Growth bound and nonlinear smoothing for the periodic derivative nonlinear Schrödinger equation

A polynomial-in-time growth bound is established for global Sobolev H s ( T ) solutions to the derivative nonlinear Schrödinger equation on the circle with s > 1 . These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodi...

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Bibliographic Details
Published in:Mathematische annalen 2024-01, Vol.388 (3), p.2289-2329
Main Authors: Isom, Bradley, Mantzavinos, Dionyssios, Stefanov, Atanas
Format: Article
Language:English
Subjects:
Cauchy problems
Mathematics
Mathematics and Statistics
Polynomials
Schrodinger equation
Smoothing
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Summary:A polynomial-in-time growth bound is established for global Sobolev H s ( T ) solutions to the derivative nonlinear Schrödinger equation on the circle with s > 1 . These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-022-02492-8