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

TL;DR

This paper establishes the first local-in-time a-priori estimates in $H^{-1}(\

Contribution

It introduces novel a-priori estimates for non-integrable perturbations of KdV in $H^{-1}$, applicable to water wave models with uneven bottoms.

Findings

First estimate for non-integrable KdV perturbations in $H^{-1}$

Applicable to models of long water waves with uneven bottoms

Uses bootstrap argument with renormalized perturbation determinant

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.