# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020102
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## 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 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 & Control Theory, doi: 10.3934/eect.2020102
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 16 (1995), 379-390.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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