December  2010, 28(4): 1635-1654. doi: 10.3934/dcds.2010.28.1635

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

1. 

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan, Japan

2. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received  October 2009 Revised  February 2010 Published  June 2010

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.
Citation: Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
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