Local ill-posedness of the 1D Zakharov system

TL;DR

This paper investigates the 1D Zakharov system and demonstrates local ill-posedness for certain Sobolev space exponents, extending the understanding of the system's behavior beyond well-posedness regimes.

Contribution

It establishes local ill-posedness results for the 1D Zakharov system outside previously known well-posedness conditions, using advanced harmonic analysis techniques.

Findings

Identifies specific exponent pairs where the system is ill-posed.

Extends the analysis of Zakharov system to new Sobolev space regimes.

Utilizes techniques from nonlinear Schrödinger equation analysis.

Abstract

Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system 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 Schroedinger equation.