• Previous Article
    Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
  • CPAA Home
  • This Issue
  • Next Article
    Global solutions of a two-dimensional Riemann problem for the pressure gradient system
doi: 10.3934/cpaa.2021005

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

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

* Corresponding author

Received  January 2020 Revised  October 2020 Published  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.

Citation: Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 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. CarberyC. 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. CollianderM. KeelG. StaffilaniH. 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. HaniB. PausaderN. 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 IsazaL. 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. KenigG. 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. PlanchonN. 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. CarberyC. 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. CollianderM. KeelG. StaffilaniH. 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. HaniB. PausaderN. 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 IsazaL. 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. KenigG. 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. PlanchonN. 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

[1]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[2]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009

[3]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[4]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101

[5]

Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511

[6]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[7]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[8]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[9]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[10]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021

[11]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[12]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[13]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[14]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[17]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[18]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[19]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[20]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (16)
  • HTML views (65)
  • Cited by (0)

Other articles
by authors

[Back to Top]