The defocusing quintic NLS in four space dimensions

Dodson, Benjamin; Miao, Changxing; Murphy, Jason; Zheng, Jiqiang

doi:10.1016/j.anihpc.2016.05.004

Dodson, Benjamin ; Miao, Changxing ; Murphy, Jason ; Zheng, Jiqiang

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 759-787.

Résumé

We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

MR Zbl

DOI : 10.1016/j.anihpc.2016.05.004

Mots clés : Nonlinear Schrödinger equation, Concentration compactness, Scattering, Interaction Morawetz inequality

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     author = {Dodson, Benjamin and Miao, Changxing and Murphy, Jason and Zheng, Jiqiang},
     title = {The defocusing quintic {NLS} in four space dimensions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {759--787},
     publisher = {Elsevier},
     volume = {34},
     number = {3},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.05.004},
     zbl = {1367.35154},
     mrnumber = {3633744},
     language = {en},
     url = {http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.004/}
}

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Dodson, Benjamin; Miao, Changxing; Murphy, Jason; Zheng, Jiqiang. The defocusing quintic NLS in four space dimensions. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 759-787. doi : 10.1016/j.anihpc.2016.05.004. http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.004/

Bibliographie
Cité par

[1] Bégout, P.; Vargas, A. Mass concentration phenomena for the $L^{2}$ -critical nonlinear Schrödinger equation, Trans. Am. Math. Soc., Volume 359 (2007) no. 11, pp. 5257–5282 (MR2327030) | DOI | MR | Zbl

[2] Bourgain, J. Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Am. Math. Soc., Volume 12 (1999) no. 1, pp. 145–171 (MR1626257) | DOI | MR | Zbl

[3] Carles, R.; Keraani, S. On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The $L^{2}$ -critical case, Trans. Am. Math. Soc., Volume 359 (2007) no. 1, pp. 33–62 (MR2247881) | DOI | MR | Zbl

[4] Cazenave, T. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, 2003 (MR2002047) | DOI | MR | Zbl

[5] Cazenave, T.; Weissler, F. The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$ , Nonlinear Anal., Volume 14 (1990) no. 10, pp. 807–836 (MR1055532) | DOI | MR | Zbl

[6] Christ, M.; Weinstein, M. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., Volume 100 (1991) no. 1, pp. 87–109 (MR1124294) | DOI | MR | Zbl

[7] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $R^{3}$ , Commun. Pure Appl. Math., Volume 57 (2004) no. 8, pp. 987–1014 (MR2053757) | DOI | MR | Zbl

[8] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^{3}$ , Ann. Math. (2), Volume 167 (2008) no. 3, pp. 767–865 (MR2415387) | DOI | MR | Zbl

[9] Dodson, B. Global well-posedness and scattering for the defocusing, $L^{2}$ -critical nonlinear Schrödinger equation when $d \geq 3$ , J. Am. Math. Soc., Volume 25 (2012) no. 2, pp. 429–463 (MR2869023) | DOI | MR | Zbl

[10] Dodson, B. Global well-posedness and scattering for the defocusing, $L^{2}$ -critical nonlinear Schrödinger equation when $d = 2$ (Preprint) | arXiv | Zbl

[11] Dodson, B. Global well-posedness and scattering for the defocusing, $L^{2}$ critical, nonlinear Schrödinger equation when $d = 1$ , Am. J. Math., Volume 138 (2016) no. 2, pp. 531–569 (MR3483476) | DOI | MR | Zbl

[12] Dodson, B. Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., Volume 285 (2015), pp. 1589–1618 (MR3406535) | DOI | MR | Zbl

[13] Dodson, B. Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension $d = 4$ for initial data below a ground state threshold (Preprint) | arXiv

[14] Ginibre, J.; Velo, G. Smoothing properties and retarded estimates for some dispersive evolution equations, Comment. Phys.-Math., Volume 144 (1992) no. 1, pp. 163–188 (MR1151250) | MR | Zbl

[15] Grillakis, G. On nonlinear Schrödinger equations, Commun. Partial Differ. Equ., Volume 25 (2000) no. 9–10, pp. 1827–1844 (MR1778782) | MR | Zbl

[16] Holmer, J.; Roudenko, S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comment. Phys.-Math., Volume 282 (2008) no. 2, pp. 435–467 (MR2421484) | MR | Zbl

[17] Keel, M.; Tao, T. Endpoint Strichartz estimates, Am. J. Math., Volume 120 (1998) no. 5, pp. 955–980 (MR1646048) | DOI | MR | Zbl

