Geodesic
Remark on well-posedness and ill-posedness for the KdV equation
Electronic journal of differential equations, Tome 2010 (2010)
We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space

$ \| \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 $a >0$.
Classification : 35Q55
Keywords: KdV equation, well-posedness, ill-posedness, Cauchy problem, Fourier restriction norm, low regularity 
@article{EJDE_2010__2010__a92,
     author = {Kato,  Takamori},
     title = {Remark on well-posedness and ill-posedness for the {KdV} equation},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1201.35170},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a92/}
}
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Kato,  Takamori. Remark on well-posedness and ill-posedness for the KdV equation. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a92/