Bounds for KdV and the 1-d cubic NLS equation in rough function spaces

Herbert Koch[1]

  • [1] Mathematisches Institut Universität Bonn

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-10
  • ISSN: 2266-0607

Abstract

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We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time H s bounds in terms of the H s size of the initial data for s - 1 4 (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in H - 1 (joint work with T. Buckmaster [2]).

How to cite

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Koch, Herbert. "Bounds for KdV and the 1-d cubic NLS equation in rough function spaces." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-10. <http://eudml.org/doc/251156>.

@article{Koch2011-2012,
abstract = {We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^\{s\}$ bounds in terms of the $H^s$ size of the initial data for $s \ge -\frac\{1\}\{4\}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^\{-1\}$ (joint work with T. Buckmaster [2]).},
affiliation = {Mathematisches Institut Universität Bonn},
author = {Koch, Herbert},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {nonlinear Schrödinger equation; Korteweg-de Vries equation},
language = {eng},
pages = {1-10},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Bounds for KdV and the 1-d cubic NLS equation in rough function spaces},
url = {http://eudml.org/doc/251156},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Koch, Herbert
TI - Bounds for KdV and the 1-d cubic NLS equation in rough function spaces
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 10
AB - We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^{s}$ bounds in terms of the $H^s$ size of the initial data for $s \ge -\frac{1}{4}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster [2]).
LA - eng
KW - nonlinear Schrödinger equation; Korteweg-de Vries equation
UR - http://eudml.org/doc/251156
ER -

References

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