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On a nonlocal Zakharov equation. (English) Zbl 0830.35123
This article deals with some equations introduced by Zakharov to describe Langmuir plasma turbulence: \[ i \dot \varphi + \Delta \varphi = - B (n \varphi), \quad \lambda^{- 2} \ddot n - \Delta n = \Delta |\varphi |^2, \] where \(B = \nabla \Delta^{-1} \nabla\). The initial value problem \(\varphi (x,0) = \varphi_0 (x),\) \(n (x,0) = n_0 (x),\) \(n_t (x,0) = n_1(x)\) is investigated. The existence and the uniqueness for the Cauchy problem is proved. Moreover, the case when \(\lambda\) tends to \(\infty\) is studied.

35Q55 NLS equations (nonlinear Schrödinger equations)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
Full Text: DOI
[1] Zakharov, V.E., Collapse and self-focusing of Langmuir waves, ()
[2] Zakharov, V.E., Collapse of Langmuir waves, Soviet phys. JETP, 35, 5, 908-914, (1972)
[3] Colin, T., Sur une équation de Schrödinger non linéaire et non locale intervenant en physique des plasmas, C. r. acad. sci. Paris, serie I, 314, 449-453, (1992) · Zbl 0755.35127
[4] Added, H.; Added, S., Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. funct. analysis, 79, 183-210, (1988) · Zbl 0655.76044
[5] Added, H.; Added, S., Existence globale de solutions fortes pour LES équations de la turbulence de Langmuir en dimension 2, C. r. acad. sci. Paris, serie I, 299, 551-554, (1984) · Zbl 0575.35080
[6] Sulem, C.; Sulem, P.L., Quelques résultats de régularité pour LES équations de la turbulence de Langmuir, C. r. acad. sci. Paris, 289, 173-176, (1979) · Zbl 0431.35077
[7] Schochet, S.H.; Weinstein, M.I., The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Communs math. phys., 106, 569-580, (1986) · Zbl 0639.76054
[8] Ozawa, T.; Tsutsumi, Y., Existence and smooting effect of solutions for the Zakharov equations, (1990)
[9] Colin, T., On a nonlocal, nonlinear Schrödinger equation occuring in plasma physics, (1992), Publication CMLA, ENS Cachan 9208 Amsterdam
[10] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations I: the Cauchy problem, J. funct. analysis, 32, 1-32, (1979) · Zbl 0396.35028
[11] Ginibre, J.; Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. math. pures appl., 64, 363-401, (1985) · Zbl 0535.35069
[12] Strichartz, R.S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., 44, 705-714, (1977) · Zbl 0372.35001
[13] Yajima, K., Existence of solutions for Schrödinger evolution equations, Communs math. phys., 110, 415-426, (1987) · Zbl 0638.35036
[14] KATO T., Nonlinear Schrödinger equations, Lecture Notes for Physics, Vol. 345, Springer, Berlin
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