Energy and local energy bounds for the 1-d cubic NLS equation in H -1 4
Koch, Herbert ; Tataru, Daniel
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 955-988 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We consider the cubic nonlinear Schrödinger equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a priori local in time H s bounds in terms of the H s size of the initial data for s⩾-1 4. This improves earlier results of Christ, Colliander and Tao [3] and of the authors (Koch and Tataru, 2007 [13]). The new ingredients are a localization in space and local energy decay, which we hope to be of independent interest.

@article{AIHPC_2012__29_6_955_0,
     author = {Koch, Herbert and Tataru, Daniel},
     title = {Energy and local energy bounds for the 1-d cubic NLS equation in $ {H}^{-\frac{1}{4}}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {955-988},
     doi = {10.1016/j.anihpc.2012.05.006},
     zbl = {1280.35137},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_6_955_0}
}

Koch, Herbert; Tataru, Daniel. Energy and local energy bounds for the 1-d cubic NLS equation in $ {H}^{-\frac{1}{4}}$. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 955-988. doi : 10.1016/j.anihpc.2012.05.006. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_6_955_0/

[1] S.A. Akhmanov, R.V. Khokhlov, A.P. Sukhorukov, Self-focusing and self-trapping of intense light beams in a nonlinear medium, Zh. Eksp. Teor. Fiz. 50 (1966), 1537-1549

[2] Michael Christ, personal communication.

[3] Michael Christ, James Colliander, Terrence Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, arXiv:math.AP/0612457 | MR 2376575 | Zbl 1136.35087

[4] Michael Christ, James Colliander, Terrence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 no. 6 (2003), 1235-1293 | MR 2018661 | Zbl 1048.35101

[5] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 no. 2 (1993), 295-368 | MR 1207209 | Zbl 0771.35042

[6] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc. 126 no. 2 (1998), 523-530 | MR 1425123 | Zbl 0910.35115

[7] Martin Hadac, Sebastian Herr, Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 3 (2009), 917-941 | Numdam | MR 2526409 | Zbl 1169.35372

[8] Shan Jin, C. David Levermore, David W. Mclaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 no. 5 (1999), 613-654 | MR 1670048 | Zbl 0935.35148

[9] S. Kamvissis, Long time behavior for semiclassical NLS, Appl. Math. Lett. 12 no. 8 (1999), 35-57 | MR 1751356 | Zbl 0978.35061

[10] Spyridon Kamvissis, Kenneth D.T.-R. Mclaughlin, Peter D. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Ann. of Math. Stud. vol. 154, Princeton University Press, Princeton, NJ (2003) | MR 1999840 | Zbl 1057.35063

[11] Carlos E. Kenig, Gustavo Ponce, Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 no. 3 (2001), 617-633 | MR 1813239 | Zbl 1034.35145

[12] Herbert Koch, Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 no. 2 (2005), 217-284 | MR 2094851 | Zbl 1078.35143

[13] Herbert Koch, Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 2007 no. 16 (2007) | MR 2353092 | Zbl 1169.35055

[14] Tadahiro Oh, Catherine Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below L 2 , arXiv:1007.2073v2 (2010) | MR 2892769 | Zbl 1258.35184

[15] Junkichi Satsuma, Nobuo Yajima, Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Progr. Theoret. Phys. Suppl. 55 (1974), 284-306 | MR 463733

[16] Terence Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math. vol. 106 (2006) | MR 2233925 | Zbl 1106.35001

[17] Laurent Thomann, Instabilities for supercritical Schrödinger equations in analytic manifolds, J. Differential Equations 245 no. 1 (2008), 249-280 | MR 2422717 | Zbl 1157.35107