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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 $\mathbb{T}$

 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.

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

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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. 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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