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

* Corresponding author

Received  April 2021 Revised  August 2021 Published  December 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.

Citation: Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4307-4319. doi: 10.3934/cpaa.2021161
References:
[1]

I. BejenaruZ. GuoS. 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. BejenaruS. HerrJ. 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. CollianderJ. 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. FangH. 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. GinibreY. 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. GuoS. LeeK. 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. GuoK. 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. BejenaruZ. GuoS. 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. BejenaruS. HerrJ. 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. CollianderJ. 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. FangH. 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. GinibreY. 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. GuoS. LeeK. 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. GuoK. 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) $
[1]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

[2]

Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021147

[3]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259

[4]

Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1707-1736. doi: 10.3934/dcds.2020338

[5]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019

[6]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087

[7]

Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811

[8]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[9]

Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061

[10]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036

[11]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[12]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029

[13]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

[14]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[15]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

[16]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[17]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[18]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[19]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[20]

Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021030

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (109)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]