Geodesic
Global solutions for 1D cubic defocusing dispersive equations: Part I
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both small and localized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data.In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for $L^2$ initial data which are small and nonlocalized. Our main structural assumption is that our nonlinearity is defocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team.In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1 There, by scaling, our result also admits a large data counterpart. 
@article{10_1017_fmp_2023_30,
     author = {Mihaela Ifrim and Daniel Tataru},
     title = {Global solutions for {1D} cubic defocusing dispersive equations: {Part} {I}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {11},
     year = {2023},
     doi = {10.1017/fmp.2023.30},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.30/}
}
TY  - JOUR
AU  - Mihaela Ifrim
AU  - Daniel Tataru
TI  - Global solutions for 1D cubic defocusing dispersive equations: Part I
JO  - Forum of Mathematics, Pi
PY  - 2023
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.30/
DO  - 10.1017/fmp.2023.30
LA  - en
ID  - 10_1017_fmp_2023_30
ER  - 
%0 Journal Article
%A Mihaela Ifrim
%A Daniel Tataru
%T Global solutions for 1D cubic defocusing dispersive equations: Part I
%J Forum of Mathematics, Pi
%D 2023
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.30/
%R 10.1017/fmp.2023.30
%G en
%F 10_1017_fmp_2023_30
Mihaela Ifrim; Daniel Tataru. Global solutions for 1D cubic defocusing dispersive equations: Part I. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.30

[1] Alazard, T. and Delort, J.-M., ‘Global solutions and asymptotic behavior for two dimensional gravity water waves’, Ann. Sci. Éc. Norm. Supér. (4) 48(5) (2015), 1149–1238.Google Scholar

[2] Borghese, M., Jenkins, R. and Mclaughlin, K. D. T.-R., ‘Long time asymptotic behavior of the focusing nonlinear Schrödinger equation’, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35(4) (2018), 887–920.Google Scholar | DOI

[3] Colliander, J., Grillakis, M. and Tzirakis, N., ‘Tensor products and correlation estimates with applications to nonlinear Schrödinger equations’, Comm. Pure Appl. Math. 62(7) (2009), 920–968.Google Scholar | DOI

[4] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation’, Math. Res. Lett. 9(5–6) (2002), 659–682.Google Scholar | DOI

[5] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ’, Comm. Pure Appl. Math. 57(8) (2004), 987–1014.Google Scholar | DOI

[6] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ’, Ann. of Math. (2) 167(3) (2008), 767–865.Google Scholar | DOI

[7] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Resonant decompositions and the -method for the cubic nonlinear Schrödinger equation on ’, Discrete Contin. Dyn. Syst. 21(3) (2008), 665–686.Google Scholar | DOI

[8] Deift, P. and Zhou, X., ‘Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space’, Comm. Pure Appl. Math. 56(8) (2003), 1029–1077. Dedicated to the memory of Jürgen K. Moser.Google Scholar | DOI

[9] Delort, J.-M., ‘Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations’, Ann. Inst. Fourier (Grenoble) 66(4) (2016), 1451–1528.Google Scholar | DOI

[10] Dodson, B., ‘Global well-posedness and scattering for the defocusing, critical, nonlinear Schrödinger equation when ’, Amer. J. Math. 138(2) (2016), 531–569.Google Scholar | DOI

[11] Dodson, B., ‘Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation’, Ann. PDE 3(1) (2017), Paper No. 5, 35.Google Scholar | DOI

[12] Hayashi, N. and Naumkin, P. I., ‘Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations’, Amer. J. Math. 120(2) (1998), 369–389.Google Scholar | DOI

[13] Hayashi, N. and Naumkin, P. I., ‘Large time asymptotics for the fractional nonlinear Schrödinger equation’, Adv. Differential equations 25(1–2) (2020), 31–80.Google Scholar | DOI

[14] Ifrim, M. and Tataru, D., ‘Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension’, Nonlinearity 28(8) (2015), 2661–2675.Google Scholar | DOI

[15] Ifrim, M. and Tataru, D., ‘Two dimensional water waves in holomorphic coordinates II: Global solutions’, Bull. Soc. Math. France 144(2) (2016), 369–394.Google Scholar | DOI

[16] Ifrim, M. and Tataru, D., ‘The lifespan of small data solutions in two dimensional capillary water waves’, Arch. Ration. Mech. Anal. 225(3) (2017), 1279–1346.Google Scholar | DOI

[17] Ifrim, M. and Tataru, D., ‘Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation’, Ann. Sci. Éc. Norm. Supér. (4) 52(2) (2019), 297–335.Google Scholar | DOI

[18] Kato, J. and Pusateri, F., ‘A new proof of long-range scattering for critical nonlinear Schrödinger equations’, Differential Integral equations 24(9–10) (2011), 923–940.Google Scholar | DOI

[19] Koch, H. and Tataru, D., ‘Energy and local energy bounds for the 1-d cubic NLS equation in ’, Ann. Inst. H. Poincaré Anal. Non Linéaire 29(6) (2012), 955–988.Google Scholar | DOI

[20] Lindblad, H., Lührmann, J. and Soffer, A., ‘Asymptotics for 1D Klein–Gordon equations with variable coefficient quadratic nonlinearities’, Arch. Ration. Mech. Anal. 241(3) (2021), 1459–1527.Google Scholar | DOI

[21] Lindblad, H. and Soffer, A., ‘Scattering and small data completeness for the critical nonlinear Schrödinger equation’, Nonlinearity 19(2) (2006), 345–353.Google Scholar | DOI

[22] Planchon, F. and Vega, L., ‘Bilinear virial identities and applications’, Ann. Sci. Éc. Norm. Supér. (4) 42(2) (2009), 261–290.Google Scholar | DOI

[23] Ryckman, E. and Visan, M., ‘Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ’, Amer. J. Math. 129(1) (2007), 1–60.Google Scholar | DOI

[24] Tao, T., ‘Global regularity of wave maps. II. Small energy in two dimensions’, Comm. Math. Phys. 224(2) (2001), 443–544.Google Scholar | DOI

[25] Tao, T., ‘Global well-posedness of the Benjamin–Ono equation in ’, J. Hyperbolic Differ. Equ. 1(1) (2004), 27–49.Google Scholar | DOI

Cité par Sources :