GLOBAL CONVERGENCE OF AN ISENTROPIC EULER-POISSON SYSTEM IN R<sup>+</sup>&#215;R<sup>d</sup>

Huimin Tian; Yue-Jun Peng; Lingling Zhang

doi:10.11948/2018.710

2018 Volume 8 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(3): 710-726

Next Article Previous Article

Huimin Tian, Yue-Jun Peng, Lingling Zhang. GLOBAL CONVERGENCE OF AN ISENTROPIC EULER-POISSON SYSTEM IN R+×Rd[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 710-726. doi: 10.11948/2018.710

Citation:

Huimin Tian, Yue-Jun Peng, Lingling Zhang. GLOBAL CONVERGENCE OF AN ISENTROPIC EULER-POISSON SYSTEM IN R⁺×R^d[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 710-726. doi: 10.11948/2018.710

GLOBAL CONVERGENCE OF AN ISENTROPIC EULER-POISSON SYSTEM IN R⁺×R^d

1 Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China;
2 Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal, 63000 Clermont-Ferrand, France

Fund Project:

Abstract

Abstract

We prove the global-in-time convergence of an Euler-Poisson system near a constant equilibrium state in the whole space Rd, as physical parameters tend to zero. The result follows from the uniform global existence of smooth solutions by means of energy estimates together with compactness arguments. For this purpose, we establish uniform estimates for div u and curl u instead of ∇u.
- Euler-Poisson system /
- uniform global smooth solution /
- energy estimate /
- compactness and convergence
MSC: 35B40;35L45;35L60

FullText(HTML)

References

[1]	G. Alì, Global existence of smooth solutions of the N-dimensional EulerPoisson model, SIAM J. Appl. Math., 2003, 35(2), 389-422. Google Scholar
[2]	G. Alì, L. Chen, A. Jüngel and Y. J. Peng, The zero-electron-mass limit in the hydrodynamic model for plasmas, Nonlinear Anal. TMA, 2010, 72(12), 4415-4427. Google Scholar
[3]	J.Y. Chemin, Fluides Parfaits Incompressibles, Astérisque, 1995, 230. Google Scholar
[4]	F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New York, 1984, 1. Google Scholar
[5]	S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Part. Diff. Equations, 2000, 25(5-6), 1099-1113. Google Scholar
[6]	D. Gérard-Varet, D. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 2013, 62(2), 359-402. Google Scholar
[7]	P. Germain, N. Masmoudi and B. Pausader, Nonneutral global solutions for the electron Euler-Poisson system in three dimensions, SIAM J. Math. Anal., 2013, 45(1), 267-278. Google Scholar
[8]	Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R³⁺¹, Comm. Math. Phys., 1998, 195(2), 249-265. Google Scholar
[9]	Y. Guo and B. Pausader, Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys., 2011, 303(1), 89-125. Google Scholar
[10]	L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 2003, 192(1), 111-133. Google Scholar
[11]	A. D. Ionescu and B. Pausader, The Euler-Poisson system in 2D:global stability of the constant equilibrium solution, Int. Math. Res. Not., 2013, 2013(4), 761-826. Google Scholar
[12]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 1975, 58(3), 181-205. Google Scholar
[13]	P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM Regional Conf. Lecture, Philadelphia, 1973, 11. Google Scholar
[14]	C. M. Liu and Y. J. Peng, Convergence of non-isentropic Euler-Poisson systems for all time, J. Math. Pures Appl., in press. DOI:10.1016/j.matpur.2017.07.017. Google Scholar
[15]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. Google Scholar
[16]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. Google Scholar
[17]	T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluids Dynamics, Publications Mathématiques d'Orsay, 78-02, Université Paris-Sud, Orsay, 1978. Google Scholar
[18]	Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pures Appl., 2015, 103(1), 39-67. Google Scholar
[19]	Y. J. Peng, Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters, SIAM J. Appl. Math., 2015, 27, 1355-1376. Google Scholar
[20]	Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 2005, 41(2), 141-160. Google Scholar
[21]	J. Simon, Compact sets in the space Lp(0; T; B), Ann. Mat. Pura Appl., 1987, 146(1), 65-96. Google Scholar
[22]	S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Part. Diff. Equations, 2004, 29(3-4), 419-456. Google Scholar
[23]	J. Xu and T. Zhang, Zero-electron-mass limit of Euler-Poisson equations, Discrete Contin. Dyn. Syst., 2013, 33(10), 4743-4768. Google Scholar