The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices
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 \(s<5/8\).
scientific article; zbMATH DE number 495434
scientific article