Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 6 (2015), 1455-1495.

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

Luc Molinet and Stéphane Vento

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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.

Article information

Source
Anal. PDE, Volume 8, Number 6 (2015), 1455-1495.

Dates
Received: 13 January 2015
Revised: 24 April 2015
Accepted: 21 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843151

Digital Object Identifier
doi:10.2140/apde.2015.8.1455

Mathematical Reviews number (MathSciNet)
MR3397003

Zentralblatt MATH identifier
1330.35384

Subjects
Primary: 35E15: Initial value problems 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness

Keywords
Benjamin–Ono equation intermediate long wave equation dispersion generalized Benjamin–Ono equation well-posedness unconditional uniqueness

Citation

Molinet, Luc; Vento, Stéphane. Improvement of the energy method for strongly nonresonant dispersive equations and applications. Anal. PDE 8 (2015), no. 6, 1455--1495. doi:10.2140/apde.2015.8.1455. https://projecteuclid.org/euclid.apde/1510843151


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