Norm inflation for the Zakharov system

Abstract

The Cauchy problem for the classical Zakharov system is shown to be ill-posed in the sense of norm inflation in a range of Sobolev spaces $H^s({\mathbb{R}}^d)\times H^l({\mathbb{R}}^d)$ for all dimensions $d$. This proves several results on well-posedness, which includes existence of solutions, uniqueness and continuous dependence on the initial data, to be sharp up to endpoints.

arXiv comment19 pages, 1 figure; typos corrected, new figure