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doi: 10.3934/cpaa.2021161
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Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $

Zijun Chen 1,, and  Shengkun Wu 2,

1. 

School of Mathematics, Monash University, VIC 3800, Australia
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

* Corresponding author

Received  April 2021 Revised  August 2021 Early access September 2021

Fund Project: The second author is partially supported by the Chinese Scholarship Council (No. 201906050022)

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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, (1996), 515–546. doi: 10.1155/S1073792896000359.  Google Scholar

[5]

T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geq 4$, preprint, arXiv: 1912.05820v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Z. Guo and K. Nakanishi, The Zakharov system in 4D radial energy space below the ground state, preprint, arXiv: 1810.05794. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.   Google Scholar

[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.  Google Scholar

[16]

H. Pecher, Global solutions with infinite energy for the one-dimensional Zakharov system, Electron. J. Differ. Equ., 2005 (2005), 1-18.   Google Scholar

[17]

A. Sanwal, Local well-posedness for the Zakharov system in dimension $d \leq 3$, preprint, arXiv: 2103.09259. Google Scholar

[18]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, (1996), 515–546. doi: 10.1155/S1073792896000359.  Google Scholar

[5]

T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geq 4$, preprint, arXiv: 1912.05820v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Z. Guo and K. Nakanishi, The Zakharov system in 4D radial energy space below the ground state, preprint, arXiv: 1810.05794. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.   Google Scholar

[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.  Google Scholar

[16]

H. Pecher, Global solutions with infinite energy for the one-dimensional Zakharov system, Electron. J. Differ. Equ., 2005 (2005), 1-18.   Google Scholar

[17]

A. Sanwal, Local well-posedness for the Zakharov system in dimension $d \leq 3$, preprint, arXiv: 2103.09259. Google Scholar

[18]

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

Figure 1.  Region of regularity $ (s, l) $
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