具有非线性阻尼的Navier-Stokes-Voigt方程的拉回吸引子

摘要/Abstract

摘要：

该文研究具有非线性阻尼的非自治Navier-Stokes-Voigt方程的长时间动力学.首先，利用Galerkin方法证明了整体弱解的存在唯一性.然后，利用能量方法建立解过程的一致渐近紧性，从而证明了拉回吸引子的存在性.此外，还建立了固定有界集族上的吸引子与满足缓增条件的集族上的吸引子之间的关系.

关键词: 拉回吸引子, 非自治, Navier-Stokes-Voigt方程, 非线性阻尼

Abstract:

In this paper, we are concerned with the long-time behavior of solutions to the non-autonomous Navier-Stokes-Voigt equations with nonlinear damping. Firstly, we prove the existence and uniqueness of global weak solutions by the Galerkin method. Then, we focus on studying the existence of pullback attractors by using an energy method, which is more simple than the weak continuous method to establish the uniformly asymptotical compactness. In addition, some relationships between the attractors for the universe of fixed bounded sets and those associated to a universe given by another tempered condition are established.

Key words: Pullback attractors, Non-autonomous, Navier-Stokes-Voigt equation, Nonlinear damping

中图分类号:

O175.26

彭小明,郑筱筱,尚亚东. 具有非线性阻尼的Navier-Stokes-Voigt方程的拉回吸引子[J]. 数学物理学报, 2021, 41(2): 357-369.

Xiaoming Peng,Xiaoxiao Zheng,Yadong Shang. Pullback Attractors for Navier-Stokes-Voigt Equations with Nonlinear Damping[J]. Acta mathematica scientia,Series A, 2021, 41(2): 357-369.

