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

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 $$ | varphi |_{H^{s,a}}=| langle xi angle^{s-a} |xi|^a widehat{varphi} |_{L_{xi}^2}. $$ 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 $(s,a) eq (-3/4,-3/4)$. 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 $s< max{-3/4,-a-3/2}$, $aleq -3/2$ or $a >0$.

Keywords

• KdV equation
• well-posedness
• ill-posedness
• Cauchy problem
• Fourier restriction norm
• low regularity