We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in H 1 (T). This leads to a global wellposedness result in H 1 (T) for the Benjamin-Ono equation.

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In this note we study the generalized 2D Zakharov-Kuznetsov equations ∂ t u + ∆∂ x u + u k ∂ x u = 0 for k ≥ 2. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces H s (R 2 ) for s > 1/4 if k = 2, s > 5/12 if k = 3 and s > 1 − 2/k if k ≥ 4.

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