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June  2019, 39(6): 3521-3533. doi: 10.3934/dcds.2019145

On the periodic Zakharov-Kuznetsov equation

Felipe Linares 1, , Mahendra Panthee 2, , Tristan Robert 3,, and  Nikolay Tzvetkov 3,

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
2. 

IMECC-UNICAMP, 13083-859, Campinas, São Paulo, Brazil
3. 

Université de Cergy-Pontoise, Cergy-Pontoise, F-95000, UMR 8088 du CNRS, France

* Corresponding author: Tristan Robert

Received  September 2018 Revised  November 2018 Published  February 2019

Fund Project: FL was partially supported by FAPERJ and CNPq Brasil, MP was partially supported by FAPESP (2016/25864-6) and CNPq (305483/2014-5) Brasil.

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on $ \mathbb T^2 $. We prove the local well-posedness for given data in $ H^s( \mathbb T^2) $ whenever $ s> 5/3 $. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the $ \mathbb R^2 $ and $ \mathbb R\times \mathbb T $ settings.

Keywords: Dispersive equation, well-posedness, Strichartz estimate.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B05.
Citation: 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
References:
[1]

J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.

[2]

J. Bourgain, On the Cauchy problem for the Kadomstev-Petviashvili equation, Geometric and Functional Analysis, 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[3]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605.  doi: 10.1353/ajm.2004.0016.

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov–Kuznetsov equation, Diff. Eq., 31 (1995), 1002-1012. 

[5]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Disc. Cont. Dyn. Syst., 34 (2014), 2061-2068.  doi: 10.3934/dcds.2014.34.2061.

[6]

A. D. Ionescu and C. E. Kenig, Local and global well-posedness of periodic KP-I equations, Ann. of Math. Stud., 163 (2007), 181-211. 

[7]

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.

[8]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[9]

C. E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. I. H. Poincaré AN, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[10]

C. E. Kenig and K. D. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Letters, 10 (2003), 879-895.  doi: 10.4310/MRL.2003.v10.n6.a13.

[11]

C. E. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space, Trans. Amer. Math. Soc., 367 (2015), 2551-2612.  doi: 10.1090/S0002-9947-2014-05982-5.

[12]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 4 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[13]

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.

[14]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbb R)$, Int. Math. Res. Not., (2003), 1449–1464. doi: 10.1155/S1073792803211260.

[15]

H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., (2005), 1833–1847. doi: 10.1155/IMRN.2005.1833.

[16]

H. Koch and N. Tzvetkov, On finite energy solutions of the KP-I equation, Mathematische Zeitschrift, 258 (2008), 55-68.  doi: 10.1007/s00209-007-0156-x.

[17]

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.

[18]

F. Linares, A. Pastor and J.-C. Saut, Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton, Comm. PDE, 35 (2010), 1674-1689.  doi: 10.1080/03605302.2010.494195.

[19]

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.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[21]

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

[22]

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

[23]

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

[24]

T. Robert, On the Cauchy problem for the periodic fifth-order KP-I equation, preprint, arXiv: 1805.02052.

[25]

T. Robert, Remark on the semilinear ill-posedness for a periodic higher order KP-I equation, C. R. Acad. Sci. Paris, 356 (2018), 891-898.  doi: 10.1016/j.crma.2018.06.002.

[26]

E. M. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993. doi: 978-0691032165.

[27]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series 32. Princeton, NJ: Princeton University, 1971.

[28]

L. Vega, Restriction Theorems and the Schrödinger multiplier on the torus, Partial Differential Equations with Minimal Smoothness and Applications (Chicago 1990), IMA Vol. Math. Appl. 42 Springer-Verlag, New York (1992), 199–211. doi: 10.1007/978-1-4612-2898-1_18.

[29]

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

show all references

References:
[1]

J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.

[2]

J. Bourgain, On the Cauchy problem for the Kadomstev-Petviashvili equation, Geometric and Functional Analysis, 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[3]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605.  doi: 10.1353/ajm.2004.0016.

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov–Kuznetsov equation, Diff. Eq., 31 (1995), 1002-1012. 

[5]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Disc. Cont. Dyn. Syst., 34 (2014), 2061-2068.  doi: 10.3934/dcds.2014.34.2061.

[6]

A. D. Ionescu and C. E. Kenig, Local and global well-posedness of periodic KP-I equations, Ann. of Math. Stud., 163 (2007), 181-211. 

[7]

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.

[8]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[9]

C. E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. I. H. Poincaré AN, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[10]

C. E. Kenig and K. D. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Letters, 10 (2003), 879-895.  doi: 10.4310/MRL.2003.v10.n6.a13.

[11]

C. E. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space, Trans. Amer. Math. Soc., 367 (2015), 2551-2612.  doi: 10.1090/S0002-9947-2014-05982-5.

[12]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 4 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[13]

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.

[14]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbb R)$, Int. Math. Res. Not., (2003), 1449–1464. doi: 10.1155/S1073792803211260.

[15]

H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., (2005), 1833–1847. doi: 10.1155/IMRN.2005.1833.

[16]

H. Koch and N. Tzvetkov, On finite energy solutions of the KP-I equation, Mathematische Zeitschrift, 258 (2008), 55-68.  doi: 10.1007/s00209-007-0156-x.

[17]

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.

[18]

F. Linares, A. Pastor and J.-C. Saut, Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton, Comm. PDE, 35 (2010), 1674-1689.  doi: 10.1080/03605302.2010.494195.

[19]

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.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[21]

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

[22]

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

[23]

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

[24]

T. Robert, On the Cauchy problem for the periodic fifth-order KP-I equation, preprint, arXiv: 1805.02052.

[25]

T. Robert, Remark on the semilinear ill-posedness for a periodic higher order KP-I equation, C. R. Acad. Sci. Paris, 356 (2018), 891-898.  doi: 10.1016/j.crma.2018.06.002.

[26]

E. M. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993. doi: 978-0691032165.

[27]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series 32. Princeton, NJ: Princeton University, 1971.

[28]

L. Vega, Restriction Theorems and the Schrödinger multiplier on the torus, Partial Differential Equations with Minimal Smoothness and Applications (Chicago 1990), IMA Vol. Math. Appl. 42 Springer-Verlag, New York (1992), 199–211. doi: 10.1007/978-1-4612-2898-1_18.

[29]

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

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