ON THE KOLMOGOROV ENTROPY OF THE WEAK GLOBAL ATTRACTOR OF 3D NAVIER-STOKES EQUATIONS: I

. One particular metric that generates the weak topology on the weak global attractor A w of three dimensional incompressible Navier-Stokes equations is introduced and used to obtain an upper bound for the Kolmogorov entropy of A w . This bound is expressed explicitly in terms of the physical parameters of the ﬂuid ﬂow.


1.
Introduction. In the study of the incompressible Navier-Stokes equations (NSEs), a frequently discussed problem is the understanding of the dynamics and the asymptotic behaviors of the solutions ( [3,5,9,10,17,18]). The asymptotic behaviors are often considered to be related with the so-called global attractor, which can usually be characterized as the compact set attracting all bounded sets with respect to the appropriate topology of some (phase) space H (see [17,18]).
In the two-dimensional case, the strong global attractor A in the strong topology of the phase space can be easily defined due to the well-posedness of the incompressible NSEs. However, in three-dimensional case, one has to resort to the weak topology and define a weaker version of the global attractor, called weak global attractor, denoted by A w (see [11]). Further discussion of interesting topological properties of A w can be found in [8].
One signifincant difference between A and A w is the complexities of these two geometric objects. It can be shown that A has finite Hausdorff and fractal dimensions (see [2,12]). However, to the best knowledge of the authors, no estimates are known regarding either the fractal dimension or the Hausdorff dimension of A w . One can even show that the interior of A w is empty (see [1,7]). It remains an open question to find a way to quantify the complexities of A w . The concept of -entropy was introduced by Kolmogorov [13] to measure the complexities for totally bounded sets in a metric space. Kolmogorov -entropy can be viewed as a quantification of 2. The Navier-Stokes equations and Leray-Hopf weak solutions. The threedimensional incompressible Navier-Stokes equations (NSEs) in Eulerian formulation are written as where, the variable u = (u 1 , u 2 , u 3 ) denotes the velocity vector field, f assumed to be time-independent represents the mass density of volume force applied to the fluid, the parameter ν > 0 is the kinematic viscosity, and p is the kinematic pressure. The space variable is denoted by x = (x 1 , x 2 , x 3 ) and the time variable by t. We assume that the flow is periodic with period L in each spatial direction x i , i = 1, 2, 3, and that the averages of the flow velocity and of the force f over Let us introduce the space H (respectively, V ) as the subspace of L 2 (Ω) 3 (respectively, H 1 (Ω) 3 ) which is the closure of the set of all R 3 -valued trigonometric polynomials v such that THE KOLMOGOROV ENTROPHY OF WEAK ATTRACTOR FOR 3D-NSE   2341 Denote the inner products in H and V respectively by and the associated norms by |u| = (u, u) 1/2 , ||u|| = ((u, u)) 1/2 . The phase space H can be identified with its dual space H , then it is easy to see that V ⊂ H ⊂ V , with the injections being continuous and each space dense in the following one. Let P : L 2 (Ω) 3 → L 2 (Ω) 3 be the orthogonal projection (called the Helmholtz-Leray projection) with range H, and define the Stokes operator as A = −P∆ (= −∆, under periodic boundary conditions), which is positive, self-adjoint with a compact inverse. As a consequence, the space H has an orthonormal basis {w j } ∞ j=1 consisting of eigenfunctions of A; namely, Aw j = λ j w j , with 0 < λ 1 = κ 2 0 := (2π/L) 2 ≤ λ 2 ≤ λ 3 ≤ · · · (see [3]). For any σ ∈ R, the operator A σ is defined as for v in the domain of A σ , denoted by We also recall the orthogonal projection P K : The NSEs can be written as a differential equation (which will be referred to as the NSE) in the real Hilbert space H in the following form where the bilinear operator B and the driving force g are defined as B(u, v) = P((u · ∇)v) and g = Pf.
The following inequalities will be needed in this paper known, respectively, as Poincaré and Agmon inequalities, with c A being a nondimensional constant. The definition of weak solutions used in this paper is the one given in [8].
Definition 2.1. A (Leray-Hopf) weak solution on a time interval J ⊂ R is defined as a function u = u(t) on J with values in H satisfying the following properties: iii. u ∈ C(J; H w ), which means u is weakly continuous in H or equivalently for every v ∈ H, the function t → (u(t), v) is continuous from J into R, where H w is the space H endowed with its weak topology; iv. u satisfies (1) in the distribution sense on J with values in V ; v. For almost all t in J, u satisfies the following energy inequality: for all t in J with t > t . The allowed times t are characterized as the points where u are continuous from the right in H and the set of all t is of total Lebesgue measure and denoted by J (u). vi. If J is closed and bounded on the left, with its left end point denoted by t 0 , then the solution is continuous in H at t 0 from the right, i.e., t 0 ∈ J (u).
From now on, a weak solution will always mean a Leray-Hopf weak solution.
An important nondimensional parameter associated with the strength of the driving force g is the Grashof number ( [8]) A related nondimensional parameter that will be used in our paper is By Poincaré inequality (2), one has where the equality occurs only if g is an eigenvector of A.

3.
Weak global attractor and Kolmogorov -entropy. The weak global attractor A w , introduced in [11], is defined as the set of all points in H each of which belongs to a global weak solution uniformly bounded in H on R, i.e.
and A w is a totally bounded set in H w . Therefore the weak topology of H is metrizable on A w . Recall that a set F in the metric space (X, d) is totally bounded if, for any η > 0, there exists finitely many open balls of radius η whose union covers F . It is well known that all compact sets are totally bounded and that a metric space is compact if and only if it is complete and totally bounded. We choose the metric function that generates the weak topology on A w as for u, v ∈ A w . Two other metric functions that will be used are and Note that the metric functions d s , d n and d w do not have physical dimensions.

ON THE KOLMOGOROV ENTROPHY OF WEAK ATTRACTOR FOR 3D-NSE 2343
For a metric space (X, d), let B d (x 0 , ρ) denote the ball with radius ρ centered at The following definitions were introduced by Kolmogorov in [15].
Definition 3.1. Suppose F is a totally bounded, non-empty set in a metric space (X, d), and let > 0 be any real number.
(i). The system γ of sets U ⊂ X is said to be an -covering of the set F , if the diameter of any U ∈ γ, sup u0,u1∈U d(u 0 , u 1 ), is no greater than 2 and where G * is defined in (5).

YONG YANG AND BINGSHENG ZHANG
It follows that Choosing t = t 0 + 1 mνκ 2 0 , we obtain from (11) that Therefore, there exists at least one point t 1 ∈ [t 0 , t 0 + 1 where the inequality (10) is satisfied.
For any δ > 0, let us introduce and Using Lemma 4.1, we can get a property about F δ .

Lemma 4.2.
For any u 0 ∈ A w and δ > 0, there exists u 1 ∈ F δ satisfying Proof. Let u(·) ∈ A w be the weak solution of NSE with initial value u(0) = u 0 , then (6) gives that |u(t)| ≤ G * νκ −1/2 0 , for all t ≥ 0. From NSE, we see that For the first term of the right hand side, we have and, the third term can be estimated by using (3); indeed, the inequality Using Lemma 4.1, we could choose t 1 ∈ [0, δ νκ 2 0 ], such that u 1 := u(t 1 ) ∈ F δ . Then (14) implies that Remark 2. Lemma 4.2 can be also stated as follows, for any δ > 0, where r δ is given in (13).
This remark has the following consequence, Lemma 4.3. For any δ > 0, and r > 0. If C δ can be covered with b r balls B dw (v i , r), i = 1, ..., b r , then A w can be covered with m balls B dw (u ji , 3r + r δ ), i = 1, ..., m, where m ≤ b r and u ji ∈ F δ , for i = 1, · · · , m.
Proof. First consider the ball B dw (v 1 , r), if F δ ∩ B dw (v 1 , r) = Ø, we choose u j1 ∈ F δ such that u j1 ∈ B dw (v 1 , r). Otherwise, we begin with the next ball B dw (v 2 , r). Suppose we have dealt with B dw (v i , r), i = 1, ..., j and obtained u jn , n = 1, ..., l. For B dw (v j+1 , r), if there is some point u jn , 1 ≤ n ≤ l such that u jn ∈ B dw (v j+1 , r) or there does not exist any other points of F δ in B dw (v j+1 , r), we will consider the next ball B dw (v j+2 , r). Otherwise we can choose one point in F δ , denoted as u j l+1 which is contained in B dw (v j+1 , r) and is different from u jn , n = 1, ..., l. After having processed all b r balls, we can get a set {u jn , n = 1, ..., m} ⊂ F δ .
Clearly, the number m is finite and m ≤ b r . We claim that A w ⊂ ∪ m i=1 B dw (u ji , 3r + r δ ). Indeed, for each u ∈ A w , by Remark 2, there exist u j such that u j ∈ F δ and u ∈ B dw (u j , r δ ). Furthermore, there exist v n and u j l such that u j ∈ B dw (v n , r) and u j l ∈ B dw (v n , r). It follows that That is u ∈ B dw (u j l , 2r + r δ ) ⊂ B dw (u j l , 3r + r δ ). This completes the proof.
In the next section, we will give an estimate on b r , the number of balls with radius r > 0 covering C δ .

