February  2022, 11(1): 41-65. doi: 10.3934/eect.2020102

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

* Corresponding author: Jianhua Wu

Received  January 2020 Revised  August 2020 Published  February 2022 Early access  November 2020

Fund Project: The first author is supported in part by National Natural Science Foundation of China (Nos. 11601417), Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2018JM1047, 2019JM-283) and Postdoctoral Fund in Shaanxi Province of China (No. 2016BSHEDZZ112). The second author was supported by the National Natural Science Foundation of China (No. 11771262)

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^+ $.

Citation: Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations and Control Theory, 2022, 11 (1) : 41-65. doi: 10.3934/eect.2020102
References:
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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. ChepyzhovM. 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. ChepyzhovV. 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. ChepyzhovV. 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. ChepzhovM. 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. FangL.-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. FucciB. 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. QinX. 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. QiuY. 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. SunD. 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. SunD. 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. ZhongM.-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. ChepyzhovM. 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. ChepyzhovV. 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. ChepyzhovV. 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. ChepzhovM. 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. FangL.-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. FucciB. 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. QinX. 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. QiuY. 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. SunD. 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. SunD. 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. ZhongM.-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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