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Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients
Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces
1. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation ,, \emph{Comput. Math. Math. Phys.}, 37 (1997), 348.
|
[2] |
A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992).
|
[3] |
C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension ,, \emph{Adv. Math. Sci. Appl.}, 4 (1994), 465.
|
[4] |
C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model ,, \emph{Nonlinear Anal.}, 42 (2000), 1499.
doi: 10.1016/S0362-546X(99)00188-1. |
[5] |
C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model ,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 198.
doi: 10.1002/cpa.10056. |
[6] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 17.
doi: 10.3934/dcdss.2009.2.17. |
[7] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays ,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181.
doi: 10.1098/rspa.2003.1166. |
[8] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays ,, \emph{J. Differential Equations}, 205 (2004), 271.
doi: 10.1016/j.jde.2004.04.012. |
[9] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces ,, \emph{J. Math. Pures Appl.}, 90 (2008), 469.
doi: 10.1016/j.matpur.2008.07.001. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces ,, \emph{Nonlinearity}, 22 (2009), 351.
doi: 10.1088/0951-7715/22/2/006. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms ,, \emph{Sb. Math}, 192 (2001), 11.
doi: 10.1070/SM2001v192n01ABEH000534. |
[12] |
V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).
|
[13] |
V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor ,, \emph{J. Dynam. Differential Equations}, 19 (2007), 655.
doi: 10.1007/s10884-007-9077-y. |
[14] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 55.
doi: 10.3934/dcdss.2009.2.55. |
[15] |
T. Colin, The cauchy problem and the continuous limit for the multilayer model in geophysical fluid dynamics ,, \emph{SIAM J. Math. Anal.}, 28 (1997), 516.
doi: 10.1137/S0036141095291269. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307.
doi: 10.1007/BF02219225. |
[17] |
A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991). Google Scholar |
[18] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations ,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 159.
doi: 10.3934/dcds.2007.17.159. |
[19] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization ,, \emph{Numer. Algorithms}, 14 (1997), 141.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211.
|
[21] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications ,, \emph{Nonlinearity}, 5 (1992), 237.
|
[22] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007.
|
[23] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces ,, \emph{J. Differential Equations}, 230 (2006), 196.
doi: 10.1016/j.jde.2006.07.009. |
[24] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces ,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 701.
doi: 10.3934/dcds.2005.13.701. |
[25] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean ,, \emph{Nonlinear Anal.}, 68 (2008), 3550.
doi: 10.1016/j.na.2007.03.046. |
[26] |
T. Tachim Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor ,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 265.
doi: 10.3934/dcds.2012.32.265. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). Google Scholar |
[28] |
P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992). Google Scholar |
[29] |
R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation ,, \emph{Appl. Anal}, 70 (1998), 147.
doi: 10.1080/00036819808840682. |
[30] |
H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations ,, \emph{Nonlinearity}, 22 (2009), 667.
doi: 10.1088/0951-7715/22/3/008. |
[31] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68., Appl. Math. Sci., (1988).
doi: 10.1007/978-1-4684-0313-8. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001).
|
[33] |
Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations ,, \emph{Dyn. Syst.}, 23 (2008), 1.
doi: 10.1080/14689360701611821. |
show all references
References:
[1] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation ,, \emph{Comput. Math. Math. Phys.}, 37 (1997), 348.
|
[2] |
A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992).
|
[3] |
C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension ,, \emph{Adv. Math. Sci. Appl.}, 4 (1994), 465.
|
[4] |
C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model ,, \emph{Nonlinear Anal.}, 42 (2000), 1499.
doi: 10.1016/S0362-546X(99)00188-1. |
[5] |
C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model ,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 198.
doi: 10.1002/cpa.10056. |
[6] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 17.
doi: 10.3934/dcdss.2009.2.17. |
[7] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays ,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181.
doi: 10.1098/rspa.2003.1166. |
[8] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays ,, \emph{J. Differential Equations}, 205 (2004), 271.
doi: 10.1016/j.jde.2004.04.012. |
[9] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces ,, \emph{J. Math. Pures Appl.}, 90 (2008), 469.
doi: 10.1016/j.matpur.2008.07.001. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces ,, \emph{Nonlinearity}, 22 (2009), 351.
doi: 10.1088/0951-7715/22/2/006. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms ,, \emph{Sb. Math}, 192 (2001), 11.
doi: 10.1070/SM2001v192n01ABEH000534. |
[12] |
V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).
|
[13] |
V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor ,, \emph{J. Dynam. Differential Equations}, 19 (2007), 655.
doi: 10.1007/s10884-007-9077-y. |
[14] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 55.
doi: 10.3934/dcdss.2009.2.55. |
[15] |
T. Colin, The cauchy problem and the continuous limit for the multilayer model in geophysical fluid dynamics ,, \emph{SIAM J. Math. Anal.}, 28 (1997), 516.
doi: 10.1137/S0036141095291269. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307.
doi: 10.1007/BF02219225. |
[17] |
A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991). Google Scholar |
[18] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations ,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 159.
doi: 10.3934/dcds.2007.17.159. |
[19] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization ,, \emph{Numer. Algorithms}, 14 (1997), 141.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211.
|
[21] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications ,, \emph{Nonlinearity}, 5 (1992), 237.
|
[22] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007.
|
[23] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces ,, \emph{J. Differential Equations}, 230 (2006), 196.
doi: 10.1016/j.jde.2006.07.009. |
[24] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces ,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 701.
doi: 10.3934/dcds.2005.13.701. |
[25] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean ,, \emph{Nonlinear Anal.}, 68 (2008), 3550.
doi: 10.1016/j.na.2007.03.046. |
[26] |
T. Tachim Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor ,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 265.
doi: 10.3934/dcds.2012.32.265. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). Google Scholar |
[28] |
P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992). Google Scholar |
[29] |
R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation ,, \emph{Appl. Anal}, 70 (1998), 147.
doi: 10.1080/00036819808840682. |
[30] |
H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations ,, \emph{Nonlinearity}, 22 (2009), 667.
doi: 10.1088/0951-7715/22/3/008. |
[31] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68., Appl. Math. Sci., (1988).
doi: 10.1007/978-1-4684-0313-8. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001).
|
[33] |
Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations ,, \emph{Dyn. Syst.}, 23 (2008), 1.
doi: 10.1080/14689360701611821. |
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