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The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. (English) Zbl 0787.35090
The paper considers the initial value problem for the Korteweg-de Vries equation \[ u_ t+(u^ 2)_ x+u_{xxx}=0,\;t,x \in \mathbb{R},\;u(x,0)=u_ 0(x), \tag{1} \] for data in classical Sobolev spaces of negative order, i.e. \(u_ 0 \in H^{-s}(\mathbb{R})\) with \(s \geq 0\). The main result shows that the problem (1) is locally well posed for \(s<5/8\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35K30 Initial value problems for higher-order parabolic equations
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