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COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 1, Number 1, March 2002

Website: http://AIMsciences.org pp. 35–50

GLOBAL WELL-POSEDNESS OF WEAK SOLUTIONS FOR THE LAGRANGIAN AVERAGED NAVIER-STOKES EQUATIONS ON BOUNDED DOMAINS

D. Coutand Department of Mathematics University of California Davis, CA 95616

J. Peirce Department of Mathematics University of California Davis, CA 95616

S. Shkoller Department of Mathematics University of California Davis, CA 95616

Abstract. In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. The LANS-α equations are able to accurately reproduce the large-scale motion (at scales larger than α > 0) of the Navier-Stokes equations while filtering or averaging over the motion of the fluid at scales smaller than α, an a priori fixed spatial scale. We prove the global well-posedness of weak H 1 solutions for the case of no-slip boundary conditions in three dimensions, generalizing the periodic-box results of [8]. We make use of the new formulation of the LANS-α equations on bounded domains given in [20] and [14], which reveals the additional boundary conditions necessary to obtain well-posedness. The uniform estimates yield global attractors; the bound for the dimension of the global attractor in 3D exactly follows the periodic box case of [8]. In 2D, our bound is α-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.

1. Introduction. The Lagrangian averaged Navier-Stokes (LANS-α) equations1 for an incompressible viscous fluid moving in a bounded fluid container Ω ⊂ Rn with smooth (at least C 3 ) boundary ∂Ω may be written as the following system of partial differential equations: 1991 Mathematics Subject Classification. 35Q35. Key words and phrases. Lagrangian averaged Navier-Stokes, Turbulence, Model. 1 Some authors had previously referred to the LANS-α equations as the Viscous Camassa-Holm equations.

35

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D. COUTAND, J. PEIRCE, AND S. SHKOLLER

∂t u + ∇u u + U α (u) = −(1 − α2 ∆)−1 grad p − νAu + F ,

(1a)

div u = 0 ,

(1b)

u = 0 on ∂Ω ,

(1c)

u(0, x) = u0 (x) ,

(1d)

U α (u) = α2 (1 − α2 ∆)−1 Div(∇u · ∇uT + ∇u · ∇u − ∇uT · ∇u).

(2)

where We use u(t, x) to denote the large-scale (or averaged) velocity field of the fluid, assumed to have constant density. The pressure function p(t, x) is determined (modulo constants) from the incompressibility constraint (1b). ν denotes the kinematic viscosity of the fluid, and α > 0 is the spatial scale at which fluid motion is filtered, i.e. spatial scales smaller than α are averaged out. The additional term F (x) ∈ H 2 ∩ H01 represents the external force acting on the system and is assumed, for simplicity, to be time-independent. We let A := −P ∆ denote the Stokes operator, with P the Leray projector onto divergence-free vector fields. It has been a longstanding problem in fluid dynamics to derive a model for the large scale motion of a fluid that averages or course-grains the small, computationally unresolvable, scales of the Navier-Stokes equations. The LANS-α equations provide one such averaged model, and have been studied rather extensively from both the mathematical, as well numerical, points of view. We refer the interested reader to Chen et al. [2, 3, 4, 5] and Mohseni et al. [17] for numerical simulations of both forced and decaying isotropic turbulence using the LANS-α model. The inviscid (ν = 0) version of (1), known as the Lagrangian averaged Euler (LAE-α) or Euler-α equations, was first given by Holm, Marsden, and Ratiu [10] in the case that Ω = Rn as T

∂t v + ∇u v + [∇u] · v = − grad p, div u = 0,

(3)

where the variable

v = (1 − α2 ∆)u may be thought of as the momentum. Foias, Holm, and Titi [8] first added viscous dissipation to (3); they argued on physical grounds that the momentum v rather than the velocity u, need be diffused. By assuming periodic boundary conditions, they obtained the following form of the LANS-α equations: ∂t v + ∇u v − α2 [∇u] · ∆u = − grad p + ν∆v + f, div u = 0, u(0, x) = u0 (x), T

(4)

with f taken in L2 . While yielding the correct equations on a periodic box, the question of how to appropriately prescribe boundary data in the no-slip u = 0 case remained open. Specifically, inversion of the dissipative term ν∆v = ν(1 − α2 ∆)u, a fourth-order operator, requires further constraints than simply u = 0 on ∂Ω. Shkoller [20] and Marsden and Shkoller [14] supplied the additional boundary condition by reformulating (4) as the system of equations (1) which we first wrote down. In the formulation (1), it is clear that if u = 0 on ∂Ω, then Au = 0 on ∂Ω

