Journal of the Mathematical Society of Japan

On unconditional well-posedness for the periodic modified Korteweg–de Vries equation

Luc MOLINET, Didier PILOD, and Stéphane VENTO

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We prove that the modified Korteweg–de Vries equation is unconditionally well-posed in $H^s({\mathbb{T}})$ for $s\ge 1/3$. For this we gather the smoothing effect first discovered by Takaoka and Tsutsumi with an approach developed by the authors that combines the energy method, with Bourgain's type estimates, improved Strichartz estimates and the construction of modified energies.


The first and third authors were partially supported by the French ANR project GEODISP. The second author was partially supported by CNPq/Brazil, grants 303051/2016–7 and 431231/2016–8.

Article information

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 147-201.

Received: 20 December 2016
Revised: 20 July 2017
First available in Project Euclid: 26 October 2018

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Zentralblatt MATH identifier

Primary: 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness 35E15: Initial value problems 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B45: A priori estimates 35D30: Weak solutions

periodic modified Korteweg–de Vries equation unconditional uniqueness well-posedness modified energy


MOLINET, Luc; PILOD, Didier; VENTO, Stéphane. On unconditional well-posedness for the periodic modified Korteweg–de Vries equation. J. Math. Soc. Japan 71 (2019), no. 1, 147--201. doi:10.2969/jmsj/76977697.

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