New bounds on the high Sobolev norms of the 1d NLS solutions

Diego Berti; Fabrice Planchon; Nicola Visciglia; Nikolay Tzvetkov

New bounds on the high Sobolev norms of the 1d NLS solutions

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arxiv 2407.17993 v2 pith:Y75JAWF2 submitted 2024-07-25 math.AP

Diego Berti , Fabrice Planchon , Nikolay Tzvetkov , Nicola Visciglia This is my paper · ORCID

classification math.AP

keywords boundscasehighnormssobolevsolutionsallowsbourgain

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abstract

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D 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.

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Asymptotically full measure sets of almost-periodic solutions for the NLS equation
math.AP 2024-12 conditional novelty 7.0 of 10

Most small Gevrey initial data for a 1D NLS with random convolution potential give global almost-periodic solutions whose Gevrey norm stays bounded for all time.

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