Unconditional uniqueness for the Benjamin-Ono equation

TL;DR

This paper proves the unconditional uniqueness of solutions to the Benjamin-Ono equation for low regularity initial data using gauge transformations, normal form reductions, and refined Strichartz estimates, extending results below the critical regularity threshold.

Contribution

It introduces a novel approach combining gauge transformations and normal form reductions to establish uniqueness below the regularity threshold s=1/6.

Findings

Uniqueness established for initial data in H^s with s<1/6

Refined Strichartz estimates enable analysis below critical regularity

Nonlinear smoothing property observed at the same regularity level

Abstract

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in $H^{s}$, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold $s=1/6$. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity.