Abstract. In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below H 1 for a large class of one-dimensional dispersive equations with a dispersion that is greater or equal to the one of the Benjamin-Ono equation. Since this is done without using a gauge transform, this enables us to prove strong convergence results for solutions of viscous versions of these equations towards the purely dispersive solutions.

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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.