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

37

as well. This follows since each term in the inviscid equations identically vanishes on the boundary, thanks to the inversion of (1 − α2 ∆) with Dirichlet boundary conditions. The viscous term in the formulation (1) was obtained by treating the Lagrangian trajectory as a stochastic process, and replacing deterministic time derivatives with backward-in-time mean stochastic derivatives, exactly following the usual procedure for obtaining the viscous dissipation term in the Navier-Stokes equations as done by Chorin [6] and Peskin [18]. The term U α (u), given in (2), provides a regularization to the Navier-Stokes equations which is not dispersive, in character. In fact this regularization is geometric in nature, and arises as geodesic flow of an H 1 right-invariant Riemannian metric on the Hilbert group of volume-preserving diffeomorphisms of the fluid container (see [13, 19, 20]); as such, this regularizer yields an a priori L∞ −H 1 estimate in three-dimensions (in fact, in n dimensions for n ≥ 2), and one is thus tempted to ask whether the LANS-α system is globally well-posed (even though, as is clear from (1a), no additional artificial viscosity is being added to the Navier-Stokes equations). In the case of the periodic box, Foias, Holm, and Titi [8] proved the global well-posedness of H 1 weak solutions in dimension three, but as we noted, their formulation (4) did not provide the obvious extension to bounded domains. Using the formulation (1), Marsden and Shkoller [14] proved the global well-posedness of classical solutions in dimension three in the case of no-slip boundary data. In this paper, we give a rather straightforward extension of that result to the H 1 weak solutions of Foias, Holm, and Titi [8]. In particular, we prove the global (in time) well-posedness and regularity of weak solutions to the LANS-α equations for initial data in the class {u ∈ H01 | div u = 0}. The analogous two-dimensional result follows trivially (as it is already known for the original Navier-Stokes equations). This leads to the existence of a nonempty, compact, convex, and connected global H 1 attractor in both two- and three-dimensions. In three-dimensions, the global attractor has the identical bound as that obtained by Foias, Holm, and Titi [8] for periodic boundary conditions. This upper bound depends on 1/α and consequently tends to infinity as α tends to zero. Due to the difference in the LiebThirring inequality in two- and three-dimensions, the two-dimensional bound for the global attractor is α-independent. In fact, the global attractor of the 2D LANSα equations is similar to the bound for the global attractor of the 2D Navier-Stokes equations. The remainder of this paper is organized into two sections. In Section 2, we establish the global well-posedness result, while Section 3 is devoted to issues concerning the global attractors. 2. Global well-posedness. In this section, we establish the global existence of unique H 1 weak solutions to the LANS-α equations on bounded domains Ω ∈ R3 . Rather than using the standard Galerkin method, we instead take a sequence of classical solutions, via the existence result in [14], and prove that this sequence converges in C([0, T ], H 1 ) to a H 1 weak solution of (1) for all T > 0. 2.1. Notation and some interpolating inequalities. We will be working in the following Hilbert spaces: V s = H s ∩ H01 and Vµs = {u ∈ V s | div u = 0} for s ≥ 1, and V˙ µs = {u ∈ Vµs | Au = 0 on ∂Ω} for s ≥ 3. We endow Vµ1 with the following

38

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

scalar product:


f, g1,α =

Ω

f · g + α2 Def f : Def g dx,

  where Def u = 12 ∇u + (∇u)T is the rate of deformation tensor, · denotes the usual dot product, and : is the contraction of two indices, e.g. a : b = aij bij . For any integer s ≥ 0, we set  |Ds u|Lp = |Dβ u|Lp Ds u = {Dβ u : |β| = s}, |β|=s

where β = (β1 , . . . , βn ) denotes a multi-index, and Dβ = ∂1β1 · · · ∂nβn .

|β| = β1 + · · · + βn ,

Throughout the paper, we let C > 0 denote a generic constant. For simplicity in notation, we will write u(t) = u(t, ·). We shall need some standard interpolation inequalities, which follow from the Gagliardo-Nirenberg inequalities [1]: Suppose   1 m i 1 1 = +a − + (1 − a) p n r n q where 1/m ≤ a ≤ 1 (if m − n − n/r is an integer ≥ 1, only a < 1 is allowed). Then for f : Ω → Rn , (5) |Di f |Lp ≤ C|Dm f |aLr · |f |1−a Lq . Two specific cases of (5) we will make use of in dimension three are, 3/4

1/4

|v|L4 ≤ 4|Dv|L2 |v|L2 , 1−i/m

|Di v|L2 ≤ C|v|L2

(6) i/m

|Dm v|L2 .

(7)

2.2. Three equivalent forms of the LANS-α equations. Three equivalent forms of the LANS-α equations will be useful to us. LANS-1: ∂t u + ∇u u + U α (u) = −νAu − (1 − α2 ∆)−1 grad p + (1 − α2 ∆)−1 f div u(t, x) = 0, u = 0 on ∂Ω, u(0, x) = u0 (x),

(8)

where f ∈ L2 . (It is convenient to replace F in equation (1a) with (1 − α2 ∆)−1 f ; there is no loss in generality as (1 − α2 ∆) with domain H 2 ∩ H01 is an isomorphism.) The Stokes operator Au = −P ∆u, is the Leray projection of −∆u onto divergencefree vector fields and has domain D(A) = H 2 ∩ H01 . As we noted above, when u = 0 on ∂Ω, then Au must also equal zero on ∂Ω. LANS-2: This form is equivalent to LANS-1 in view of our remark that LANS-1 implies Au = 0 on ∂Ω:   ∂t (1 − α2 ∆)u + ∇u (1 − α2 ∆)u − α2 ∇uT · ∆u = ν(1 − α2 ∆)Au − grad p + f, (9) together with the constraint div u(t, x) = 0 and boundary data u = Au = 0 on ∂Ω. Note that when the domain Ω is the period box T3 , the Stokes operator is given by −∆, and (9) reduces to (4), the LANS-α equation used by Foias, Holm, and Titi [8].

