We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the 1D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.
New bounds on the high Sobolev norms of the 1D NLS solutions
Planchon F.;Tzvetkov N.;Visciglia N.
2025-01-01
Abstract
We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the 1D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.File in questo prodotto:
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