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Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis

A bilinear estimate with applications to the KdV equation. (English) Zbl 0848.35114

J. Am. Math. Soc. 9, No. 2, 573-603 (1996).
The authors start the work with a wide panoramic view of the results related to the initial value problem for the Korteweg-de Vries equation. The main result of the paper is to have found the lower best index \(s\), such that we have local well posedness in \(H^s(\mathbb{R})\): Existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on \(|u_0|_{H^s}\). The value \(s> -3/4\), is the optimal one provided by the used method. The result improves a previous one of the same authors. The method combines oscillatory integral estimates with bilinear estimates for \(\partial_x(u^2/2)\) in the Bourgain function spaces associated with the index \(s\). The estimates extend to the periodic case.
Reviewer: L.Vazquez (Madrid)

Cited in 11 Reviews
Cited in 367 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35G25 Initial value problems for nonlinear higher-order PDEs
35D99 Generalized solutions to partial differential equations

Keywords:

Schrödinger equation; initial value problem; well-posedness
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\textit{C. E. Kenig} et al., J. Am. Math. Soc. 9, No. 2, 573--603 (1996; Zbl 0848.35114)
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References:

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