Improvement of the energy method for strongly non resonant dispersive equations and applications

TL;DR

This paper introduces a new method to establish local well-posedness for strongly non resonant dispersive equations, achieving results below H^1 and enabling convergence analysis without gauge transforms.

Contribution

The authors develop a novel approach to prove well-posedness for a broad class of dispersive equations without gauge transforms, extending the theory below H^1.

Findings

Unconditional well-posedness below H^1 for certain dispersive equations.

New convergence results for viscous solutions towards dispersive solutions.

Applicability to equations with dispersion at least as strong as Benjamin-Ono.

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.