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

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


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