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Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise
The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations
Departamento de Matemáticas, Universidad Nacional de Colombia, A. A. 3840 Medellín, Colombia |
$\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\}\, , $ |
$0 < \alpha\leq1$, $D_x^{\alpha}$ |
$(D_x^{\alpha}f)\widehat{\;\;}(\xi, \eta): = |\xi|^{\alpha}\widehat{f}(\xi, \eta)\, , ~~~~~~~~~~~~~~~~~~~~~~~~(0.1)$ |
$\mathcal H$ |
$H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-\alpha)$ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
T. Kato, Quasilinear equations of evolution, with applications to PDE, Lecture Notes in Mathematics, vol. 448, Springer, Berlin, (1975), 25–70. |
[9] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
B. Kim, Three-dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusets Institute of Technology, Department of Mathematics, Cambridge, MA, 2006. |
[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. |
[18] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2015.
doi: 10.1007/978-1-4939-2181-2. |
[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. |
[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. |
[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. |
[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.
|
[23] |
G. Ponce,
On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations, 4 (1991), 527-542.
|
[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. |
[25] |
J. C. Saut,
Sur quelques gééalisations de l'éuation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.
|
[26] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, Regional Conference Series in Mathematics, Number 106, AMS, 2006.
doi: 10.1090/cbms/106. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
T. Kato, Quasilinear equations of evolution, with applications to PDE, Lecture Notes in Mathematics, vol. 448, Springer, Berlin, (1975), 25–70. |
[9] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
B. Kim, Three-dimensional Solitary Waves in Dispersive Wave Systems, PhD thesis, Massachusets Institute of Technology, Department of Mathematics, Cambridge, MA, 2006. |
[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. |
[18] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2015.
doi: 10.1007/978-1-4939-2181-2. |
[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. |
[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. |
[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. |
[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.
|
[23] |
G. Ponce,
On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations, 4 (1991), 527-542.
|
[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. |
[25] |
J. C. Saut,
Sur quelques gééalisations de l'éuation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.
|
[26] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, Regional Conference Series in Mathematics, Number 106, AMS, 2006.
doi: 10.1090/cbms/106. |
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