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A note on global existence for the Zakharov system on $ \mathbb{T} $
Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA 02139, USA |
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.
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. |
| [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. |
| [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. |
| [4] |
J. Bourgain,
Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
H. Takaoka,
Well-posedness for the Zakharov system with the periodic boundary condition, Differential Integral Equations, 12 (1999), 789-810.
|
| [15] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
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. |
| [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. |
| [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. |
| [4] |
J. Bourgain,
Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
H. Takaoka,
Well-posedness for the Zakharov system with the periodic boundary condition, Differential Integral Equations, 12 (1999), 789-810.
|
| [15] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
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