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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  1. S. A. Akhmanov, R.V. Khokhlov, and A. P. Sukhorukov. Self-focusing and self-trapping of intense light beams in a nonlinear medium. Zh. Eksp. Teor. Fiz., 50:1537–1549, 1966. 
  2. T. Buckmaster and H. Koch. The korteweg-de-vries equation at h - 1 regularity. arXiv:1112.4657, 2011. Zbl1331.35300
  3. Michael Christ, James Colliander, and Terrence Tao. A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. Preprint arXiv:math.AP/0612457. Zbl1136.35087
  4. Michael Christ, James Colliander, and Terrence Tao. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math., 125(6):1235–1293, 2003. Zbl1048.35101MR2018661
  5. P. Deift and X. Zhou. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2), 137(2):295–368, 1993. Zbl0771.35042MR1207209
  6. E. Grenier. Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc., 126(2):523–530, 1998. Zbl0910.35115MR1425123
  7. Zihua Guo. Global well-posedness of Korteweg-de Vries equation in H - 3 / 4 ( ℝ ) . J. Math. Pures Appl. (9), 91(6):583–597, 2009. Zbl1173.35110MR2531556
  8. Shan Jin, C. David Levermore, and David W. McLaughlin. The semiclassical limit of the defocusing NLS hierarchy. Comm. Pure Appl. Math., 52(5):613–654, 1999. Zbl0935.35148MR1670048
  9. S. Kamvissis. Long time behavior for semiclassical NLS. Appl. Math. Lett., 12(8):35–57, 1999. Zbl0978.35061MR1751356
  10. Spyridon Kamvissis, Kenneth D. T.-R. McLaughlin, and Peter D. Miller. Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation, volume 154 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2003. Zbl1057.35063MR1999840
  11. T. Kappeler and P. Topalov. Global wellposedness of KdV in H - 1 ( 𝕋 , ℝ ) . Duke Math. J., 135(2):327–360, 2006. Zbl1106.35081MR2267286
  12. Thomas Kappeler, Peter Perry, Mikhail Shubin, and Peter Topalov. The Miura map on the line. Int. Math. Res. Not., (50):3091–3133, 2005. Zbl1089.35058MR2189502
  13. Carlos E. Kenig, Gustavo Ponce, and Luis Vega. On the ill-posedness of some canonical dispersive equations. Duke Math. J., 106(3):617–633, 2001. Zbl1034.35145MR1813239
  14. H. Koch and D. Tataru. Energy and local energy bounds for the 1-d cubic NLS equation in H - 1 / 4 . arxiv:1012.0148, 2010. 
  15. Herbert Koch and Daniel Tataru. A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Int. Math. Res. Not. IMRN, 16:Art. ID rnm053, 36, 2007. Zbl1169.35055MR2353092
  16. Yvan Martel and Frank Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal., 157(3):219–254, 2001. Zbl0981.35073MR1826966
  17. F. Merle and L. Vega. L 2 stability of solitons for KdV equation. Int. Math. Res. Not., (13):735–753, 2003. Zbl1022.35061MR1949297
  18. Luc Molinet. A note on ill posedness for the KdV equation. Differential Integral Equations, 24(7-8):759–765, 2011. Zbl1249.35292MR2830706
  19. Junkichi Satsuma and Nobuo Yajima. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Progr. Theoret. Phys. Suppl. No. 55, pages 284–306, 1974. MR463733
  20. Laurent Thomann. Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Differential Equations, 245(1):249–280, 2008. Zbl1157.35107MR2422717

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