Energy and local energy bounds for the 1-d cubic NLS equation in H -1 4
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, p. 955-988
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},
     publisher = {Elsevier},
     volume = {29},
     number = {6},
     year = {2012},
     pages = {955-988},
     doi = {10.1016/j.anihpc.2012.05.006},
     zbl = {1280.35137},
     language = {en},
     url = {http://http://www.numdam.org/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) no. 6, pp. 955-988. doi : 10.1016/j.anihpc.2012.05.006. http://www.numdam.org/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