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Attractors for nonautonomous nonhomogeneous Navier-Stokes equations. (English) Zbl 0908.35098

The Navier-Stokes equations are considered in a smooth bounded domain Ω 2

u t-νΔu+(u·)u+p=f,divu=0inΩ,u=ϕonΩ,

where f(x,t) and ϕ(x,t) are quasiperiodic in t. It means that f(·,t)=f(·,α 1 t,,α k t), ϕ(·,t)=ϕ(·,α 1 t,,α k t), where f(·,ω 1 ,,ω k ) and ϕ(·,ω 1 ,,ω k ) are 2π-periodic in each argument ω i , the {α i } being rationally independent. The authors construct the family of processes associated to these equations and prove the existence of the uniform attractor. The following estimate of the Hausdorff dimension of the attractor 𝒜 k is obtained

dim𝒜 k k+c(G 1 +Re 3/2 )+c ' (G 1 +G 2 +Re 3/2 )k 1/2

where G 1 and G 2 depend of f,ϕ and Ω, Re is the Reynolds number, c and c ' are nondimensional constants independent of Re,G 1 ,G 2 . The approach of V. V. Chepyzhov and M. I. Vishik [|J. Math. Pures Appl. 73, 279-333 (1994; Zbl 0838.58021)] is used.

35Q30Stokes and Navier-Stokes equations
37C70Attractors and repellers, topological structure
76D05Navier-Stokes equations (fluid dynamics)
35-99Partial differential equations (PDE) (MSC2000)