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.