# American Institute of Mathematical Sciences

January  2013, 12(1): 503-518. doi: 10.3934/cpaa.2013.12.503

## Incompressible type euler as scaling limit of compressible Euler-Maxwell equations

 1 College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China, China 2 College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District, Beijing 100022

Received  December 2010 Revised  April 2011 Published  September 2012

In this paper, we study the convergence of time-dependent Euler-Maxwell equations to incompressible type Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the local existence of smooth solutions to the limit equations is proved by an iterative scheme. Moreover, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and the symmetric hyperbolic property of the systems.
Citation: Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503
##### References:
 [1] F. Chen, "Introduction to Plasma Physics and Controlled Fusion,", Vol. 1, (1984). Google Scholar [2] Andreas Dinklage et al, "Plasma Physics, Lect. Notes Phys,", 670, (2005). Google Scholar [3] V. E. Golant, A. P. Zhilinski and I. E. Sakharov, "Fundamentals of Plasma Physics,", John Wiley and Sons, (1980). doi: 01.44227. Google Scholar [4] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar [5] S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 31 (2006), 571. Google Scholar [6] L. Hsiao, P. Markowichand S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors,, J. Differential Equations, 192 (2003), 111. doi: 10.1016/S0022-0396(03)00063-9. Google Scholar [7] M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system,, J. Nonlinear Sci., 11 (2001), 193. doi: 10.1007/s00332-001-0004-9. Google Scholar [8] Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations,, Asymptot. Anal., 41 (2005), 141. Google Scholar [9] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics,, Comm. Partial Differential Equations, 25 (2000), 1099. doi: 10.1080/03605300008821542. Google Scholar [10] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity,, Comm. Partial Differential Equations, 29 (2004), 419. doi: 10.1081/PDE-120030403. Google Scholar [11] Pierre Crispel and Pierre Degond, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit,, J. Comput. Phys., 223 (2006), 208. doi: 10.1016/j.jcp.2006.09.004. Google Scholar [12] G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations,, Trans. Theory Stat. Phys., 29 (2000), 311. doi: 10.1080/00411450008205877. Google Scholar [13] J. W. Jerome, The Cauchy problem for compressible hydrodynamic-Maxwell systems: A local theory for smooth solutions,, Differential and Integral Equations, 16 (2003), 1345. Google Scholar [14] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations,, Chin. Ann. Math., 28(B) (2007), 583. doi: 10.1007/s11401-005-0556-3. Google Scholar [15] Y. J. Peng and S. Wang, Rigorous derivayion of incompressible e-MHD equations from compressible Euler-Maxwell equations,, SIAM J. MATH. ANAL., 40 (2008), 540. doi: 10.1137/070686056. Google Scholar [16] J. W. Yang and S. Wang, Convergence of the nonisentropic Euler-Maxwell equations to compressible Euler-Poisson equations,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3267863. Google Scholar [17] J. W. Yang and S. Wang, The non-relativistic limit of Euler-Maxwell equations for two-fluid plasma,, Nonlinear Anal., 72 (2010), 1829. doi: 10.1016/j.na.2009.09.024. Google Scholar [18] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Eule equations,, Comm. Partial Differential Equations, 33 (2008), 349. doi: 10.1080/03605300701318989. Google Scholar [19] W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms,, J. Differential Equations, 155 (1999), 89. doi: 10.1006/jdeq.1998.3584. Google Scholar [20] Y. J. Peng, Y.-G. Wang and W.-A. Yong, Quasi-neutral limit of the non-isentropic the Euler-Poisson system,, Procedings of the Royal Society of Edinburgh, 136A (2006), 1013. doi: 10.1017/S0308210500004856. Google Scholar [21] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, Spectral Theory and Differential Equations (Proc. Sympos., (1974), 25. doi: 10.1007/BFb0067080. Google Scholar [22] A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,", Applied Mathematical Sciences, (1984). Google Scholar [23] S. Klainerman and A. Majda, Compressible and incompressible fluids,, Comm. Pure Appl. Math., XXXV (1982), 629. doi: 10.1002/cpa.3160350503. Google Scholar [24] S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar

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##### References:
 [1] F. Chen, "Introduction to Plasma Physics and Controlled Fusion,", Vol. 1, (1984). Google Scholar [2] Andreas Dinklage et al, "Plasma Physics, Lect. Notes Phys,", 670, (2005). Google Scholar [3] V. E. Golant, A. P. Zhilinski and I. E. Sakharov, "Fundamentals of Plasma Physics,", John Wiley and Sons, (1980). doi: 01.44227. Google Scholar [4] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar [5] S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 31 (2006), 571. Google Scholar [6] L. Hsiao, P. Markowichand S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors,, J. Differential Equations, 192 (2003), 111. doi: 10.1016/S0022-0396(03)00063-9. Google Scholar [7] M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system,, J. Nonlinear Sci., 11 (2001), 193. doi: 10.1007/s00332-001-0004-9. Google Scholar [8] Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations,, Asymptot. Anal., 41 (2005), 141. Google Scholar [9] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics,, Comm. Partial Differential Equations, 25 (2000), 1099. doi: 10.1080/03605300008821542. Google Scholar [10] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity,, Comm. Partial Differential Equations, 29 (2004), 419. doi: 10.1081/PDE-120030403. Google Scholar [11] Pierre Crispel and Pierre Degond, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit,, J. Comput. Phys., 223 (2006), 208. doi: 10.1016/j.jcp.2006.09.004. Google Scholar [12] G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations,, Trans. Theory Stat. Phys., 29 (2000), 311. doi: 10.1080/00411450008205877. Google Scholar [13] J. W. Jerome, The Cauchy problem for compressible hydrodynamic-Maxwell systems: A local theory for smooth solutions,, Differential and Integral Equations, 16 (2003), 1345. Google Scholar [14] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations,, Chin. Ann. Math., 28(B) (2007), 583. doi: 10.1007/s11401-005-0556-3. Google Scholar [15] Y. J. Peng and S. Wang, Rigorous derivayion of incompressible e-MHD equations from compressible Euler-Maxwell equations,, SIAM J. MATH. ANAL., 40 (2008), 540. doi: 10.1137/070686056. Google Scholar [16] J. W. Yang and S. Wang, Convergence of the nonisentropic Euler-Maxwell equations to compressible Euler-Poisson equations,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3267863. Google Scholar [17] J. W. Yang and S. Wang, The non-relativistic limit of Euler-Maxwell equations for two-fluid plasma,, Nonlinear Anal., 72 (2010), 1829. doi: 10.1016/j.na.2009.09.024. Google Scholar [18] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Eule equations,, Comm. Partial Differential Equations, 33 (2008), 349. doi: 10.1080/03605300701318989. Google Scholar [19] W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms,, J. Differential Equations, 155 (1999), 89. doi: 10.1006/jdeq.1998.3584. Google Scholar [20] Y. J. Peng, Y.-G. Wang and W.-A. Yong, Quasi-neutral limit of the non-isentropic the Euler-Poisson system,, Procedings of the Royal Society of Edinburgh, 136A (2006), 1013. doi: 10.1017/S0308210500004856. Google Scholar [21] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, Spectral Theory and Differential Equations (Proc. Sympos., (1974), 25. doi: 10.1007/BFb0067080. Google Scholar [22] A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,", Applied Mathematical Sciences, (1984). Google Scholar [23] S. Klainerman and A. Majda, Compressible and incompressible fluids,, Comm. Pure Appl. Math., XXXV (1982), 629. doi: 10.1002/cpa.3160350503. Google Scholar [24] S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar
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