Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 24, pp. 1-22.

Title: Local ill-posedness of the 1D Zakharov system

Author: Justin Holmer (Univ. of California, Berkeley, CA, USA)

Abstract:
 Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for
 the Zakharov system
$$\displaylines{
 i\partial_tu + \Delta u = nu \cr
 \partial_t^2 n - \Delta n = \Delta |u|^2 \cr
 u(x,0)=u_0(x), \cr
 n(x,0)=n_0(x), \quad \partial_tn(x,0)=n_1(x)
}$$
 where $u=u(x,t)\in \mathbb{C}$, $n=n(x,t)\in \mathbb{R}$,
  $x\in \mathbb{R}$, and $t\in \mathbb{R}$. The proof was made
 for any dimension $d$, in the inhomogeneous Sobolev spaces
 $(u,n)\in H^k(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ for a range
 of exponents $k$, $s$ depending on $d$.  Here we restrict to dimension
 $d=1$ and present a few results establishing local ill-posedness
 for exponent pairs $(k,s)$ outside of the well-posedness regime.  
 The techniques employed are rooted in the work of Bourgain (1993),
 Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and Christ-Colliander-Tao (2003)
 applied to the nonlinear Schrodinger equation.
 
Submitted March 24, 2006. Published February 12, 2007.
Math Subject Classifications: 35Q55, 35Q51, 35R25.
Key Words: Zakharov system; Cauchy problem; local well-posedness;
           local ill-posedness.