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 & Continuous Dynamical Systems - A, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061
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.

[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. BejenaruS. 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. BennettA. 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. 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.

[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. KenigG. 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. 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.

[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. 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.

[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. BejenaruS. 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. BennettA. 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. 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.

[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. KenigG. 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. 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.

[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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