[18] Kenig, C.; Merle, F. Global well-posedness, scattering, and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006) no. 3, pp. 645–675 (MR2257393) | DOI | MR | Zbl

[19] Kenig, C.; Merle, F. Scattering for ${\dot{H}}^{1 / 2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Am. Math. Soc., Volume 362 (2010) no. 4, pp. 1937–1962 (MR2574882) | DOI | MR | Zbl

[20] Keraani, S. On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Differ. Equ., Volume 175 (2001) no. 2, pp. 353–392 (MR1855973) | DOI | MR | Zbl

[21] Keraani, S. On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., Volume 235 (2006) no. 1, pp. 171–192 (MR2216444) | DOI | MR | Zbl

[22] Killip, R.; Tao, T.; Visan, M. The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc., Volume 11 (2009) no. 6, pp. 1203–1258 (MR2557134) | MR | Zbl

[23] Killip, R.; Visan, M. Energy-supercritical NLS: critical ${\dot{H}}^{s}$ -bounds imply scattering, Commun. Partial Differ. Equ., Volume 35 (2010) no. 6, pp. 945–987 (MR2753625) | DOI | MR | Zbl

[24] Killip, R.; Visan, M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Am. J. Math., Volume 132 (2010) no. 2, pp. 361–424 (MR2654778) | MR | Zbl

[25] Killip, R.; Visan, M. Nonlinear Schrödinger equations at critical regularity, Clay Math. Proc., Volume 17 (2013), pp. 325–437 (MR3098643) | MR | Zbl

[26] Killip, R.; Visan, M. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, Volume 5 (2012) no. 4, pp. 855–885 (MR3006644) | DOI | MR | Zbl

[27] Killip, R.; Visan, M.; Zhang, X. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, Volume 1 (2008) no. 2, pp. 229–266 (MR2472890) | DOI | MR | Zbl

[28] Merle, F.; Vega, L. Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not. (1998) no. 8, pp. 399–425 (MR1628235) | DOI | MR | Zbl

[29] Miao, C.; Murphy, J.; Zheng, J. The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., Volume 267 (2014) no. 6, pp. 1662–1724 (MR3237770) | DOI | MR | Zbl

[30] Murphy, J. Intercritical NLS: critical ${\dot{H}}^{s}$ -bounds imply scattering, SIAM J. Math. Anal., Volume 46 (2014) no. 1, pp. 939–997 (MR3166962) | DOI | MR | Zbl

[31] Murphy, J. The ${\dot{H}}^{1 / 2}$ -critical NLS in high dimensions, Discrete Contin. Dyn. Syst., Ser. A, Volume 34 (2014) no. 2, pp. 733–748 (MR3094603) | DOI | MR | Zbl

[32] Murphy, J. The radial defocusing nonlinear Schrödinger equation in three space dimensions, Commun. Partial Differ. Equ., Volume 40 (2015) no. 2, pp. 265–308 (MR3277927) | DOI | MR | Zbl

[33] Ryckman, E.; Visan, M. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $R^{1 + 4}$ , Am. J. Math., Volume 129 (2007) no. 1, pp. 1–60 (MR2288737) | DOI | MR | Zbl

[34] Shao, S. Maximizers for the Strichartz and Sobolev–Strichartz inequalities for the Schrödinger equation, Electron. J. Differ. Equ., Volume 3 (2009), pp. 13 pp. (MR2471112) | MR | Zbl

[35] Strichartz, R.S. Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Volume 44 (1977) no. 3, pp. 705–714 (MR0512086) | DOI | MR | Zbl

[36] Tao, T. Global well-posedness and scattering for higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, N.Y. J. Math., Volume 11 (2005), pp. 57–80 (MR2154347) | MR | Zbl

[37] Tao, T.; Visan, M.; Zhang, X. Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., Volume 140 (2007) no. 1, pp. 165–202 (MR2355070) | MR | Zbl

[38] Visan, M. The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, UCLA, 2006 (Ph.D thesis MR2709575) | MR

[39] Visan, M. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., Volume 138 (2007) no. 2, pp. 281–374 (MR2318286) | DOI | MR | Zbl

[40] Visan, M. Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not. (2012) no. 5, pp. 1037–1067 (MR2899959) | DOI | MR | Zbl

[41] Visan, M. Dispersive equations, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, vol. 45, Birkhäuser/Springer, Basel, 2014

[42] Xie, J.; Fang, D. Global well-posedness and scattering for the defocusing ${\dot{H}}^{s}$ -critical NLS, Chin. Ann. Math., Ser. B, Volume 34 (2013) no. 6, pp. 801–842 (MR3122297) | MR | Zbl

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