Geodesic
Remark on well-posedness and ill-posedness for the KdV equation
Electronic Journal of Differential Equations, Tome 2010 (2010).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     author = {Kato, Takamori},
     title = {Remark on well-posedness and ill-posedness for the {KdV} equation},
     journal = {Electronic Journal of Differential Equations},
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     volume = {2010},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a292/}
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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__a292/