参考文献 33

1	Oskolkov A P . The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap Nauchn Semin LOMI, 1973, 38, 98- 136
2	Cao Y , Lunasin E M , Titi E S . Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models. Commun Math Sci, 2006, 4 (4): 823- 848 doi: 10.4310/CMS.2006.v4.n4.a8
3	Kalantarov V K , Titi E S . Global attractors and determining modes for the 3D Navier-Stokes-Voigt equations. Chin Ann Math Ser B, 2009, 30 (6): 697- 714 doi: 10.1007/s11401-009-0205-3
4	Kalantarov V K , Levant B , Titi E S . Gevrey regularity of the global attractor of the 3d Navier-Stokes-Voigt equations. J Nonlinear Sci, 2009, 19 (2): 133- 152 doi: 10.1007/s00332-008-9029-7
5	Yue G C , Zhong C K . Attractors for autonomous and nonautonomous 3d Navier-Stokes-Voigt equations. Discrete Contin Dyn Syst Ser B, 2011, 16 (3): 985- 1002
6	Dou Y W , Yang X G , Qin Y M . Remarks on uniform attractors for the 3d nonautonomous Navier-Stokes-Voigt equations. Bound Value Probl, 2011, 2011 (1): 49 doi: 10.1186/1687-2770-2011-49
7	García-luengo J , Marín-Rubio P , Real J . Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations. Nonlinearity, 2012, 25 (4): 905- 930 doi: 10.1088/0951-7715/25/4/905
8	Zelati M C , Gal C G . Singular limits of Voigt models in fluid dynamics. J Math Fluid Mech, 2015, 17 (2): 233- 259 doi: 10.1007/s00021-015-0201-1
9	Yang X G , Li L , Lu Y J . Regularity of uniform attractor for 3d non-autonomous Navier-Stokes-Voigt equation. Appl Math Comput, 2018, 334, 11- 29
10	Niche C J . Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum. J Differential Equations, 2016, 260 (8): 4440- 4453
11	Anh C T , Trang P T . Decay rate of solutions to 3d Navier Stokes Voigt equations in $H^m$ spaces. Appl Math Lett, 2016, 61, 1- 7 doi: 10.1016/j.aml.2016.04.015
12	Qin Y , Yang X , Liu X . Averaging of a 3d Navier Stokes Voigt equation with singularly oscillating forces. Nonlinear Anal RWA, 2012, 13 (2): 893- 904 doi: 10.1016/j.nonrwa.2011.08.025
13	Anh C T , Trang P T . On the regularity and convergence of solutions to the 3d Navier Stokes Voigt equations. Comput Math Appl, 2017, 73 (4): 601- 615 doi: 10.1016/j.camwa.2016.12.023
14	Celebi A O , Kalantarov V K , Polat M . Global attractors for 2d Navier Stokes Voigt equations in an unbounded domain. Appl Anal, 2009, 88 (3): 381- 392 doi: 10.1080/00036810902766682
15	Anh C T , Trang P T . Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013, 143 (2): 223- 251 doi: 10.1017/S0308210511001491
16	Cai X J , Jiu Q S . Weak and strong solutions for the incompressible Navier-Stokes equations with damping. J Math Anal Appl, 2008, 343 (2): 799- 809 doi: 10.1016/j.jmaa.2008.01.041
17	Zhang Z J , Wu X L , Lu M . On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping. J Math Anal Appl, 2011, 377 (1): 414- 419 doi: 10.1016/j.jmaa.2010.11.019
18	Zhou Y . Regularity and uniqueness for the 3d incompressible Navier Stokes equations with damping. Appl Math Lett, 2012, 25 (11): 1822- 1825 doi: 10.1016/j.aml.2012.02.029
19	Jia Y , Zhang X W , Dong B Q . The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping. Nonlinear Anal RWA, 2011, 12 (3): 1736- 1747 doi: 10.1016/j.nonrwa.2010.11.006
20	Jiang Z H , Zhu M X . The large time behavior of solutions to 3d Navier-Stokes equations with nonlinear damping. Math Methods Appl Sci, 2012, 35 (1): 97- 102 doi: 10.1002/mma.1540
21	Song X L , Hou Y R . Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete Contin Dyn Syst, 2011, 31 (1): 239- 252 doi: 10.3934/dcds.2011.31.239
22	Song X L , Hou Y R . Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping. J Math Anal Appl, 2015, 422 (1): 337- 351 doi: 10.1016/j.jmaa.2014.08.044
23	Song X L, Liang F, Wu J H. Pullback ${\cal D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping. Bound Value Probl, 2016, Article number: 145
24	Pardo D , José V , Ángel G . Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete Contin Dyn Syst, 2019, 24 (8): 3569- 3590
25	Li F , You B . Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete Contin Dyn Syst, 2020, 25 (1): 55- 80
26	Liu H , Lin L , Sun C F , Xiao Q K . The exponential behavior and stabilizability of the stochastic 3d Navier-Stokes equations with damping. Reviews in Mathematical Physics, 2019, 31 (7): 1950023 doi: 10.1142/S0129055X19500235
27	Di Plinio F , Giorgini A , Pata V , Temam R . Navier-Stokes-Voigt equations with memory in 3d lacking instantaneous kinematic viscosity. J Nonlinear Sci, 2018, 28, 653- 686 doi: 10.1007/s00332-017-9422-1
28	Wang X Q , Qin Y M . Three-dimensional Navier-Stokes-Voigt equation with a memory and the Brinkman-Forchheimer damping term. Math Methods Appl Sci, 2019, doi: 10.1002/mma.5710
29	Lv W B , Lu L Q , Wu S H . Decay characterization of the solutions to the Navier-Stokes-Voigt equations with damping. J Math Phys, 2020, 61, 081508 doi: 10.1063/1.5096875
30	Rosa R . The global attractor for the 2d Navier-Stokes flow on some unbounded domains. Nonlinear Anal, 2001, 32 (1): 71- 85
31	Marín-Rubio P , Real J . On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems. Nonlinear Anal, 2009, 71 (9): 3956- 3963 doi: 10.1016/j.na.2009.02.065
32	García-Luengo J , Marín-Rubio P , Real J . Pullback attractors in V for non-autonomous 2d-Navier-Stokes equations and their tempered behaviour. J Differential Equations, 2012, 252 (8): 4333- 4356 doi: 10.1016/j.jde.2012.01.010
33	Lions J L. Quelques Méthodes de Résolution des Problémes aux Limites Nonlineaires. Paris: Dunod, 1969

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