On the Cauchy problem for the Kadomtsev-Petviashvili equation. (English) Zbl 0787.35086

Bourgain, J.

Geom. Funct. Anal. 3, No. 4, 315-341 (1993).

The author gives a global result for the Cauchy problem of the Kadomtsev- Petviashvili equation. The solution is globally well-posed for initial data in \(L^ 2(\mathbb{R}^ 2/\mathbb{Z}^ 2)\). The result is obtained on extending a local result (obtained by applying a Picard fixed point argument) to a global one with the help of the \(L^ 2\)-conservation law.
The method is analogous to the one used for the Schrödinger and Korteweg-de Vries equations, and it is based on an analysis of multiple Fourier series.

Reviewer: L.Vazquez (Madrid)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
31B10	Integral representations, integral operators, integral equations methods in higher dimensions

Keywords:

wellposedness; Fourier analysis; periodic solution; Picard theorem; local solution; global solution; Cauchy problem; Kadomtsev-Petviashvili equation; multiple Fourier series

References:

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