Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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).
    Mathematics Subject Classification: Primary: 35Q55.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    J. BourgainRefinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 1998, 253-283. doi: 10.1155/S1073792898000191.


    J. Bourgain and J. CollianderOn wellposedness of the Zakharov system, Internat. Math. Res. Notices, 1996, 515-546. doi: 10.1155/S1073792896000359.


    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.


    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.


    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.


    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.


    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.


    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.


    N. KishimotoLocal well-posedness for the Zakharov system on multidimensional torus, to appear in J. Anal. Math., arXiv:1109.3527.


    N. Kishimoto and M. MaedaConstruction of blow-up solutions for Zakharov system on $\mathbbT ^2$, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, arXiv:1109.3528.


    H. PecherGlobal rough solutions for the Zakharov system in two spatial dimensions, preprint, arXiv:1203.2173.


    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.


    M. I. WeinsteinNonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.


    V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

  • 加载中

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint