A Note on C^2 Ill-posedness Results for the Zakharov System in Arbitrary Dimension

Abstract

This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces H^k(\R^d)×H^l(\R^d)×H^l−1(\R^d).We recall the well-posedness and ill-posedness results known to date and establish new ill-posedness results.We prove C^2 ill-posedness for some new indices (k, l) ∈ \R^2. Moreover, our results are valid in arbitrary dimension. We believe that our detailed proofs are built on a methodical approach and can be adapted to obtain similar results for other systems and equations.