American Institute of Mathematical Sciences

February  2014, 34(2): 733-748. doi: 10.3934/dcds.2014.34.733

The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions

 1 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, United States

Received  January 2013 Revised  February 2013 Published  August 2013

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 4.$ In the spirit of Kenig and Merle [10], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.
Citation: Jason Murphy. The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 733-748. doi: 10.3934/dcds.2014.34.733
References:
 [1] P. Bégout and A. Vargas, Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 359 (2007), 5257. doi: 10.1090/S0002-9947-07-04250-X. Google Scholar [2] R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The $L^2$-critical case,, Trans. Amer. Math. Soc., 359 (2007), 33. doi: 10.1090/S0002-9947-06-03955-9. Google Scholar [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A. Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar [5] M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, J. Funct. Anal., 100 (1991), 87. doi: 10.1016/0022-1236(91)90103-C. Google Scholar [6] J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations,, Comm. Math. Phys., 144 (1992), 163. doi: 10.1007/BF02099195. Google Scholar [7] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation,, Comm. Math. Phys., 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar [8] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar [9] C. E. Kenig and 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. doi: 10.1007/s00222-006-0011-4. Google Scholar [10] C. E. Kenig and F. Merle, Scattering for $\dotH^{1/2)$ bounded solutions to the cubic, defocusing NLS in 3 dimensions,, Trans. Amer. Math. Soc. 362 (2010), 362 (2010), 1937. doi: 10.1090/S0002-9947-09-04722-9. Google Scholar [11] S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations,, J. Diff. Eq., 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar [12] S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation,, J. Funct. Anal., 235 (2006), 171. doi: 10.1016/j.jfa.2005.10.005. Google Scholar [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data,, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203. doi: 10.4171/JEMS/180. Google Scholar [14] R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity,, to appear in proceedings of the Clay summer school, (2008). Google Scholar [15] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, Amer. J. Math., 132 (2010), 361. doi: 10.1353/ajm.0.0107. Google Scholar [16] R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering,, Comm. Partial Differential Equations, 35 (2010), 945. doi: 10.1080/03605301003717084. Google Scholar [17] R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, Proc. Amer. Math. Soc., 139 (2011), 1805. doi: 10.1090/S0002-9939-2010-10615-9. Google Scholar [18] J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation,, J. Funct. Anal., 30 (1978), 245. doi: 10.1016/0022-1236(78)90073-3. Google Scholar [19] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2$D$,, Int. Math. Res. Not., (1998), 399. doi: 10.1155/S1073792898000270. Google Scholar [20] J. Murphy, Inter-critical NLS: Critical $\dotH^s$-bounds imply scattering,, , (). Google Scholar [21] S. Shao, Maximizers for the Strichartz inequalities and Sobolev-Strichartz inequalities for the Schrödinger equation,, Electron. J. Differential Equations, (2009). Google Scholar [22] R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar [23] T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS,, Forum Math., 20 (2008), 881. doi: 10.1515/FORUM.2008.042. Google Scholar [24] T. Tao, M. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions,, Duke Math. J., 140 (2007), 165. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar

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References:
 [1] P. Bégout and A. Vargas, Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 359 (2007), 5257. doi: 10.1090/S0002-9947-07-04250-X. Google Scholar [2] R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The $L^2$-critical case,, Trans. Amer. Math. Soc., 359 (2007), 33. doi: 10.1090/S0002-9947-06-03955-9. Google Scholar [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A. Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar [5] M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, J. Funct. Anal., 100 (1991), 87. doi: 10.1016/0022-1236(91)90103-C. Google Scholar [6] J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations,, Comm. Math. Phys., 144 (1992), 163. doi: 10.1007/BF02099195. Google Scholar [7] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation,, Comm. Math. Phys., 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar [8] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar [9] C. E. Kenig and 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. doi: 10.1007/s00222-006-0011-4. Google Scholar [10] C. E. Kenig and F. Merle, Scattering for $\dotH^{1/2)$ bounded solutions to the cubic, defocusing NLS in 3 dimensions,, Trans. Amer. Math. Soc. 362 (2010), 362 (2010), 1937. doi: 10.1090/S0002-9947-09-04722-9. Google Scholar [11] S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations,, J. Diff. Eq., 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar [12] S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation,, J. Funct. Anal., 235 (2006), 171. doi: 10.1016/j.jfa.2005.10.005. Google Scholar [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data,, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203. doi: 10.4171/JEMS/180. Google Scholar [14] R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity,, to appear in proceedings of the Clay summer school, (2008). Google Scholar [15] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, Amer. J. Math., 132 (2010), 361. doi: 10.1353/ajm.0.0107. Google Scholar [16] R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering,, Comm. Partial Differential Equations, 35 (2010), 945. doi: 10.1080/03605301003717084. Google Scholar [17] R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, Proc. Amer. Math. Soc., 139 (2011), 1805. doi: 10.1090/S0002-9939-2010-10615-9. Google Scholar [18] J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation,, J. Funct. Anal., 30 (1978), 245. doi: 10.1016/0022-1236(78)90073-3. Google Scholar [19] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2$D$,, Int. Math. Res. Not., (1998), 399. doi: 10.1155/S1073792898000270. Google Scholar [20] J. Murphy, Inter-critical NLS: Critical $\dotH^s$-bounds imply scattering,, , (). Google Scholar [21] S. Shao, Maximizers for the Strichartz inequalities and Sobolev-Strichartz inequalities for the Schrödinger equation,, Electron. J. Differential Equations, (2009). Google Scholar [22] R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar [23] T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS,, Forum Math., 20 (2008), 881. doi: 10.1515/FORUM.2008.042. Google Scholar [24] T. Tao, M. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions,, Duke Math. J., 140 (2007), 165. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar
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