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 & 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. doi: doi:10.1155/S1073792898000191. Google Scholar [2] J. Bourgain, "Global Solutions Of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999). Google Scholar [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477. doi: doi:10.1007/BF01258601. Google Scholar [4] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807. doi: doi:10.1016/0362-546X(90)90023-A. Google Scholar [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). Google Scholar [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. Google Scholar [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. doi: doi:10.1002/cpa.20029. Google Scholar [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. doi: doi:10.3934/dcds.2008.21.665. Google Scholar [9] J. Colliander and T. Roy, Bootstrapped Morawetz Estimates and Resonant Decomposition for Low Regularity Global solutions of Cubic NLS on $\mathbfR^{2}$,, preprint, (). Google Scholar [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. doi: doi:10.3934/cpaa.2007.6.1023. Google Scholar [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. doi: doi:10.1142/S0219891607001161. Google Scholar [12] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math, 120 (1998), 955. doi: doi:10.1353/ajm.1998.0039. Google Scholar [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data,, preprint, 11 (2009), 1203. doi: doi:10.4171/JEMS/180. Google Scholar [14] C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge University Press, (1993). Google Scholar [15] E. M. Stein, "Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton University Press, (1993). Google Scholar [16] T. Tao, "Nonlinear Dispersive Equations,", Published for the Conference Board of the Mathematical Sciences, (2006). Google Scholar [17] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkh\, (1991). Google Scholar [18] M. E. Taylor, "Partial Differential Equations I - III,", Springer-Verlag, (1996). Google Scholar [19] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, Funkcial. Ekvac., 30 (1987), 115. Google Scholar

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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. doi: doi:10.1155/S1073792898000191. Google Scholar [2] J. Bourgain, "Global Solutions Of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999). Google Scholar [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477. doi: doi:10.1007/BF01258601. Google Scholar [4] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807. doi: doi:10.1016/0362-546X(90)90023-A. Google Scholar [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). Google Scholar [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. Google Scholar [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. doi: doi:10.1002/cpa.20029. Google Scholar [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. doi: doi:10.3934/dcds.2008.21.665. Google Scholar [9] J. Colliander and T. Roy, Bootstrapped Morawetz Estimates and Resonant Decomposition for Low Regularity Global solutions of Cubic NLS on $\mathbfR^{2}$,, preprint, (). Google Scholar [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. doi: doi:10.3934/cpaa.2007.6.1023. Google Scholar [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. doi: doi:10.1142/S0219891607001161. Google Scholar [12] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math, 120 (1998), 955. doi: doi:10.1353/ajm.1998.0039. Google Scholar [13] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data,, preprint, 11 (2009), 1203. doi: doi:10.4171/JEMS/180. Google Scholar [14] C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge University Press, (1993). Google Scholar [15] E. M. Stein, "Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton University Press, (1993). Google Scholar [16] T. Tao, "Nonlinear Dispersive Equations,", Published for the Conference Board of the Mathematical Sciences, (2006). Google Scholar [17] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkh\, (1991). Google Scholar [18] M. E. Taylor, "Partial Differential Equations I - III,", Springer-Verlag, (1996). Google Scholar [19] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, Funkcial. Ekvac., 30 (1987), 115. Google Scholar
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