American Institute of Mathematical Sciences

Advanced
March  2018, 38(3): 1479-1504. doi: 10.3934/dcds.2018061

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

Shinya Kinoshita

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Received  May 2017 Revised  October 2017 Published  December 2017

This paper is concerned with the Cauchy problem of the Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the nonlinear version of the classical Loomis-Whitney inequality.

Keywords: Well-posedness, Cauchy problem, low regularity, bilinear estimate, Strichartz estimate.
Mathematics Subject Classification: 35Q55, 35A01.
Citation: Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061
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]

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.

[3]

I. Bejenaru, S. Herr and D. Tataru, A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam, 26 (2010), 707-728. 

[4] P. M. Bellan, Fundamentals of Plasmas Physics, Cambridge, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511807183.
[5]

J. Bennett, A. Carbery and J. Wright, A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett., 12 (2005), 443-457.  doi: 10.4310/MRL.2005.v12.n4.a1.

[6]

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.

[7]

J. Holmer, Local ill-posedness of the 1D Zakharov system, Electron. J. Diff. Equations, 24 (2007), 22pp. 

[8]

I. Kato, Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions, Comm. Pure. Appl. Anal., 15 (2016), 2247-2280.  doi: 10.3934/cpaa.2016036.

[9]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[10]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.

[11]

H. Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math.J., 72 (1993), 503-539.  doi: 10.1215/S0012-7094-93-07219-5.

[12]

H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math, 118 (1996), 1-16.  doi: 10.1353/ajm.1996.0002.

[13]

N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math., 172 (2008), 535-583.  doi: 10.1007/s00222-008-0110-5.

[14]

T. Ozawa, K. Tsutaya and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann., 313 (1999), 127-140.  doi: 10.1007/s002080050254.

[15]

S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Diff. Eq., 16 (2011), 667-690. 

[16]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, AMS, 2006.

[17]

K. Tsugawa, Time local well-posedness of the coupled system of nonlinear wave equations with different propagation speeds, Surikaisekikenkyusho Kokyuroku, 1235 (2001), 61-90. 

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]

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.

[3]

I. Bejenaru, S. Herr and D. Tataru, A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam, 26 (2010), 707-728. 

[4] P. M. Bellan, Fundamentals of Plasmas Physics, Cambridge, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511807183.
[5]

J. Bennett, A. Carbery and J. Wright, A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett., 12 (2005), 443-457.  doi: 10.4310/MRL.2005.v12.n4.a1.

[6]

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.

[7]

J. Holmer, Local ill-posedness of the 1D Zakharov system, Electron. J. Diff. Equations, 24 (2007), 22pp. 

[8]

I. Kato, Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions, Comm. Pure. Appl. Anal., 15 (2016), 2247-2280.  doi: 10.3934/cpaa.2016036.

[9]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[10]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.

[11]

H. Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math.J., 72 (1993), 503-539.  doi: 10.1215/S0012-7094-93-07219-5.

[12]

H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math, 118 (1996), 1-16.  doi: 10.1353/ajm.1996.0002.

[13]

N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math., 172 (2008), 535-583.  doi: 10.1007/s00222-008-0110-5.

[14]

T. Ozawa, K. Tsutaya and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann., 313 (1999), 127-140.  doi: 10.1007/s002080050254.

[15]

S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Diff. Eq., 16 (2011), 667-690. 

[16]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, AMS, 2006.

[17]

K. Tsugawa, Time local well-posedness of the coupled system of nonlinear wave equations with different propagation speeds, Surikaisekikenkyusho Kokyuroku, 1235 (2001), 61-90. 

[1]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[2]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903
[3]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[4]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[5]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[6]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46
[7]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
[8]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[9]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure and Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[10]

Wenming Hu, Huicheng Yin. Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1891-1919. doi: 10.3934/cpaa.2019088
[11]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[12]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081
[13]

Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3683-3702. doi: 10.3934/cpaa.2021126
[14]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114
[15]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[16]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[17]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[18]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[19]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036
[20]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (220)
  • HTML views (324)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy