doi: 10.3934/dcdsb.2022020
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Received  August 2021 Revised  November 2021 Early access February 2022

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

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.

Citation: Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022020
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. BreinierC. 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. DemouliniD. 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. FeireislC. 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. GwiazdaA. 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. JiangQ.-C. JuH.-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. JuH.-L. LiY. 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. WangJ.-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.

show all references

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. BreinierC. 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. DemouliniD. 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. FeireislC. 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. GwiazdaA. 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. JiangQ.-C. JuH.-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. JuH.-L. LiY. 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. WangJ.-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.

[1]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086

[2]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[3]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[4]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

[5]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[6]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[7]

Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201

[8]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[9]

Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448

[10]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085

[11]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011

[12]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[13]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046

[14]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic and Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

[15]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003

[16]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101

[17]

Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1

[18]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013

[19]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292

[20]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345

2021 Impact Factor: 1.497

Article outline

[Back to Top]