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September  2020, 19(9): 4285-4325. doi: 10.3934/cpaa.2020194

On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation

Universidade Federal do Piauí, Campus Universitário Ministro Petrônio Portella, Ininga, 64049-550, Teresina-PI, Brazil

Received  September 2019 Revised  March 2020 Published  June 2020

Fund Project: The author was supported by CNPq 140383/2013-1

In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual $ L^{2}( \mathbb R^2) $-based Sobolev spaces $ H^s( \mathbb R^2) $ for $ s>\frac{5}{4} $ which coincides with the best available result in the literature proved employing more complicated tools.

Citation: A. C. Nascimento. On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4285-4325. doi: 10.3934/cpaa.2020194
References:
[1]

B. Bajvsank and R. Coifman, On singular integrals, in Proc. Symp. Pure Math., American Mathematical Society, Providence, RI, (1966), 1–17.

[2]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. 

[3]

J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.

[4]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. London. Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.197A.0035.

[5]

A. P. Calderón, Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA, 53 (1965), 1092-1099.  doi: 10.1073/pnas.53.A.1092.

[6]

A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differ. Equ., 261 (2016), 2041-2067.  doi: 10.1016/j.jde.2016.04.022.

[7]

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.

[8]

L. DawsonH. McGahagan and G. Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2081-2090.  doi: 10.1090/S0002-9939-08-09355-6.

[9]

A. Esfahani and A. Pastor, Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation, Proc. Amer. Math. Soc., 139 (2011), 943-956.  doi: 10.1090/S0002-9939-2010-10532-4.

[10]

A. Esfahani and A. Pastor, Instability of solitary wave solutions for the generalized BO-ZK equation, J. Differ. Equ., 247 (2009), 3181-3201.  doi: 10.1016/j.jde.2009.09.014.

[11]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg-type inequality and applications, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 171-185.  doi: 10.1007/s00574-016-0017-5.

[12]

A. Esfahani and A. Pastor, On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations, Bull. Lond. Math. Soc., 43 (2011), 1130-1140.  doi: 10.1112/blms/bdr048.

[13]

A. EsfahaniA. Pastor and J. L. Bona, Stability and decay properties of solitary-wave solutions for the generalized BO-ZK equation, Adv. Differ. Equ., 20 (2015), 801-834. 

[14]

G. B. Folland, Introduction to Partial Differential Equations, 2$^{nd}$ edition, Princeton University Press, Princeton, NJ, 1995.

[15]

L. Grafakos and O. Seungly, The Kato-Ponce Inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.

[16]

R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Commun. Partial Differ. Equ., 11 (1986), 1031-1081.  doi: 10.1080/03605308608820456.

[17]

A. D. IonescuC. Kenig and D. Tataru, Global well-posedness of the initial value problem for the KP I equation in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[18]

P. IsazaF. Linares and G. Ponce, On the propagation of regularities in solutions of the Benjamin-Ono equation, J. Funct. Anal., 270 (2016), 976-1000.  doi: 10.1016/j.jfa.201A.11.009.

[19]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 48 (2016), 1006-1024.  doi: 10.1137/15M1012098.

[20]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation, Commun. Partial Differ. Equ., 40 (2015), 1336-1364.  doi: 10.1080/03605302.2014.985794.

[21]

M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of two-dimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations, Chaos, 15 (2005), Art. 037104. doi: 10.1063/1.1877892.

[22]

T. Kato, On the Cauchy Problem for the (Generalized) Korteweg-de Vries Equation, in Advances in Mathematics Supplementary Studies, Studies in Applied Mathematics, Vol. 8, London, Academic Press, (1983), 93–128.

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891–907. doi: 10.1002/cpa.3160410704.

[24]

C. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[25]

C. Kenig and K. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Lett., 10 (2003), 879-895.  doi: 10.4310/MRL.2003.v10.n6.a13.

[26]

C. E. KenigF. LinaresG. Ponce and L. Vega, On the regularity of solutions to the k-generalized Korteweg-de Vries equation, Proc. Amer. Math. Soc., 146 (2018), 3759-3766.  doi: 10.1090/proc/13506.

[27]

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: 10.2307/2939277.

