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

Previous Article
Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows
DCDS-S Home
This Issue
Next Article
Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere

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

Abstract
Related Papers

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 and 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.
[2]	A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
[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.
[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.
[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.
[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.
[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.
[8]	E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
[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.
[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.
[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.
[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.
[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.
[8]	E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[2]	Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365
[3]	Felipe Linares, Mahendra Panthee, Tristan Robert, Nikolay Tzvetkov. On the periodic Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3521-3533. doi: 10.3934/dcds.2019145
[4]	Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
[5]	Mo Chen, Lionel Rosier. Exact controllability of the linear Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3889-3916. doi: 10.3934/dcdsb.2020080
[6]	Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1067-1103. doi: 10.3934/dcds.2021147
[7]	Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
[8]	Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005
[9]	Felipe Linares, Gustavo Ponce. On special regularity properties of solutions of the Zakharov-Kuznetsov equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1561-1572. doi: 10.3934/cpaa.2018074
[10]	Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061
[11]	Felipe Linares, Jean-Claude Saut. The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 547-565. doi: 10.3934/dcds.2009.24.547
[12]	Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[13]	Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[14]	Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[15]	Robert Schippa. On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5189-5215. doi: 10.3934/dcds.2020225
[16]	Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4307-4319. doi: 10.3934/cpaa.2021161
[17]	Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[18]	Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[19]	Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[20]	Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Zhaohi Huo Yueling Jia Qiaoxin Li
on Google Scholar
Zhaohi Huo Yueling Jia Qiaoxin Li

[Back to Top]