Abstract
The main purpose of this paper is to provide an effective procedure to study rigorously the relationship between unipolar and bipolar Euler-Poisson systems in the perspective of mass. Based on the fact that the mass of an electron is far less than that of an ion, we amplify this property by letting \(m_e/m_i\rightarrow 0\) and using two different singular limits to illustrate it, which are the zero-electron mass limit and the infinity-ion mass limit. We use the method of asymptotic expansions to handle the problem and find that the limiting process from bipolar to unipolar systems is actually the process of decoupling, but not the vanishing of equations of the corresponding the other particle.
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References
Alì, G., Chen, L.: The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data. Nonlinearity 24(10), 2745–2761 (2011)
Alì, G., Chen, L., Jüngel, A., Peng, Y.-J.: The zero-electron-mass limit in the hydrodynamic model for plasmas. Nonlinear Anal. 72(12), 4415–4427 (2010)
Besse, C., Degond, P., Deluzet, F., Claudel, J., Gallice, G., Tessieras, C.: A model hierarchy for ionospheric plasma modeling. Math. Models Methods Appl. Sci. 14(3), 393–415 (2004)
Chen, F.: Introduction to Plasma Physics and Controlled Fusion, volume 1. PlenumPress, (1984)
Goudon, T., Jüngel, A., Peng, Y.-J.: Zero-mass-electrons limits in hydrodynamic models for plasmas. Appl. Math. Lett. 12(4), 75–79 (1999)
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(1):83–118, (2000)
Jüngel, A., Peng, Y.-J.: Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited. Z. Angew. Math. Phys. 51(3), 385–396 (2000)
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975)
Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves, vol. 11. SIAM Regional Conf. Lecture, Philadelphia (1973)
Li, Y., Peng, Y.-J., Xi, S.: Rigorous derivation of a Boltzmann relation from isothermal Euler-Poisson systems. J. Math. Phys. 59(12), 14 (2018). (123501)
Liu, C., Guo, Z., Peng, Y.-J.: Global stability of large steady-states for an isentropic Euler-Maxwell system in \(\mathbb{R} ^3\). Commun. Math. Sci. 17(7), 1841–1860 (2019)
Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables, vol. 53. Springer-Verlag, New York (1984)
Peng, Y.-J.: Stability of non-constant equilibrium solutions for Euler-Maxwell equations. J. Math. Pures Appl. 103(1(9)), 39–67 (2015)
Peng, Y.-J.: Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters. SIAM J. Math. Anal. 47(2), 1355–1376 (2015)
Peng, Y.-J., Wang, S.: Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations. Chin. Ann. Math. Ser. B 28(5), 583–602 (2007)
Peng, Y.-J., Wang, S.: Convergence of compressible Euler-Maxwell equations to incompressible Euler equations. Comm. Partial Differ. Equ. 33(1–3), 349–376 (2008)
Peng, Y.-J., Wang, S.: Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations. SIAM J. Math. Anal. 40(2), 540–565 (2008)
Peng, Y.-J., Wang, S.: Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete Contin. Dynam. Syst. 23(1–2), 415–433 (2009)
Rishbeth, H., Garriott, O.K.: Introduction to ionospheric physics. Academic Press, New York and London (1969)
Xu, J., Yong, W.-A.: Zero-electron-mass limit of hydrodynamic models for plasmas. Proc. Roy. Soc. Edinburgh Sect. A 141(2), 431–447 (2011)
Xu, J., Zhang, T.: Zero-electron-mass limit of Euler-Poisson equations. Discrete Contin. Dynam. Syst. 33(10), 4743–4768 (2013)
Yang, J., Wang, S.: The non-relativistic limit of Euler-Maxwell equations for two-fluid plasma. Nonlinear Anal. 72(3–4), 1829–1840 (2010)
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We would like to thank the National Natural Science Foundation of China 12201360, Natural Science Foundation of Shandong Province ZR2020QA016.
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Xi, S., Zhao, L. From Bipolar Euler-Poisson System to Unipolar Euler-Poisson One in the Perspective of Mass. J. Math. Fluid Mech. 26, 10 (2024). https://doi.org/10.1007/s00021-023-00838-z
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DOI: https://doi.org/10.1007/s00021-023-00838-z