# On the Cauchy Problem for the Zakharov System

@article{Ginibre1997OnTC,
title={On the Cauchy Problem for the Zakharov System},
author={Jean Ginibre and Yoshio Tsutsumi and Giorgio Velo},
journal={Journal of Functional Analysis},
year={1997},
volume={151},
pages={384-436}
}
• Published 15 December 1997
• Mathematics
• Journal of Functional Analysis
Abstract We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data ( u (0), n (0), ∂ t n (0))∈ H k ⊕ H lscr; ⊕ H l−1 , in arbitrary space dimension ν . We define a natural notion of criticality according to which the critical values of ( k , l) are ( ν /2−3/2, ν /2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of ( k , l). The results…
432 Citations
On the 2 d Zakharov system with L 2 Schrödinger data
• Mathematics
• 2009
We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L ×H−1/2 ×H−3/2. This is the space of optimal regularity in the sense that the
ON THE INITIAL VALUE PROBLEM FOR THE ZAKHAROV SYSTEM IN 2
• Mathematics
• 2009
We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L × H−1/2 × H−3/2. This is the space of optimal regularity in the sense that the
Smoothing and global attractors for the Zakharov system on the torus
• Mathematics
• 2012
In this paper we consider the Zakharov system with periodic boundary con- ditions in dimension one. In the rst part of the paper, it is shown that for xed initial data in a Sobolev space, the
Local well-posedness of the Cauchy problem for the degenerate Zakharov system
X iv :2 10 3. 05 34 0v 1 [ m at h. A P] 9 M ar 2 02 1 LOCAL WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE DEGENERATE ZAKHAROV SYSTEM ISAO KATO Abstract. The aim of this paper is to investigate
A remark on the local well-posedness for a coupled system of mKdV type equations in H^s × H^k
• X. Carvajal
• Mathematics
Differential Equations & Applications
• 2020
. We consider the initial value problem associated to a system consisting modi ﬁ ed Korteweg-de Vries type equations and using only bilinear estimates of the type J 1 J L 2 x L 2 t , where J is the
On the unboundedness of higher regularity Sobolev norms of solutions for the critical Schrödinger–Debye system with vanishing relaxation delay
• Mathematics
• 2015
We consider the Schrödinger–Debye system in Rn, for n  =  3,4. Developing on previously known local well-posedness results, we start by establishing global well-posedness in H1(R3)×L2(R3) for a broad
Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and more dimensions
• Mathematics
• 2016
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension d ≥ 5 with initial datum (u, ∂tu, n, ∂tn)|t=0 ∈ H (R) × H(R)× Ḣ(R)× Ḣ(R). The critical value of s is sc = d/2− 2.

## References

SHOWING 1-10 OF 22 REFERENCES
A bilinear estimate with applications to the KdV equation
• Mathematics
• 1996
u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence
On the Zakharov and Zakharov-Schulman Systems
• Mathematics
• 1995
Abstract We consider the initial value problem for the Zakharov system [formula] which models the long wave Langmuir turbulence in a plasma. Using the standard iteration scheme in the original system
On the well-posedness of Benney's interaction equation of short and long waves
• Mathematics
• 1996
where S and L denote the short wave envelope and the long wave profile, respectively; cg, cl, ↵, and are real constants. When = 0, this equation is a decoupled system and the first equation in (1.2)
The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence
• Mathematics
• 1986
AbstractWe consider the initial value problem for the Zakharov equations \begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x)
Existence and Smoothing Effect of Solutions for the Zakharov Equations
• Physics, Mathematics
• 1992
where E is a function from R* xR% to C9 n is a function from R^ xR% to R and 1<^JV<^3. (1. !)-(!. 3) describe the long wave Langmuir turbulence in a plasma (see [20]). E(t, x) denotes the slowly
The initial value problem for a class of nonlinear dispersive equations
• Mathematics
• 1990
We consider the initial value problem for a (generalized) equation which arises in the study of propagation of unidirectional nonlinear, dispersive waves. The aim is to study the local and global
Generalized Strichartz Inequalities for the Wave Equation
• Mathematics
• 1995
Abstract We make a synthetic exposition of the generalized Strichartz inequalities for the wave equation obtained in [6] together with the limiting cases recently obtained in [13] with as simple
Nonlinear scalar field equations, I existence of a ground state
• Mathematics
• 1983
1. The Main Result; Examples . . . . . . . . . . . . . . . . . . . . . . . 316 2. Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 319 3. The Constrained Minimization Method .
Scattering theory in the energy space for a class of nonlinear Schrödinger equations
• Mathematics
• 1985
On etudie le comportement asymptotique par rapport au temps des solutions et la theorie de la diffusion pour l'equation de Schrodinger non lineaire dans R n , n≥2. On demontre l'existence des