Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 142, pp. 1-15.

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

Author: Takamori Kato (Nagoya Univ., Japan)

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 08, 2010.
Math Subject Classifications: 35Q55.
Key Words: KdV equation; well-posedness; ill-posedness; Cauchy problem;
           Fourier restriction norm; low regularity.