doi: 10.3934/dcds.2020338

Local well-posedness for the Klein-Gordon-Zakharov system in 3D

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Received  May 2020 Revised  August 2020 Published  October 2020

We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by Bejenaru-Herr for the Zakharov system and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.

Citation: Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020338
References:
[1]

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

[2]

I. BejenaruS. Herr and D. Tataru, A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam., 26 (2010), 707-728.  doi: 10.4171/RMI/615.  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. 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

[5]

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.  Google Scholar

[6]

S. Kinoshita, Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D, Discr. Cont. Dynamical systems, 38 (2018), 1479-1504.  doi: 10.3934/dcds.2018061.  Google Scholar

[7]

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.  Google Scholar

[8]

T. OzawaK. 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.  Google Scholar

[9]

S. Selberg, An isotropic bilinear $L^2$ estimate related to the 3D wave equation, Int. Math. Res. Not., 2008 (2008), 63 pp. doi: 10.1093/imrn/rnn107.  Google Scholar

show all references

References:
[1]

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

[2]

I. BejenaruS. Herr and D. Tataru, A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam., 26 (2010), 707-728.  doi: 10.4171/RMI/615.  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. 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

[5]

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.  Google Scholar

[6]

S. Kinoshita, Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D, Discr. Cont. Dynamical systems, 38 (2018), 1479-1504.  doi: 10.3934/dcds.2018061.  Google Scholar

[7]

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.  Google Scholar

[8]

T. OzawaK. 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.  Google Scholar

[9]

S. Selberg, An isotropic bilinear $L^2$ estimate related to the 3D wave equation, Int. Math. Res. Not., 2008 (2008), 63 pp. doi: 10.1093/imrn/rnn107.  Google Scholar

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[3]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[4]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230

[5]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[6]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[7]

Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002

[8]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[9]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[10]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[11]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006

[12]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[13]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[14]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[16]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[17]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[18]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[19]

Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262

[20]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

2019 Impact Factor: 1.338

Article outline

[Back to Top]