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.