The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response
Numerical study of a family of dissipative KdV equations
January 2013, 12(1): 503-518. doi: 10.3934/cpaa.2013.12.503
Incompressible type euler as scaling limit of compressible Euler-Maxwell equations
Jianwei Yang 1, , Ruxu Lian 1, and Shu Wang 2,
College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China, China
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.
Keywords: Euler-Maxwell equations, non-relativistic limit., incompressible type Euler equations, quasineutral limit.
Mathematics Subject Classification: Primary: 35B40, 35C20, 35L60; Secondary: 35Q3.
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
F. Chen, "Introduction to Plasma Physics and Controlled Fusion,", Vol. 1, (1984). Google Scholar
Andreas Dinklage et al, "Plasma Physics, Lect. Notes Phys,", 670, (2005). Google Scholar
V. E. Golant, A. P. Zhilinski and I. E. Sakharov, "Fundamentals of Plasma Physics,", John Wiley and Sons, (1980). doi: 01.44227. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,", Applied Mathematical Sciences, (1984). Google Scholar
S. Klainerman and A. Majda, Compressible and incompressible fluids,, Comm. Pure Appl. Math., XXXV (1982), 629. doi: 10.1002/cpa.3160350503. Google Scholar
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
Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure & Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963
Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure & Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365
Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415
Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086
Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108
Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1
Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743
Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549
Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275
Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127
Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013
Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335
Andrea Natale, François-Xavier Vialard. Embedding Camassa-Holm equations in incompressible Euler. Journal of Geometric Mechanics, 2019, 11 (2) : 205-223. doi: 10.3934/jgm.2019011
La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981
Yachun Li, Qiufang Shi. Global existence of the entropy solutions to the isentropic relativistic Euler equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 763-778. doi: 10.3934/cpaa.2005.4.763
Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221
Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
María J. Martín, Jukka Tuomela. 2D incompressible Euler equations: New explicit solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4547-4563. doi: 10.3934/dcds.2019187
Jianwei Yang Ruxu Lian Shu Wang