  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.

Analysis of PDEs