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

39

LANS-3: This form is the analog of the Navier-Stokes equations written in terms of the Helmholtz-Hodge projection:   (10) ∂t u + νAu + P α ∇u u + U α (u) − (1 − α2 ∆)−1 f = 0, where for s ≥ 1, P α is the Stokes projector defined below. Definition 2.1. For s ≥ 1, we let P α : V s → Vµs denote the Stokes projector, a continuous ·, ·1,α -orthogonal idempotent operator (see Proposition 1 of [20]). It is defined as P α (w) = w − (1 − α2 ∆)−1 grad p = v, where (v, p) solve the Stokes problem: given w ∈ V s , there is a unique vector field v ∈ Vµs and a function p (unique up to an additive constant) such that (1 − α2 ∆)v + grad p = (1 − α2 ∆)w, div v = 0, v = 0 on ∂Ω. 2.3. Results. We begin with two elementary lemmas whose proofs are relegated to the appendix. Lemma 2.1. V˙ µ4 is dense in Vµ1 . Lemma 2.2. Let λ1 = inf1 Then for all v ∈ Vµ2 ,

v∈Vµ

|∇v|2L2 be the smallest eigenvalue of the Stokes operator. |v|2L2

  |∇v|2L2 + α2 |Av|2L2 ≥ λ1 |v|2L2 + α2 |∇v|2L2

Definition 2.2. Let f ∈ L2 (Ω) and u0 ∈ Vµ1 . For any T > 0, a function u ∈ C([0, T ]; Vµ1 ) ∩ L2 ([0, T ]; D(A)) 2 2 ∞ 1 with du dt ∈ L ([0, T ]; L )∩L ((0, T ); Vµ ) and u(0) = u0 is said to be a weak solution to the LANS-α equations with initial data u0 in the interval [0, T ] provided



d (1 − α2 ∆)u, w dt



 + ν (1 − α2 ∆)Au, w D(A)

D(A)

+



B(u, (1 − α2 ∆)u), w

D(A)

= (f, w)

(12)

for every w ∈ D(A) and for almost every t ∈ [0, T ] with B(u, v) = ∇u v + ∇uT · v; moreover, u(0) = u0 in Vµ1 . Here, the equation (12) is understood in the following sense: for every t0 , t ∈ [0, T ],
t   2 u(t), (1 − α ∆)w D(A) + B(u(s), (1 − α2 ∆)u(s)), w D(A) ds t0




= −

t

t0

 ν Au(s), (1 − α2 ∆)w D(A) ds +




t

(f, w) ds. t0

For our proof, we shall make use of the following Lemma 2.3 (Theorem 5.2 of [14]). For u0 ∈ V˙ µs , s ∈ [3, 5), and f ∈ C ∞ , there exists a unique solution u to equation (8) in C([0, ∞), V˙ µs ) ∩ C ∞ ((0, ∞) × Ω). We can now state our main result.

40

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

Theorem 2.1. For f ∈ L2 (Ω) and u0 ∈ Vµ1 , there exists a unique weak solution u ∈ C([0, ∞), Vµ1 ) ∩ L∞ ((s, ∞), V˙ µ3 ),

∀s > 0

to equation (8). The solution depends continuously on the initial data u0 .  ∞ ∞ ˙4 such that u0 → u0 in H 1 (by Proof: Choose {u }∞ =1 ∈ Vµ and {f }=1 ∈ C  2 Lemma 2.1) and f → f in L . By Lemma 2.3, for each  ∈ N, there exists a unique solution u of the LANS-α equations in C([0, ∞), V˙ µ4 ) ∩ C ∞ ((0, ∞) × Ω). Clearly, this solution also satisfies the weak formulation:
t    u (t), (1 − α2 ∆)w D(A) + B(u (s), (1 − α2 ∆)u (s)), w D(A) ds t0




= −

t t0

 ν Au (s), (1 − α2 ∆)w D(A) ds +




t

(f  , w) ds,

t0

for all w ∈ D(A). In order to show the limit as  → ∞ is also a weak solution, we need to develop the appropriate energy estimates. An H 1 Estimate. Let a = ∇u
u + U α (u ) − (1 − α2 ∆)−1 f  . Taking the ·, ·1,α inner product of LANS-3 together with u ,

d   u ,u + νAu , u 1,α + P α (a ), u 1,α = 0. dt 1,α By the definition of ·, ·1,α ,

   d   1 d  2 u ,u |u |L2 + α2 |∇u |2L2 +ν |∇u |2L2 + α2 |Au |2L2 . +νAu , u 1,α = dt 2 dt 1,α Notice that the previous integration by parts used the additional boundary condition Au = 0. Using the properties of P α , we get P α (a ), u 1,α = −(f  , u ). Therefore, 

 1 d 2 |u |L2 + α2 |∇u |2L2 + ν |∇u |2L2 + α2 |Au |2L2 = (f  , u ). 2 dt By Poincar´e’s inequality and Young’s inequality, 

 d 2 |u |L2 + α2 |∇u |2L2 + ν |∇u |2L2 + α2 |Au |2L2 ≤ K1 dt where K1 =

|f
|2L2 νλ1

(13)

(14)

. By Lemma 2.2, 



d |u |2L2 + α2 |∇u |2L2 + νλ1 |u |2L2 + α2 |∇u |2L2 ≤ K1 . dt Using Gronwall’s inequality, we get the H 1 estimate   K1 (1 − e−νλ1 t ) |u (t)|2L2 + α2 |∇u (t)|2L2 ≤ e−νλ1 t |u (0)|2L2 + α2 |∇u (0)|2L2 + νλ1 K1 ≤ k1 := |u0 |2L2 + α2 |∇u (0)|2L2 + (15) νλ1 where λ1 is defined in Lemma 2.2. Therefore we have proved that u ∈ L∞ ([0, ∞), Vµ1 ) independently of .

