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January  2017, 37(1): 449-483. doi: 10.3934/dcds.2017019

Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

Francis Ribaud 1, and  Stéphane Vento 2,

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

Received  January 2016 Revised  September 2016 Published  November 2016

Fund Project: The second author is partially supported by the french ANR project GEODISP.

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation
$ u_t-D_x^α u_{x} + u_{xyy} = uu_x,\,\,\,\,\,\, (t,x,y)∈\mathbb{R}^3, 1≤ α≤ 2,$
is locally well-posed in the spaces
$E^s$
,
$s > \frac{2}{\alpha } - \frac{3}{4}$
, endowed with the norm
$\|f{{\|}_{{{E}^{s}}}}=\|{{\left\langle {{\left| \xi \right|}^{\alpha }}+{{\mu }^{2}} \right\rangle }^{s}}\hat{f}{{\|}_{{{L}^{2}}({{\mathbb{R}}^{2}})}}.$
 As a consequence, we get the global well-posedness in the energy space
$E^{1/2}$
 as soon as
$α>\frac 85$
. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru [10] combined with new Strichartz estimates and a modified energy.
Keywords: BO-ZK equation, Cauchy problem, local well-posedness.
Mathematics Subject Classification: Primary:35A01, 35Q53;Secondary:35Q6.
Citation: Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
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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