May  2022, 42(5): 2257-5593. doi: 10.3934/dcds.2021190

On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

Departamento de Matemáticas, Universidad Nacional de Colombia, Sede–Bogotá, Colombia

Received  June 2021 Revised  October 2021 Published  May 2022 Early access  December 2021

In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type
$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $
in anisotropic Sobolev spaces, where
$ 1\le \alpha \le 1 $
,
$ \mathscr{H} $
is the Hilbert transform and
$ D_{x}^{\alpha} $
is the fractional derivative, both with respect to
$ x $
.
Citation: Jorge Morales Paredes, Félix Humberto Soriano Méndez. On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2257-5593. doi: 10.3934/dcds.2021190
References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2]

M. J. Ablowitz and H. Segur, Long internal waves in fluids of great depth, Stud. Appl. Math., 62 (1980), 249-262.  doi: 10.1002/sapm1980623249.

[3]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.

[4]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X.

[5]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[6]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.

[7]

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.

[8]

A. CastroD. Córdoba and F. Gancedo, Singularity formations for a surface wave model, Nonlinearity, 23 (2010), 2835-2847.  doi: 10.1088/0951-7715/23/11/006.

[9]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolut ion equations, Comm. Math. Phys., 144 (1992), 163–188, URL http://projecteuclid.org/euclid.cmp/1104249221. doi: 10.1007/BF02099195.

[10]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Soviet Physics Doklady, 15 (1970), 539. 

[11]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; Dedicated to Konrad Jörgens), vol. 448 of Lecture Notes in Math., Springer, Berlin, 1975, 25–70.

[12]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[13]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[14]

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

[15]

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

[16]

C. E. KenigY. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005.

[17]

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

[18]

B. Kim, Three-Dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusetts Institute of Technology. Dept. of Mathematics., Cambridge, MA, 2006.

[19]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.  doi: 10.1016/j.physd.2014.12.004.

[20]

K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.

[21]

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

[22]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.

[23]

D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinet. Relat. Models, 6 (2013), 989-1009.  doi: 10.3934/krm.2013.6.989.

[24]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.  doi: 10.1137/080739173.

[25]

F. LinaresD. Pilod and J.-C. Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4195-4221.  doi: 10.1137/110828277.

[26]

F. LinaresD. Pilod and J.-C. Saut, Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858. 

[27]

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

[28]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2009, https://link.springer.com/book/10.1007/978-0-387-84899-0.

[29]

J. D. C. Lizarazo Osorio, El Problema de Cauchy de la Clase de Ecuaciones de Dispersión Generalizada de Benjamin-Ono Bidimensionales, PhD thesis, Universidad Nacional de Colombia, Bogotá, Colombia, 2018, https://repositorio.unal.edu.co/handle/unal/63728

[30]

L. MolinetJ.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[31]

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

[32]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305–349, http://projecteuclid.org/euclid.cmp/1104270835. doi: 10.1007/BF02101705.

[33]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.

[34]

F. Sánchez Salazar, El Problema de Cauchy Asociado a una Ecuación del Tipo RBO-ZK, PhD thesis, Universidad Nacional de Colombia, 2015, URL https://repositorio.unal.edu.co/handle/unal/54116.

[35]

J.-C. Saut, Remarks on the generalized kadomtsev-petviashvili equations, Indiana University Mathematics Journal, 42 (1993), 1011–1026, URL http://www.jstor.org/stable/24897132. doi: 10.1512/iumj.1993.42.42047.

[36] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. 
[37]

G. B. Whitham, Variational methods and applications to water waves, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 299 (1967), 6-25.  doi: 10.1098/rspa.1967.0119.

[38]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.

[39]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Soviet Physics JETP, 29 (1974), 594-597. 

show all references

References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2]

M. J. Ablowitz and H. Segur, Long internal waves in fluids of great depth, Stud. Appl. Math., 62 (1980), 249-262.  doi: 10.1002/sapm1980623249.

[3]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.

[4]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X.

[5]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[6]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.

[7]

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.

[8]

A. CastroD. Córdoba and F. Gancedo, Singularity formations for a surface wave model, Nonlinearity, 23 (2010), 2835-2847.  doi: 10.1088/0951-7715/23/11/006.

[9]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolut ion equations, Comm. Math. Phys., 144 (1992), 163–188, URL http://projecteuclid.org/euclid.cmp/1104249221. doi: 10.1007/BF02099195.

[10]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Soviet Physics Doklady, 15 (1970), 539. 

[11]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; Dedicated to Konrad Jörgens), vol. 448 of Lecture Notes in Math., Springer, Berlin, 1975, 25–70.

[12]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[13]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[14]

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

[15]

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

[16]

C. E. KenigY. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005.

[17]

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

[18]

B. Kim, Three-Dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusetts Institute of Technology. Dept. of Mathematics., Cambridge, MA, 2006.

[19]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.  doi: 10.1016/j.physd.2014.12.004.

[20]

K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.

[21]

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

[22]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.

[23]

D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinet. Relat. Models, 6 (2013), 989-1009.  doi: 10.3934/krm.2013.6.989.

[24]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.  doi: 10.1137/080739173.

[25]

F. LinaresD. Pilod and J.-C. Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4195-4221.  doi: 10.1137/110828277.

[26]

F. LinaresD. Pilod and J.-C. Saut, Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858. 

[27]

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

[28]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2009, https://link.springer.com/book/10.1007/978-0-387-84899-0.

[29]

J. D. C. Lizarazo Osorio, El Problema de Cauchy de la Clase de Ecuaciones de Dispersión Generalizada de Benjamin-Ono Bidimensionales, PhD thesis, Universidad Nacional de Colombia, Bogotá, Colombia, 2018, https://repositorio.unal.edu.co/handle/unal/63728

[30]

L. MolinetJ.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[31]

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

[32]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305–349, http://projecteuclid.org/euclid.cmp/1104270835. doi: 10.1007/BF02101705.

[33]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.

[34]

F. Sánchez Salazar, El Problema de Cauchy Asociado a una Ecuación del Tipo RBO-ZK, PhD thesis, Universidad Nacional de Colombia, 2015, URL https://repositorio.unal.edu.co/handle/unal/54116.

[35]

J.-C. Saut, Remarks on the generalized kadomtsev-petviashvili equations, Indiana University Mathematics Journal, 42 (1993), 1011–1026, URL http://www.jstor.org/stable/24897132. doi: 10.1512/iumj.1993.42.42047.

[36] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. 
[37]

G. B. Whitham, Variational methods and applications to water waves, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 299 (1967), 6-25.  doi: 10.1098/rspa.1967.0119.

[38]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.

[39]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Soviet Physics JETP, 29 (1974), 594-597. 

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