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Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force. (English) Zbl 0931.35124

The authors consider a viscous incompressible Newtonian fluid in a bounded open domain D 3 with smooth boundary D, described by the classical Navier-Stokes equations

u t+(u·)u+p=νΔu+f+g tin[0,)×D,
divu=0in[0,)×D,u=0in[0,)×D,u(0,x)=u 0 (x),xD,

where u is the velocity field, p the pressure field, ν>0 the kinematic viscosity, and f+(g/t) the body force. Let X=L loc 2 [0,;H), and W(g)X be the space of generalized weak solutions. Let G be the space of all gC 0 (;V) that have polynomial growth at - and satisfy the property

lim inf α lim sup t 1 t -t 0 z α (s,g) L 4 8 ds=0,

where z α (t,g)=g(t)- - t (A+α)e -(t-s)(A+α) g(s)ds. Let

B(M 0 ,g)=uW(g): 0 1 u ( s ) 2 dsM 0

be defined for all constants M 0 >0 and all gG. Let M(·) be a function, defined on the backward orbit of g (the set {θ -τ g;τ0}) with values in (0,), so that the subexponential growth condition is fulfilled,

lim τ+ log + M(θ-τg) τ=0·(1)

For gG and a function M(·) satisfying the assumption (1) let us define the set

𝒜g , M ( · )= τ0 tτ Φ(t,θ -t g)BM (θ -t g) , θ -t g ¯

and let for gG

𝒜(g)= M(·) 𝒜(g,M(·)) ¯·(2)

Let 𝒟 be the set of all measurable multifunctions contained in a ball B(M(g),g), so that (1) is fulfilled with P Q -probability 1. There exists an α>0 and a {θ t } t -invariant set G α C 0 (;V) of full P Q -measure consisting of functions g of polynomial growth so that

2C * lim τ± 1 τ -τ 0 |z α (0,θ s g)| L 4 8 ds=β α ·

The main theorem of this paper says that for sufficiently large α>0 we have the following:

(i) The attractor 𝒜(g) defined by (2) for gG α (for gC 0 (,V)G α we set 𝒜(g)={0}) defines a ¯ C 0 (-,;V) P Q -measurable multifunction.

(ii) 𝒜(g) is the unique attractor in 𝒟.

(iii) For any D𝒟, ε>0,

P Q d Φ(t,g)D(g) ¯ , 𝒜 (θ t g) > ε0fort·

MSC:
35Q30Stokes and Navier-Stokes equations
35R60PDEs with randomness, stochastic PDE
35B41Attractors (PDE)
37L30Attractors and their dimensions, Lyapunov exponents