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September  2020, 40(9): 5189-5215. doi: 10.3934/dcds.2020225

On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations

Robert Schippa

Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Analysis, Englerstrasse 2, 76131 Karlsruhe, Germany

Received  July 2019 Revised  October 2019 Published  June 2020

Fund Project: Financial support by the German Research Foundation (DFG) through the IRTG 2235 and CRC 1173, Project-ID 258734477, is gratefully acknowledged

A family of dispersive equations is considered, which links a higher-dimensional Benjamin-Ono equation and the Zakharov-Kuznetsov equation. For these fractional Zakharov-Kuznetsov equations new well-posedness results are proved using transversality and time localization to small frequency dependent time intervals.

Keywords: Benjamin-Ono equation, Zakharov-Kuznetsov equation, local well-posedness, short-time Fourier restriction norm method.
Mathematics Subject Classification: Primary: 35Q53, 35B30.
Citation: Robert Schippa. On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5189-5215. doi: 10.3934/dcds.2020225
References:
[1]

T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-562.   Google Scholar

[2]

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.  Google Scholar

[3]

J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, (1998), 253–283. doi: 10.1155/S1073792898000191.  Google Scholar

[4]

M. Christ, J. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math. (N.S.), 32 (2012), 51-75.  doi: 10.14510/lm-ns.v32i1.32.  Google Scholar

[5]

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

[6]

Z. H. Guo, Local well-posedness and a priori bounds for the modified Benjamin-Ono equation, Adv. Differential Equations, 16 (2011), 1087-1137.   Google Scholar

[7]

Z. H. Guo, Local well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces, J. Differential Equations, 252 (2012), 2053-2084.  doi: 10.1016/j.jde.2011.10.012.  Google Scholar

[8]

Z. H. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656–1729. doi: 10.1093/imrn/rnw271.  Google Scholar

[9]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[10]

M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971-972.  doi: 10.1016/j.anihpc.2010.01.006.  Google Scholar

[11]

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.  Google Scholar

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J. Hickman, F. Linares, O. G. Riaño, K. M. Rogers and J. Wright, On a higher dimensional version of the Benjamin-Ono equation, SIAM J. Math. Anal., 51 (2019), 4544-4569.  doi: 10.1137/19M1241970.  Google Scholar

[13]

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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955–980, http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039.  Google Scholar

[15]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math. (2), 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.  Google Scholar

[16]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, e-prints, arXiv: 1905.01490. Google Scholar

[17]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

[18]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.  Google Scholar

[19]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s({\Bbb R})$, Int. Math. Res. Not., (2003), 1449–1464. doi: 10.1155/S1073792803211260.  Google Scholar

[20]

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.  Google Scholar

[21]

F. Linares, M. Panthee, T. Robert and N. Tzvetkov, On the periodic Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 3521-3533.  doi: 10.3934/dcds.2019145.  Google Scholar

[22]

M. Mariş, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085.  doi: 10.1016/S0362-546X(01)00880-X.  Google Scholar

[23]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.  doi: 10.1137/S0036141001385307.  Google Scholar

[24]

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

[25]

D. E. Pelinovsky and V. I. Shrira, Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows, Physics Letters A, 206 (1995), 195-202.  doi: 10.1016/0375-9601(95)00608-6.  Google Scholar

[26]

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

[27]

F. Ribaud and S. Vento, Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 37 (2017), 449-483.  doi: 10.3934/dcds.2017019.  Google Scholar

[28]

J.-C. Saut, Benjamin-Ono and intermediate long wave equations: Modeling, IST and PDE, Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, Fields Inst. Commun., Springer, New York, 83 (2019), 95-160.   Google Scholar

[29]

R. Schippa, On shorttime bilinear Strichartz estimates and applications to the Shrira equation, Nonlinear Anal., 198 (2020), 111910. doi: 10.1016/j.na.2020.111910.  Google Scholar

[30]

R. Schippa, On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$, e-prints, arXiv: 1704.07174. Google Scholar

[31] C. D. Sogge, Fourier Integrals in Classical Analysis, Second edition, Cambridge Tracts in Mathematics, 210. Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781316341186.  Google Scholar
[32]

V. Zakharov and E. Kuznetsov, On three dimensional solitons, J. Exp. Theor. Phys., 39 (1974), 285-286.   Google Scholar

show all references

References:
[1]

T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-562.   Google Scholar

[2]

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.  Google Scholar

[3]

J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, (1998), 253–283. doi: 10.1155/S1073792898000191.  Google Scholar

[4]

M. Christ, J. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math. (N.S.), 32 (2012), 51-75.  doi: 10.14510/lm-ns.v32i1.32.  Google Scholar

[5]

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

[6]

Z. H. Guo, Local well-posedness and a priori bounds for the modified Benjamin-Ono equation, Adv. Differential Equations, 16 (2011), 1087-1137.   Google Scholar

[7]

Z. H. Guo, Local well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces, J. Differential Equations, 252 (2012), 2053-2084.  doi: 10.1016/j.jde.2011.10.012.  Google Scholar

[8]

Z. H. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656–1729. doi: 10.1093/imrn/rnw271.  Google Scholar

[9]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[10]

M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971-972.  doi: 10.1016/j.anihpc.2010.01.006.  Google Scholar

[11]

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.  Google Scholar

[12]

J. Hickman, F. Linares, O. G. Riaño, K. M. Rogers and J. Wright, On a higher dimensional version of the Benjamin-Ono equation, SIAM J. Math. Anal., 51 (2019), 4544-4569.  doi: 10.1137/19M1241970.  Google Scholar

[13]

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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955–980, http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039.  Google Scholar

[15]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math. (2), 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.  Google Scholar

[16]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, e-prints, arXiv: 1905.01490. Google Scholar

[17]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

[18]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.  Google Scholar

[19]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s({\Bbb R})$, Int. Math. Res. Not., (2003), 1449–1464. doi: 10.1155/S1073792803211260.  Google Scholar

[20]

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.  Google Scholar

[21]

F. Linares, M. Panthee, T. Robert and N. Tzvetkov, On the periodic Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 3521-3533.  doi: 10.3934/dcds.2019145.  Google Scholar

[22]

M. Mariş, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085.  doi: 10.1016/S0362-546X(01)00880-X.  Google Scholar

[23]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.  doi: 10.1137/S0036141001385307.  Google Scholar

[24]

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

[25]

D. E. Pelinovsky and V. I. Shrira, Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows, Physics Letters A, 206 (1995), 195-202.  doi: 10.1016/0375-9601(95)00608-6.  Google Scholar

[26]

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

[27]

F. Ribaud and S. Vento, Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 37 (2017), 449-483.  doi: 10.3934/dcds.2017019.  Google Scholar

[28]

J.-C. Saut, Benjamin-Ono and intermediate long wave equations: Modeling, IST and PDE, Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, Fields Inst. Commun., Springer, New York, 83 (2019), 95-160.   Google Scholar

[29]

R. Schippa, On shorttime bilinear Strichartz estimates and applications to the Shrira equation, Nonlinear Anal., 198 (2020), 111910. doi: 10.1016/j.na.2020.111910.  Google Scholar

[30]

R. Schippa, On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$, e-prints, arXiv: 1704.07174. Google Scholar

[31] C. D. Sogge, Fourier Integrals in Classical Analysis, Second edition, Cambridge Tracts in Mathematics, 210. Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781316341186.  Google Scholar
[32]

V. Zakharov and E. Kuznetsov, On three dimensional solitons, J. Exp. Theor. Phys., 39 (1974), 285-286.   Google Scholar

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