A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918309361876992 |
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| author | Ifrim, Mihaela Laurens, Thierry |
| author_facet | Ifrim, Mihaela Laurens, Thierry |
| contents | We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of the sharp well-posedness theory for KdV. In particular, we show that our analysis applies to models for long waves in a shallow channel of water with an uneven bottom.
The proof of our main result is based upon a bootstrap argument for the renormalized perturbation determinant coupled with a local smoothing norm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$ Ifrim, Mihaela Laurens, Thierry Analysis of PDEs 35Q53 We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of the sharp well-posedness theory for KdV. In particular, we show that our analysis applies to models for long waves in a shallow channel of water with an uneven bottom. The proof of our main result is based upon a bootstrap argument for the renormalized perturbation determinant coupled with a local smoothing norm. |
| title | A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$ |
| topic | Analysis of PDEs 35Q53 |
| url | https://arxiv.org/abs/2502.04614 |