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arXiv · 2206.08719

Norm inflation for the derivative nonlinear Schr\"odinger equation

Yuzhao Wang·Younes Zine
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Abstract

In this note, we study the ill-posedness problem for the derivative nonlinear Schr\"odinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.

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Yuzhao Wang, Younes Zine. 2022-06-17. Norm inflation for the derivative nonlinear Schr\"odinger equation. https://arxiv.org/abs/2206.08719

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