Local ill-posedness of the 1D Zakharov system

Holmer, Justin

Electronic Journal of Differential Equations, Tome 2007 (2007).

Résumé

Summary: 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.

Détail

Classification : 35Q55, 35Q51, 35R25
Keywords: Zakharov system, Cauchy problem, local well-posedness, local ill-posedness