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November  2016, 15(6): 2247-2280. doi: 10.3934/cpaa.2016036

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions

Isao Kato 1,

1. 

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Received  January 2016 Revised  July 2016 Published  September 2016

We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n,$ $ \partial_t n)|_{t=0} \in H^{s+1}(R^d) \times H^s(R^d) \times\dot{H}^s(R^d) \times \dot{H}^{s-1}(R^d)$. The critical value of $s$ is $s_c=d/2-2$. By the radial Strichartz estimates and $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 4$ for radial initial datum. For non-radial initial datum, we prove that the local well-posedness hold at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$.
Keywords: V^2$ type Bourgain spaces., Scattering, $U^2, low regularity, Cauchy problem, well-posedness, bilinear estimate, radial Strichartz estimate.
Mathematics Subject Classification: Primary: 35Q55, 35B40; Secondary: 35A01, 35A0.
Citation: Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036
References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations,, \emph{GAFA}, 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar

[2]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, \emph{Indi. Univ. Math. J.}, 62 (2013), 991. doi: 10.1512/iumj.2013.62.4970. Google Scholar

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J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar

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J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, \emph{J. Funct. Anal.}, 133 (1995), 50. doi: 10.1006/jfan.1995.1119. Google Scholar

[5]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry,, \emph{Int. Math. Res. Not.}, 9 (2014), 2327. Google Scholar

[6]

Z. Guo, K. Nakanishi and S. Wang, Small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry,, \emph{Math. Res. Nett.}, 21 (2014), 733. doi: 10.4310/MRL.2014.v21.n4.a8. Google Scholar

[7]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation,, \emph{J. Anal. Math.}, 124 (2014), 1. doi: 10.1007/s11854-014-0025-6. Google Scholar

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M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, \emph{Ann. I. H. Poincar\'e AN}, 26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

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M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space"[Ann. I. H. Poincaré AN, 26 (2009), 917-941],, \emph{Ann. I. H. Poincar\'e AN}, 27 (2010), 971. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

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S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(T^3)$,, \emph{Duke. Math. J.}, 159 (2011), 329. doi: 10.1215/00127094-1415889. Google Scholar

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H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data,, \emph{Comm. Pure. Appl. Anal.}, 13 (2014), 1563. doi: 10.3934/cpaa.2014.13.1563. Google Scholar

[12]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications,, \emph{Adv. Stud. Pure. Math.}, 23 (1994), 223. Google Scholar

[13]

I. Kato and K. Tsugawa, Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions, preprint,, \arXiv{1512.00551}., (). Google Scholar

[14]

J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications,, \emph{J. Math. Pures Appl.}, 95 (2011), 48. doi: 10.1016/j.matpur.2010.10.001. Google Scholar

[15]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, \emph{J. Amer. Soc.}, 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955. Google Scholar

[17]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators,, \emph{Comm. Pure. Appl. Math.}, 58 (2005), 217. doi: 10.1002/cpa.20067. Google Scholar

[18]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces,, \emph{Int. Math. Res. Not.}, 16 (2007). doi: 10.1093/imrn/rnm053. Google Scholar

[19]

H. Lindblad, Counterexamples to local existence for semi-linear wave equations,, \emph{Amer. J. Math.}, 118 (1996), 1. Google Scholar

[20]

H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, \emph{J. Funct. Anal.}, 130 (1995), 357. doi: 10.1006/jfan.1995.1075. Google Scholar

[21]

S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, \emph{Math. Ann.}, 322 (2002), 603. doi: 10.1007/s002080200008. Google Scholar

[22]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the Dirac equation,, \emph{Rev. Mat. Iberoamericana.}, 19 (2003), 179. doi: 10.4171/RMI/342. Google Scholar

[23]

N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 975. doi: 10.1142/S0219891605000683. Google Scholar

[24]

N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to a singular nonlinear Schrödinger system,, \emph{Ann. I. H. Poincar\'e AN}, 27 (2010), 1073. doi: 10.1016/j.anihpc.2010.02.002. Google Scholar

[25]

T. Ozawa, K. Tsutaya and Y. Tsutsumi, Normal form and global solutions for the Klein-Gordon-Zakharov equations,, \emph{Ann. I. H. Poincar\'e AN}, 12 (1995), 459. Google Scholar

[26]

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,, \emph{Math. Ann.}, 313 (1999), 127. doi: 10.1007/s002080050254. Google Scholar

[27]

