Center for diffusion of mathematic journals

 
 
 
 

Séminaire Laurent Schwartz — EDP et applications

Table of contents for this volume | Previous article | Next article
Herbert Koch
Bounds for KdV and the 1-d cubic NLS equation in rough function spaces
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exp. No. 11, 10 p., doi: 10.5802/slsedp.8
Article PDF

Résumé - 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]).

Bibliography

[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.
[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.  Zbl 1136.35087
[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.  MR 2018661 |  Zbl 1048.35101
[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.  MR 1207209 |  Zbl 0771.35042
[6] E. Grenier. Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Amer. Math. Soc., 126(2):523–530, 1998.  MR 1425123 |  Zbl 0910.35115
[7] Zihua Guo. Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\mathbb{R})$. J. Math. Pures Appl. (9), 91(6):583–597, 2009.  MR 2531556 |  Zbl 1173.35110
[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.  MR 1670048 |  Zbl 0935.35148
[9] S. Kamvissis. Long time behavior for semiclassical NLS. Appl. Math. Lett., 12(8):35–57, 1999.  MR 1751356 |  Zbl 0978.35061
[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.  MR 1999840 |  Zbl 1057.35063
[11] T. Kappeler and P. Topalov. Global wellposedness of KdV in $H^{-1}(\mathbb{T},\mathbb{R})$. Duke Math. J., 135(2):327–360, 2006.  MR 2267286 |  Zbl 1106.35081
[12] Thomas Kappeler, Peter Perry, Mikhail Shubin, and Peter Topalov. The Miura map on the line. Int. Math. Res. Not., (50):3091–3133, 2005.  MR 2189502 |  Zbl 1089.35058
[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.  MR 1813239 |  Zbl 1034.35145
[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.  MR 2353092 |  Zbl 1169.35055
[16] Yvan Martel and Frank Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal., 157(3):219–254, 2001.  MR 1826966 |  Zbl 0981.35073
[17] F. Merle and L. Vega. $L^2$ stability of solitons for KdV equation. Int. Math. Res. Not., (13):735–753, 2003.  MR 1949297 |  Zbl 1022.35061
[18] Luc Molinet. A note on ill posedness for the KdV equation. Differential Integral Equations, 24(7-8):759–765, 2011.  MR 2830706 |  Zbl 1249.35292
[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.  MR 463733
[20] Laurent Thomann. Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Differential Equations, 245(1):249–280, 2008.  MR 2422717 |  Zbl 1157.35107
Copyright Cellule MathDoc 2019 | Credit | Site Map