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March  2021, 20(3): 1039-1058. doi: 10.3934/cpaa.2021005

Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation

Raphaël Côte and  Frédéric Valet ,

IRMA UMR 7501, Université de Strasbourg, CNRS, F-67000 Strasbourg, France

* Corresponding author

Received  January 2020 Revised  October 2020 Published  March 2021 Early access  January 2021

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions $ u $ with initial data $ u_0 \in H^s $ are known to be global if $ s \ge 1 $. We prove that for any integer $ s \ge 2 $, $ \| u(t) \|_{H^s} $ grows at most polynomially in $ t $ for large times $ t $. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies.

It is inspired by analoguous results by Staffilani [21] on the non linear Schrödinger and Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces and a careful study of the variation of the $ H^s $ norm.

Keywords: Zakharov-Kuznetsov equation, weak turbulence, growth of high Sobolev norms.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B65, 35Q35.
Citation: Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005
References:
[1]

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

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[3]

A. Carbery, C. E. Kenig and S. N. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbb{R}^3$, T. Am. Math. Soc., 5 (2013), 2367-2407.  doi: 10.1090/S0002-9947-2012-05685-6.  Google Scholar

[4]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[5]

A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differentsial~ nye Uravneniya, 31 (1995), 1070-1081.  Google Scholar

[6]

J. Ginibre, Le probléme de Cauchy pour des edp semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 796 (1995), 163-187.  Google Scholar

[7]

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.  Google Scholar

[8]

D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Commun. Math. Phys., 324 (2013), 961-993.  doi: 10.1007/s00220-013-1825-8.  Google Scholar

[9]

Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, 3 (2015), 1-63.  doi: 10.1017/fmp.2015.5.  Google Scholar

[10]

J. Pedro Isaza, L. Jorge Mejía and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation, J. Differ. Equ., 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002.  Google Scholar

[11]

J. Pedro Isaza and L. Jorge Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differ. Equ., 68 (2003), 1-12.   Google Scholar

[12]

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

[13]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, arXiv: 1911.13265. doi: 10.3934/dcds.2018061.  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, in Studies in phase space analysis with applications to PDEs, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[15]

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.  Google Scholar

[16]

F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 4 (2011), 1060-1085.  doi: 10.1016/j.jfa.2010.11.005.  Google Scholar

[17]

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

[18]

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

[19]

F. Planchon, N. Tzvetkov and N. Visciglia, On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds, Anal. PDE, 10 (2017), 1123-1147.  doi: 10.2140/apde.2017.10.1123.  Google Scholar

[20]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb R$, Indiana Univ. Math. J., 60 (2011), 1487-1516.  doi: 10.1512/iumj.2011.60.4399.  Google Scholar

[21]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

[22]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz, 66 (1974), 594-597.   Google Scholar

show all references

References:
[1]

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

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[3]

A. Carbery, C. E. Kenig and S. N. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbb{R}^3$, T. Am. Math. Soc., 5 (2013), 2367-2407.  doi: 10.1090/S0002-9947-2012-05685-6.  Google Scholar

[4]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[5]

A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differentsial~ nye Uravneniya, 31 (1995), 1070-1081.  Google Scholar

[6]

J. Ginibre, Le probléme de Cauchy pour des edp semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 796 (1995), 163-187.  Google Scholar

[7]

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.  Google Scholar

[8]

D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Commun. Math. Phys., 324 (2013), 961-993.  doi: 10.1007/s00220-013-1825-8.  Google Scholar

[9]

Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, 3 (2015), 1-63.  doi: 10.1017/fmp.2015.5.  Google Scholar

[10]

J. Pedro Isaza, L. Jorge Mejía and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation, J. Differ. Equ., 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002.  Google Scholar

[11]

J. Pedro Isaza and L. Jorge Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differ. Equ., 68 (2003), 1-12.   Google Scholar

[12]

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

[13]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, arXiv: 1911.13265. doi: 10.3934/dcds.2018061.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 4 (2011), 1060-1085.  doi: 10.1016/j.jfa.2010.11.005.  Google Scholar

[17]

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

[18]

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

[19]

F. Planchon, N. Tzvetkov and N. Visciglia, On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds, Anal. PDE, 10 (2017), 1123-1147.  doi: 10.2140/apde.2017.10.1123.  Google Scholar

[20]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb R$, Indiana Univ. Math. J., 60 (2011), 1487-1516.  doi: 10.1512/iumj.2011.60.4399.  Google Scholar

[21]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

[22]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz, 66 (1974), 594-597.   Google Scholar

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