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

41

An H 2 Estimate. Since Au is divergence-free, we take the L2 inner product of LANS-2 together with Au to get    1 d |∇u |2L2 + α2 |Au |2L2 + ν |Au |2L2 + α2 |∇Au |2L2 + 2 dt +(B(u , (1 − α2 ∆)u ), Au ) = (f  , Au ). (16) To bound the nonlinear term, note |(∇u
u , Au )|

≤

|∇u |L2 |u |L4 |Au |L4 ≤ Ck1 |u |H 1 |Au |L4 3/4

1/4

≤ Ck12 |∇Au |L2 |Au |L2

να2 ν |∇Au |2L2 + |Au |2L2 + C(|u |H 1 ), (17) 4 4 by the Gagliardo-Nirenberg inequality (2.2) and a repeated use of Young’s inequality. The boundary conditions, the div u = 0 constraint, and one integration by parts lead to (18) |(∇u
∆u , Au )| = 0. ≤

For the third term, using (5) and (6) we have the estimate, |(∇u T ∆u , Au )|

≤

7/4

1/4

|∇∆u |L2 |u |L4 |Au |L4 ≤ C|u |H 1 |∇Au |L2 |Au |L2 23/12

1/12

13/12

23/12

≤ Ck1 |∇Au |L2 |u |L2 ≤ Ck1 |∇Au |L2 ν |∇Au |2L2 + C(|u |H 1 ). ≤ (19) 4 Again Young’s inequality implies 1 ν |(f  , Au )| ≤ |f  |L2 |Au |L2 ≤ |f  |2L2 + |Au |2L2 . ν 4 Therefore by (17)-(19), the equation (16) becomes    1 d |∇u |2L2 + α2 |Au |2L2 + ν |Au |2L2 + α2 |∇Au |2L2 2 dt ≤ |(B(u , (1 − α2 ∆)u ), Au )| + |(f  , Au )| να2 ν 1 |∇Au |2L2 + |Au |2L2 + |f  |2L2 + C. ≤ 2 2 ν Hence by Lemma 2.2 we have 

 d |∇u |2L2 + α2 |Au |2L2 + νλ1 |∇u |2L2 + α2 |Au |2L2 ≤ K2 , (20) dt   d  |∇u(t)|2L2 + α2 |Au(t)|2L2 eνλ1 t ≤ K2 eνλ1 t . dt where K2 = ν2 |f  |2L2 + C. Since|Au|2L2 is not necessarily bounded at t = 0, for every s1 > 0 we integrate both sides from [s1 , t],   |∇u(t)|2L2 + α2 |Au(t)|2L2 ≤ |∇u(s1 )|2L2 + α2 |Au(s1 )|2L2 eνλ1 (s1 −t) + νλ1 K2 . (21) Thus, independently of , u ∈ L∞ ((s1 , ∞), Vµ2 (Ω)), for all s1 > 0 An H 3 Estimate. Since A2 u is not necessarily equal to zero on ∂Ω, we do not use it to derive an H 3 -estimate. To get this estimate we will make use of the

42

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

Ladyzhenskaya method [11]. Let ut denote ∂t u, and differentiate the equations LANS-2 with respect to time to get ∂t (1 − α2 ∆)ut

+ ∇u
t (1 − α2 ∆)u + ∇u
(1 − α2 ∆)ut − α2 (∇ut )T · ∆u − α2 (∇ut )T · ∆ut = −grad pt − ν(1 − α2 ∆)Aut .

Noting that ut ∈ D(A), we take the L2 inner product with ut to get 1 d  2 |ut |L2 2 dt

+

   α2 |∇ut |2L2 + ν |∇ut |2L2 + α2 |Aut |2L2

≤ |(ut , ∇u
t u )| + α2 |(∆ut , ∇u
ut + ∇u
t u )|.

(22)

Before estimating the right hand side, we would like a bound for |ut |2L2 which will be used in the later computations. For each , u satisfies the equation LANS-3 and consequently, |ut |2L2

≤ ν|(Au , ut )| + |(∇u
u , ut )|

+|(U α (u ), ut )| + |((1 − α2 ∆)−1 f  , ut )|.

(23)

The first term is simply ν|(Au , ut )| ≤ 18 |ut |2L2 + 2ν 2 |Au |2L2 by Young’s inequality. Using (6) and Young’s inequality, the second term becomes, |(∇u
u , ut )|

1/4

3/4

≤ 16|∇u |L2 |u |L2 |ut |L2 |Au|L2 8/5

2/5

8/5

≤ ν 2 |Au |2L2 + C|∇u |L2 |u |L2 |ut |L2 1 ≤ ν 2 |Au |2L2 + |ut |2L2 + Ck110 , 8

where k1 is the time independent H 1 bound (15). Let v  = (1 − α2 ∆)−1 ut . Then, by integration by parts, the third term of (23) is |(U α (u ), ut )|

≤ α2 |(∆u · ∇u , v  )| ≤ α2 |u |2H 2 |v  |H 1 α2  2 |v |H 1 + α2 |u |4H 2 . ≤ 4

By definition, v  − α2 ∆v  = ut . Taking the L2 inner product of both sides with v  we have an H 1 estimate 1 1 |v  |2L2 + α2 |∇v  |2L2 = (ut , v  ) ≤ |ut |2L2 + |v  |2L2 . 2 2 Hence |v  |2H 1 ≤

