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Averaging of 2D Navier-Stokes equations with singularly oscillating forces. (English) Zbl 1170.35345

The authors consider the non-autonomous 2D Navier-Stokes equations with nonslip boundary condition for ρ[0,1), ε>0

v t-νΔv+(v·)v+p=g 0 (x,t)+ε -ρ g 1 (x,t/ε),divv=0,xΩ,v| Ω =0,(1)

where v(x,t)=(v 1 ,v 2 ) is the velocity vector field, p is the pressure, ν=const>0 is the kinematic viscosity. Along with (1) the averaged Navier-Stokes equations are considered

v t-νΔv+(v·)v+p=g 0 (x,t),divv=0,xΩ,v| Ω =0·(2)

Under suitable assumptions on the external force, the uniform boundedness of the related uniform global attractors A ε is established for the equations (1). A 0 is the attractor to the equations (2). It is proved that A ε converge to A 0 as ε0 + in the standard Hausdorff semidistance in the space H(Ω).


MSC:
35B41Attractors (PDE)
35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE