On the growth of Sobolev norms for the cubic Szegő equation

Patrick Gérard[1]; Sandrine Grellier[2]

  • [1] Université Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, UMR 8628 France
  • [2] MAPMO-UMR 6628 Département de Mathématiques Université d’Orleans 45067 Orléans Cedex 2 France

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • Volume: 13, Issue: 2, page 1-20
  • ISSN: 2266-0607

Abstract

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We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.

How to cite

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Gérard, Patrick, and Grellier, Sandrine. "On the growth of Sobolev norms for the cubic Szegő equation." Séminaire Laurent Schwartz — EDP et applications 13.2 (2014-2015): 1-20. <http://eudml.org/doc/275684>.

@article{Gérard2014-2015,
abstract = {We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.},
affiliation = {Université Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, UMR 8628 France; MAPMO-UMR 6628 Département de Mathématiques Université d’Orleans 45067 Orléans Cedex 2 France},
author = {Gérard, Patrick, Grellier, Sandrine},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {Hankel operators; inverse spectral problems; compressed shift operator; action-angle variables},
language = {eng},
number = {2},
pages = {1-20},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On the growth of Sobolev norms for the cubic Szegő equation},
url = {http://eudml.org/doc/275684},
volume = {13},
year = {2014-2015},
}

TY - JOUR
AU - Gérard, Patrick
AU - Grellier, Sandrine
TI - On the growth of Sobolev norms for the cubic Szegő equation
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 13
IS - 2
SP - 1
EP - 20
AB - We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.
LA - eng
KW - Hankel operators; inverse spectral problems; compressed shift operator; action-angle variables
UR - http://eudml.org/doc/275684
ER -

References

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