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