Zero-electron-mass limit of Euler-Poisson equations

Jiang Xu; Ting Zhang

doi:10.3934/dcds.2013.33.4743

Article Contents

2013, Volume 33, Issue 10: 4743-4768. Doi: 10.3934/dcds.2013.33.4743

This issue Previous Article Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms Next Article Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom

Zero-electron-mass limit of Euler-Poisson equations

Jiang Xu^1, and
Ting Zhang^2,

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
2.
Department of Mathematics, Zhejiang University, Hangzhou 310027

Received: September 30, 2012

Revised: December 31, 2012

Published: September 2013

Abstract Related Papers Cited by

Abstract

We study the limit of vanishing ratio of the electron mass to the ion mass (zero-electron-mass limit) in the scaled Euler-Poisson equations. As the first step of this justification, we construct the uniform global classical solutions in critical Besov spaces with the aid of ``Shizuta-Kawashima" skew-symmetry. Then we establish frequency-localization estimates of Strichartz-type for the equation of acoustics according to the semigroup formulation. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for ill-preparedinitial data) in a refined way as the scaled electron-mass tends to zero. In comparison with the classical zero-mach-number limit in [7,23], we obtain different dispersive estimates due to the coupled electric field.

Keywords:

Mathematics Subject Classification: Primary: 35B25, 35L45; Secondary: 35M20.

Citation:

Related Papers

Cited by

References

[1]	T. Alazard, Low mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.doi: 10.1007/s00205-005-0393-2.
[2]	G. Alì, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422.doi: 10.1137/S0036141001393225.
[3]	G. Alì and L. Chen, The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data, Nonlinearity, 24 (2011), 2745-2761.doi: 10.1088/0951-7715/24/10/005.
[4]	G. Alì, L. Chen, A. Jüngel and Y. J. Peng, The zero-electron-mass limit in the hydrodynamic model for plasmas, Nonlinear Anal. TMA, 72 (2010), 4415-4427.doi: 10.1016/j.na.2010.02.016.
[5]	J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations," Oxford Lecture Series in Mathematics and its Applications, 32. The Clarendon Press, Oxford University Press, Oxford, 2006
[6]	L. Chen, X. Q. Chen and C. L. Zhang, Vanishing electron mass limit in the bipolar Euler-Poisson system, Nonlinear Anal. RWA, 12 (2011), 1002-1012.doi: 10.1016/j.nonrwa.2010.08.023.
[7]	R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4), 35 (2002), 27-75.doi: 10.1016/S0012-9593(01)01085-0.
[8]	P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3 (1990), 25-29.doi: 10.1016/0893-9659(90)90130-4.
[9]	D. Y. Fang, J. Xu and T. Zhang, Global exponential stability of classical solutions to the hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 17 (2007), 1507-1530.doi: 10.1142/S0218202507002364.
[10]	T. Goudon, A. Jüngel and Y. J. Peng, Zero-mass-electrons limits in hydrodynamic models for plasmas, Appl. Math. Lett., 12 (1999), 75-79.doi: 10.1016/S0893-9659(99)00038-5.
[11]	Y. Guo and W. Strauss, Stability of semiconductor states with Insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30.doi: 10.1007/s00205-005-0369-2.
[12]	D. W. Hewett, Low-frequency electro-magnetic (Darwin) applications in plasma simulation, Comput. Phys. Commun., 84 (1994), 243-277.
[13]	L. Hsiao, S. Jiang and P. Zhang, Global existence and exponential stablity of smooth solutions to a full hydrodynamic model to semiconductors, Monatshefte f$\ddotu$r Mathematik, 136 (2002), 269-285.doi: 10.1007/s00605-002-0485-0.
[14]	D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. Mat. Iberoamericana, 15 (1999), 1-36.doi: 10.4171/RMI/248.
[15]	A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-electron-mass limits in the drift-diffusion equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 17 (2000), 83-118.doi: 10.1016/S0294-1449(99)00101-8.
[16]	F. Kazeminezhad, J. M. Dawson, J. N. Leboeuf, R. Sydora and D. Holland, A vlasov particle ion zero mass electron model for plasma simulations, J. Comput. Phys., 102 (1992), 277-296.doi: 10.1016/0021-9991(92)90372-6.
[17]	S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.doi: 10.1002/cpa.3160340405.
[18]	S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.doi: 10.1002/cpa.3160350503.
[19]	S. Kawashima and W. A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal., 174 (2004), 345-364.doi: 10.1007/s00205-004-0330-9.
[20]	C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dyn. Syst.-A, 5 (1999), 449-455.doi: 10.3934/dcds.1999.5.449.
[21]	P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equations, Arch. Ration. Mech. Anal., 129 (1995), 129-145.doi: 10.1007/BF00379918.
[22]	P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.doi: 10.1007/978-3-7091-6961-2.
[23]	G. Métivier and S. Schochet, The incompressible limit of the Non-Isentropic euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.doi: 10.1007/PL00004241.
[24]	S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. PDE, 29 (2004), 419-456.doi: 10.1081/PDE-120030403.
[25]	J. Xu and W.-A. Yong, Relaxation-time limits of non-isentropic hydrodynamic models for semiconductors, J. Diff. Equs., 247 (2009), 1777-1795.doi: 10.1016/j.jde.2009.06.018.
[26]	J. Xu and W.-A. Yong, Zero-electron-mass limit of hydrodynamic models for plasmas, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 431-447.doi: 10.1017/S0308210510000119.
[27]	W.-A. Yong, Diffusive relaxation limit of multi-dimensional isentropic hydrodynamic models for semiconductor, SIAM J. Appl. Math., 64 (2004), 1737-1748.doi: 10.1137/S0036139903427404.
[28]	W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172 (2004), 247-266.doi: 10.1007/s00205-003-0304-3.