Global well-posedness and scattering for the focusing nonlinear Schrödinger equation in the nonradial case

• Autorzy: Han P.
• Języki publikacji: EN

Abstrakty

EN

The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: [formula]. Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem proposed by Kenig and Merle [Invent. Math. 166 (2006), 645-675].

Słowa kluczowe

EN

critical energy; focusing Schrodinger equation; global well-posedness; scattering

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 3
• Strony: 487--504
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Han P.

• Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Beijing 100190, China,

Bibliografia

• [1] J. Bourgain, Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145–171.
• [2] D. Cao, P. Han, Inhomogeneous critical nonlinear Schrödinger equations with a harmonic potential, J. Math. Phys. 51 (2010), 043505, 24 pp.
• [3] R. Carles, S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The L2-critical case, Trans. Amer. Math. Soc. 359 (2007), 33–6.
• [4] T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
• [5] T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and Its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.
• [6] T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear Anal. 14 (1990), 807-836.
• [7] T. Cazenave, F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987 ), 18–29, Lecture Notes in Math., 1394, Springer, Berlin, (1989).
• [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoke, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3, Ann. Math. 167 (2008), 767–865.
• [9] T. Duyckaerts, J. Holmer, S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett. 15 (2008), 1233–1250.
• [10] T. Duyckaerts, F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), 1787–1840.
• [11] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979), 1–71.
• [12] R.T. Glassey, On the blowup of nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794–1797.
• [13] M.G. Grillakis, On nonlinear Schrödinger equations, Commun. Partial Differ. Equations.25 (2000), 1827–1844.
• [14] T. Hmidi, S. Keraani, Remarks on the blowup for the L2-critical nonlinear Schrödinger equations, SIAM J. Math. Anal. 38 (2006), 1035–1047.
• [15] J. Holmer, S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX 2007, No. 1, Art. ID abm004, 31 pp.
• [16] J. Holmer, S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), 435–467.
• [17] D. Li, X. Zhang, Dynamics for the energy critical nonlinear Schröinger equation in high dimensions, J. Funct. Anal. 256 (2009), 1928–1961.
• [18] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations. 175 (2001), 353–392.
• [19] C.E. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645–675.
• [20] J. Krieger, W. Schlag, Non-generic blow-up solutions for the critical focusing NLS in 1-D, J. Eur. Math. Soc. 11 (2009), 1–125.
• [21] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.
• [22] R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, arXiv:0804.1018v1 [math.AP].
• [23] F. Merle, Nonexistence of minimal blow-up solutions of equation iut = Δu-k(x)/u/4/N u in RN, Ann. Inst. Henri Poincaré Physique Thérique 64 (1996), 33–85.
• [24] T. Ogawa, Y. Tsutsumi, Blow-up of H1-solution for the nonlinear Schrödinger equation, J. Differ. Equations 92 (1991), 317–330.
• [25] T. Ogawa, Y. Tsutsumi, Blow-up of H1-solution for the nonlinear Schrödinger equation with critic power nonlinearity, Proc. Amer. Math. Soc. 111 (1991), 487–496.
• [26] E. Ryckman, M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R1+4, Amer. J. Math. 129 (2007), 1–60.
• [27] W. Schlag, Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. Math. 169 (2009), 139–227.
• [28] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353–372.
• [29] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math. 11 (2005), 57–80.
• [30] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007), 281–374.
• [31] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–576.