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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, 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.  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), 107-156. 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), 483-491. 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), 851-873. 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), 705-749. 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), 665-686. 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), 265-281. 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), 349-389.  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, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 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), 908-914. 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), 1063-1089. 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), 107-156. 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), 483-491. 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), 851-873. 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), 705-749. 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), 665-686. 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), 265-281. 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), 349-389.  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, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 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), 908-914. Google Scholar

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