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Lifespan of solutions to a damped plate equation with logarithmic nonlinearity
Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China |
2. | College of Science, Xi'an University of Science and Technology, Xi'an, 710054, China |
We consider a non-autonomous two-dimensional Newton-Boussinesq equation with singularly oscillating external forces depending on a small parameter $ \varepsilon $. We prove the existence of the uniform attractor $ A^\varepsilon $ when the Prandtl number $ P_r>1 $. Furthermore, under suitable translation-compactness and divergence type condition assumptions on the external forces, we obtain the uniform (with respect to $ \varepsilon $) boundedness of the related uniform attractors $ A^\varepsilon $ as well as the convergence of the attractor $ A^\varepsilon $ to the attractor $ A^0 $ as $ \varepsilon\rightarrow 0^+ $.
References:
[1] |
C. T. Anh and N. D. Toan,
Nonclassical diffusion equations on $\mathbb{R}^N$ with singularly oscillating external forces, Appl. Math. Lett., 38 (2014), 20-26.
doi: 10.1016/j.aml.2014.06.008. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
V. V. Chepyzhov, M. Conti and V. Pata,
Averaging of equations of viscoelasticity with singularly oscillating external forces, J. Math. Pures. Appl. (9), 108 (2017), 841-868.
doi: 10.1016/j.matpur.2017.05.007. |
[4] |
V. V. Chepyzhov, V. Pata and M. I. Vishik,
Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370.
doi: 10.1088/0951-7715/22/2/006. |
[5] |
V. V. Chepyzhov, V. Pata and M. I. Vishik,
Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl. (9), 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[7] |
V. V. Chepzhov and M. I. Vishik,
Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, ESAIM Control Optim. Calc. Var., 8 (2002), 467-487.
doi: 10.1051/cocv:2002056. |
[8] |
V. V. Chepyzhov and M. I. Vishik,
Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dynam. Differential Equations, 19 (2007), 655-684.
doi: 10.1007/s10884-007-9077-y. |
[9] |
V. V. Chepzhov, M. I. Vishik and W. L. Wendland,
On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38.
doi: 10.3934/dcds.2005.12.27. |
[10] |
M. Efendiev and S. Zelik,
Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenenization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 961-989.
doi: 10.1016/S0294-1449(02)00115-4. |
[11] |
M. Efendiev and S. Zelik,
The regular attractor for the reaction-diffusion systems with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differential Equations, 8 (2003), 673-732.
|
[12] |
S.-M. Fang, L.-Y. Jin and B.-L. Guo,
Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations, Appl. Math. Mech. (English Ed.), 31 (2010), 405-414.
doi: 10.1007/s10483-010-0401-9. |
[13] |
G. Fucci, B. Wang and P. Singh,
Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013.
doi: 10.1016/j.na.2008.02.098. |
[14] |
B. Guo,
Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 16 (1995), 379-390.
|
[15] |
B. L. Guo,
Spectral method for solving the two-dimensional Newton-Boussinesq equations, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 208-218.
doi: 10.1007/BF02006004. |
[16] |
J. K. Hale, Asymptotic behavior of dissipative systems, in Dynamics of Infinite-Dimensional Systems, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer, Berlin, 1987,123–128. |
[17] |
Y. Hou and K. Li,
The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain, Nonlinear Anal., 58 (2004), 609-630.
doi: 10.1016/j.na.2004.02.031. |
[18] |
H. Ma and Q. Zhang,
Global existence and uniqueness of Yudovich's solutions to the 3D Newton-Boussinesq system, Appl. Anal., 97 (2018), 1814-1827.
doi: 10.1080/00036811.2017.1343463. |
[19] |
Y. Qin, X. Yang and X. Liu,
Averaging of 3D Navier-Stokes-Voight equation with singularly oscillating forces, Nonlinear Anal. Real World Appl., 13 (2012), 893-904.
doi: 10.1016/j.nonrwa.2011.08.025. |
[20] |
H. Qiu, Y. Du and Z. Yao,
A note on the regularity criterion of the two-dimensional Newton-Boussinesq equation, Nonlinear Anal. Real World Appl., 12 (2011), 2012-2015.
doi: 10.1016/j.nonrwa.2010.12.017. |
[21] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[22] |
X.-L. Song and Y.-R. Hou,
Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.
