Electron. J. Diff. Eqns., Vol. 2007(2007), No. 24, pp. 1-22.

Local ill-posedness of the 1D Zakharov system

Justin Holmer

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.

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Justin Holmer
Department of Mathematics
University of California
Berkeley, CA 94720-3840, USA
email: holmer@math.berkeley.edu

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