Global existence and long-time behavior of smooth solutions of two fluid Euler-Maxwell equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 737–759 (2012)
Peng, Y.J., Wang, S.: Rigorous derivation of incompressible e-MHD equations from compressible Euler–Maxwell equations. SIAM J. Math. Anal. 40, 540–565 (2008)
Peng, Y.J., Wang, S., Gu, Q.L.: Relaxation limit and global existence of smooth solutions of compressible Euler–Maxwell equations. SIAM J. Math. Anal. 43, 944–970 (2011)
Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura. Appl. 146, 65–96 (1987)
Ueda, Y., Wang, S., Kawashima, S.: Dissipative structure of the regularity type and time asymptotic decay of solutions for the Euler–Maxwell system. SIAM J. Math. Anal. 44, 2002–2017 (2012)
Wasiolek, V.: Uniform global existence and convergence of Euler–Maxwell systems with small parameters. Commun. Pure Appl. Anal. 15, 2007–2021 (2016)
Wang, S., Feng, Y.H., Li, X.: The asymptotic behavior of global smooth solutions of bipolar non-isentropic compressible Euler–Maxwell system for plasma. SIAM J. Math. Anal. 44, 3429–3457 (2012)
Xi, S., Zhao, L.: From bipolar Euler-Poisson system to unipolar Euler-Poisson system in the perspective of mass. arXiv:2002.10867
Xu, J.: Global classical solutions to the compressible Euler–Maxwell equations. SIAM J. Math. Anal. 43, 2688–2718 (2011)
Xu, J., Xiong, J., Kawashima, S.: Global well-posedness in critical Besov spaces for two-fluid Euler–Maxwell equations. SIAM J. Math. Anal. 45, 1422–1447 (2013)
Xu, J., Yong, W.A.: Zero-electron-mass limit of hydrodynamic models for plasmas. Proc. R. Soc. Edinburgh Sect. A. 141, 431–447 (2011)
Xu, J., Zhang, T.: Zero-electron-mass limit of Euler–Poisson equations. Discrete Contin. Dyn. Syst. 33, 4743–4768 (2013)
Yang, J., Wang, S., Wang, F.: Approximation of a compressible Euler–Poisson equations by a non-isentropic Euler–Maxwell equations. Appl. Math. Comp. 219, 6142–6151 (2013)
Zhao, L.: The rigorous derivation of unipolar Euler-Maxwell system for electrons from bipolar Euler-Maxwell system by infinity-ion-mass limit. preprint (2020)
Acknowledgements
The authors are grateful to the referee for the comments. The first author would like to express his sincere gratitude to Professor Yue-Jun Peng of Université Clermont Auvergne for excellent directions in France. The authors are supported by the the BNSF (1164010, 1132006), NSFC (11771031, 11531010, 11831003), NSF of Qinghai Province (2017-ZJ-908), NSF of Henan Province (162300410084), the Key Research Fund of Henan Province (16A110019), the general project of scientific research project of the Beijing education committee of China(KM202111232008), the research fund of Beijing Information Science and Technology University (2025029).
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Feng, YH., Li, X. & Wang, S. The global convergence of non-isentropic Euler–Maxwell equations via Infinity-Ion-Mass limit. Z. Angew. Math. Phys. 72, 28 (2021). https://doi.org/10.1007/s00033-020-01459-6
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DOI: https://doi.org/10.1007/s00033-020-01459-6
Keywords
- Two fluids non-isentropic Euler–Maxwell equations
- The infinity-ion-mass limit
- Local convergence
- Global convergence