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Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces. (English) Zbl 1083.35094

The authors prove he existence of attractors for the 2d-Navier-Stokes equations with an exterior force f(t) by using the theory of processes by Babin, Vishik and others. A process is based on a Banach space X, an index set Σ (itself a topological vector space). A process is a collection U σ (t,τ) of nonlinar operators, acting on X, labeled by σΣ such that

U σ (t,s)U σ (s,τ)=U σ (t,τ),tsτ,U σ (τ,τ)=Id,τ,σΣ·(1)

Σ is called the symbol space, σ a symbol. Concepts such as uniform attractor, uniform absorbing set etc. are now introduced. E.g., B 0 X is uniformly absorbing if given C and a bounded set BX there is T 0 =T 0 (τ,B)τ such that

σΣ U σ (t,τ)BB 0 fortT 0 ·(2)

The authors now prove a number of preparatory lemmas concerning properties of processes. These results are then applied to the 2d-Navier-Stokes equation on a smooth bounded domain Ω. To this end this equation is put into standard abstract form

t u+νAu+B(u,u)=f(t),u(0)=u 0 (3)

based on the Hilbert spaces

H={uL 2 (Ω) 2 ,div(u)=0,u·n | Ω =0},norm||,V={uH 0 1 (Ω) 2 ,div(u)=0},norm·

The exterior force f(t)=σ(t), t is taken as symbol of the system (3), resp. of the induced process; one assumes

sup t t t+1 |f(s)| 2 ds<·(4)

After recalling global existence and uniqueness of solutions of (3) the authors proceed to prove the existence of a uniform attractor A 0 of (3) and investigate its properties. The relevant Theorem 3.3 states among others (expressed somewhat losely) that if f(t), t has an additional property called “normal” then the A 0 associated with f(t), t coincides with the uniform attractor A b associated with f(t+b), t , for any b. Further results of this type are obtained (Theorems 4.1, 4.2).


MSC:
35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE
35B41Attractors (PDE)
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)