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arXiv · 2311.08142

Intermediate long wave equation in negative Sobolev spaces

Andreia Chapouto·Justin Forlano·Guopeng Li·Tadahiro Oh·Didier Pilod
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Abstract

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity $s = -\frac 12$ as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the $H^s$-norm of a solution to ILW for $ - \frac 12 < s < 0$. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s < - \frac 12$. Our results apply to both the real line case and the periodic case.

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Andreia Chapouto, Justin Forlano, Guopeng Li, Tadahiro Oh, Didier Pilod. 2023-11-14. Intermediate long wave equation in negative Sobolev spaces. https://arxiv.org/abs/2311.08142

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