Upper Bounds on the Number of Determining Modes ... - UCI Math

Indiana University Mathematics Journal, Vol. 42, No. 3, (1993).

Upper Bounds on the Number of Determining Modes, Nodes, and Volume Elements for the Navier-Stokes Equations Don A. Jones1 and Edriss S. Titi2 ; 3

Dedicated to Professor Ciprian Foias on the Occasion of his 60th Birthday

Abstract

In this paper we present improved upper bounds on the number of determining Fourier modes, determining nodes and volume elements for the Navier-Stokes equations. The bounds presented here seem to agree with some of the heuristic physical estimates conjectured by Manley and Treve.

1 Introduction The conventional theory of turbulence asserts that turbulent ows are monitored by a nite number of degrees of freedom. The notions of determining modes, [5], [6] determining nodes [7], [8], [9], and determining volume elements [8],[10] are rigorous attempts to identify those parameters that control turbulent ows (see Sections 3, 4 and 5 for the de nitions of determining modes, nodes and volume elements respectively). Since the problem of global existence and uniqueness of strong solutions to the 3D Navier-Stokes Equations (NSE) is still an open question, most of the research on estimating these parameters has been concentrated on the 2D case. Here we will give an improvement of the previously reported estimates on the number of determining modes, nodes and volume elements [5], [9] and [10] respectively. In this paper we will consider only the case with periodic boundary conditions. The same techniques, however, work for other boundary conditions as well, but they lead to larger upper bounds. It is not clear yet whether this is merely a mathematical technicality or if Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA. Department of Mathematics, University of California, Irvine, CA 92717, USA. 3 Center of Applied Mathematics, Cornell University, Ithaca, NY 14853, USA.

1 2

1

it is due to the eect of the physical boundaries. We will denote by G the Grashof number (see Section 2), which plays an analogous role as the Reynolds number, and which will be our bifurcation parameter. It is known that if the Grashof number is small enough, then the NSE possess a unique, globally stable, steady state solution (cf. [19]). It is expected that as the Grashof number G increases, this steady state goes through a sequence of bifurcations that lead to chaotic dynamics (see however [17]). Therefore, it is natural to use the Grashof number G as a parameter for measuring the complexity of the dynamics. There have been many studies to estimate the number of degrees of freedom of the solutions for the NSE in terms of the Grashof number. A sharp upper bound for the fractal, as well as the Hausdor dimension of the global attractor was established in [2] (see also references therein). This upper bound is of the order of G2=3(1+log G)1=3. Also, it has been reported in [5] that there exists an upper bound for the number of determining Fourier modes for the 2D NSE, with periodic boundary conditions, of the order of G(1 + log G)1=2. In [9], we obtain an upper bound for the number of determining nodes of the order G2(1 + log G), and for the number of determining volume elements, [10], of the order G2. In this paper we slightly improve the estimate on the number of determining modes and obtain an upper bound of the order G. Similarly, we improve our previous estimates on the number of determining nodal values and of determining volume elements to G. These new estimates are in agreement with the heuristic estimates, which are based on physical arguments, that have been conjectured by Manley and Treve ([5], [16], [20]). The question as to the sharpness of these estimates remains open. However, in practice experimentalists take their measurements at one point (node), or sometimes at few, but small number of points in the physical domain. Then they rely on the generic imbedding theorem of Takens [18] to unfold the attractor and to understand the characteristics of the dynamics. It is worth mentioning, however, that Takens' theorem is only a generic theorem and that there are some interesting physical examples in uid ows where it degenerates. Such examples occur, for instance, in the presence of certain symmetries [4]. It has been recently shown, however, by [13] that for the 1D complex Ginzburg-Landau equation, two nodes are determining if they are suciently close. The idea of the proof of [13] is valid for many other 1D dissipative parabolic PDEs (see for instance [3]). Also, a straight forward extension of the proof given in [13] has been reported for the 2D NSE, and the results imply that there is a small closed determining curve, [15]. This result holds under the assumption that the solutions of the NSE are spatially analytic. However, one can show, in a straight forward manner, that these results are also valid if the solutions enjoy the unique continuation property, which is a weaker assumption on the space of solutions. The results of [13] indicate that the minimal number of determining nodes may not have anything to do with the dimension of the global attractor. On the other hand, if one takes enough nodes, one can parameterize the inertial manifold (when it exists) in terms of the nodal values of the solution, and hence, show that the dynamics of the nodal values is equivalent to that of the PDE [8]. Even though we still do not know whether the 2D NSE has an inertial manifold, one can show that there is a parameterization of the global attractor for the 2D NSE, with periodic boundary conditions, in terms of the nodal values of the solutions on the attractor [21]. The proof of the last statement uses ideas from [8], [11] and [12]. The latter statement gives an armative answer to the assertion conjectured by Foias and Temam [7]: there exists a nite number of nodes such that the solutions on the global attractor, for the 2D NSE, are determined in a unique fashion by their values at these nodes. 2

We organize the paper as follows. In Section 2 we recall some of the basic properties of the 2D NSE with periodic boundary conditions. In Section 3 we recall the notion of determining modes. We also show how to eliminate the logarithmic term found in the upper bound for the number of determining modes in [5]. In Section 4 we recall the notion of determining nodes and give an upper bound for their number. Though the idea for the proof is essentially the same as in [9], there are some major dierences in our estimates. A similar analysis of the number of determining volume elements is given in Section 5. To make the paper self-contained we include an appendix that contains the proof of certain interpolation estimates that are needed here. This paper is dedicated to Professor Ciprian Foias on the occasion of his 60th birthday to thank him for his endless support and encouragement to us and to express our admiration for his scienti c work in the various elds of the mathematical sciences.

