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September  2013, 33(9): 4095-4122. doi: 10.3934/dcds.2013.33.4095

## Resonant decomposition and the $I$-method for the two-dimensional Zakharov system

 1 Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan

Received  March 2012 Revised  December 2012 Published  March 2013

The initial value problem of the Zakharov system on a two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some `resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).
Citation: Nobu Kishimoto. Resonant decomposition and the $I$-method for the two-dimensional Zakharov system. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4095-4122. doi: 10.3934/dcds.2013.33.4095
##### References:
 [1] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020. [3] J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 1998, 253-283. doi: 10.1155/S1073792898000191. [4] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, 1996, 515-546. doi: 10.1155/S1073792896000359. [5] F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal., 9 (2010), 483-491. doi: 10.3934/cpaa.2010.9.483. [6] J. Ceccon and M. Montenegro, Optimal $L^p$-Riemannian Gagliardo-Nirenberg inequalities, Math. Z., 258 (2008), 851-873. doi: 10.1007/s00209-007-0202-8. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [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 $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686. doi: 10.3934/dcds.2008.21.665. [9] D. Fang, H. Pecher and S. Zhong, Low regularity global well-posedness for the two-dimensional Zakharov system, Analysis (Munich), 29 (2009), 265-281. doi: 10.1524/anly.2009.1018. [10] L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. II, Comm. Math. Phys., 160 (1994), 349-389. [11] N. Kishimoto, Local well-posedness for the Zakharov system on multidimensional torus, to appear in J. Anal. Math., arXiv:1109.3527. [12] N. Kishimoto and M. Maeda, Construction of blow-up solutions for Zakharov system on $\mathbbT ^2$, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, arXiv:1109.3528. [13] H. Pecher, Global rough solutions for the Zakharov system in two spatial dimensions, preprint, arXiv:1203.2173. [14] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,'' CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006. [15] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. [16] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

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##### References:
 [1] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020. [3] J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 1998, 253-283. doi: 10.1155/S1073792898000191. [4] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, 1996, 515-546. doi: 10.1155/S1073792896000359. [5] F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal., 9 (2010), 483-491. doi: 10.3934/cpaa.2010.9.483. [6] J. Ceccon and M. Montenegro, Optimal $L^p$-Riemannian Gagliardo-Nirenberg inequalities, Math. Z., 258 (2008), 851-873. doi: 10.1007/s00209-007-0202-8. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [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 $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686. doi: 10.3934/dcds.2008.21.665. [9] D. Fang, H. Pecher and S. Zhong, Low regularity global well-posedness for the two-dimensional Zakharov system, Analysis (Munich), 29 (2009), 265-281. doi: 10.1524/anly.2009.1018. [10] L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. II, Comm. Math. Phys., 160 (1994), 349-389. [11] N. Kishimoto, Local well-posedness for the Zakharov system on multidimensional torus, to appear in J. Anal. Math., arXiv:1109.3527. [12] N. Kishimoto and M. Maeda, Construction of blow-up solutions for Zakharov system on $\mathbbT ^2$, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, arXiv:1109.3528. [13] H. Pecher, Global rough solutions for the Zakharov system in two spatial dimensions, preprint, arXiv:1203.2173. [14] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,'' CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006. [15] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. [16] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
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