We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space , which is defined by the norm
We obtain the local well-posedness in with , and . The proof is based on Kishimoto's work  which proved the sharp well-posedness in the Sobolev space . Moreover we prove ill-posedness when , or .
Submitted August 19, 2010. Published October 8, 2010.
Math Subject Classifications: 35Q55.
Key Words: KdV equation; well-posedness; ill-posedness; Cauchy problem; Fourier restriction norm; low regularity.
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| Takamori Kato |
Graduate School of Mathematics, Nagoya University
Chikusa-ku, Nagoya, 464-8602, Japan
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