Figure 1.
Region of regularity $ (s, l) $
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December
2021, 20(12): 4307-4319.
doi: 10.3934/cpaa.2021161
Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $
| 1. | School of Mathematics, Monash University, VIC 3800, Australia |
| 2. | College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China |
The Zakharov system in dimension $ d = 2,3 $ is shown to have a local unique solution for any initial values in the space $ H^{s} \times H^{l} \times H^{l-1} $, where a new range of regularity $ (s, l) $ is given, especially at the line $ s-l = -1 $. The result is obtained mainly by the normal form reduction and the Strichartz estimates.
Keywords:
Zakharov,
local well-posedness,
normal form reduction,
Strichartz estimate,
contraction mapping principle.
Mathematics Subject Classification:
Primary: 35Q55, 35L70.
Citation:
Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure and Applied Analysis,
2021, 20
(12)
: 4307-4319.
doi: 10.3934/cpaa.2021161
![]()
References:
| [1] |
I. Bejenaru, Z. Guo, S. Herr and K. Nakanishi,
Well-posedness and scattering for the Zakharov system in four dimensions, Analysis & PDE, 8 (2015), 2029-2055.
doi: 10.2140/apde.2015.8.2029. |
| [2] |
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. |
| [3] |
I. Bejenaru and S. Herr,
Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506.
doi: 10.1016/j.jfa.2011.03.015. |
| [4] |
J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, (1996), 515–546.
doi: 10.1155/S1073792896000359. |
| [5] |
T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geq 4$, preprint, arXiv: 1912.05820v2. |
| [6] |
J. Colliander, J. Holmer and N. Tzirakis,
Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638.
doi: 10.1090/S0002-9947-08-04295-5. |
| [7] |
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. |
| [8] |
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. |
| [9] |
Z. Guo, S. Lee, K. Nakanishi and C. Wang,
Generalized Strichartz estimates and scattering for 3D Zakharov system, Commun. Math. Phys., 331 (2014), 239-259.
doi: 10.1007/s00220-014-2006-0. |
| [10] |
Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not., (2014), 2327–2342.
doi: 10.1093/imrn/rns296. |
| [11] |
Z. Guo and K. Nakanishi, The Zakharov system in 4D radial energy space below the ground state, preprint, arXiv: 1810.05794. |
| [12] |
Z. Guo, K. Nakanishi and S. Wang,
Global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case, Commun. Partial Differ. Equ., 39 (2014), 1158-1184.
doi: 10.1080/03605302.2013.836715. |
| [13] |
I. Kato and K. Tsugawa,
Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions, Differ. Integral Equ., 30 (2017), 763-794.
|
| [14] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
| [15] |
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. |
| [16] |
H. Pecher,
Global solutions with infinite energy for the one-dimensional Zakharov system, Electron. J. Differ. Equ., 2005 (2005), 1-18.
|
| [17] |
A. Sanwal, Local well-posedness for the Zakharov system in dimension $d \leq 3$, preprint, arXiv: 2103.09259. |
| [18] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
show all references
References:
| [1] |
I. Bejenaru, Z. Guo, S. Herr and K. Nakanishi,
Well-posedness and scattering for the Zakharov system in four dimensions, Analysis & PDE, 8 (2015), 2029-2055.
doi: 10.2140/apde.2015.8.2029. |
| [2] |
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. |
| [3] |
I. Bejenaru and S. Herr,
Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506.
doi: 10.1016/j.jfa.2011.03.015. |
| [4] |
J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, (1996), 515–546.
doi: 10.1155/S1073792896000359. |
| [5] |
T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geq 4$, preprint, arXiv: 1912.05820v2. |
| [6] |
J. Colliander, J. Holmer and N. Tzirakis,
Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638.
doi: 10.1090/S0002-9947-08-04295-5. |
| [7] |
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. |
| [8] |
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. |
| [9] |
Z. Guo, S. Lee, K. Nakanishi and C. Wang,
Generalized Strichartz estimates and scattering for 3D Zakharov system, Commun. Math. Phys., 331 (2014), 239-259.
doi: 10.1007/s00220-014-2006-0. |
| [10] |
Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not., (2014), 2327–2342.
doi: 10.1093/imrn/rns296. |
| [11] |
Z. Guo and K. Nakanishi, The Zakharov system in 4D radial energy space below the ground state, preprint, arXiv: 1810.05794. |
| [12] |
Z. Guo, K. Nakanishi and S. Wang,
Global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case, Commun. Partial Differ. Equ., 39 (2014), 1158-1184.
doi: 10.1080/03605302.2013.836715. |
| [13] |
I. Kato and K. Tsugawa,
Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions, Differ. Integral Equ., 30 (2017), 763-794.
|
| [14] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
| [15] |
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. |
| [16] |
H. Pecher,
Global solutions with infinite energy for the one-dimensional Zakharov system, Electron. J. Differ. Equ., 2005 (2005), 1-18.
|
| [17] |
A. Sanwal, Local well-posedness for the Zakharov system in dimension $d \leq 3$, preprint, arXiv: 2103.09259. |
| [18] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
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