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

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

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.
Citation: Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

A. CarberyC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012. 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]

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. Google Scholar

[9]

S. HerrA. D. IonescuC. 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. Google Scholar

[10]

A. D. IonescuC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

C.E. KenigG. 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. Google Scholar

[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. Google Scholar

[16]

D. LannesF. 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. Google Scholar

[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. 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]

L. MolinetJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. Google Scholar

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. Google Scholar

[2]

A. CarberyC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012. 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]

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. Google Scholar

[9]

S. HerrA. D. IonescuC. 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. Google Scholar

[10]

A. D. IonescuC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

C.E. KenigG. 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. Google Scholar

[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. Google Scholar

[16]

D. LannesF. 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. Google Scholar

[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. 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]

L. MolinetJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. Google Scholar

[1]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

[2]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[3]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[4]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[5]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[6]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[7]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[8]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[9]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[10]

Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061

[11]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036

[12]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

[13]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[14]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521

[15]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605

[16]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

[17]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

[18]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737

[19]

Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671

[20]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (23)
  • HTML views (21)
  • Cited by (0)

Other articles
by authors

[Back to Top]