1  2 2α2 |ut |L2

and therefore

|(U α (u ), ut )| ≤

1 2 |u | 2 + α2 |u |4H 2 . 8 tL

The last term of (23) can be estimated in a similar way 1 2 1 |v |L2 + 2|f  |2L2 ≤ |ut |2L2 + 2|f  |2L2 . 8 8 Combining the above estimates, we obtain the bound |((1 − α2 ∆)−1 f  , ut )| = |(f  , v  )| ≤

|ut |2L2 ≤ 6ν 2 |Au |2L2 + 2α2 |u |2H 2 + 4|f  |2L2 + Ck110 , (24)   From the H 2 estimate (21) we conclude that ut ∈ L∞ (s1 , ∞), L2 for all s1 > 0. Using this result, we will now estimate each of the terms on the right hand side of

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

43

(22). Using (5)-(7) and Young’s inequality, we get |(ut , ∇u
t u )|

+ α2 |(∆ut , ∇u
ut + ∇u
t u )|

3/4 5/4 15/8 1/8 ≤ C |∇ut |L2 |ut |L2 |u |H 2 + α2 |Aut |L2 |ut |L2 |u |H 1 +  3/4 1/4 + α2 |Aut |L2 |∇ut |L2 |u |L2 |u |H 1

ν να2 |∇ut |2L2 + |Aut |2L2 + Cs1 , 2 2 where Cs1 is a constant multiple of the H 2 bound depending only on the lower time bound s1 > 0. Therefore (22) becomes    d  2 |ut |L2 + α2 |∇ut |2L2 + ν |∇ut |2L2 + α2 |Aut |2L2 ≤ Cs1 . dt By Lemma 2.2,    d  2 |ut |L2 + α2 |∇ut |2L2 + νλ1 |ut |2L2 + α2 |∇ut |2L2 ≤ Cs1 dt   d   2 |ut |L2 + α2 |∇ut |2L2 eνλ1 t ≤ Cs1 eνλ1 t . dt For every s2 > 0, we integrate from [s2 , t] and obtain the H 1 bound of ut ,   |ut (t)|2L2 + α2 |∇ut (t)|2L2 ≤ |ut (s2 )|2L2 + α2 |∇ut (s2 )|2L2 eνλ1 (s2 −t) + νλ1 Cs1 ≤

Hence, independently of , ut ∈ L∞ ((s2 , ∞), Vµ1 ), for all s2 > 0. Let s = min{s1 , s2 }. By standard Sobolev inequalities, we also have ∇u
u belongs L∞ ((s, ∞), H 3 ). Therefore, |P α (∇u
u )|H 3 ≤ |∇u
u |H 3 , implies that for all s > 0, P α (∇u
u ) belongs to L∞ ((s, ∞), H 3 ), independently of . From Lemma 5.1 in [14] and our H 2 estimate (21), we infer that U α (u ) is contained in L∞ ((s, ∞), H 1+σ ) for 0 < σ < 1/3 and consequently P α (U α (u )) belongs L∞ ((s, ∞), H 1+σ ), independently of . Thus using the equation LANS-3 and the results for ut , we have that for any s > 0,   νAu = −P α ∇u
u + U α (u ) − (1 − α2 ∆)−1 f  + ut belongs to L∞ ((s, ∞), Vµ1 ), independently of . By elliptic regularity of the Stokes operator A, we conclude that u ∈ L∞ ((s, ∞), V˙ µ3 ). Moreover for all T > 0, (24) implies ut ∈ L2 ([0, T ], H) (independently of ) and therefore the classical compactness theorem (see for instance [7, 12]) enables us to
conclude that there is a subsequence u such that for all T > 0, u u u




→

u

weakly in

L2 ([0, T ], D(A)),




→

u

strongly in

L2 ([0, T ], Vµ1 ),




→

u

in

C([0, T ], H),

44

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

where H = {u ∈ L2 | div u = 0}. It is straightforward to verify that u satisfies the weak formulation associated with f . Indeed, from an identical argument provided in [8], we may conclude that u ∈ C([0, ∞), Vµ1 ). Furthermore, we infer that u(0) = u0 . Hence we are lead to the existence of weak solutions for the LANS-α equation. The uniqueness and continuous dependence on initial data of weak solutions can be proved in the same classical way as was done in [8] on Page 14 for the periodic case. We remark that a standard contraction mapping argument can be used to show the existence of LANS-α solutions u in C([0, T ], Vµs ) for s > 1. In the absence of forcing or when f is C ∞ , the solution u of the LANS-α equations is instantly regularized so that u ∈ C ∞ ((0, ∞) × Ω). 3. Estimating the dimension of the global attractor. Theorem 2.1 is sufficient to define the semi-group S(t) by S(t) : u0 ∈ Vµ1 → u(t) ∈ Vµ1 . We will now show the uniform compactness property of the operator S and the existence of an H 2 absorbing set. By Theorem I.1.1 in [21], this implies the existence of a maximal compact global H 1 attractor. 3.1. Absorbing sets and attractor. In this section we prove the existence of an absorbing set in Vµ2 . Usually proving the existence of absorbing sets amounts to proving a priori estimates; for the LANS-α equations, these estimates are established in Section 2.3. Thus (21) implies lim sup cα2 |u|2H 2 ≤ ρ22 := νλ1 K2 = 2λ1 |f |2L2 + C(|u|H 1 ). t→+∞

