A-priori bounds for KdV equation below $H^{-3/4}$

TL;DR

This paper establishes new a-priori bounds for solutions to the KdV equation on the real line, extending the known bounds to initial data in Sobolev spaces with regularity as low as $H^{-4/5}$, thus broadening the understanding of solution behavior below previous regularity thresholds.

Contribution

It provides the first a-priori bounds for KdV solutions in Sobolev spaces with regularity below $H^{-3/4}$, specifically down to $H^{-4/5}$, advancing the theory of low-regularity solutions.

Findings

Proved local in time $H^s$ bounds for $s geq -4/5$

Extended the known regularity threshold for a-priori bounds

Demonstrated control of solutions with rough initial data

Abstract

We consider the Korteweg-de Vries Equation (KdV) on the real line, and prove that the smooth solutions satisfy a-priori local in time $H^s$ bound in terms of the $H^s$ size of the initial data for $s\geq -4/5$.