[28]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Commun. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[29]

C. E. Kenig and S. N. Ziesler, Maximal function estimates with applications to a modified Kadomstev-Petviashvili equation, commun. Pure Appl. Anal., 4 (2005), 45-91.  doi: 10.3934/cpaa.200A.4.45.

[30]

H. Koch and N. Tzvetkov, On the Local Well-Posedness of the Benjamin-Ono equation in $H^{s}(\mathbb R)$, Int. Math. Res. Not., 26 (2003), 1449-1464.  doi: 10.1155/S1073792803211260.

[31]

J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kuznetsov lateral dispersion, Chaos, 16 (2006), Art. 043103. doi: 10.1063/1.2355555.

[32]

D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23-100.  doi: 10.4171/rmi/1049.

[33]

F. Linares, H. Miyazaki and G. Ponce, On a class of solutions to the generalized KdV type equation, Commun. Contemp. Math., (2019). doi: 10.1142/S0219199718500566.

[34]

F. LinaresD. Pilod and J. C. Saut, The Cauchy problem for the fractional Kadomtsev-Petviashvili equations, SIAM J. Math. Anal., 50 (2018), 3172-3209.  doi: 10.1137/17M1145379.

[35]

F. LinaresG. Ponce and D. L. Smith, On the regularity of solutions to a class of nonlinear dispersive equations, Math. Ann., 369 (2017), 797-837.  doi: 10.1007/s00208-016-1452-8.

[36]

F. Linares and G. Ponce, On special regularity Properties of Solutions of the Zakharov-Kuznetsov Equation, Commun. Pure Appl. Anal., 17 (2018), 1561-1572.  doi: 10.3934/cpaa.2018074.

[37]

A. J. Mendez, On the propagation of regularity for solutions of the dispersion generalized Benjamin-Ono equation, to appear in Anal. Partial Differ. Equ., arXiv: 1901.00823.

[38]

A. J. Mendez, On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation, preprint, arXiv: 1902.08296.

[39]

L. MolinetJ. 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.

[40]

C. MuscaluJ. PipherT. Tau and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.

[41]

A. C. Nascimento, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, J. Math. Anal. Appl., 478 (2019), 156-181.  doi: 10.1016/j.jmaa.2019.0A.024.

[42]

A. C. Nascimento, On the properties in the solutions of a Kadomtsev-Petviashvili-Benjamin-Ono(KP-BO) equation, preprint.

[43]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Jpn., 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.

[44]

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.

[45]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286. 

show all references

References:
[1]

B. Bajvsank and R. Coifman, On singular integrals, in Proc. Symp. Pure Math., American Mathematical Society, Providence, RI, (1966), 1–17.

[2]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. 

[3]

J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.

[4]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. London. Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.197A.0035.

[5]

A. P. Calderón, Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA, 53 (1965), 1092-1099.  doi: 10.1073/pnas.53.A.1092.

[6]

A. Cunha and A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, J. Differ. Equ., 261 (2016), 2041-2067.  doi: 10.1016/j.jde.2016.04.022.

[7]

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.

[8]

L. DawsonH. McGahagan and G. Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2081-2090.  doi: 10.1090/S0002-9939-08-09355-6.

[9]

A. Esfahani and A. Pastor, Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation, Proc. Amer. Math. Soc., 139 (2011), 943-956.  doi: 10.1090/S0002-9939-2010-10532-4.

[10]

A. Esfahani and A. Pastor, Instability of solitary wave solutions for the generalized BO-ZK equation, J. Differ. Equ., 247 (2009), 3181-3201.  doi: 10.1016/j.jde.2009.09.014.

[11]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg-type inequality and applications, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 171-185.  doi: 10.1007/s00574-016-0017-5.

[12]

A. Esfahani and A. Pastor, On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations, Bull. Lond. Math. Soc., 43 (2011), 1130-1140.  doi: 10.1112/blms/bdr048.

[13]

A. EsfahaniA. Pastor and J. L. Bona, Stability and decay properties of solitary-wave solutions for the generalized BO-ZK equation, Adv. Differ. Equ., 20 (2015), 801-834. 

[14]

G. B. Folland, Introduction to Partial Differential Equations, 2$^{nd}$ edition, Princeton University Press, Princeton, NJ, 1995.

[15]

L. Grafakos and O. Seungly, The Kato-Ponce Inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.

[16]

R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Commun. Partial Differ. Equ., 11 (1986), 1031-1081.  doi: 10.1080/03605308608820456.

[17]

A. D. IonescuC. Kenig and D. Tataru, Global well-posedness of the initial value problem for the KP I equation in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[18]

P. IsazaF. Linares and G. Ponce, On the propagation of regularities in solutions of the Benjamin-Ono equation, J. Funct. Anal., 270 (2016), 976-1000.  doi: 10.1016/j.jfa.201A.11.009.

[19]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 48 (2016), 1006-1024.  doi: 10.1137/15M1012098.

[20]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation, Commun. Partial Differ. Equ., 40 (2015), 1336-1364.  doi: 10.1080/03605302.2014.985794.

[21]

M. C. Jorge, G. Cruz-Pacheco, L. Mier-y-Teran-Romero and N. F. Smyth, Evolution of two-dimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations, Chaos, 15 (2005), Art. 037104. doi: 10.1063/1.1877892.

[22]

T. Kato, On the Cauchy Problem for the (Generalized) Korteweg-de Vries Equation, in Advances in Mathematics Supplementary Studies, Studies in Applied Mathematics, Vol. 8, London, Academic Press, (1983), 93–128.

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891–907. doi: 10.1002/cpa.3160410704.

[24]

C. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[25]

C. Kenig and K. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Lett., 10 (2003), 879-895.  doi: 10.4310/MRL.2003.v10.n6.a13.

[26]

C. E. KenigF. LinaresG. Ponce and L. Vega, On the regularity of solutions to the k-generalized Korteweg-de Vries equation, Proc. Amer. Math. Soc., 146 (2018), 3759-3766.  doi: 10.1090/proc/13506.

[27]

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: 10.2307/2939277.

[28]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Commun. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[29]

C. E. Kenig and S. N. Ziesler, Maximal function estimates with applications to a modified Kadomstev-Petviashvili equation, commun. Pure Appl. Anal., 4 (2005), 45-91.  doi: 10.3934/cpaa.200A.4.45.

[30]

H. Koch and N. Tzvetkov, On the Local Well-Posedness of the Benjamin-Ono equation in $H^{s}(\mathbb R)$, Int. Math. Res. Not., 26 (2003), 1449-1464.  doi: 10.1155/S1073792803211260.

[31]

J. C. Latorre, A. A. Minzoni, N. F. Smyth and C. A. Vargas, Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kuznetsov lateral dispersion, Chaos, 16 (2006), Art. 043103. doi: 10.1063/1.2355555.

[32]

D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23-100.  doi: 10.4171/rmi/1049.

[33]

F. Linares, H. Miyazaki and G. Ponce, On a class of solutions to the generalized KdV type equation, Commun. Contemp. Math., (2019). doi: 10.1142/S0219199718500566.

[34]

F. LinaresD. Pilod and J. C. Saut, The Cauchy problem for the fractional Kadomtsev-Petviashvili equations, SIAM J. Math. Anal., 50 (2018), 3172-3209.  doi: 10.1137/17M1145379.

[35]

F. LinaresG. Ponce and D. L. Smith, On the regularity of solutions to a class of nonlinear dispersive equations, Math. Ann., 369 (2017), 797-837.  doi: 10.1007/s00208-016-1452-8.

[36]

F. Linares and G. Ponce, On special regularity Properties of Solutions of the Zakharov-Kuznetsov Equation, Commun. Pure Appl. Anal., 17 (2018), 1561-1572.  doi: 10.3934/cpaa.2018074.

[37]

A. J. Mendez, On the propagation of regularity for solutions of the dispersion generalized Benjamin-Ono equation, to appear in Anal. Partial Differ. Equ., arXiv: 1901.00823.

[38]

A. J. Mendez, On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation, preprint, arXiv: 1902.08296.

[39]

L. MolinetJ. 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.

[40]

C. MuscaluJ. PipherT. Tau and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.

[41]

A. C. Nascimento, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, J. Math. Anal. Appl., 478 (2019), 156-181.  doi: 10.1016/j.jmaa.2019.0A.024.

[42]

A. C. Nascimento, On the properties in the solutions of a Kadomtsev-Petviashvili-Benjamin-Ono(KP-BO) equation, preprint.

[43]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Jpn., 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.

[44]

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.

[45]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286. 

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