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Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations. (English) Zbl 0787.35097
In this first part the author develops a harmonic analysis method to solve the nonlinear periodic (in the space variables) Schrödinger equation (NLSE) $i\partial_ tu+\Delta_ xu+u | u |^{p- 2}=0,\quad (p \geq 3),\quad u(x,0)=\varphi(x).$ The method is based on solving the equivalent integral equation by Picard’s fixed point method, where the nonlinearity is controlled in the iteration process by the Strichartz’s inequality.
The main idea here is to try to adjust this approach to the periodic case. In such a case, the estimates are local in time and hence, at this stage, only a local solution is obtained. In order to obtain the global solution, the local results must be combined with the conservation laws. The main feature of the approach followed here is an analysis on multiple Fourier series.
As an illustration of the results, we have that, if $$n=3$$, the NLSE has a global unique solution for $$4<p<6$$ and sufficiently small $$H^ 1$$-data. For dimensions $$n \geq 4$$, only local wellposedness statements are obtained.
[For part II, see the review below].
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