We conclude that the ball BVµ2 (0, ρ2 ) of Vµ2 , denoted B2 , is an absorbing set in Vµ2 for the semi-group S(t). We choose ρ2  > ρ2 and denote B0 the ball BVµ1 (0, ρ2  ). If B is any bounded set of Vµ1 , then from (21), S(t)B ⊂ B0 for t ≥ t0 (B, ρ2  ), where t0 = s1 +

|∇u(s1 )|2L2 + α2 |Au(s1 )|2L2 1 log . νλ1 ρ2  2 − ρ22

At the same time, this result proves the uniform compactness of S(t); any bounded set B of Vµ1 is included in such a ball, and for u0 ∈ Vµ1 and t ≥ t0 , u(t) belongs to B2 which is bounded in H 2 and relatively compact in H 1 . Consequently, by Theorem I.1.1 in [21], the LANS-α equations have a nonempty compact, convex, and connected global attractor in H 1 , Aα = ∩s>0 (∪t≥s S(t)B2 ). 3.2. Dimension of the attractor. The dimension of the three-dimensional attractor for the LANS-α equations follows the arguments of Theorem 6 in [8] (stated below), and leads to the same bound as in the periodic case. It is important to note that the bound tends to infinity as α → 0. Theorem 3.1 (Theorem 6 of [8]). The Hausdorff and fractal dimensions of the global attractor of the LANS-α equations, dH (Aα ) and dF (Aα ), respectively, satisfy:   2/3  3/8  1 1 α α  4/3 3/2 , ,G dH (A ) ≤ dF (A ) ≤ c max G α4 λ21 α6 λ31

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

where G =

|f |L2

3/4

ν 2 λ1

45

is the Grashoff number, and c > 0 is a constant depending only

of the shape of Ω. Next we estimate the dimension of the attractor in two-dimensions and prove that this dimension can be bounded independently of α by the Grashoff number. Hence, in two-dimensions, we find a similar bound as for the Navier-Stokes equations. Theorem 3.2. The Hausdorff and fractal dimensions of the global attractor of the LANS-α equations, dH (A) and dF (A), respectively, satisfy: dH (A) ≤ dF (A) ≤ C  G, |f |L2 is the Grashoff number, and C  > 0 is a constant depending only ν 2 λ1 on the shape of Ω. where G =

Proof: We can write equation the LANS-α equations as   ∂t u = F (u) + P α (1 − α2 ∆)−1 f u(0)

= u0 ∈ Vµ1

with F (u) = −νAu − P α [∇u u + U α (u)]. Taking the first variation of the above equation we obtain the linearized equation ∂t U U (0)

= F  (u(t))U = U0 ∈

(25)

Vµ1 .

Indeed, it can be proved by classical arguments that if u is a solution of the LANS-α equations given by Theorem 2.1, then the initial- and boundary-value problem (25) possesses a unique solution U ∈ C([0, T ]; Vµ1 ) ∩ L2 ([0, T ]; D(A)),

∀T > 0.

Furthermore, it can also be proved that for every t > 0, the function u0 → S(t)u0 is Fr´echet differentiable in Vµ1 at u0 with differential L(t, u0 ) : ξ ∈ Vµ1 → U (t) ∈ Vµ1 , where U is the solution to (25) with U (0) = ξ. Let 1 ≤ N ∈ N and U1 (0), . . . , UN (0) be linearly independent vectors in Vµ1 . Let U1 (t), . . . , UN (t) be the corresponding solutions of (25) with initial conditions U1 (0), . . . , UN (0). Let PN (t) be the orthonormal projector of Vµ1 on PN (t)Vµ1 = Span [U1 (t), . . . , UN (t)]. Then for each N ∈ N,
t |U1 (t) ∧ · · · ∧ UN (t)|ΛN Vµ1 = |U1 (0) ∧ · · · ∧ UN (0)|ΛN Vµ1 exp Tr F  (u(τ )) ◦ PN (τ ) dτ. 

0

Let TN (t) = Tr PN (t)F  (u(t))|PN (t)Vµ1 . Recall that the vector space Vµ1 is endowed with the inner product
u · v + α2 Def u : Def v dx. u, v1,α = Ω

(ϕj (t))N j=1

be an orthonormal basis, with respect to the inner At a given time t, let product ·, ·1,α , of the space PN (t)Vµ1 . Define ψj = (ϕj , α∂1 ϕj , . . . , α∂N ϕj )

46

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

∂ where ∂i = ∂x is the i-th partial derivative. Notice that by this definition, since i ϕi , ϕj 1,α = δij for i, j = 1 . . . N , then (ψj , ψk ) = δjk for j, k = 1, . . . , N . Also define

ρ(x, t)

N 

=

ψj (x, t) · ψj (x, t)

(26)

j=1 N    ϕj (x, t) · ϕj (x, t) + α2 Def ϕj (x, t) : Def ϕj (x, t)

=

j=1

and qN (t) =

N
 j=1

Ω

Def ψj : Def ψj dx.

(27)

In the following calculations, {Ci }i∈N denotes positive constants depending only on the shape of Ω. By the Lieb-Thirring inequality in two-dimensions, we have
ρ2 dx ≤ Co qN (t). Ω

TN estimate. In the calculations below we will suspend the dependence on x and t. Using the linearized equation (4.21) and remembering ϕj = Aϕj = 0 on ∂Ω, we obtain TN (t)

=

N  j=1

= −ν

F  (u(t)) · ϕj (·, t), ϕj (·, t)1,α N 
 Ω

j=1

+

N  




−

+α




≤

Ω

|Aϕj |2 dx


∇u (ϕj − α ∆ϕj ) · ϕj dx −

∇u · ∆ϕj · ϕj dx + α T

Ω




2

Ω

j=1 2

Def ϕj : Def ϕj dx + α2

2


Ω

∇ϕTj

Ω



∇ϕj (u − α2 ∆u) · ϕj dx 

· ∆u · ϕj dx

−Co νqN (t) + I1 + I2 + I3 + I4

Now we obtain estimates of Ii in terms of qN (t).
I1 = −α2 ui ∂i ∆ϕj · ϕj dx
Ω ui ∆ϕj · ∂i ϕj dx = α2 Ω  