T. Schottdorf, Global existence without decay for quadratic Klein-Gordon equations, preprint,, \arXiv{1209.1518}., (). Google Scholar

[28]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, in \emph{AMS}, (2006). doi: 10.1090/cbms/106. Google Scholar

[29]

D. Tataru, Local and global results for wave maps I,, \emph{Comm. Part. Diff. Eq.}, 23 (1998), 1781. doi: 10.1080/03605309808821400. Google Scholar

show all references

References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations,, \emph{GAFA}, 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar

[2]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, \emph{Indi. Univ. Math. J.}, 62 (2013), 991. doi: 10.1512/iumj.2013.62.4970. Google Scholar

[3]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar

[4]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, \emph{J. Funct. Anal.}, 133 (1995), 50. doi: 10.1006/jfan.1995.1119. Google Scholar

[5]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry,, \emph{Int. Math. Res. Not.}, 9 (2014), 2327. Google Scholar

[6]

Z. Guo, K. Nakanishi and S. Wang, Small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry,, \emph{Math. Res. Nett.}, 21 (2014), 733. doi: 10.4310/MRL.2014.v21.n4.a8. Google Scholar

[7]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation,, \emph{J. Anal. Math.}, 124 (2014), 1. doi: 10.1007/s11854-014-0025-6. Google Scholar

[8]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, \emph{Ann. I. H. Poincar\'e AN}, 26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

[9]

M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space"[Ann. I. H. Poincaré AN, 26 (2009), 917-941],, \emph{Ann. I. H. Poincar\'e AN}, 27 (2010), 971. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

[10]

S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(T^3)$,, \emph{Duke. Math. J.}, 159 (2011), 329. doi: 10.1215/00127094-1415889. Google Scholar

[11]

H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data,, \emph{Comm. Pure. Appl. Anal.}, 13 (2014), 1563. doi: 10.3934/cpaa.2014.13.1563. Google Scholar

[12]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications,, \emph{Adv. Stud. Pure. Math.}, 23 (1994), 223. Google Scholar

[13]

I. Kato and K. Tsugawa, Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions, preprint,, \arXiv{1512.00551}., (). Google Scholar

[14]

J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications,, \emph{J. Math. Pures Appl.}, 95 (2011), 48. doi: 10.1016/j.matpur.2010.10.001. Google Scholar

[15]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, \emph{J. Amer. Soc.}, 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955. Google Scholar

[17]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators,, \emph{Comm. Pure. Appl. Math.}, 58 (2005), 217. doi: 10.1002/cpa.20067. Google Scholar

[18]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces,, \emph{Int. Math. Res. Not.}, 16 (2007). doi: 10.1093/imrn/rnm053. Google Scholar

[19]

H. Lindblad, Counterexamples to local existence for semi-linear wave equations,, \emph{Amer. J. Math.}, 118 (1996), 1. Google Scholar

[20]

H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, \emph{J. Funct. Anal.}, 130 (1995), 357. doi: 10.1006/jfan.1995.1075. Google Scholar

[21]

S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, \emph{Math. Ann.}, 322 (2002), 603. doi: 10.1007/s002080200008. Google Scholar

[22]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the Dirac equation,, \emph{Rev. Mat. Iberoamericana.}, 19 (2003), 179. doi: 10.4171/RMI/342. Google Scholar

[23]

N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 975. doi: 10.1142/S0219891605000683. Google Scholar

[24]

N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to a singular nonlinear Schrödinger system,, \emph{Ann. I. H. Poincar\'e AN}, 27 (2010), 1073. doi: 10.1016/j.anihpc.2010.02.002. Google Scholar

[25]

T. Ozawa, K. Tsutaya and Y. Tsutsumi, Normal form and global solutions for the Klein-Gordon-Zakharov equations,, \emph{Ann. I. H. Poincar\'e AN}, 12 (1995), 459. Google Scholar

[26]

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,, \emph{Math. Ann.}, 313 (1999), 127. doi: 10.1007/s002080050254. Google Scholar

[27]

T. Schottdorf, Global existence without decay for quadratic Klein-Gordon equations, preprint,, \arXiv{1209.1518}., (). Google Scholar

[28]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, in \emph{AMS}, (2006). doi: 10.1090/cbms/106. Google Scholar

[29]

D. Tataru, Local and global results for wave maps I,, \emph{Comm. Part. Diff. Eq.}, 23 (1998), 1781. doi: 10.1080/03605309808821400. Google Scholar

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