Vol. 8, No. 6, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Improvement of the energy method for strongly nonresonant dispersive equations and applications

Luc Molinet and Stéphane Vento

Vol. 8 (2015), No. 6, 1455–1495
DOI: 10.2140/apde.2015.8.1455
Abstract

We propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly nonresonant dispersive equations. As an example, we obtain unconditional well-posedness of the Cauchy problem in the energy space for a large class of one-dimensional dispersive equations with a dispersion that is greater than the one of the Benjamin–Ono equation. At the level of dispersion of the Benjamin–Ono, we also prove the well-posedness in the energy space but without unconditional uniqueness. Since we do not use a gauge transform, this enables us in all cases to prove strong convergence results in the energy space for solutions of viscous versions of these equations towards the purely dispersive solutions. Finally, it is worth noting that our method of proof works on the torus as well as on the real line.

Keywords
Benjamin–Ono equation, intermediate long wave equation, dispersion generalized Benjamin–Ono equation, well-posedness, unconditional uniqueness
Mathematical Subject Classification 2010
Primary: 35E15, 35Q53, 35A02
Milestones
Received: 13 January 2015
Revised: 24 April 2015
Accepted: 21 May 2015
Published: 5 September 2015
Authors
Luc Molinet
Laboratoire de Mathématiques et Physique Théorique, CNRS (UMR 7350)
Université François Rabelais
Fédération Denis Poisson
Faculté des Sciences et Techniques
Parc de Grandmont
37200 Tours
France
Stéphane Vento
Laboratoire Analyse, Géométrie et Applications, CNRS (UMR 7539)
Université Paris 13
Institut Galilée
99 avenue J. B. Clément
93430 Villetaneuse
France