Global well-posedness for the ILW equation in $H^s(\mathbb{T})$ for $s>-\frac12$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910989708951552 |
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| author | Gassot, Louise Laurens, Thierry |
| author_facet | Gassot, Louise Laurens, Thierry |
| contents | We prove that the intermediate long wave (ILW) equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for $s > -\frac12$. The previous record for well-posedness was $s\geq 0$, and the system is known to be ill-posed for $s<-\frac12$. We then demonstrate that the solutions of ILW converge to those of the Benjamin--Ono equation in $H^s(\mathbb{T})$ in the infinite-depth limit.
Our methods do not rely on the complete integrability of ILW, but rather treat ILW as a perturbation of the Benjamin--Ono equation by a linear term of order zero. To highlight this, we establish a general well-posedness result for such perturbations, which also applies to the Smith equation for continental-shelf waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness for the ILW equation in $H^s(\mathbb{T})$ for $s>-\frac12$ Gassot, Louise Laurens, Thierry Analysis of PDEs We prove that the intermediate long wave (ILW) equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for $s > -\frac12$. The previous record for well-posedness was $s\geq 0$, and the system is known to be ill-posed for $s<-\frac12$. We then demonstrate that the solutions of ILW converge to those of the Benjamin--Ono equation in $H^s(\mathbb{T})$ in the infinite-depth limit. Our methods do not rely on the complete integrability of ILW, but rather treat ILW as a perturbation of the Benjamin--Ono equation by a linear term of order zero. To highlight this, we establish a general well-posedness result for such perturbations, which also applies to the Smith equation for continental-shelf waves. |
| title | Global well-posedness for the ILW equation in $H^s(\mathbb{T})$ for $s>-\frac12$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.05149 |