Exponential sums and nonlinear Schrödinger equations. (English) Zbl 0787.35096

Bourgain, J.

Geom. Funct. Anal. 3, No. 2, 157-178 (1993).

A global result is given for the Cauchy problem, in space dimension \(n=4\), of the Schrödinger equation \[ i \partial_ tu+\Delta u+uF(| u |^ 2)=0,\;u \text{ periodic in } x,\;u(x,0)=\varphi(x), \] \(\varphi \in H^ 2(\mathbb{R}^ 4/ \mathbb{Z}^ 4)\), \(\| \varphi \|_ 2\) small, \(F(z)\leq cz^{1/2}\), \(| F'(z)| \leq Cz^{-1/2}\), \(| F''(z)| \leq Cz^{-3/2}\).
The global wellposedness is derived from the local wellposedness and the conservation laws. The local result is based on Picard’s fixed point method, using the associated integral equation.

Reviewer: L.Vazquez (Madrid)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

nonlinear Schrödinger equations; Fourier analysis; Cauchy problem; global wellposedness; local wellposedness; Picard’s fixed point method

References:

[1]	[B]J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Preprint I.H.E.S., September 1992, and Part I in this issue.