Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

TL;DR

This paper proves local well-posedness for the KP-I initial value problem with small initial data in a weighted energy space, correcting a previous proof and strengthening the main results.

Contribution

It provides a correct proof of local well-posedness for KP-I with weighted initial data, addressing and fixing an error in prior work by Colliander, Kenig, and Staffilani.

Findings

Established local well-posedness for KP-I with weighted initial data.

Identified and corrected a key estimate error in previous literature.

Strengthened the main result of the prior GAFA paper.

Abstract

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper.