# American Institute of Mathematical Sciences

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

 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.

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.

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##### 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.
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