Revista Matemática Iberoamericana


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Volume 34, Issue 4, 2018, pp. 1563–1608
DOI: 10.4171/rmi/1036

Published online: 2018-12-03

Unconditional uniqueness for the modified Korteweg–de Vries equation on the line

Luc Molinet[1], Didier Pilod[2] and Stéphane Vento[3]

(1) Université François Rabelais, Tours, France
(2) University of Bergen, Norway
(3) Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France

We prove that the modified Korteweg–de Vries (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s > 1/3$. Our method of proof combines the improvement of the energy method introduced recently by the first and third authors with the construction of a modified energy. Our approach also yields a priori estimates for the solutions of mKdV in $H^s(\mathbb R)$, for $s > 0$, and enables us to construct weak solutions at this level of regularity.

Keywords: Modified Korteweg–de Vries equation, unconditional uniqueness, well-posedness, modified energy

Molinet Luc, Pilod Didier, Vento Stéphane: Unconditional uniqueness for the modified Korteweg–de Vries equation on the line. Rev. Mat. Iberoam. 34 (2018), 1563-1608. doi: 10.4171/rmi/1036