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Global solutions of the Euler-Maxwell two-fluid system in 3D. (English) Zbl 1345.35075
The results obtained in this article are related to the Euler-Maxwell system, that represents the two-fluid model describing plasma dynamics. This model is a system of nonlinear hyperbolic conservation laws with no dissipation and no relaxation effects. The authors develop a method that can be used for complicated physical coupled systems, at least in dimension 3. The global dynamics of the solutions to the Euler-Maxwell system is analyzed by means of dispersive analysis combined with energy estimates, relying on the Fourier transform. It is proved that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions for the 3D Euler-Maxwell system. The proposed method can be extended to other quasilinear problems in 3D and the constructed solutions represent the first smooth, nontrivial global solutions.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 35Q60 PDEs in connection with optics and electromagnetic theory 35B65 Smoothness and regularity of solutions to PDEs 35L65 Hyperbolic conservation laws
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##### References:
 [1] Y. Guo, A. D. Ionescu, and B. Pausader, ”Global solutions of certain plasma fluid models in three-dimension,” J. Math. Phys., vol. 55, p. 12, 2014. · Zbl 1308.76340 [2] S. Alinhac, ”Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux,” Invent. Math., vol. 111, iss. 3, pp. 627-670, 1993. · Zbl 0798.35129 [3] J. A. Bittencourt, Fundamentals of Plasma Physics, 3rd ed., New York: Springer-Verlag, 2004. · Zbl 1084.76001 [4] S. Cordier and E. Grenier, ”Quasineutral limit of an Euler-Poisson system arising from plasma physics,” Comm. Partial Differential Equations, vol. 25, iss. 5-6, pp. 1099-1113, 2000. · Zbl 0978.82086 [5] G. Chen, J. W. Jerome, and D. Wang, ”Compressible Euler-Maxwell equations,” in Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, 2000, pp. 311-331. · Zbl 1019.82023 [6] D. Christodoulou, ”Global solutions of nonlinear hyperbolic equations for small initial data,” Comm. Pure Appl. Math., vol. 39, iss. 2, pp. 267-282, 1986. · Zbl 0612.35090 [7] D. Christodoulou, ”The formation of shocks in 3-dimensional fluids,” in Recent Advances in Nonlinear Partial Differential Equations and Applications, Providence, RI: Amer. Math. Soc., 2007, vol. 65, pp. 17-30. · Zbl 1138.35060 [8] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton, NJ: Princeton Univ. Press, 1993, vol. 41. · Zbl 0827.53055 [9] J. -L. Delcroix and A. Bers, Physique des plasmas, Paris: InterEditions/CNRS Editions, 1994. [10] J. Delort and D. Fang, ”Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data,” Comm. Partial Differential Equations, vol. 25, iss. 11-12, pp. 2119-2169, 2000. · Zbl 0979.35101 [11] J. Delort, D. Fang, and R. Xue, ”Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions,” J. Funct. Anal., vol. 211, iss. 2, pp. 288-323, 2004. · Zbl 1061.35089 [12] P. Degond, F. Deluzet, and D. Savelief, ”Numerical approximation of the Euler-Maxwell model in the quasineutral limit,” J. Comput. Phys., vol. 231, iss. 4, pp. 1917-1946, 2012. · Zbl 1244.82009 [13] D. Gérard-Varet, D. Han-Kwan, and F. Rousset, ”Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries,” Indiana Univ. Math. J., vol. 62, iss. 2, pp. 359-402, 2013. · Zbl 1417.35119 [14] P. Germain, ”Global existence for coupled Klein-Gordon equations with different speeds,” Ann. Inst. Fourier $$($$Grenoble$$)$$, vol. 61, iss. 6, pp. 2463-2506 (2012), 2011. · Zbl 1255.35162 [15] P. Germain and N. Masmoudi, ”Global existence for the Euler-Maxwell system,” Ann. Sci. Éc. Norm. Supér., vol. 47, iss. 3, pp. 469-503, 2014. · Zbl 1311.35195 [16] P. Germain, N. Masmoudi, and B. Pausader, ”Nonneutral global solutions for the electron Euler-Poisson system in three dimensions,” SIAM J. Math. Anal., vol. 45, iss. 1, pp. 267-278, 2013. · Zbl 1282.35285 [17] P. Germain, N. Masmoudi, and J. Shatah, ”Global solutions for 3D quadratic Schrödinger equations,” Int. Math. Res. Not., vol. 2009, iss. 3, pp. 414-432, 2009. · Zbl 1156.35087 [18] P. Germain, N. Masmoudi, and J. Shatah, ”Global solutions for the gravity water waves equation in dimension 3,” Ann. of Math., vol. 175, iss. 2, pp. 691-754, 2012. · Zbl 1241.35003 [19] P. Germain, N. Masmoudi, and J. Shatah, ”Global existence for capillary water waves,” Comm. Pure Appl. Math., vol. 68, iss. 4, pp. 625-687, 2015. · Zbl 1314.35100 [20] Y. Guo, ”Smooth irrotational flows in the large to the Euler-Poisson system in $$\mathbb R^{3+1}$$,” Comm. Math. Phys., vol. 195, iss. 2, pp. 249-265, 1998. · Zbl 0929.35112 [21] Y. Guo and B. Pausader, ”Global smooth ion dynamics in the Euler-Poisson system,” Comm. Math. Phys., vol. 303, iss. 1, pp. 89-125, 2011. · Zbl 1220.35129 [22] Y. Guo and X. Pu, ”KdV limit of the Euler-Poisson system,” Arch. Ration. Mech. Anal., vol. 211, iss. 2, pp. 673-710, 2014. · Zbl 1283.35110 [23] Y. Guo and S. A. Tahvildar-Zadeh, ”Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics,” in Nonlinear Partial Differential Equations, Providence, RI: Amer. Math. Soc., 1999, vol. 238, pp. 151-161. · Zbl 0973.76100 [24] S. Gustafson, K. Nakanishi, and T. Tsai, ”Scattering theory for the Gross-Pitaevskii equation in three dimensions,” Commun. Contemp. Math., vol. 11, iss. 4, pp. 657-707, 2009. · Zbl 1180.35481 [25] A. D. Ionescu and B. Pausader, ”The Euler-Poisson system in 2D: global stability of the constant equilibrium solution,” Int. Math. Res. Not., vol. 2013, iss. 4, pp. 761-826, 2013. · Zbl 1320.35270 [26] A. D. Ionescu and B. Pausader, ”Global solutions of quasilinear systems of Klein-Gordon equations in 3D,” J. Eur. Math. Soc. $$($$JEMS$$)$$, vol. 16, iss. 11, pp. 2355-2431, 2014. · Zbl 1316.35180 [27] A. D. Ionescu and F. Pusateri, ”Global solutions for the gravity water waves system in 2d,” Invent. Math., vol. 199, iss. 3, pp. 653-804, 2015. · Zbl 1325.35151 [28] J. Jang, ”The two-dimensional Euler-Poisson system with spherical symmetry,” J. Math. Phys., vol. 53, iss. 2, p. 023701, 2012. · Zbl 1274.76383 [29] J. Jang, D. Li, and X. Zhang, ”Smooth global solutions for the two-dimensional Euler Poisson system,” Forum Math., vol. 26, iss. 3, pp. 645-701, 2014. · Zbl 1298.35148 [30] F. John, ”Blow-up of solutions of nonlinear wave equations in three space dimensions,” Manuscripta Math., vol. 28, iss. 1-3, pp. 235-268, 1979. · Zbl 0406.35042 [31] F. John and S. Klainerman, ”Almost global existence to nonlinear wave equations in three space dimensions,” Comm. Pure Appl. Math., vol. 37, iss. 4, pp. 443-455, 1984. · Zbl 0599.35104 [32] T. Kato, ”The Cauchy problem for quasi-linear symmetric hyperbolic systems,” Arch. Rational Mech. Anal., vol. 58, iss. 3, pp. 181-205, 1975. · Zbl 0343.35056 [33] S. Klainerman, ”Long time behaviour of solutions to nonlinear wave equations,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Warsaw, 1984, pp. 1209-1215. · Zbl 0581.35052 [34] S. Klainerman, ”Uniform decay estimates and the Lorentz invariance of the classical wave equation,” Comm. Pure Appl. Math., vol. 38, iss. 3, pp. 321-332, 1985. · Zbl 0635.35059 [35] S. Klainerman, ”Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions,” Comm. Pure Appl. Math., vol. 38, iss. 5, pp. 631-641, 1985. · Zbl 0597.35100 [36] S. Klainerman, ”The null condition and global existence to nonlinear wave equations,” in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Providence, RI: Amer. Math. Soc., 1986, vol. 23, pp. 293-326. · Zbl 0599.35105 [37] D. Lannes, F. Linares, and J. Saut, ”The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation,” in Studies in Phase Space Analysis with Applications to PDEs, New York: Springer-Verlag, 2013, vol. 84, pp. 181-213. · Zbl 1273.35263 [38] D. Li and Y. Wu, ”The Cauchy problem for the two dimensional Euler-Poisson system,” J. Eur. Math. Soc. $$($$JEMS$$)$$, vol. 16, iss. 10, pp. 2211-2266, 2014. · Zbl 1308.35220 [39] H. Lindblad and I. Rodnianski, ”The weak null condition for Einstein’s equations,” C. R. Math. Acad. Sci. Paris, vol. 336, iss. 11, pp. 901-906, 2003. · Zbl 1045.35101 [40] H. Lindblad and I. Rodnianski, ”The global stability of Minkowski space-time in harmonic gauge,” Ann. of Math., vol. 171, iss. 3, pp. 1401-1477, 2010. · Zbl 1192.53066 [41] Y. Peng, ”Global existence and long-time behavior of smooth solutions of two-fluid Euler-Maxwell equations,” Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 29, iss. 5, pp. 737-759, 2012. · Zbl 1251.35159 [42] X. Pu, ”Dispersive limit of the Euler-Poisson system in higher dimensions,” SIAM J. Math. Anal., vol. 45, iss. 2, pp. 834-878, 2013. · Zbl 1291.35306 [43] J. Shatah, ”Normal forms and quadratic non-linear Klein-Gordon equations,” Comm. Pure Appl. Math., vol. 38, iss. 5, pp. 685-696, 1985. · Zbl 0597.35101 [44] T. C. Sideris, ”Formation of singularities in three-dimensional compressible fluids,” Comm. Math. Phys., vol. 101, iss. 4, pp. 475-485, 1985. · Zbl 0606.76088 [45] J. C. H. Simon, ”A wave operator for a non-linear Klein-Gordon equation,” Lett. Math. Phys., vol. 7, iss. 5, pp. 387-398, 1983. · Zbl 0539.35007 [46] B. Texier, ”Derivation of the Zakharov equations,” Arch. Ration. Mech. Anal., vol. 184, iss. 1, pp. 121-183, 2007. · Zbl 1370.35249
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