KdV on an incoming tide

TL;DR

This paper proves global well-posedness of the KdV equation for initial data with step-like asymptotics plus low-regularity perturbations, extending previous results and providing new proofs for classical theorems.

Contribution

It introduces a general well-posedness framework for KdV with exotic spatial asymptotics and establishes global solutions for step-like initial data with low regularity.

Findings

Global well-posedness for step-like initial data plus $H^{-1}$ perturbations.

New proof of the Bona--Smith theorem using low-regularity methods.

Extension of well-posedness results to exotic spatial asymptotics.

Abstract

Given smooth step-like initial data $V(0,x)$ on the real line, we show that the Korteweg--de Vries equation is globally well-posed for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. The proof uses our general well-posedness result for exotic spatial asymptotics. As a prerequisite, we show that KdV is globally well-posed for $H^3(\mathbb{R})$ perturbations of step-like initial data. In the case $V \equiv 0$, we obtain a new proof of the Bona--Smith theorem using the low-regularity methods that established the sharp well-posedness of KdV in $H^{-1}$.