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arXiv · 2502.04614

A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$

Mihaela Ifrim·Thierry Laurens
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Abstract

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.

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Mihaela Ifrim, Thierry Laurens. 2025-02-07. A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$. https://arxiv.org/abs/2502.04614

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