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

Luc Molinet (1) , Stéphane Vento (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LAGA - Laboratoire Analyse, Géométrie et 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 $ 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.

Mots clés: dispersive equations; resonance relation; local well-posedness; unconditional uniqueness

Domaines: Equations aux dérivées partielles [math.AP]

Luc Molinet, Stéphane Vento. Improvement of the energy method for strongly non resonant dispersive equations and applications. Analysis & PDE, 2015, 8 (6), pp.1455-1495. ⟨hal-01064252⟩