#### 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