Local well-posedness in low regularity of the MKDV equation with periodic boundary condition

Kenji Nakanishi; Hideo Takaoka; Yoshio Tsutsumi

doi:10.3934/dcds.2010.28.1635

Article Contents

2010, Volume 28, Issue 4: 1635-1654. Doi: 10.3934/dcds.2010.28.1635

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Local well-posedness in low regularity of the MKDV equation with periodic boundary condition

1.
Department of Mathematics, Kyoto University, Kyoto 606-8502
2.
Department of Mathematics, Hokkaido University, Sapporo 060-0810

Received: September 30, 2009

Revised: January 31, 2010

Published: November 2010

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Abstract

We study the local well-posedness in low regularity of the Cauchy problem for the mKdV equation on one-dimensional torus by modifying the Fourier restriction method due to Bourgain. We show the local well-posedness in $H^s$, $s > 1/3$. In the case $s > 1/4$, we prove the local existence of solution in $H^s$ and moreover the well-posedness in $H^s$ under a certain additional assumption on initial data. For the proof, we modify the Fourier restriction norm to take into account the oscillation of the phase of solution, which is caused by the nonlinear interaction.

Keywords:

Key words and phrases} Local well-posedness in low regularity,
mKdV equation,
phase correction of solution.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B10, 35B30.

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