-
Previous Article
Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
- DCDS-B Home
- This Issue
-
Next Article
From approximate synchronization to identical synchronization in coupled systems
Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D
Mathematics Departement, Lebanese University, Hadat-Lebanon |
In this paper, we consider the Zakharov-Kuznetsov equation in 3D, with a dissipative term of order $ 0 < \alpha \leq 2 $ in the $ x $ direction. We prove that the problem is locally well-posed in $ H^{s}( { I\!\!R}^3) $, for $ s > 1-\frac{\alpha}{2} $, and by an a priori energy estimate, we prove that the problem is globally well-posed in $ H^{1}( { I\!\!R}^3) $.
References:
[1] |
O. V. Besov, V. P. Il'in and S. M. Nikolskii, Integral Representations of Functions and Imbedding Theorems, Vol. 1, New York-Toronto, Ont.-London, 1978. |
[2] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[3] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. Part II: The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[4] |
J. Bourgain,
On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[5] |
M. Darwich,
On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation, J. Differential Equations, 253 (2012), 1584-1603.
doi: 10.1016/j.jde.2012.05.013. |
[6] |
A. V. Faminskiĭ,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
|
[7] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Séminaire Bourbaki, Vol. 1994/95, Astérisque, 237 (1996), 163-187.
|
[8] |
J. Ginibre, Y. 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. |
[9] |
J. Ginibre and G. Velo,
Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[10] |
A. Grünrock and S. Herr,
The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 34 (2014), 2061-2068.
doi: 10.3934/dcds.2014.34.2061. |
[11] |
H. Hirayama, Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space, J. Differential Equations, 267 (2019), 4089–4116, arXiv: 1806.01039.
doi: 10.1016/j.jde.2019.04.030. |
[12] |
R. J. Iório Jr. and W. V. L. Nunes,
On equations of KP-type, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 725-743.
doi: 10.1017/S0308210500021740. |
[13] |
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. |
[14] |
C. E. Kenig, G. Ponce and L. Vega,
On the (generalized) Korteweg-de-Vries equation, Duke Math. J., 59 (1989), 585-610.
doi: 10.1215/S0012-7094-89-05927-9. |
[15] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de-Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[16] |
E. A. Kuznetsov and V. E. Zakharov,
Three-dimensional solitons, Zhurnal Eksperimental'noi i Teroreticheskoi Fiziki, 66 (1974), 594-597.
|
[17] |
F. Linares and A. Pastor,
Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.
doi: 10.1137/080739173. |
[18] |
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. |
[19] |
L. Molinet,
The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation, Differential Integral Equations, 13 (2000), 189-216.
|
[20] |
L. Molinet,
On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations, J. Differential Equations, 152 (1999), 30-74.
doi: 10.1006/jdeq.1998.3522. |
[21] |
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. |
[22] |
L. Molinet and F. Ribaud,
On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005.
doi: 10.1155/S1073792802112104. |
[23] |
E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids, 13 (1970), 1432–1434.
doi: 10.1063/1.1693097. |
[24] |
F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289–2304. |
[25] |
J. C. Saut,
Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.
doi: 10.1512/iumj.1993.42.42047. |
[26] |
N. Tzvetkov,
Remark on the local regularity of Kadomtsev-Petviashvili-II equation, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 709-712.
doi: 10.1016/S0764-4442(98)80035-9. |
[27] |
G. Zihua and W. Baoxiang,
Global well-posedness and inviscid limit for the Korteweg-de-Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.
doi: 10.1016/j.jde.2009.03.006. |
show all references
References:
[1] |
O. V. Besov, V. P. Il'in and S. M. Nikolskii, Integral Representations of Functions and Imbedding Theorems, Vol. 1, New York-Toronto, Ont.-London, 1978. |
[2] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[3] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. Part II: The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[4] |
J. Bourgain,
On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[5] |
M. Darwich,
On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation, J. Differential Equations, 253 (2012), 1584-1603.
doi: 10.1016/j.jde.2012.05.013. |
[6] |
A. V. Faminskiĭ,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
|
[7] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Séminaire Bourbaki, Vol. 1994/95, Astérisque, 237 (1996), 163-187.
|
[8] |
J. Ginibre, Y. 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. |
[9] |
J. Ginibre and G. Velo,
Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[10] |
A. Grünrock and S. Herr,
The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 34 (2014), 2061-2068.
doi: 10.3934/dcds.2014.34.2061. |
[11] |
H. Hirayama, Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space, J. Differential Equations, 267 (2019), 4089–4116, arXiv: 1806.01039.
doi: 10.1016/j.jde.2019.04.030. |
[12] |
R. J. Iório Jr. and W. V. L. Nunes,
On equations of KP-type, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 725-743.
doi: 10.1017/S0308210500021740. |
[13] |
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. |
[14] |
C. E. Kenig, G. Ponce and L. Vega,
On the (generalized) Korteweg-de-Vries equation, Duke Math. J., 59 (1989), 585-610.
doi: 10.1215/S0012-7094-89-05927-9. |
[15] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de-Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[16] |
E. A. Kuznetsov and V. E. Zakharov,
Three-dimensional solitons, Zhurnal Eksperimental'noi i Teroreticheskoi Fiziki, 66 (1974), 594-597.
|
[17] |
F. Linares and A. Pastor,
Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.
doi: 10.1137/080739173. |
[18] |
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. |
[19] |
L. Molinet,
The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation, Differential Integral Equations, 13 (2000), 189-216.
|
[20] |
L. Molinet,
On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations, J. Differential Equations, 152 (1999), 30-74.
doi: 10.1006/jdeq.1998.3522. |
[21] |
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. |
[22] |
L. Molinet and F. Ribaud,
On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005.
doi: 10.1155/S1073792802112104. |
[23] |
E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids, 13 (1970), 1432–1434.
doi: 10.1063/1.1693097. |
[24] |
F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289–2304. |
[25] |
J. C. Saut,
Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.
doi: 10.1512/iumj.1993.42.42047. |
[26] |
N. Tzvetkov,
Remark on the local regularity of Kadomtsev-Petviashvili-II equation, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 709-712.
doi: 10.1016/S0764-4442(98)80035-9. |
[27] |
G. Zihua and W. Baoxiang,
Global well-posedness and inviscid limit for the Korteweg-de-Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.
doi: 10.1016/j.jde.2009.03.006. |
[1] |
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 |
[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] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205 |
[8] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 |
[17] |
Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097 |
[18] |
Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061 |
[19] |
Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036 |
[20] |
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 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]