Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor

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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

Xueli Song ^1,2, and Jianhua Wu ^1,,

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^+ $.

Keywords: Newton-Boussinesq equation, singularly oscillating force, uniform attractor, translation bounded function, convergence of uniform attractor.

Mathematics Subject Classification: Primary: 35B41, 35B40; Secondary: 35Q35.

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:

[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.
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[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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