Local ill-posedness of the 1D Zakharov system

الأصل · EN

Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system for any dimension d, in the inhomogeneous Sobolev spaces (u,n)∈ Hᵏ(Rᵈ)× Hˢ(Rᵈ) 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.