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

Raphaël Côte; Frédéric Valet

doi:10.3934/cpaa.2021005

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

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 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, 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
[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

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

[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]	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
[3]	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
[4]	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
[5]	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
[6]	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
[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]	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
[10]	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
[11]	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
[12]	Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[13]	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
[14]	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
[15]	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
[16]	Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[17]	Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[18]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[19]	Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors