Abstract
This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MÉTIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.
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References
Added H., Added S. (1984) Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2. C. R. Math. Acad. Sci. 299, 551–554
Bony J.M. (1981) Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non-linéaires. Ann. Sci. École Norm. Sup. (4) 14, 209–246
Bourgain J., Colliander J. (1996) On wellposedness of the Zakharov system. Int. Math. Res. Not 1, 515–546
Cheverry C. (2004) Propagation of oscillations in real vanishing viscosity limit. Comm. Math. Phys. 247, 655–695
Cheverry C., Guès O., Métivier G. (2003) Oscillations fortes sur un champ linéairement dégénéré. Ann. Sci. École Norm. Sup. (4) 36, 691–745
Colin M., Colin T. (2004) On a quasilinear Zakharov system describing laser-plasma interactions. Differential Integral Equations 17, 297–330
Colin T., Ebrard G., Gallice G., Texier B. (2004) Justification of the Zakharov model from Klein-Gordon-waves systems. Comm. Partial Differential Equations 29:1365–1401
Colin, T., Métivier, G.: Instabilities in Zakharov equations for laser propagation in a plasma. Preprint
Grenier E. (1997) Pseudo-differential estimates of singular perturbations. Comm. Pure Appl. Math. 50, 821–865
Delcroix, J.-L., Bers, A.: Physique des Plasmas. Vols. 1 and 2, InterEditions-Editions du CNRS, 1994
Ginibre J., Tsutsumi Y., Velo G. (1997) On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151, 384–436
Joly J.-L., Métivier G., Rauch J. (1998) Diffractive nonlinear geometric optics with rectification. Indiana Univ. Math. J. 47:1167–1241
Joly J.-L., Métivier G., Rauch J. (2000) Transparent nonlinear geometric optics and -Bloch equations. J. Differential Equations 166, 175–250
Lannes D. (2006) Sharp estimates for pseudo-differential operators with limited regularity and commutators. J. Funct. Anal. 232, 495–539
Linares F., Ponce G., Saut J.-C. (2005) On a degenerate Zakharov system. Bull. Braz. Math. Soc. 36, 1–23
Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175, (2005)
Musher S., Rubenchik A., Zakharov V. (1985) Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Rep. 129, 285–366
Ozawa T., Tsutsumi Y. (1992) Existence and smoothing effect of solution for the Zakharov equation. Publ. Res. Inst. Math. Sci. 28, 329–361
Serre, D.: Oscillations nonlinéaires de haute fréquence, dim = 1. Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. XII (Paris, 1991–1993), 190–210. Harlow, Longman Sci. Tech., 1994
Sone, Y.: Kinetic Theory and Fluid Dynamics. Modeling and Simulation in Science, Engineering and Technology, Birhaüser, 2002
Sulem C., Sulem P.-L. (1979) Quelques résultats de régularité pour les équations de la turbulence de Langmuir. C. R. Math. Acad. Sci. Paris. 289:A173–A176
Sulem, C., Sulem, P.-L.: The nonlinear Schrödinger equation: self-focusing and wave collapse. Applied Math. Sciences 139, Springer Verlag, 1999
Schochet S., Weinstein M. (1986) The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys. 106, 569–580
Takata, S., Aoki, K.: The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: asymptotic analysis of the Boltzmann equation. The Sixteenth International Conference on Transport Theory, Atlanta, 1999, Transport Theory Statist. Phys. 30, 205–237 (2001)
Taylor, M.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics Vol. 100, Birkhaüser Boston, 1991
Texier B. (2004) The short wave limit for nonlinear, symmetric hyperbolic systems. Adv. Differential Equations 9, 1–52
Texier B. (2005) WKB asymptotics for the Euler-Maxwell equations. Asymptot. Anal. 42, 211–250
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Communicated by A. Bressan
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Texier, B. Derivation of the Zakharov Equations. Arch Rational Mech Anal 184, 121–183 (2007). https://doi.org/10.1007/s00205-006-0034-4
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DOI: https://doi.org/10.1007/s00205-006-0034-4