Global well-posedness and scattering for the defocusing $\dot H^s$-critical NLS

Abstract

The authors consider the scattering phenomena of the defocusing $\dot H^s$-critical NLS. It is shown that if a solution of the defocusing NLS remains bounded in the critical homogeneous Sobolev norm on its maximal interval of existence, then the solution is global and scatters.

Keywords

Nonlinear / Schrödinger / equation / Scattering / Global / well-posedness

References

[1] Bourgain J. Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc., 1999, 12: 145-171

[2] Cazenave T. Semilinear Schrödinger Equations, 2003, New York: New York University, Courant Institute of Mathematical Sciences

[3] Cazenave T, Weissler F B. The Cauchy problem for the critical nonlinear Schrödinger equation in H ^s. Nonlinear Anal., Theory, Methods Appl., 1990, 14: 807-836

[4] Cazenave T, Weissler F B. Rapidly decaying solutions of the nonlinear Schrödinger equation. Comm. Math. Phys., 1992, 147(1): 75-100

[5] Cazenave T, Fang D, Han Z. Continuous dependence for NLS in fractional order spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2011, 28(1): 135-147

[6] Christ M, Weinstein M. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal., 1991, 100: 87-109

[7] Colliander J, Keel M, Staffilani G Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3. Comm. Pure Appl. Math., 2004, 57: 987-1014

[8] Colliander J, Keel M, Staffilani G Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ³. Ann. Math., 2008, 167: 767-865

[9] Dodson B. Global well-posedness and scattering for the defocusing, L ²-critical, nonlinear Schrödinger equation when d ≥ 3. J. Amer. Math. Soc., 2012, 25(2): 429-463

[10] Dodson, B., Global well-posedness and scattering for the defocusing, L ²-critical, nonlinear Schrödinger equation when d = 2, to appear, arXiv:1006.1375

[11] Dodson, B., Global well-posedness and scattering for the defocusing, L ²-critical, nonlinear Schrödinger equation when d = 1, to appear, arXiv:1010.0040

[12] Duyckaerts T, Holmer J, Roudenko S. Scattering for the non-radial 3D cubic nonlinear Schrödinger equations. Math. Res. Lett., 2008, 15: 1233-1250

[13] Fang D, Xie J, Cazenave T. Scattering for the focusing energy-subcritical NLS. Sci. China Math., 2011, 54(10): 2037-2062

[14] Gérard P. Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var., 1998, 3: 213-233

[15] Grillakis M. On nonlinear Schrödinger equations. Comm. Part. Diff. Eq., 2000, 25(9–10): 1827-1844

[16] Holmer, J. and Roudenko, S., On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express, 2007, 2007, artical ID abm004. DOI: 10.1093/amrx/abm004

[17] Holmer J, Roudenko S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equations. Commun. Math. Phys., 2008, 282: 435-467

[18] Kato T. An L ^{q, r}-theory for nonlinear Schrödinger equations. spectral and scattering theory and applications, Advanced Studies in Pure Mathematics, 1994, 23: 223-238

[19] Keel M, Tao T. Endpoint Strichartz estimates. Amer. J. Math., 1998, 120: 955-980

[20] Kenig C E, Merle F. Global well-posedness, scattering and blow up for the energycritical, focusing, nonlinear Schrödinger equation in the radial case. Invent. Math., 2006, 166: 645-675

[21] Kenig C E, Merle F. Scattering for $\dot H^{\tfrac{1}{2}}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc., 2010, 362(4): 1937-1962

[22] Keraani S. On the defect of compactness for the Strichartz estimates for the Schrödinger equations. J. Diff. Eq., 2001, 175: 353-392

[23] Killip R, Visan M. Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering. Comm. Part. Diff. Eq., 2010, 35: 945-987

[24] Killip R, Visan M. Nonlinear Schrödinger equations at critical regularity. Proceedings of Clay Summer School, 2008 1-12

[25] Killip R, Visan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Amer. J. Math., 2010, 132(2): 361-424

[26] Killip R, Visan M, Zhang X. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE, 2008, 1(2): 229-266

[27] Strauss W A. Existence of solitary waves in higher dimensions. Commun. Math. Phys., 1977, 55(2): 149-162

[28] Tao T, Visan M, Zhang X. Minimal-mass blowup solutions of the mass-critical NLS. Forum Math., 2008, 20(5): 881-919

[29] Triebel H. Interpolation Theory, Function Spaces, Differential Operators, 1978, Amsterdan: North-Holland

[30] Visan M. Global well-posedness and scattering for the defocusing cubic NLS in four dimensions. Int. Math. Res. Not., 2012, 5: 1037-1067

[31] Visan M. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J., 2007, 138: 281-374

[32] Weinstein M. Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys., 1982, 87(4): 567-576