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doi: 10.3934/dcds.2020338

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

Hartmut Pecher

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.

Keywords: Klein-Gordon-Zakharov, local well-posedness, low regularity, Fourier restriction norm method, bilinear estimates.
Mathematics Subject Classification: 35Q55, 35A01.
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. 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

[2]

I. Bejenaru, S. 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. 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

[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. 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.  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. 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

[2]

I. Bejenaru, S. 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. 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

[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. 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.  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

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