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December  2016, 9(6): 1797-1851. doi: 10.3934/dcdss.2016075

Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$

Zhaohi Huo 1, , Yueling Jia 2, and  Qiaoxin Li 3,

1. 

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing, 100094
3. 

China Academy of Aerospace Aerodynamics, Beijing 100074, China

Received  July 2015 Revised  September 2016 Published  November 2016

The Cauchy problem of the 3D Zakharov-Kuznetsov equation $$ u_{t}+\partial_{\tilde{x}_{*,1}}\Delta u +(u^2)_{\tilde{x}_{*,1}}=0, (x,t)\in \mathbb{R}^3 \times \mathbb{R}, \ x=(\tilde{x}_{*,1},\tilde{x}_{*,2},\tilde{x}_{*,3});$$ is considered. It is shown that it is globally well-posed in energy space $H^1(\mathbb{R}^3)$. It answer an open problem: Is it globally well-posed in energy space $H^1 (\mathbb{R}^3)$ for 3D Z-K equtation [10,12,13]?
    Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$.
    The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.
Keywords: Zakharov-Kuznetsov equation, dyadic $X_{s, Well-posedness, b}$ spaces in higher dimension..
Mathematics Subject Classification: Primary: 35E15; Secondary: 35Q5.
Citation: Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
References:
[1]

J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, part II: the KdV equation, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.  Google Scholar

[2]

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A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798. doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

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A. D. Ionescu, C. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304. doi: 10.1007/s00222-008-0115-0.  Google Scholar

[5]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[6]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[7]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. Google Scholar

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D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, 181-213, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[10]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.  Google Scholar

[11]

F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085. doi: 10.1016/j.jfa.2010.11.005.  Google Scholar

[12]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371. doi: 10.1016/j.anihpc.2013.12.003.  Google Scholar

[13]

F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304. doi: 10.1137/110850566.  Google Scholar

[14]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.  Google Scholar

show all references

References:
[1]

J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, part II: the KdV equation, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.  Google Scholar

[2]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.  Google Scholar

[3]

A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798. doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

[4]

A. D. Ionescu, C. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304. doi: 10.1007/s00222-008-0115-0.  Google Scholar

[5]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[6]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[7]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. Google Scholar

[9]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, 181-213, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[10]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.  Google Scholar

[11]

F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085. doi: 10.1016/j.jfa.2010.11.005.  Google Scholar

[12]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371. doi: 10.1016/j.anihpc.2013.12.003.  Google Scholar

[13]

F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304. doi: 10.1137/110850566.  Google Scholar

[14]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.  Google Scholar

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