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Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation. (English) Zbl 0787.35098
[For part I, cf. the review above.]
The author studies the initial value problem for the periodic, in space, Korteweg-de Vries equation and some of its variants $\partial_ tu+\partial_{xxx}u+u^ k \partial_ xu=0,\;u(x,0)=\varphi(x)\quad (k \geq 1),\;u\text{ periodic in } x.$ It is established a local result by Picard’s theorem, verifying a contracting property for the associated integral equation in a suitable space, mainly using Fourier analysis estimates. Combining the existence of local solutions with the $$L^ 2$$- conservation law, it is obtained the global solution (after regularization).
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