doi: 10.3934/dcds.2011.31.239. |
[23] |
X.-L. Song and Y.-R. Hou,
Pullback $D$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009.
doi: 10.3934/dcds.2012.32.991. |
[24] |
X.-L. Song and J.-H. Wu,
Existence of global attractors for two-dimensional Newton-Boussinesq equation, Nonlinear Anal., 157 (2017), 1-19.
doi: 10.1016/j.na.2017.03.002. |
[25] |
C. Sun, D. Cao and J. Duan,
Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761.
doi: 10.3934/dcdsb.2008.9.743. |
[26] |
C. Sun, D. Cao and J. Duan,
Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[27] |
T. Tachim Medjo,
A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ Model with oscillating external force and its global attractor, Commun. Pure Appl. Anal., 10 (2011), 415-433.
doi: 10.3934/cpaa.2011.10.415. |
[28] |
T. Tachim Medjo,
A non-autonomous two-phase flow model with oscillating external force and its global attractor, Nonlinear Anal., 75 (2012), 226-243.
doi: 10.1016/j.na.2011.08.024. |
[29] |
T. Tachim Medjo,
Averaging of a 3D primitive equations with oscillating external forces, Appl. Anal., 92 (2013), 869-900.
doi: 10.1080/00036811.2011.640628. |
[30] |
T. Tachim Medjo,
Non-autonomous 3D primitive equations with oscillating external force and its global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 265-291.
doi: 10.3934/dcds.2012.32.265. |
[31] |
T. Tachim Medjo,
Non-autonomous planetary 3D geostrophic equations with oscillating external force and its global attractor, Nonlinear Anal. Real World Appl., 12 (2011), 1437-1452.
doi: 10.1016/j.nonrwa.2010.10.004. |
[32] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[33] |
M. I. Vishik and V. V. Chepyzhov,
Attractors of dissipative hyperbolic equations with singularly oscillating external forces, Math. Notes, 79 (2006), 483-504.
doi: 10.1007/s11006-006-0054-2. |
[34] |
B. Wang and R. Jones,
Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902.
doi: 10.1016/j.na.2010.01.026. |
[35] |
R. Wang and Y. Li,
Asymptotic autonomy of kernel sections for Newton-Boussinesq equations on unbounded zonary domains, Dyn. Partial Differ. Equ., 16 (2019), 295-316.
doi: 10.4310/DPDE.2019.v16.n3.a4. |
[36] |
C. Zhao and Y. Li,
$H^2$-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal., 56 (2004), 1091-1103.
doi: 10.1016/j.na.2003.11.006. |
[37] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
show all references
References:
[1] |
C. T. Anh and N. D. Toan,
Nonclassical diffusion equations on $\mathbb{R}^N$ with singularly oscillating external forces, Appl. Math. Lett., 38 (2014), 20-26.
doi: 10.1016/j.aml.2014.06.008. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
V. V. Chepyzhov, M. Conti and V. Pata,
Averaging of equations of viscoelasticity with singularly oscillating external forces, J. Math. Pures. Appl. (9), 108 (2017), 841-868.
doi: 10.1016/j.matpur.2017.05.007. |
[4] |
V. V. Chepyzhov, V. Pata and M. I. Vishik,
Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370.
doi: 10.1088/0951-7715/22/2/006. |
[5] |
V. V. Chepyzhov, V. Pata and M. I. Vishik,
Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl. (9), 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[7] |
V. V. Chepzhov and M. I. Vishik,
Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, ESAIM Control Optim. Calc. Var., 8 (2002), 467-487.
doi: 10.1051/cocv:2002056. |
[8] |
V. V. Chepyzhov and M. I. Vishik,
Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dynam. Differential Equations, 19 (2007), 655-684.
doi: 10.1007/s10884-007-9077-y. |
[9] |
V. V. Chepzhov, M. I. Vishik and W. L. Wendland,
On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38.
doi: 10.3934/dcds.2005.12.27. |
[10] |
M. Efendiev and S. Zelik,
Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenenization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 961-989.
doi: 10.1016/S0294-1449(02)00115-4. |
[11] |
M. Efendiev and S. Zelik,
The regular attractor for the reaction-diffusion systems with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differential Equations, 8 (2003), 673-732.
|
[12] |
S.-M. Fang, L.-Y. Jin and B.-L. Guo,
Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations, Appl. Math. Mech. (English Ed.), 31 (2010), 405-414.
