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September  2019, 18(5): 2473-2489. doi: 10.3934/cpaa.2019112

A note on global existence for the Zakharov system on $ \mathbb{T} $

E. Compaan

Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA 02139, USA

Received  June 2018 Revised  January 2019 Published  April 2019

Fund Project: The first author is supported by NSF MSPRF #1704865.

We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in $ L^2 $-based Sobolev spaces. We also obtain global well-posedness in $ H^{\frac12+} \times L^2 $, which is sharp (up to endpoints) in the class of $ L^2 $-based Sobolev spaces.

Keywords: Zakharov system, global well-posedness, high-low method, I-method, probabilistic well-posedness.
Mathematics Subject Classification: Primary: 35Q55, 35Q40.
Citation: E. Compaan. A note on global existence for the Zakharov system on $ \mathbb{T} $. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2473-2489. doi: 10.3934/cpaa.2019112
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, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.  doi: 10.1215/S0012-7094-94-07607-2.  Google Scholar

[3]

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

[4]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.   Google Scholar

[5]

J. Bourgain, Refinement of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Not., (1998), 253–283. doi: 10.1155/S1073792898000191.  Google Scholar

[6]

J. Bourgain, A remark on normal forms and the "I-method" for periodic NLS, J. Anal. Math., 94 (2004), 125-157.  doi: 10.1007/BF02789044.  Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eq., 2001 (2001), 1-7.   Google Scholar

[8]

J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(T)$, Duke Math. J., 161 (2012), 367-414.  doi: 10.1215/00127094-1507400.  Google Scholar

[9]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[10]

N. Kishimoto, Local well-posedness for the Zakharov system on the multidimensional torus, J. Anal. Math., 119 (2013), 213-253.  doi: 10.1007/s11854-013-0007-0.  Google Scholar

[11]

N. Kishimoto, Resonant decomposition and the I-method for the two-dimensional Zakharov system, Discrete Contin. Dyn. Syst., 33 (2013), 4095-4122.  doi: 10.3934/dcds.2013.33.4095.  Google Scholar

[12]

J. Lebowitz, H. Rose and E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687.  doi: 10.1007/BF01026495.  Google Scholar

[13]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2.  Google Scholar

[14]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Differential Integral Equations, 12 (1999), 789-810.   Google Scholar

[15]

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, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.  doi: 10.1215/S0012-7094-94-07607-2.  Google Scholar

[3]

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

[4]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.   Google Scholar

[5]

J. Bourgain, Refinement of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Not., (1998), 253–283. doi: 10.1155/S1073792898000191.  Google Scholar

[6]

J. Bourgain, A remark on normal forms and the "I-method" for periodic NLS, J. Anal. Math., 94 (2004), 125-157.  doi: 10.1007/BF02789044.  Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eq., 2001 (2001), 1-7.   Google Scholar

[8]

J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(T)$, Duke Math. J., 161 (2012), 367-414.  doi: 10.1215/00127094-1507400.  Google Scholar

[9]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[10]

N. Kishimoto, Local well-posedness for the Zakharov system on the multidimensional torus, J. Anal. Math., 119 (2013), 213-253.  doi: 10.1007/s11854-013-0007-0.  Google Scholar

[11]

N. Kishimoto, Resonant decomposition and the I-method for the two-dimensional Zakharov system, Discrete Contin. Dyn. Syst., 33 (2013), 4095-4122.  doi: 10.3934/dcds.2013.33.4095.  Google Scholar

[12]

J. Lebowitz, H. Rose and E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687.  doi: 10.1007/BF01026495.  Google Scholar

[13]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2.  Google Scholar

[14]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Differential Integral Equations, 12 (1999), 789-810.   Google Scholar

[15]

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

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