# American Institute of Mathematical Sciences

January  2011, 10(1): 127-140. doi: 10.3934/cpaa.2011.10.127

## Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation

 1 Department of Mathematics, University of California - Berkeley, 970 Evans Hall, Number 3840; Berkeley, CA 94720-3840, USA Government

Received  January 2010 Revised  March 2010 Published  November 2010

We prove global well-posedness for the cubic, defocusing, nonlinear Schrödinger equation on $R^2$ with data $u_0 \in H^s(R^2)$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].
Citation: Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127
##### References:
 [1] J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 5 (1998), 253-283. doi: doi:10.1155/S1073792898000191. [2] J. Bourgain, "Global Solutions Of Nonlinear Schrödinger Equations," American Mathematical Society, Providence, 1999. [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: doi:10.1007/BF01258601. [4] 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: doi:10.1016/0362-546X(90)90023-A. [5] J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $R^2$, Int. Math. Res. Not. IMRN, 23 (2007), Art. ID rnm090, 30. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-686. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $R^3$, Commun. Pure Appl. Anal., 57 (2004), 987-1014. doi: doi:10.1002/cpa.20029. [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686. doi: doi:10.3934/dcds.2008.21.665. [9] J. Colliander and T. Roy, Bootstrapped Morawetz Estimates and Resonant Decomposition for Low Regularity Global solutions of Cubic NLS on $\mathbf{R}^{2}$, preprint, arXiv:0811.1803. [10] D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions, Commun. Pure Appl. Anal., 6 (2007), 1023-1041. doi: doi:10.3934/cpaa.2007.6.1023. [11] Y. F. Fang and M. G. Grillakis, On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $R^{2+1}$, J. Hyperbolic Differ. Equ., 4 (2007), 233-257. doi: doi:10.1142/S0219891607001161. [12] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math, 120 (1998), 955-980. doi: doi:10.1353/ajm.1998.0039. [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, preprint, arXiv:0707.3188v2, Journal of the European Mathematical Society, 11 (2009) 1203-1258. doi: doi:10.4171/JEMS/180. [14] C. D. Sogge, "Fourier Integrals in Classical Analysis," Cambridge University Press, Cambridge, 1993. [15] E. M. Stein, "Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, NJ, 1993. [16] T. Tao, "Nonlinear Dispersive Equations," Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. [17] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhäuser, Boston, 1991. [18] M. E. Taylor, "Partial Differential Equations I - III," Springer-Verlag, New York, 1996. [19] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

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##### References:
 [1] J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 5 (1998), 253-283. doi: doi:10.1155/S1073792898000191. [2] J. Bourgain, "Global Solutions Of Nonlinear Schrödinger Equations," American Mathematical Society, Providence, 1999. [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: doi:10.1007/BF01258601. [4] 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: doi:10.1016/0362-546X(90)90023-A. [5] J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $R^2$, Int. Math. Res. Not. IMRN, 23 (2007), Art. ID rnm090, 30. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-686. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $R^3$, Commun. Pure Appl. Anal., 57 (2004), 987-1014. doi: doi:10.1002/cpa.20029. [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686. doi: doi:10.3934/dcds.2008.21.665. [9] J. Colliander and T. Roy, Bootstrapped Morawetz Estimates and Resonant Decomposition for Low Regularity Global solutions of Cubic NLS on $\mathbf{R}^{2}$, preprint, arXiv:0811.1803. [10] D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions, Commun. Pure Appl. Anal., 6 (2007), 1023-1041. doi: doi:10.3934/cpaa.2007.6.1023. [11] Y. F. Fang and M. G. Grillakis, On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $R^{2+1}$, J. Hyperbolic Differ. Equ., 4 (2007), 233-257. doi: doi:10.1142/S0219891607001161. [12] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math, 120 (1998), 955-980. doi: doi:10.1353/ajm.1998.0039. [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, preprint, arXiv:0707.3188v2, Journal of the European Mathematical Society, 11 (2009) 1203-1258. doi: doi:10.4171/JEMS/180. [14] C. D. Sogge, "Fourier Integrals in Classical Analysis," Cambridge University Press, Cambridge, 1993. [15] E. M. Stein, "Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, NJ, 1993. [16] T. Tao, "Nonlinear Dispersive Equations," Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. [17] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhäuser, Boston, 1991. [18] M. E. Taylor, "Partial Differential Equations I - III," Springer-Verlag, New York, 1996. [19] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
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