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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 $H^s(\bbfR)$: 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.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35G25Initial value problems for nonlinear higher-order PDE
35D99Generalized solutions of PDE
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