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September  2020, 25(9): 3715-3724. doi: 10.3934/dcdsb.2020087

Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D

Mohamad Darwich

Mathematics Departement, Lebanese University, Hadat-Lebanon

Received  March 2019 Revised  October 2019 Published  September 2020 Early access  April 2020

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) $.

Keywords: Zakharov-Kuznetsov equation, initial value problem, well-posedness, bilinear Strichartz estimates, Bourgain's spaces.
Mathematics Subject Classification: primary: 35A01, 35Q53; secondary: 35Q60.
Citation: 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
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.

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