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May  2019, 18(3): 1177-1203. doi: 10.3934/cpaa.2019057

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Eddye Bustamante , José Jiménez Urrea and  Jorge Mejía

Departamento de Matemáticas, Universidad Nacional de Colombia, A. A. 3840 Medellín, Colombia

Received  March 2018 Revised  August 2018 Published  November 2018

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation
$\left. \begin{array}{rl} u_t+D_x^{\alpha} u_x +\mathcal Hu_{yy} +uu_x &\hspace{-2mm} = 0, \qquad\qquad (x, y)\in\mathbb R^2, \; t\in\mathbb R, \\ u(x, y, 0)&\hspace{-2mm} = u_0(x, y), \end{array} \right\}\, , $
where
$0 < \alpha\leq1$, $D_x^{\alpha}$
 denotes the operator defined through the Fourier transform by
$(D_x^{\alpha}f)\widehat{\;\;}(\xi, \eta): = |\xi|^{\alpha}\widehat{f}(\xi, \eta)\, , ~~~~~~~~~~~~~~~~~~~~~~~~(0.1)$
and
$\mathcal H$
 denotes the Hilbert transform with respect to the variable x, is locally well posed in the Sobolev space
$H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-\alpha)$
.
Keywords: Benjamin Ono equation.
Mathematics Subject Classification: 35Q53.
Citation: Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057
References:
[1]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Notes 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. App. Math., 62 (1980), 249-262.  doi: 10.1002/sapm1980623249.  Google Scholar

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B. Akers and P. Milewski, A model equation for wave packet solitary waves arising from capillary-gravity flows, Studies in Applied Mathematics, 122 (2009), 249-274.  doi: 10.1111/j.1467-9590.2009.00432.x.  Google Scholar

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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

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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.  doi: 10.1016/j.jde.2016.04.022.  Google Scholar

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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.  Google Scholar

[7]

A. Esfahani and A. Pastor, Two dimensional solitary waves in shear flows, Calc. Var. Partial Differential Equations, 57 (2018), 57-102.  doi: 10.1007/s00526-018-1383-1.  Google Scholar

[8]

T. Kato, Quasilinear equations of evolution, with applications to PDE, Lecture Notes in Mathematics, vol. 448, Springer, Berlin, (1975), 25–70.  Google Scholar

[9]

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

[10]

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

[11]

C. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. I.H. PoincaréAN, 21 (2004), 87-838.  doi: 10.1016/j.anihpc.2003.12.002.  Google Scholar

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C. Kenig and K. D. 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.  Google Scholar

[13]

C. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Mathematical Journal, 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.  Google Scholar

[14]

C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar

[15]

C. Kenig, G. 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

[16]

B. Kim, Three-dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusets Institute of Technology, Department of Mathematics, Cambridge, MA, 2006.  Google Scholar

[17]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbb R)$, IMRN International Mathematics Research Notices, 26 (2003), 1449–1464. doi: 10.1155/S1073792803211260.  Google Scholar

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2015. doi: 10.1007/978-1-4939-2181-2.  Google Scholar

[19]

F. Linares, D. Pilod and J. C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, Siam J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.  Google Scholar

[20]

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

[21]

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

[22]

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.   Google Scholar

[23]

G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations, 4 (1991), 527-542.   Google Scholar

[24]

G. Preciado and F. Soriano, On the Cauchy problem of a two-dimensional Benjamin-Ono equation, arXiv:1503.04290v1 [Math.AP] 14 Mar 2015. doi: 10.12732/ijam.v26i6.1.  Google Scholar

[25]

J. C. Saut, Sur quelques gééalisations de l'éuation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.   Google Scholar

[26]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, Regional Conference Series in Mathematics, Number 106, AMS, 2006. doi: 10.1090/cbms/106.  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 Notes 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. App. Math., 62 (1980), 249-262.  doi: 10.1002/sapm1980623249.  Google Scholar

[3]

B. Akers and P. Milewski, A model equation for wave packet solitary waves arising from capillary-gravity flows, Studies in Applied Mathematics, 122 (2009), 249-274.  doi: 10.1111/j.1467-9590.2009.00432.x.  Google Scholar

[4]

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

[5]

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.  doi: 10.1016/j.jde.2016.04.022.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Esfahani and A. Pastor, Two dimensional solitary waves in shear flows, Calc. Var. Partial Differential Equations, 57 (2018), 57-102.  doi: 10.1007/s00526-018-1383-1.  Google Scholar

[8]

T. Kato, Quasilinear equations of evolution, with applications to PDE, Lecture Notes in Mathematics, vol. 448, Springer, Berlin, (1975), 25–70.  Google Scholar

[9]

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

[10]

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

[11]

C. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. I.H. PoincaréAN, 21 (2004), 87-838.  doi: 10.1016/j.anihpc.2003.12.002.  Google Scholar

[12]

C. Kenig and K. D. 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.  Google Scholar

[13]

C. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Mathematical Journal, 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.  Google Scholar

[14]

C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar

[15]

C. Kenig, G. 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

[16]

B. Kim, Three-dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusets Institute of Technology, Department of Mathematics, Cambridge, MA, 2006.  Google Scholar

[17]

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbb R)$, IMRN International Mathematics Research Notices, 26 (2003), 1449–1464. doi: 10.1155/S1073792803211260.  Google Scholar

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2015. doi: 10.1007/978-1-4939-2181-2.  Google Scholar

[19]

F. Linares, D. Pilod and J. C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, Siam J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.  Google Scholar

[20]

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

[21]

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

[22]

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.   Google Scholar

[23]

G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations, 4 (1991), 527-542.   Google Scholar

[24]

G. Preciado and F. Soriano, On the Cauchy problem of a two-dimensional Benjamin-Ono equation, arXiv:1503.04290v1 [Math.AP] 14 Mar 2015. doi: 10.12732/ijam.v26i6.1.  Google Scholar

[25]

J. C. Saut, Sur quelques gééalisations de l'éuation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.   Google Scholar

[26]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, Regional Conference Series in Mathematics, Number 106, AMS, 2006. doi: 10.1090/cbms/106.  Google Scholar

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