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On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
On the periodic Zakharov-Kuznetsov equation
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 |
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.
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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