5.
Kolmogorov -entropy of the weak global attractor.

5.1.
Covering of C δ . According to Lemma 4.3, for any given δ > 0, in order to get a covering of the weak global attractor A w , it suffices to find a covering of δ } with balls of radius r > 0 in the metirc d w .
Proof. For any u 1 , u 2 ∈ C δ , denote u = u 1 − u 2 . By the definition of d w in (7), where K ≥ 1 is an integer, and I denotes the identity operator. Consequently, if K is chosen to be large enough such that then The inequality (15) is equivalent to A sufficient condition to guarantee that (17) holds is Taking the first integer K ≥ 1 that satisfies (18), then By (16), it follows that for any r−covering of C δ P K H, we can find a 2r−covering of C δ having the same number of sets. This completes the proof.
A special finite covering of the set C δ P K H with respect to the metric d s , defined in (9), and an upper bound of the cardinal number of this covering are given in the following lemma.
Lemma 5.2. For any η > 0 and integer K ≥ 1, we have, where S ⊂ C δ and the cardinal of S satisfies the estimate card (S) ≤

ON THE KOLMOGOROV ENTROPHY OF WEAK ATTRACTOR FOR 3D-NSE 2347
Proof. It follows from the definition of C δ in (12) and Poincaré inequality (2) that Notice that P K H is a Banach space of finite dimension. For fixed R > 0, let u 1 , · · · , u Nη (N η is called metric entropy, which is an upper bound for covering number) be a maximum set of points in B ds (0, R), the ball of radius R > 0 in P K H with |u i − u j | > η, for i = j, then the closed balls of radius η/2 centered at the u j s are disjoint, and their union lies within the ball of radius R + η/2 centered at the origin. Consequently, and thus, The result follows by applying (20) with R = G * 1 + 1/δ. Due to Lemma 5.2 and Lemma 5.3, the following covering of C δ using the metric d w can be obtained,

5.2.
Kolmogorov -entropy. Now, for any fixed > 0, based on the above lemmas, we are ready to the get a estimate on the Kolmogorov −entropy of the weak attractor A w in the space H endowed with the weak topology generated by the metric d w .
Theorem 5.4. The Kolmogorov −entropy for the weak global attractor A w , endowed with the weak topology generated by d w , of 3D Navier-Stokes equations is bounded above by the following explicit formula, H (A w ) ≤ 2 4π 3 ( √ 3 2 + 1 + ln ln β) 3 − 1 ln( √ 2β + 1), Remark 6. If the natural metric d n , defined in (8), is used for the weak topology on A w , the above arguments will lead to an upper bound on the Kolmogoroventropy that implies df (A w ) ≤ ∞. Hence, if one consider A w endowed with the weak topology generated by the metric d n , the techniques used in this paper will not provide good estimate on H (A w ).