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Improvement of the
energy method for strongly nonresonant dispersive equations
and applications
Luc Molinet and Stéphane Vento |
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Vol. 8 (2015), No. 6, 1455–1495
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DOI: 10.2140/apde.2015.8.1455
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35E15, 35Q53, 35A02
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Milestones
Received: 13 January 2015
Revised: 24 April 2015
Accepted: 21 May 2015
Published: 5 September 2015
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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 |
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| Stéphane Vento | |
| Laboratoire Analyse, Géométrie et
Applications, CNRS (UMR 7539) Université Paris 13 Institut Galilée 99 avenue J. B. Clément 93430 Villetaneuse France |
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