Norm inflation for the Zakharov system

TL;DR

This paper demonstrates that the classical Zakharov system exhibits norm inflation in certain Sobolev spaces, establishing sharp ill-posedness results and clarifying the limits of well-posedness for the system.

Contribution

It proves the ill-posedness of the Zakharov system via norm inflation in Sobolev spaces, refining the understanding of its well-posedness boundaries.

Findings

Norm inflation occurs in Sobolev spaces for the Zakharov system.

Well-posedness results are sharp up to endpoints.

The study clarifies the limits of existence, uniqueness, and continuous dependence for solutions.

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.