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An overdetermined problem associated to the Finsler Laplacian
Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation
IRMA UMR 7501, Université de Strasbourg, CNRS, F-67000 Strasbourg, France |
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 [
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. |
[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. |
[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. |
[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. |
[5] |
A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differentsial~ nye Uravneniya, 31 (1995), 1070-1081. |
[6] |
J. Ginibre, Le probléme de Cauchy pour des edp semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 796 (1995), 163-187. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
V. E. Zakharov and E. A. Kuznetsov,
On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz, 66 (1974), 594-597.
|
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. |
[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. |
[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. |
[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. |
[5] |
A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differentsial~ nye Uravneniya, 31 (1995), 1070-1081. |
[6] |
J. Ginibre, Le probléme de Cauchy pour des edp semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 796 (1995), 163-187. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
V. E. Zakharov and E. A. Kuznetsov,
On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz, 66 (1974), 594-597.
|
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