doi: 10.1007/s10483-010-0401-9. |
[13] |
G. Fucci, B. Wang and P. Singh,
Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013.
doi: 10.1016/j.na.2008.02.098. |
[14] |
B. Guo,
Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 16 (1995), 379-390.
|
[15] |
B. L. Guo,
Spectral method for solving the two-dimensional Newton-Boussinesq equations, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 208-218.
doi: 10.1007/BF02006004. |
[16] |
J. K. Hale, Asymptotic behavior of dissipative systems, in Dynamics of Infinite-Dimensional Systems, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer, Berlin, 1987,123–128. |
[17] |
Y. Hou and K. Li,
The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain, Nonlinear Anal., 58 (2004), 609-630.
doi: 10.1016/j.na.2004.02.031. |
[18] |
H. Ma and Q. Zhang,
Global existence and uniqueness of Yudovich's solutions to the 3D Newton-Boussinesq system, Appl. Anal., 97 (2018), 1814-1827.
doi: 10.1080/00036811.2017.1343463. |
[19] |
Y. Qin, X. Yang and X. Liu,
Averaging of 3D Navier-Stokes-Voight equation with singularly oscillating forces, Nonlinear Anal. Real World Appl., 13 (2012), 893-904.
doi: 10.1016/j.nonrwa.2011.08.025. |
[20] |
H. Qiu, Y. Du and Z. Yao,
A note on the regularity criterion of the two-dimensional Newton-Boussinesq equation, Nonlinear Anal. Real World Appl., 12 (2011), 2012-2015.
doi: 10.1016/j.nonrwa.2010.12.017. |
[21] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[22] |
X.-L. Song and Y.-R. Hou,
Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.
doi: 10.3934/dcds.2011.31.239. |
[23] |
X.-L. Song and Y.-R. Hou,
Pullback $D$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009.
doi: 10.3934/dcds.2012.32.991. |
[24] |
X.-L. Song and J.-H. Wu,
Existence of global attractors for two-dimensional Newton-Boussinesq equation, Nonlinear Anal., 157 (2017), 1-19.
doi: 10.1016/j.na.2017.03.002. |
[25] |
C. Sun, D. Cao and J. Duan,
Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761.
doi: 10.3934/dcdsb.2008.9.743. |
[26] |
C. Sun, D. Cao and J. Duan,
Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[27] |
T. Tachim Medjo,
A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ Model with oscillating external force and its global attractor, Commun. Pure Appl. Anal., 10 (2011), 415-433.
doi: 10.3934/cpaa.2011.10.415. |
[28] |
T. Tachim Medjo,
A non-autonomous two-phase flow model with oscillating external force and its global attractor, Nonlinear Anal., 75 (2012), 226-243.
doi: 10.1016/j.na.2011.08.024. |
[29] |
T. Tachim Medjo,
Averaging of a 3D primitive equations with oscillating external forces, Appl. Anal., 92 (2013), 869-900.
doi: 10.1080/00036811.2011.640628. |
[30] |
T. Tachim Medjo,
Non-autonomous 3D primitive equations with oscillating external force and its global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 265-291.
doi: 10.3934/dcds.2012.32.265. |
[31] |
T. Tachim Medjo,
Non-autonomous planetary 3D geostrophic equations with oscillating external force and its global attractor, Nonlinear Anal. Real World Appl., 12 (2011), 1437-1452.
doi: 10.1016/j.nonrwa.2010.10.004. |
[32] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[33] |
M. I. Vishik and V. V. Chepyzhov,
Attractors of dissipative hyperbolic equations with singularly oscillating external forces, Math. Notes, 79 (2006), 483-504.
doi: 10.1007/s11006-006-0054-2. |
[34] |
B. Wang and R. Jones,
Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902.
doi: 10.1016/j.na.2010.01.026. |
[35] |
R. Wang and Y. Li,
Asymptotic autonomy of kernel sections for Newton-Boussinesq equations on unbounded zonary domains, Dyn. Partial Differ. Equ., 16 (2019), 295-316.
doi: 10.4310/DPDE.2019.v16.n3.a4. |
[36] |
C. Zhao and Y. Li,
$H^2$-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal., 56 (2004), 1091-1103.
doi: 10.1016/j.na.2003.11.006. |
[37] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
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