• Previous Article
    Number of bounded distance equivalence classes in hulls of repetitive Delone sets
  • DCDS Home
  • This Issue
  • Next Article
    An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515)
doi: 10.3934/dcds.2021190
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems, 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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[39]

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

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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[39]

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

[1]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097

[2]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078

[3]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483

[4]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087

[5]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[6]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[7]

Roger P. de Moura, Ailton C. Nascimento, Gleison N. Santos. On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021022

[8]

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

[9]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019

[10]

Pedro Isaza, Juan López, Jorge Mejía. Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 887-905. doi: 10.3934/cpaa.2006.5.887

[11]

Felipe Linares, Jean-Claude Saut. The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 547-565. doi: 10.3934/dcds.2009.24.547

[12]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

[13]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727

[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]

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

[16]

Felipe Linares, Mahendra Panthee, Tristan Robert, Nikolay Tzvetkov. On the periodic Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3521-3533. doi: 10.3934/dcds.2019145

[17]

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

[18]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[19]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863

[20]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (63)
  • HTML views (49)
  • Cited by (0)

[Back to Top]