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Averaging of a 3D Navier-Stokes-Voight equation with singularly oscillating forces. (English) Zbl 1239.35128

Summary: For ρ[0,1) and ε(0,1), we investigate the uniform attractors of a 3D non-autonomous Navier-Stokes-Voight equation with singularly oscillating forces

u t -νΔu-α 2 Δu t +(u·)u+p=f 0 (t,x)+ε -ρ f 1 (t/ε,x),xΩ,·u=0,xΩ,u(t,x)| Ω =0,u(τ,x)=u τ (x),τ,

together with the averaged equations (corresponding to the limiting case ε=0)

u t -νΔu-α 2 Δu t +(u·)u+p=f 0 (t,x),xΩ,·u=0,xΩ,u(t,x)| Ω =0,u(τ,x)=u τ (x),τ·

Under suitable assumptions on the external forces, we obtain the uniform boundedness of the related uniform attractor 𝒜 ε of the first system, and the convergence of 𝒜 ε to the attractor 𝒜 ε of the second system as ε0 + .

MSC:
35Q35PDEs in connection with fluid mechanics
35B41Attractors (PDE)
76D05Navier-Stokes equations (fluid dynamics)
References:
[1]Y. Qin, X. Yang, X. Liu, Uniform attractors for a 3D non-autonomous Navier–Stokes–Voight Equations, Preprint, 2009.
[2]Babin, A. V.; Vishik, M. I.: Attractors of evolutionary equations, Studies in mathematics and its applications 25 (1992)
[3]Bonfoh, A.; Grasselli, M.; Miranville, A.: Singularly perturbed 1D Cahn–Hilliard equation revisited, Nodea nonlinear differential equations appl. 17, No. 6, 663-695 (2010)
[4]Chepyzhov, V. V.; Gatti, S.; Grasselli, M.; Miranville, A.; Pata, V.: Trajectory and global attractors for evolution equations with memory, Appl. math. Lett. 19, No. 1, 87-96 (2006) · Zbl 1082.35035 · doi:10.1016/j.aml.2005.03.007
[5]Chepyzhov, V. V.; Pata, V.; Vishik, M. I.: Averaging of non-autonomous damped wave equations with singularity oscillating forces, J. math. Pures appl. 90, 469-491 (2008)
[6]Chepyzhov, V. V.; Vishik, M. I.: Evolution equations and their trajectory attractors, J. math. Pures appl. 76, 664-913 (1997) · Zbl 0896.35032 · doi:10.1016/S0021-7824(97)89978-3
[7]Chepyzhov, V. V.; Vishik, M. I.: Attractors for equations of mathematical physics, (2001)
[8]Chueshov, I.: Introduction to the theory of infinite-dimensional dissipative systems, (2002)
[9]Evans, L. C.: Weak convergence methods for nonlinear partial differencial equations, (1990)
[10]Feifeisl, E.: Dynamics of viscous compressible fluids, (2004)
[11]Gal, C. G.; Grasselli, M.: Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system, Physica D 240, No. 7, 629-635 (2011) · Zbl 1214.37055 · doi:10.1016/j.physd.2010.11.014
[12]Grasselli, M.; Pata, V.: Attractor for a conserved phase-field system with hyperbolic heat conduction, Math. methods appl. Sci. 27, No. 16, 1917-1934 (2004) · Zbl 1072.37055 · doi:10.1002/mma.533
[13]Grasselli, M.; Pata, V.: Uniform attractors of nonautonomous dynamical systems with memory, Progr. nonlinear differential equations appl. 50, 155-178 (2002) · Zbl 1039.34074
[14]Hale, J. K.: Asymptotic behavior of dissipative systems, (1988)
[15]Hou, Y.; Li, K.: The uniform attractors for the 2D non-autonomous Navier–Stokes flow in some unbounded domain, Nonlinear anal. TMA 58, 609-630 (2004) · Zbl 1057.35031 · doi:10.1016/j.na.2004.02.031
[16]Kalantarov, V. K.: Attractors for some nonlinear problems of mathematical physics, Zap. nauchn. Sem. LOMI 152, 50-54 (1986) · Zbl 0621.35022
[17]V.K. Kalantarov, Global behavior of solutions of nonlinear equations of mathematical physics of classical and non-classical type, Postdoctoral Thesis, St. Petersburg, 1988.
[18]Kalantarov, V. K.; Titi, E. S.: Global attractors and determining models for the 3D Navier–Stokes–voight equations, Chin. ann. Math. ser. B 30, 697-714 (2009) · Zbl 1178.37112 · doi:10.1007/s11401-009-0205-3
