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