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Weak and strong attractors for the 3D Navier-Stokes system. (English) Zbl 1131.35056

The authors study the asymptotic behaviour of weak solutions of the Navier-Stokes equations

t u+νAu+B(u,u)=f,u(τ)=u τ (1)

considered on a smooth bounded domain Ω 3 , and subject to homogeneous Dirichlet boundary conditions; here A=-P Δ is the Stokes operator, while B(u,u)=P(u)u is the nonlinearity. Eq. (1) is studied in an L 2 -setting based on the familiar spaces H, V [R. Temam, Navier-Stokes equations. Theory and numerical analysis. Amsterdam etc.: North-Holland (1979; Zbl 0426.35003)]. Following these lines, a standard definition of weak solution is given. Starting point is an unproved assumption (H) imposed on the exterior force f(t):

(H) Let fL for 2 ([τ,),H); there exists a function F(α,β,γ), continuous and nondecreasing in α and nonincreasing in β, such that for any u τ V there exists a globally defined weak solution u() such that u(τ)=u τ and

u(t) L 4 (Ω) 3 F(u τ V ,τ,T)forallT>τandt[τ,T]·(2)

It then follows that the weak solutions u() provided by assumption (H) satisfy

V τ (u(t))V τ (u(s)),tsτ(3)


V τ (u(t))=1 2u(t) 2 +ν τ t (u)(r) 2 dr- τ t (f(r),u(r))dr,

and are unique in the class of solutions which satisfy (3). The authors then prove a series of theorems, all expressing properties of weak solutions provided by (H). Based on these preparations, the authors then proceed to study a class of attractors associated with these weak solutions. They set τ=0 in (1)–(3), assume that fH in (1) is autonomous and define a set valued mapping G from + ×H into the set P(H) of all nonempty subsets of H as follows:

(4) G(t,u 0 ) is the set of all u(t), where u() is a globally defined weak solution subject to (3), and with u(0)=u 0 · It is proved that G is a strict multivalued semiflow, i.e.

G(0,u 0 )=u 0 andG(t 1 ,G(t 2 ,u 0 ))=G(t 1 +t 2 ,u 0 )·

For such semiflows the notion of global attractor can be defined along established lines. The authors prove that the semiflow G has a global attractor 𝒜, which is minimal. The case of nonautonomous f(t) is more involved and requires the introduction of multivalued processes in the sense of A. V. Babin and M. I. Vishik [Attractors of evolution equations. Moscow: Nauka (1989; Zbl 0804.58003)]. The authors prove the existence of a global attractor in this more general situation. The paper concludes with a theory of attractors for multivalued processes on topological spaces, extending results obtained by V. S. Melnik and J. Valero [Set-Valued Anal. 8, No. 4, 375–403 (2000; Zbl 1063.35040)] for metric spaces.

35Q30Stokes and Navier-Stokes equations
37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35K55Nonlinear parabolic equations
37B25Lyapunov functions and stability; attractors, repellers
58C06Set-valued and function-space valued mappings on manifolds
35B40Asymptotic behavior of solutions of PDE