We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a t ϵ (∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasi-periodic solutions.

Long time dynamics of Schrödinger and wave equations on flat tori / Berti, M.; Maspero, A.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 267:2(2019), pp. 1167-1200. [10.1016/j.jde.2019.02.004]

Long time dynamics of Schrödinger and wave equations on flat tori

Berti, M.;Maspero, A.
2019

Abstract

We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a t ϵ (∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasi-periodic solutions.
267
2
1167
1200
https://www.sciencedirect.com/science/article/pii/S0022039619300701?via%3Dihub
https://arxiv.org/abs/1811.06714
Berti, M.; Maspero, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/87888
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