Ill-posedness for the Zakharov system with generalized nonlinearity

@inproceedings{Biagioni2003IllposednessFT,
  title={Ill-posedness for the Zakharov system with generalized nonlinearity},
  author={Hebe de Azevedo Biagioni and Felipe Linares},
  year={2003}
}
We study the ill-posedness question for the one-dimensional Zakharov system and a generalization of it in one and higher dimensions. Our point of reference is the criticality criteria introduced by Ginibre, Tsutsumi and Velo (1997) to establish local well-posedness. 

Figures from this paper

Ill-Posedness for the Benney system
We discuss ill-posedness issues for the initial value problem associated to the Benney system. To prove our results we use the method introduced by Kenig, Ponce and Vega [10] to show ill-posedness
Norm inflation for the Zakharov system
. We prove norm inflation in new regions of Sobolev regularities for the scalar Zakharov system in the spatial domain R d for arbitrary d ∈ N . To this end, we apply abstract considerations of
A Note on $C^2$ Ill-Posedness Results for the Zakharov System in Arbitrary Dimension
This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces $H^k(\mathbb R^d)\!\times\!H^l(\mathbb R^d)\!\times\!H^{l-1}\!(\mathbb R^d)$. We recall the
Local well-posedness for the Zakharov system in dimension d ≤ 3
  • A. Sanwal
  • Mathematics
    Discrete & Continuous Dynamical Systems
  • 2021
<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id="M1">\begin{document}$ d\leqslant 3 $\end{document}</tex-math></inline-formula> is shown to be locally

References

SHOWING 1-10 OF 19 REFERENCES
On the ill-posedness of some canonical dispersive equations
We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the
Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations
Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in Hs(R), s < 1/2. This result implies that best result
Wellposedness for Zakharov Systems with Generalized Nonlinearity
was introduced by Zakharov [7] as a model for Langmuir turbulence in a plasma. The wellposedness theory of the Zakharov system has recently been improved. Local wellposedness below the energy space
On the Cauchy Problem for the Zakharov System
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
On the Zakharov and Zakharov-Schulman Systems
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
Orbital stability of solitary waves of Zakharov system
This article is concerned with the orbital stability of solitary waves of the Zakharov system. By applying the abstract results of M. Grillakis et al. [J. Funct. Anal. 74, 160–197 (1987); 94, 308–348
Existence of solitary waves in higher dimensions
The elliptic equation Δu=F(u) possesses non-trivial solutions inRn which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear
Existence and Smoothing Effect of Solutions for the Zakharov Equations
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
...
...