Electron. J. Diff. Equ., Vol. 2010(2010), No. 142, pp. 1-15.

Remark on well-posedness and ill-posedness for the KdV equation

Takamori Kato

Abstract:
We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space $H^{s,a}(\mathbb{R})$, which is defined by the norm
$$ \| \varphi \|_{H^{s,a}}=\| \langle \xi \rangle^{s-a} |\xi|^a \widehat{\varphi} \|_{L_{\xi}^2}. $$
We obtain the local well-posedness in $H^{s,a}$ with $s \geq \max\{-3/4,-a-3/2\} $, $-3/2< a \leq 0$ and $(s,a) \neq (-3/4,-3/4)$. The proof is based on Kishimoto's work [12] which proved the sharp well-posedness in the Sobolev space $H^{-3/4}(\mathbb{R})$. Moreover we prove ill-posedness when $s< \max\{-3/4,-a-3/2\}$, $a\leq -3/2$ or $a >0$.

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
email: d08003r@math.nagoya-u.ac.jp

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