March  2011, 10(2): 397-414. doi: 10.3934/cpaa.2011.10.397

Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$

1. 

Department of Mathematics University of Toronto, 100 St. George St, Room 4072 Toronto, Ontario M5S 3G3

2. 

Department Of Mathematics, University of California, Los Angeles, CA, USA Government

Received  June 2010 Revised  October 2010 Published  December 2010

We prove global well-posedness for the $L^2$-critical cubic defocusing nonlinear Schrödinger equation on $R^2$ with data $u_0 \in H^s(R^2)$ for $ s > \frac{1}{3}$. The proof combines a priori Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of $X^{s,b}$ spaces. The second one is to control the error of the a priori Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.
Citation: J. Colliander, Tristan Roy. Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$. Communications on Pure & Applied Analysis, 2011, 10 (2) : 397-414. doi: 10.3934/cpaa.2011.10.397
References:
[1]

J. Bourgain, Refinement of Strichartz inequality and applications to $2D-NLS$ 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 critical nonlinear Schrödinger equation in $H^s$,, Non. Anal. TMA, 14 (1990), 807.   Google Scholar

[4]

J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $R^2$ ,, Int. Math. Res. Not., 23 (2007).   Google Scholar

[5]

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. Letters, 9 (2002), 659.   Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schröinger equation on $Bbb R^2$,, Discrete Contin. Dyn. Syst., 21 (2008), 665.  doi: doi:10.3934/dcds.2008.21.665.  Google Scholar

[7]

Y. Fang and M. 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.   Google Scholar

[8]

R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equations in two dimensions with radial data,, Journ. Eur. Math. Soc. (JEMS), 11 (2009), 1203.   Google Scholar

show all references

References:
[1]

J. Bourgain, Refinement of Strichartz inequality and applications to $2D-NLS$ 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 critical nonlinear Schrödinger equation in $H^s$,, Non. Anal. TMA, 14 (1990), 807.   Google Scholar

[4]

J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $R^2$ ,, Int. Math. Res. Not., 23 (2007).   Google Scholar

[5]

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. Letters, 9 (2002), 659.   Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schröinger equation on $Bbb R^2$,, Discrete Contin. Dyn. Syst., 21 (2008), 665.  doi: doi:10.3934/dcds.2008.21.665.  Google Scholar

[7]

Y. Fang and M. 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.   Google Scholar

[8]

R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equations in two dimensions with radial data,, Journ. Eur. Math. Soc. (JEMS), 11 (2009), 1203.   Google Scholar

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