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Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation
1. | Université Paris-Est Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France |
2. | Université Paris 13 Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France |
$ u_t-D_x^α u_{x} + u_{xyy} = uu_x,\,\,\,\,\,\, (t,x,y)∈\mathbb{R}^3, 1≤ α≤ 2,$ |
$E^s$ |
$s > \frac{2}{\alpha } - \frac{3}{4}$ |
$\|f{{\|}_{{{E}^{s}}}}=\|{{\left\langle {{\left| \xi \right|}^{\alpha }}+{{\mu }^{2}} \right\rangle }^{s}}\hat{f}{{\|}_{{{L}^{2}}({{\mathbb{R}}^{2}})}}.$ |
$E^{1/2}$ |
$α>\frac 85$ |
References:
[1] |
J. L. Bona and R. Smith,
The initial value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond., Ser. A, 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[2] |
A. Carbery, C. E. Kenig and S. N. Ziesler,
Restriction for homogeneous polynomial surfaces in R3, Trans. Amer. Math. Soc., 365 (2013), 2367-2407.
doi: 10.1090/S0002-9947-2012-05685-6. |
[3] |
A. Cunha and A. Pastor,
The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces, J. Math. Anal. Appl., 417 (2014), 660-693.
doi: 10.1016/j.jmaa.2014.03.056. |
[4] |
A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differential Equations, 261 (2016), 2041–2067, arXiv: 1601.02803.
doi: 10.1016/j.jde.2016.04.022. |
[5] |
A. Esfahani and A. Pastor,
Ill-posseness results for the (generalized) Benjamin-Ono-ZakharovKuznetsov equation, Proc. Amer. Math. Soc., 139 (2011), 943-956.
doi: 10.1090/S0002-9939-2010-10532-4. |
[6] |
A. V. Faminskii,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
|
[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] |
Z. Guo,
The Fourier restriction norm method for the Zakharov-Kuznetsov equation, J. Differential Equations, 252 (2012), 2053-2084.
doi: 10.1016/j.jde.2011.10.012. |
[9] |
S. Herr, A. D. Ionescu, C. E. Kenig and H. Koch,
A para-differential renormalization technique for nonlinear dispersive equations, Comm. Partial Differential Equations, 35 (2010), 1827-1875.
doi: 10.1080/03605302.2010.487232. |
[10] |
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. |
[11] |
R. J. Iorio,
On the cauchy problem for the Benjamin-Ono equation, C.P.D.E., 11 (1986), 1031-1081.
doi: 10.1080/03605308608820456. |
[12] |
M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of twodimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations,
Chaos 15 (2005), 037104, 13pp.
doi: 10.1063/1.1877892. |
[13] |
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. |
[14] |
C.E. Kenig, G. Ponce and L. Vega,
Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
doi: 10.1090/S0894-0347-1991-1086966-0. |
[15] |
H. Koch and N. Tzvetkov,
Local well-posedness of the Benjamin-Ono equation in $ H^s(\mathbb{R})$, I.M.R.N., 26 (2003), 1449-1464.
doi: 10.1155/S1073792803211260. |
[16] |
D. Lannes, F. Linares and J.-C. Saut,
The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation, Prog. Nonlinear Differ. Equ. Appl., 84 (2013), 181-213.
doi: 10.1007/978-1-4614-6348-1_10. |
[17] |
J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion,
Chaos 16 (2006), 043103, 10pp.
doi: 10.1063/1.2355555. |
[18] |
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[19] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono equation and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[20] |
L. Molinet and S. Vento,
Improvement of the energy method for strongly non resonant dispersive equations and applications, Anal. PDE, 8 (2015), 1455-1495.
doi: 10.2140/apde.2015.8.1455. |
[21] |
F. Ribaud and S. Vento,
Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.
doi: 10.1137/110850566. |
[22] |
T. Tao,
Global well-posedness of the Benjamin-Ono equation in H1($\mathbb{R}$),, J. Hyp. Diff. Eq., 1 (2004), 17-49.
doi: 10.1142/S0219891604000032. |
[23] |
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. R. Soc. Lond., Ser. A, 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[2] |
A. Carbery, C. E. Kenig and S. N. Ziesler,
Restriction for homogeneous polynomial surfaces in R3, Trans. Amer. Math. Soc., 365 (2013), 2367-2407.
doi: 10.1090/S0002-9947-2012-05685-6. |
[3] |
A. Cunha and A. Pastor,
The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces, J. Math. Anal. Appl., 417 (2014), 660-693.
doi: 10.1016/j.jmaa.2014.03.056. |
[4] |
A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differential Equations, 261 (2016), 2041–2067, arXiv: 1601.02803.
doi: 10.1016/j.jde.2016.04.022. |
[5] |
A. Esfahani and A. Pastor,
Ill-posseness results for the (generalized) Benjamin-Ono-ZakharovKuznetsov equation, Proc. Amer. Math. Soc., 139 (2011), 943-956.
doi: 10.1090/S0002-9939-2010-10532-4. |
[6] |
A. V. Faminskii,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.
|
[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] |
Z. Guo,
The Fourier restriction norm method for the Zakharov-Kuznetsov equation, J. Differential Equations, 252 (2012), 2053-2084.
doi: 10.1016/j.jde.2011.10.012. |
[9] |
S. Herr, A. D. Ionescu, C. E. Kenig and H. Koch,
A para-differential renormalization technique for nonlinear dispersive equations, Comm. Partial Differential Equations, 35 (2010), 1827-1875.
doi: 10.1080/03605302.2010.487232. |
[10] |
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. |
[11] |
R. J. Iorio,
On the cauchy problem for the Benjamin-Ono equation, C.P.D.E., 11 (1986), 1031-1081.
doi: 10.1080/03605308608820456. |
[12] |
M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of twodimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations,
Chaos 15 (2005), 037104, 13pp.
doi: 10.1063/1.1877892. |
[13] |
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. |
[14] |
C.E. Kenig, G. Ponce and L. Vega,
Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
doi: 10.1090/S0894-0347-1991-1086966-0. |
[15] |
H. Koch and N. Tzvetkov,
Local well-posedness of the Benjamin-Ono equation in $ H^s(\mathbb{R})$, I.M.R.N., 26 (2003), 1449-1464.
doi: 10.1155/S1073792803211260. |
[16] |
D. Lannes, F. Linares and J.-C. Saut,
The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation, Prog. Nonlinear Differ. Equ. Appl., 84 (2013), 181-213.
doi: 10.1007/978-1-4614-6348-1_10. |
[17] |
J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion,
Chaos 16 (2006), 043103, 10pp.
doi: 10.1063/1.2355555. |
[18] |
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[19] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono equation and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[20] |
L. Molinet and S. Vento,
Improvement of the energy method for strongly non resonant dispersive equations and applications, Anal. PDE, 8 (2015), 1455-1495.
doi: 10.2140/apde.2015.8.1455. |
[21] |
F. Ribaud and S. Vento,
Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.
doi: 10.1137/110850566. |
[22] |
T. Tao,
Global well-posedness of the Benjamin-Ono equation in H1($\mathbb{R}$),, J. Hyp. Diff. Eq., 1 (2004), 17-49.
doi: 10.1142/S0219891604000032. |
[23] |
V. E. Zakharov and E. A. Kuznetsov,
On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.
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