Unconditional uniqueness for the modified Korteweg-de Vries equation on the line

TL;DR

This paper establishes unconditional well-posedness of the modified Korteweg-de Vries (mKdV) equation in Sobolev spaces with regularity above 1/3, using advanced energy methods and constructing weak solutions.

Contribution

It introduces a novel combination of energy improvement techniques and modified energy construction to prove well-posedness and weak solutions for mKdV in low regularity spaces.

Findings

Unconditional well-posedness for s > 1/3 in H^s(R)

A priori estimates for solutions in H^s(R) for s > 0

Construction of weak solutions at low regularity

Abstract

We prove that the modified Korteweg- de Vries equation (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s> \frac 13$. Our method of proof combines the improvement of the energy method introduced recently by the first and third authors with the construction of a modified energy. Our approach also yields \textit{a priori} estimates for the solutions of mKdV in $H^s(\mathbb R)$, for $s>0$, and enables us to construct weak solutions at this level of regularity.