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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.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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##### References:
 [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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