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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 & Continuous Dynamical Systems - A, 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. doi: 10.1088/0951-7715/22/5/007. Google Scholar

[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. doi: 10.1007/BF01896020. Google Scholar

[3]

J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity,, Internat. Math. Res. Notices, 1998 (): 253. doi: 10.1155/S1073792898000191. Google Scholar

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515. doi: 10.1155/S1073792896000359. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.483. Google Scholar

[6]

J. Ceccon and M. Montenegro, Optimal $L^p$-Riemannian Gagliardo-Nirenberg inequalities,, Math. Z., 258 (2008), 851. doi: 10.1007/s00209-007-0202-8. Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. 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 $\mathbbR^2$,, Discrete Contin. Dyn. Syst., 21 (2008), 665. doi: 10.3934/dcds.2008.21.665. Google Scholar

[9]

D. Fang, H. Pecher and S. Zhong, Low regularity global well-posedness for the two-dimensional Zakharov system,, Analysis (Munich), 29 (2009), 265. doi: 10.1524/anly.2009.1018. Google Scholar

[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. Google Scholar

[11]

N. Kishimoto, Local well-posedness for the Zakharov system on multidimensional torus,, to appear in J. Anal. Math., (). Google Scholar

[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, (). Google Scholar

[13]

H. Pecher, Global rough solutions for the Zakharov system in two spatial dimensions,, preprint, (). Google Scholar

[14]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,'', CBMS Regional Conference Series in Mathematics, 106 (2006). Google Scholar

[15]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567. Google Scholar

[16]

V. E. Zakharov, Collapse of Langmuir waves,, Sov. Phys. JETP, 35 (1972), 908. Google Scholar

show all references

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. doi: 10.1088/0951-7715/22/5/007. Google Scholar

[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. doi: 10.1007/BF01896020. Google Scholar

[3]

J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity,, Internat. Math. Res. Notices, 1998 (): 253. doi: 10.1155/S1073792898000191. Google Scholar

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515. doi: 10.1155/S1073792896000359. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.483. Google Scholar

[6]

J. Ceccon and M. Montenegro, Optimal $L^p$-Riemannian Gagliardo-Nirenberg inequalities,, Math. Z., 258 (2008), 851. doi: 10.1007/s00209-007-0202-8. Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. 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 $\mathbbR^2$,, Discrete Contin. Dyn. Syst., 21 (2008), 665. doi: 10.3934/dcds.2008.21.665. Google Scholar

[9]

D. Fang, H. Pecher and S. Zhong, Low regularity global well-posedness for the two-dimensional Zakharov system,, Analysis (Munich), 29 (2009), 265. doi: 10.1524/anly.2009.1018. Google Scholar

[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. Google Scholar

[11]

N. Kishimoto, Local well-posedness for the Zakharov system on multidimensional torus,, to appear in J. Anal. Math., (). Google Scholar

[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, (). Google Scholar

[13]

H. Pecher, Global rough solutions for the Zakharov system in two spatial dimensions,, preprint, (). Google Scholar

[14]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,'', CBMS Regional Conference Series in Mathematics, 106 (2006). Google Scholar

[15]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567. Google Scholar

[16]

V. E. Zakharov, Collapse of Langmuir waves,, Sov. Phys. JETP, 35 (1972), 908. Google Scholar

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