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

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

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

Jason Murphy ^1,

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

Received January 2013 Revised February 2013 Published August 2013

Abstract
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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.

Keywords: $\dot{H}^{1/2}$-critical, Nonlinear Schrödinger equation, scattering., Lin-Strauss Morawetz inequality, concentrationcompactness.

Mathematics Subject Classification: Primary: 35Q5.

Citation: Jason Murphy. The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions. Discrete and Continuous Dynamical Systems, 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), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.
[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), 33-62. doi: 10.1090/S0002-9947-06-03955-9.
[3]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
[5]	M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.
[6]	J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195.
[7]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.
[8]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[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), 645-675. doi: 10.1007/s00222-006-0011-4.
[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), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9.
[11]	S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.
[12]	S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005.
[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), 1203-1258. doi: 10.4171/JEMS/180.
[14]	R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, to appear in proceedings of the Clay summer school "Evolution Equations,'' June 23-July 18, Eidgenössische Technische Hochschule, Zürich, 2008.
[15]	R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.
[16]	R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering, Comm. Partial Differential Equations, 35 (2010), 945-987. doi: 10.1080/03605301003717084.
[17]	R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions, Proc. Amer. Math. Soc., 139 (2011), 1805-1817. doi: 10.1090/S0002-9939-2010-10615-9.
[18]	J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263. doi: 10.1016/0022-1236(78)90073-3.
[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), 399-425. doi: 10.1155/S1073792898000270.
[20]	J. Murphy, Inter-critical NLS: Critical $\dotH^s$-bounds imply scattering,, , ().
[21]	S. Shao, Maximizers for the Strichartz inequalities and Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009), 13 pp.
[22]	R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.
[23]	T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042.
[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), 165-202. doi: 10.1215/S0012-7094-07-14015-8.

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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), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.
[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), 33-62. doi: 10.1090/S0002-9947-06-03955-9.
[3]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
[5]	M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.
[6]	J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195.
[7]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.
[8]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[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), 645-675. doi: 10.1007/s00222-006-0011-4.
[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), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9.
[11]	S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.
[12]	S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005.
[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), 1203-1258. doi: 10.4171/JEMS/180.
[14]	R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, to appear in proceedings of the Clay summer school "Evolution Equations,'' June 23-July 18, Eidgenössische Technische Hochschule, Zürich, 2008.
[15]	R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.
[16]	R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering, Comm. Partial Differential Equations, 35 (2010), 945-987. doi: 10.1080/03605301003717084.
[17]	R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions, Proc. Amer. Math. Soc., 139 (2011), 1805-1817. doi: 10.1090/S0002-9939-2010-10615-9.
[18]	J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263. doi: 10.1016/0022-1236(78)90073-3.
[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), 399-425. doi: 10.1155/S1073792898000270.
[20]	J. Murphy, Inter-critical NLS: Critical $\dotH^s$-bounds imply scattering,, , ().
[21]	S. Shao, Maximizers for the Strichartz inequalities and Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009), 13 pp.
[22]	R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.
[23]	T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042.
[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), 165-202. doi: 10.1215/S0012-7094-07-14015-8.

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