Acta Mathematica

Volume 231 (2023)

Number 1

Sharp well-posedness results of the Benjamin–Ono equation in $H^s (\mathbb{T}, \mathbb{R})$ and qualitative properties of its solutions

Pages: 31 – 88

DOI: https://dx.doi.org/10.4310/ACTA.2023.v231.n1.a2

Authors

Patrick Gérard (Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris–Saclay, Orsay, France)

Thomas Kappeler (Institut für Mathematik, Universität Zürich, Switzerland)

Peter Topalov (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)

Abstract

$\def\HSTR{H^s (\mathbb{T}, \mathbb{R})}$We prove that the Benjamin–Ono equation on the torus is globally in time well-posed in the Sobolev space $\HSTR$ for any $s \gt -\frac{1}{2}$ and ill-posed for $s \leqslant -\frac{1}{2}$. Hence the critical Sobolev exponent $s_c = -\frac{1}{2}$ of the Benjamin–Ono equation is the threshold for wellposedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin–Ono equation on the torus are orbitally stable in $\HSTR$ for any $s \gt -\frac{1}{2}$. Novel conservation laws and a non-linear Fourier transform on $\HSTR$ with $s \gt -\frac{1}{2}$ are key ingredients into the proofs of these results.

Keywords

Benjamin–Ono equation, well-posedness, critical Sobolev exponent, almost periodicity of solutions, orbital stability of traveling waves

2010 Mathematics Subject Classification

Primary 37K15. Secondary 47B35.

Received 14 April 2020

Accepted 23 May 2021

Published 29 September 2023