2 = α ∂k (ui ∂k ϕj ) · ∂i ϕj dx − ∂k ui ∂k ϕj · ∂i ϕj dx Ω   Ω

2 = α − ui ∂k ϕj · ∂ki ϕj dx − ∂k ui ∂k ϕj · ∂i ϕj dx Ω Ω

α2 2 = −α ∂k ui ∂k ϕj · ∂i ϕj dx − ui ∂i [∂k ϕj · ∂k ϕj ] dx 2 Ω
Ω = −α2 ∂k ui ∂k ϕj · ∂i ϕj dx Ω

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

47

By Young’s inequality,  

√ |∂k ϕj |2L2 |∂i ϕj |2L2 + |∂k ui |L2 ρ(u, t) Def u : Def u dx |I1 | ≤ α2 dx ≤ 2 2 Ω Ω 


≤

Ω

|∇u|2L2 dx


Co ν C qN (t) + 1 8 ν

≤

Ω

Ω

ρ(u, t)2 dx ≤ Co qN (t)1/2

Ω

|∇u|2L2 dx

|∇u|2L2 dx.

Also consider,


2 2 (ϕj )β ∆u∂β ϕj dx. I2 = − (ϕj )β ∂β (u−α ∆u)·ϕj dx = − (ϕj )β ∂β uϕj dx+α Ω

Ω

Ω

By Young’s inequality and definitions (26) and (27),
|I2 |

≤

Ω

|∇u|L2

|∇u|L2 ρ(u, t) dx +

Ω


≤

Ω

|∇u|2L2

≤ C0 qN (t)1/2

α 2




Ω

α|∆u|L2

N   |ϕj |2 2 L

2

j=1

+ α2

|∇ϕj |2L2 2

 dx

|∆u|L2 ρ(u, t) dx 
+ α2 |∆u|2L2 dx ρ2 dx

Ω

Ω

Ω




Co ν C qN (t) + 1 8 ν

≤




|ϕj |2L2 dx +

j=1


≤

N 

|∇u|2L2 + α2 |∆u|2L2 dx
Ω



 |∇u|2L2 + α2 |∆u|2L2 dx

Similarly, = −α2

I3


Ω

∂k ∇uT · ∂k ϕj · ϕj dx − α2


Ω

∇uT · ∂k ϕj · ∂k ϕk dx

α |I3 | ≤ C2 |∆u|L2 |ρ(u, t)|L2 + |∇u|L2 |ρ(u, t)|L2 2
  Co ν C qN (t) + 3 |∇u|2L2 + α2 |∆u|2L2 dx ≤ 8 ν Ω and |I4 |

≤ ≤




N    α |ϕj |2L2 + α2 |∇ϕj |2L2 dx ≤ |∆u|L2 |ρ(u, t)|L2 2 Ω j=1  
Co ν C qN (t) + 4 α2 |∆u|2L2 dx 8 ν Ω

α 2

|∆u|L2

Combining the above estimates we have established the following inequality
  C ν C |∇u|2L2 + α2 |∆u|2L2 dx. TN (t) ≤ − o qN (t) + 5 2 ν Ω

(28)

48

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

Furthermore from Lemma V.2.1 in [21] we infer that, 
N 
 qN (t) = Def ϕj : Def ϕj dx + α2 |Aϕj |2L2 dx Ω

j=1 N 

≥

Ω

Aϕj , ϕj 1,α ≥

j=1

N 

λj

j=1

since ϕj (t) is orthonormal in PN (t)Vµ1 . Hence following a result in [16], qN (t) ≥

N 

λj ≥ C6 λ1 N 2 .

j=1

Finally by (28),

1 T C 1 T TN (t) dt ≤ −νC7 λ1 N 2 + 5 |∇u|2L2 + α2 |∆u|2L2 dt. T 0 ν T 0 From (14) and (15) we infer that
1 T lim sup |∇u|2L2 + α2 |∆u|2L2 dt T →∞ T 0


1 T |∇u|2L2 + α2 |Au|2L2 dt T →∞ T 0 C  |f |2 2 C8 K1 ≤ 82 L . ν ν λ1

≤ C8 lim sup ≤

Hence lim sup T →∞

Therefore, if N >

C9



1 T




|f |2 ν 4 λ1

0

T

TN (t) dt ≤ −νC7 λ1 N 2 +

1/2

C8 |f |2L2 . ν ν 2 λ1

we conclude that 1 lim sup T T →∞


0

T

TN (t) dt < 0,

and

 |f |L2 dimH A ≤ . ν 2 λ1 Thus N provides an upper bounded for the fractal and Hausdorff dimension of the |f | 2 global attractor. With G = 2 L denoting the Grashoff number in 2D, ν λ1 α

C9



dimH Aα ≤ C9 G, which is the desired result. Appendix. Proof of Lemma 2.1 Let v ∈ Vµ1 . We will find a sequence vm ∈ V˙ µ4 which converges to v in H 1 . The proof makes use of the Lax-Milgram Theorem to provide a compact operator from Vµ1 to V˙ µ4 . Define the bilinear form E : V˙ µ4 ×V˙ µ4 → R by E[u, v] = u, vH 4 . 4 The function space V˙ is a closed subspace of H 4 and therefore a Hilbert space µ

endowed with the usual H 4 topology. Since E is defined as the usual inner product

WEAK SOLUTIONS FOR LANS-α ON BOUNDED DOMAINS

49

on H 4 , it is coercive and continuous. We define, ∗ using the Riesz-Representation 4 ˜ ˙ Theorem, the bounded linear function f ∈ Vµ by ∀v ∈ V˙ µ4

f˜(v) = (f, v)L2

where f ∈ V˙ µ4 . By the Lax-Milgram Theorem, there exists a unique u(f ) ∈ V˙ µ4 such that E[u, v] = u, vH 4 = (f, v)L2 , ∀v ∈ V˙ µ4 . This is the weak formulation of an elliptic problem Lu = f where L is an eighth order elliptic differential operator. The map L−1 : f → u is a continuous map from V˙ µ4 → V˙ µ4 . Since H 4 is compactly embedded in H 1 , L−1 is a compact operator from Vµ1 → V˙ µ4 , and consequently has a discrete spectrum of eigenfunctions {ei }i∈N . The ∞  ci ei . eigenfunctions {ei }i∈N form a Hilbert basis of Vµ1 and therefore we write v = i=1

The ellipticity of L−1 implies that {ei }i∈N are also eigenfunctions of L, allowing m  ci ei ∈ V˙ µ4 us to conclude that {ei }i∈N ∈ V˙ µ4 . Consequently, the sequence vm = i=1

converges in H 1 to v as m → ∞. Proof of Lemma 2.2 We just need to show that |Av|2L2 ≥ λ1 |∇v|2L2 (since |∇v|2L2 ≥ λ1 |v|2L2 by definition). Let {hi }i∈N be eigenfunctions associated to the Stokes operator A with corresponding eigenvalues λi . For all v ∈ Vµ2 , we write ∞ ∞   ci hi . Then Av = ci λi hi . The orthogonality property of hi implies v= i=1

(Av, v) =

∞ 

i=1

c2i λi

=

|∇v|2L2

and consequently,

i=1

|Av|2L2

=

∞ 

c2i λ2i

i=1

≥

∞ 

 c2i λi

λ1 ≥ λ1 (Av, v) = λ1 |∇v|2L2

i=1

which completes the proof. Acknowledgments. The authors would like to thank Ciprian Foias, Darryl Holm, Jerry Marsden, and Marcel Oliver for stimulating conversations. Work was partially supported by the NSF-KDI grant ATM-98-73133 and an IGPP Los Alamos National Laboratory minigrant. SS was also partially supported by the Alfred P. Sloan Foundation Research Fellowship, and the NSF grant DMS-0105004. REFERENCES [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623–727. [2] S.Y. Chen, C. Foias, D.D. Holm, E.J. Olson, , E.S. Titi, and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett, 81 (1998), 5338–5341. [3] S.Y. Chen, Foias, C., Holm, D.D., Olson, E.J., Titi, E.S., and Wynne, S., The CamassaHolm equations and turbulence in pipes and channels, Physica D, 133 (1999), 49–65. [4] S.Y. Chen, C. Foias, D.D. Holm, E.J. Olson, , E.S. Titi, and S. Wynne, The Camassa-Holm equations and turbulence in pipes and channels, Phys. Fluids, 11 (1999), 2343–2353.

50

D. COUTAND, J. PEIRCE, AND S. SHKOLLER

[5] S.Y. Chen, D.D. Holm, L.G. Margolin, and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66–83. [6] A.J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), 785– 796. [7] P. Constantine, and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988. [8] C. Foias, D.D. Holm and E.S. Titi, The three dimensional viscous Camassa-Holm Equations and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Diff. Eq., (1999), to appear. ´ models of ideal fluids with [9] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincare nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173–4177. ´ equations and semidirect [10] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincare products with applications to continuum theories, Adv. Math., 137 (1998), 1–81. [11] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon & Breach, (1963). (Revised English edn, translated from the Russian by Richard A. Silverman.) ´thodes de Re ´solution des Proble `mes aux Limites Non[12] J.-L. Lions, Quelques Me ´aires. Dunod, Paris, 1969. Line [13] J.E. Marsden, T. Ratiu and S. Shkoller, The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal., 10 (2000), 582– 599. [14] J.E. Marsden and S. Shkoller, Global well-posedness for the LANS-α equations on bounded domains, Proc. Roy. Soc. London, 359 (2001), 1449–1468. [15] J.E. Marsden, and S. Shkoller, The anisotropic Lagrangian averaged Euler equations, Arch. Ration. Mech. Anal., (2001), to appear. ´rateurs de ´finis sur la restriction de syste `mes [16] G. M´ etivier, Valeurs propres d’ope variationnels ` a des sous-espaces, J. Math. Pures. Appl., 57 (1978), 133–156. [17] K. Mohseni, S. Shkoller, B. Kosovi´c and J.E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes (LANS-α) equations for homogeneous isotropic turbulence, Phys. Fluids (2001). (Submitted.) [18] C.S. Peskin, A random-walk interpretation of the incompressible Navier-Stokes equations, Comm. Pure Appl. Math, 38 (1985), 845–852. [19] S. Shkoller Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics, J. Funct. Analysis, 160 (1998), 337–365. [20] S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geometry, 55 (2000), 145–191. [21] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. SpringerVerlag, 1997.

Received for publication April 2001. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]