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October  2013, 33(10): 4743-4768. doi: 10.3934/dcds.2013.33.4743

## Zero-electron-mass limit of Euler-Poisson equations

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

Received  October 2012 Revised  January 2013 Published  April 2013

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.
Citation: Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] D. W. Hewett, Low-frequency electro-magnetic (Darwin) applications in plasma simulation, Comput. Phys. Commun., 84 (1994), 243-277. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [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.  Google Scholar [24] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. PDE, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] D. W. Hewett, Low-frequency electro-magnetic (Darwin) applications in plasma simulation, Comput. Phys. Commun., 84 (1994), 243-277. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [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.  Google Scholar [24] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. PDE, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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