#### Vol. 5, No. 2, 2012

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

### Luc Molinet and Didier Pilod

Vol. 5 (2012), No. 2, 365–395
##### Abstract

Ionescu and Kenig proved that the Cauchy problem associated with the Benjamin–Ono equation is globally well posed in ${L}^{2}\left(ℝ\right)$. 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 ${H}^{s}\left(ℝ\right)$ for $s>\frac{1}{4}$. Note that our approach also permits us to simplify the proof of the global well-posedness in ${L}^{2}\left(\mathbb{T}\right)$ and yields unconditional well-posedness in ${H}^{\frac{1}{2}}\left(\mathbb{T}\right)$.

##### Keywords
Benjamin–Ono equation, initial value problem, gauge transformation
##### Mathematical Subject Classification 2010
Primary: 35A07, 35Q53
Secondary: 76B55
##### Milestones
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