On the quasineutral limit for the compressible Euler-Poisson equations

Jianwei Yang; Dongling Li; Xiao Yang

doi:10.3934/dcdsb.2022020

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2022, Volume 27, Issue 11: 6797-6806. Doi: 10.3934/dcdsb.2022020

This issue Previous Article Global well-posedness of the three-dimensional viscous primitive equations with bounded delays Next Article An efficient finite element method and error analysis for fourth order problems in a spherical domain

On the quasineutral limit for the compressible Euler-Poisson equations

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, Henan Province, China

^*Corresponding author: Jianwei Yang
^*Corresponding author: Jianwei Yang

Received: August 2021

Revised: November 2021

Early access: February 2022

Published: October 2022

The first author is supported by Natural Science Foundation of Henan Province (No. 202300410277)

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Abstract

In this paper, we consider the quasineutral limit of compressible Euler-Poisson equations based on the concept of dissipative measure-valued solutions. In the case of well-prepared initial data under periodic boundary condictions, we prove that dissipative measure-valued solutions of the compressible Euler-Poisson equations converge to the smooth solution of the incompressible Euler system when the Debye length tends to zero.

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Mathematics Subject Classification: Primary: 35B25; Secondary: 35Q35.

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References

[1]	Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529.
[2]	Y. Breinier, C. De Lellis and L. Székelyhidi Jr, Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys., 305 (2011), 351-361. doi: 10.1007/s00220-011-1267-0.
[3]	J. Březina and E. Feireisl, Measure-valued solutions to the complete Euler system revisited, Z. Angew. Math. Phys., 69 (2018), Paper No. 57, 17 pp. doi: 10.1007/s00033-018-0951-8.
[4]	N. Chaudhuri, On weak (measure-value)-strong uniqueness for compressible Navier-Stokes system with non-monotone pressure law, J. Math. Fluid Mech., 22 (2020), Paper No. 17, 13 pp. doi: 10.1007/s00021-019-0465-y.
[5]	S. Demoulini, D. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal., 205 (2012), 927-961. doi: 10.1007/s00205-012-0523-6.
[6]	R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.
[7]	E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 141, 20 pp. doi: 10.1007/s00526-016-1089-1.
[8]	E. Feireisl, C. Klingenberg and S. Markfelder, On the low Mach number limit for the compressible Euler system, SIAM J. Math. Anal., 51 (2019), 1496-1513. doi: 10.1137/17M1131799.
[9]	P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890. doi: 10.1088/0951-7715/28/11/3873.
[10]	S. Jiang, Q.-C. Ju, H.-L. Li and Y. Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53 (2010), 3099-3114. doi: 10.1007/s11425-010-4114-4.
[11]	Q.-C. Ju, H.-L. Li, Y. Li and S. Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9 (2010), 1577-1590. doi: 10.3934/cpaa.2010.9.1577.
[12]	T. Kato, Nonstationary flows of viscous and ideal fluids in $ \mathbb{R}^3$, J. Funct. Anal., 9 (1972), 296-305. doi: 10.1016/0022-1236(72)90003-1.
[13]	T. Kato and C.-Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28. doi: 10.1016/0022-1236(84)90024-7.
[14]	D. Kröner and W. Zajaczkowski, Measure-valued solutions of the Euler equations for ideal compressible polytropic fluids, Math. Methods Appl. Sci., 19 (1996), 235-252. doi: 10.1002/(SICI)1099-1476(199602)19:3<235::AID-MMA772>3.0.CO;2-4.
[15]	J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDE's, Chapman and Hall, London, 1996.
[16]	Š. Nečasová and T. Tang, On a singular limit for the compressible rotating Euler system, J. Math. Fluid Mech., 22 (2020), Paper No. 43, 14 pp. doi: 10.1007/s00021-020-00504-8.
[17]	J. Neustupa, Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids, Math. Nachr., 163 (1993), 217-227. doi: 10.1002/mana.19931630119.
[18]	P. Pedregal, Parametrized Measures and Variational Principles, Birkhäuser, 1997. doi: 10.1007/978-3-0348-8886-8.
[19]	X.-K. Pu, Quasineutral limit of the pressureless Euler-Poisson equation, Appl. Math. Lett., 30 (2014), 33-37. doi: 10.1016/j.aml.2013.12.008.
[20]	S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. PDE, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.
[21]	S. Wang, J.-W Yang and D. Luo, Convergence of compressible Euler-Poisson system to incompressible Euler equations, Appl. Math. Comput., 216 (2010), 3408-3418. doi: 10.1016/j.amc.2010.04.035.