The Cauchy problem for the Benjamin–Ono equation in L2 revisited

Molinet, Luc; Pilod, Didier

doi:10.2140/apde.2012.5.365

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

The Cauchy problem for the Benjamin–Ono equation in $L^2$ revisited

Luc Molinet and Didier Pilod

Vol. 5 (2012), No. 2, 365–395

DOI: 10.2140/apde.2012.5.365

Abstract

Ionescu and Kenig proved that the Cauchy problem associated with the Benjamin–Ono equation is globally well posed in $L^{2} (ℝ)$ . In this paper we give a simpler proof of Ionescu and Kenig’s result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in $H^{s} (ℝ)$ for $s > \frac{1}{4}$ . Note that our approach also permits us to simplify the proof of the global well-posedness in $L^{2} (T)$ and yields unconditional well-posedness in $H^{\frac{1}{2}} (T)$ .

Keywords

Benjamin–Ono equation, initial value problem, gauge transformation

Mathematical Subject Classification 2010

Primary: 35A07, 35Q53

Secondary: 76B55

Milestones

Received: 7 September 2010

Accepted: 15 January 2011

Published: 27 August 2012

Authors

Luc Molinet
Laboratoire de Mathématiques et Physique Théorique Université de Tours Parc Grandmont 37200 Tours France

Didier Pilod
Instituto de Matemática Universidade Federal do Rio de Janeiro Caixa Postal 68530 21945-970 Rio de Janeiro Brazil

Vol. 5, No. 2, 2012