Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations

Peng, Yue-Jun

doi:10.1016/j.anihpc.2012.04.002

Peng, Yue-Jun

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 737-759.

Résumé

We consider Cauchy problems and periodic problems for two-fluid compressible Euler–Maxwell equations arising in the modeling of magnetized plasmas. These equations are symmetrizable hyperbolic in the sense of Friedrichs but donʼt satisfy the so-called Kawashima stability condition. For both problems, we prove the global existence and long-time behavior of smooth solutions near a given constant equilibrium state. As a byproduct, we obtain similar results for two-fluid compressible Euler–Poisson equations.

MR Zbl

DOI : 10.1016/j.anihpc.2012.04.002

Classification : 35L45, 35L60, 35Q60
Mots-clés : Two-fluid flows, Euler–Maxwell equations, Partially dissipative hyperbolic systems, Global smooth solutions, Long-time behavior, Energy estimates

@article{AIHPC_2012__29_5_737_0,
     author = {Peng, Yue-Jun},
     title = {Global existence and long-time behavior of smooth solutions of two-fluid {Euler{\textendash}Maxwell} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {737--759},
     publisher = {Elsevier},
     volume = {29},
     number = {5},
     year = {2012},
     doi = {10.1016/j.anihpc.2012.04.002},
     mrnumber = {2971029},
     zbl = {1251.35159},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.002/}
}

TY  - JOUR
AU  - Peng, Yue-Jun
TI  - Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 737
EP  - 759
VL  - 29
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.002/
DO  - 10.1016/j.anihpc.2012.04.002
LA  - en
ID  - AIHPC_2012__29_5_737_0
ER  -

%0 Journal Article
%A Peng, Yue-Jun
%T Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 737-759
%V 29
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.002/
%R 10.1016/j.anihpc.2012.04.002
%G en
%F AIHPC_2012__29_5_737_0

Peng, Yue-Jun. Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 737-759. doi : 10.1016/j.anihpc.2012.04.002. http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.002/

Bibliographie
Cité par

[1] G. Alì, Global existence of smooth solutions of the N-dimensional Euler–Poisson model, SIAM J. Appl. Math. 35 (2003), 389-422 | MR | Zbl

[2] G. Alì, A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations 190 (2003), 663-685 | MR | Zbl

[3] K. Beauchard, E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal. 199 (2011), 177-227 | MR | Zbl

[4] C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice, C. Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci. 14 (2004), 393-415 | MR | Zbl

[5] S. Bianchini, B. Hanouzet, R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), 1559-1622 | MR | Zbl

[6] G. Carbou, B. Hanouzet, R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations 246 (2009), 291-319 | MR | Zbl

[7] F. Chen, Introduction to Plasma Physics and Controlled Fusion, vol. 1, Plenum Press, New York (1984)

[8] G.Q. Chen, J.W. Jerome, D.H. Wang, Compressible Euler–Maxwell equations, Transport Theory Statist. Phys. 29 (2000), 311-331 | MR | Zbl

[9] P. Degond, F. Deluzet, D. Savelief, Numerical approximation of the Euler–Maxwell model in the quasineutral limit, J. Comput. Phys. 231 (2012), 1917-1946 | MR | Zbl

[10] R.J. Duan, Global smooth flows for the compressible Euler–Maxwell system: the relaxation case, J. Hyperbolic Differ. Equ. 8 (2011), 375-413 | MR | Zbl

[11] L.C. Evans, Partial Differential Equations, Grad. Stud. Math. vol. 19, American Mathematical Society, Providence, RI (1998) | MR

[12] K.O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345-392 | MR | Zbl

[13] P. Germain, N. Masmoudi, Global existence for the Euler–Maxwell system, arXiv:1107.1595 (2011) | MR

[14] Y. Guo, Smooth irrotational flows in the large to the Euler–Poisson system in $ℝ^{3 + 1}$ , Comm. Math. Phys. 195 (1998), 249-265 | MR | Zbl

[15] Y. Guo, W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal. 179 (2006), 1-30 | MR | Zbl

[16] B. Hanouzet, R. Natalini, Global existence of smooth solutions for partial dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal. 169 (2003), 89-117 | MR | Zbl

[17] L. Hsiao, P.A. Markowich, S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations 192 (2003), 111-133 | MR | Zbl

[18] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), 181-205 | MR | Zbl

[19] S. Klainerman, 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 | MR | Zbl

[20] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York (1984) | MR | Zbl

[21] P. Marcati, 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 | MR | Zbl

[22] P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York (1990) | MR

[23] T. Nishida, Nonlinear hyperbolic equations and related topics in fluids dynamics, in: Publications Mathématiques dʼOrsay, No. 78-02, Université Paris-Sud, Orsay, 1978. | MR

[24] Y.J. Peng, S. Wang, Convergence of compressible Euler–Maxwell equations to compressible Euler–Poisson equations, Chin. Ann. Math. Ser. B 28 (2007), 583-602 | MR | Zbl

[25] Y.J. Peng, S. Wang, Convergence of compressible Euler–Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations 33 (2008), 349-376 | MR | Zbl

[26] Y.J. Peng, S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler–Maxwell equations, SIAM J. Math. Anal. 40 (2008), 540-565 | MR | Zbl

[27] Y.J. Peng, S. Wang, Asymptotic expansions in two-fluid compressible Euler–Maxwell equations with small parameters, Discrete Contin. Dyn. Syst. 23 (2009), 415-433 | MR | Zbl

[28] Y.J. Peng, S. Wang, G.L. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler–Maxwell equations, SIAM J. Math. Anal. 43 (2011), 944-970 | MR | Zbl

[29] F. Poupaud, M. Rascle, J.P. Vila, Global solutions to the isothermal Euler–Poisson system with arbitrarily large data, J. Differential Equations 123 (1995), 93-121 | MR | Zbl

[30] H. Rishbeth, O.K. Garriott, Introduction to Ionospheric Physics, Academic Press (1969)

[31] Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), 249-275 | MR | Zbl

[32] B. Texier, WKB asymptotics for the Euler–Maxwell equations, Asymptot. Anal. 42 (2005), 211-250 | MR | Zbl

[33] J. Xu, Global classical solutions to the compressible Euler–Maxwell equations, SIAM J. Math. Anal. 43 (2011), 2688-2718 | MR | Zbl

[34] W.A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), 247-266 | MR | Zbl

[35] Y.N. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (1999), 225-279 | MR | Zbl

[36] B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys. 157 (1993), 1-22 | MR | Zbl

Cité par Sources :