[19]Kloeden, P. E.; Langa, J. A.: Flattening, squeezing and the existence of random attractors, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 463, 163-181 (2007) · Zbl 1133.37323 · doi:10.1098/rspa.2006.1753
[20]Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow, (1969) · Zbl 0184.52603
[21]Ladyzhenskaya, O. A.: Attractors for semigroup and evolution equations, (1991) · Zbl 0755.47049
[22]Lu, S. S.; Wu, H.; Zhong, C. K.: Attractors for non-autonomous 2D Navier–Stokes equations with normal external forces, Discrete contin. Dyn. syst. 13, No. 3, 701-719 (2005) · Zbl 1083.35094 · doi:10.3934/dcds.2005.13.701
[23]Ma, S.; Zhong, C. K.: The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external force, Discrete contin. Dyn. syst. 18, No. 1, 53-70 (2007) · Zbl 1132.35018 · doi:10.3934/dcds.2007.18.53
[24]Ma, Q. F.; Wang, S. H.; Zhong, C. K.: Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana univ. Math. J. 51, 1541-1559 (2002) · Zbl 1028.37047 · doi:10.1512/iumj.2002.51.2255
[25]Miranville, A.; Wang, X.: Attractors for non-autonomous nonhomogenerous Navier–Stokes equations, Nonlinearity 10, No. 5, 1047-1061 (1997) · Zbl 0908.35098 · doi:10.1088/0951-7715/10/5/003
[26]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. Sem. LOMI 38, 98-136 (1973)
[27]Qin, Y.: Nonlinear parabolic–hyperbolic coupled systems and their attractors, operator theory, advances and applications, vol. 184, (2008)
[28]Qin, Y.: Universal attractor in H4 for the nonlinear one-dimensional compressible Navier–Stokes equations, J. differential equations 207, No. 1, 21-72 (2004) · Zbl 1064.35150 · doi:10.1016/j.jde.2004.08.022
[29]Qin, Y.; Liu, H.; Song, C.: Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys, Proc. roy. Soc. Edinburgh sect. A 138, No. 5, 1103-1135 (2008) · Zbl 1173.35386 · doi:10.1017/S0308210506000503
[30]Qin, Y.; Rivera, J. E. M.: Exponential stability and universal attractors for the Navier–Stokes equations of compressible fluids between two horizontal parallel plates in R3, Appl. numer. Math. 47, No. 2, 209-235 (2003) · Zbl 1140.76445 · doi:10.1016/S0168-9274(03)00067-9
[31]Qin, Y.; Rivera, J. E. M.: Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas, Proc. roy. Soc. Edinburgh sect. A 132, No. 3, 685-709 (2002) · Zbl 1006.35080 · doi:10.1017/S0308210500001840
[32]F. Ramos, E.S. Titi, Invariant measures for the 3D Navier–Stokes–Voigt equations and their Navier–Stokes limit, Preprint.
[33]Robinson, J. C.: Infinite-dimensional dynamics systems, (2001)
[34]Sell, G. R.: Global attractors for the three-dimensional Navier–Stokes equations, J. dynam. Differential equations 8, 1-33 (1996) · Zbl 0855.35100 · doi:10.1007/BF02218613
[35]Sell, G. R.; You, Y.: Dynamics of evolutionary equations, (2002)
[36]Temam, R.: Infinite dimensional dynamical systems in mechanics and physics, (1997)
[37]Temam, R.: Navier–Stokes equations, theory and numerical analysis, (1979)
[38]Zheng, S.; Qin, Y.: Universal attractors for the Navier–Stokes equations of compressible and heat-conductive fluid in bounded annular domains in rn, Arch. ration. Mech. anal. 160, No. 2, 153-179 (2001) · Zbl 1027.35098 · doi:10.1007/s002050100163
[39]Zheng, S.; Qin, Y.: Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart. appl. Math. 59, No. 3, 579-599 (2001) · Zbl 1172.35335
[40]Zhong, C. K.; Yang, M. H.; Sun, C. Y.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffusion equations, J. differential equations 223, No. 2, 367-399 (2006) · Zbl 1101.35022 · doi:10.1016/j.jde.2005.06.008
[41]Chepyzhov, V. V.; Vishik, M. I.: Averaging of 2D Navier–Stokes equations with singularity oscillating forces, Nonlinearity 22, 351-370 (2009) · Zbl 1170.35345 · doi:10.1088/0951-7715/22/2/006