## Journal of the Mathematical Society of Japan

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

#### Abstract

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.

#### Note

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

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

Dates
Revised: 20 July 2017
First available in Project Euclid: 26 October 2018

https://projecteuclid.org/euclid.jmsj/1540541018

Digital Object Identifier
doi:10.2969/jmsj/76977697

Mathematical Reviews number (MathSciNet)
MR3909918

Zentralblatt MATH identifier
07056561

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1540541018

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