• Previous Article
    Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients
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

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 $d \ge 5$.
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

[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

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

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[3]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[4]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[5]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[6]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[8]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[9]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[10]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[11]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[12]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[14]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[15]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[16]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[17]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[19]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[20]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]