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 $H^k\left({\mathrm{ℝ}}^d\right)\times H^l\left({\mathrm{ℝ}}^d\right)\times H^{l-1}\left({\mathrm{ℝ}}^d\right)$. 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 $(k,l)\in {\mathrm{ℝ}}^2$. 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.

Keywords:
Zakharov System; C² Ill-posedness