Improvement of the energy method for strongly nonresonant dispersive equations and applications

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 H¹ 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.