2 Functional Setting and Preliminary Results We consider the 2D NSE for a viscous incompressible uid in IR2 with periodic boundary conditions on the square = (0; L)  (0; L). The governing equations are @u ? 
u + (u
r)u + rp = f @t 2

in IR2  (0; 1) > > > = r
u = 0 in IR  (0; 1) (2.1) > u(x1; x2; t) = u(x1; x2 + L; t) > > ; u(x1; x2; t) = u(x1 + L; x2; t); where f = f (x; t), the volume force, and  > 0, the kinematic viscosity, are given. We denote by u = u(x; t) the velocity vector, and p = p(x; t) the pressure which are the unknowns. Further, we assume that the integrals of u and f vanish on for all time (i.e., u and f have zero mean in ). Following the notation in [1], [14], [19], we set 9

V = fu : IR2 7! IR2; vector-valued trigonometric polynomials with period L; r
u = 0; and udx = 0g: Z



Further, we set

H = the closure of V in (L2( ))2; V = the closure of V in (H 1( ))2; where H l ( ) (l = 1; 2; : : :) denote the usual L2 -Sobolev spaces. H is a Hilbert space with the inner product and norm (u; v ) =

Z



juj =

u(x)
v(x)dx;

Z

1=2





ju(x)j2dx

;

respectively, and u(x)
v (x) is the usual Euclidean scalar product. Thanks to the Poincare inequality, V is also a Hilbert space with inner product and norm ((u; v )) =

2

X

i;j =1

Z

2 Z @v 2 @ui @vi dx; kvk2 = X i dx; @x @x @x

j j j i;j =1

3

respectively. Let P denote the orthogonal projection in L2 ( )  L2( ) onto H . We denote by A the Stokes operator Au = ?P
u; (notice that in the periodic case Au = ?
u) and the bilinear operator B(u; v) = P ((u
r)v) for all u; v in D(A) = V \ (H 2( )  H 2 ( )). We recall that the operator A is a self-adjoint positive de nite operator with compact inverse. Thus there exists a complete orthonormal set wj of eigenfunctions of A such that Awj = j wj and 0 < 1  2  : : : , where 1 = ( 2L )2, and j = O(j ) for j ! 1. The NSE, (2.1), are equivalent to the functional dierential equation in H du + Au + B(u; u) = f; (2.2) dt where from now on f = Pf , and it is assumed that f satis es f 2 L1 ((0; 1); H ). That is, supt0 jf (t)j < 1, (for details see for example [1], [14], [19]). Let 1=2 Z F = lim sup jf (t; x)j2dx : t!1



Following [5] we de ne the generalized Grashof number Gr as 2 Gr = F 2 = 4L 2F 2 : 1

The generalized Grashof number will play an analogous role as the Reynolds number and will be our bifurcation parameter. In what follows all our estimates will be in terms of the generalized Grashof2 number. Notice that if f is time independent, then Gr is the Grashof number G = 4L2jf 2j . For questions related to existence, uniqueness, and regularity of solutions the reader is referred for instance to [1], [14], [19] and the references therein. We recall the following inequalities which are satis ed by B (u; v ) (cf. [1], [14], [19]). j(B(u; v); w)j  c1juj1=2kuk1=2kvkjwj1=2kwk1=2 8u; v; w 2 V: (2.3) Alternatively, we may use Agmon's inequality kukL1( )  c2juj 12 jAuj 21 8u 2 D(A); to obtain j(B(u; v); w)j  c3juj1=2jAuj1=2kvkjwj 8u 2 D(A); v; w 2 V: (2.4) In addition, the operator B enjoys the property (B (u; v ); w) = ?(B (u; w); v ), and in the 2D case with periodic boundary conditions it satis es: (B (w; w); Aw) = 0 8w 2 D(A) (2.5) (cf. [1], [19]). Dierentiating this last expression with respect to w in the direction of u, we obtain the useful identity (B (u; w); Aw) + (B (w; u); Aw) + (B (w; w); Au) = 0 (2.6) for all u; w 2 D (A) (see [2]). 4

3 Determining Modes The rst mathematically rigorous indication that the large time behavior of the solutions to the NSE has a nite number of degrees of freedom was given in [6]. More speci cally, the NSE was shown to have a nite number of determining modes. We denote by Pm the orthogonal projection onto the linear space spanned by fw1; w2; : : :; wmg, the rst m eigenfunctions of the Stokes operator A, and Qm = I ? Pm. Further, cj (j = 1; 2; : : :) will denote adequate dimensionless positive constants. Let u; v solve respectively the Navier-Stokes equations

du + Au + B(u; u) = f (t) dt u(0) = u0; dv + Av + B(v; v) = g(t) dt v(0) = v0; where f; g are given forces in L1 (0; 1; H ). A set of modes fwj gmj=1 is called determining if we have lim ju(t) ? v (t)j = 0; t!1 whenever

(3.1) (3.2)

tlim !1 jf (t) ? g (t)j = 0;

and

lim jPm u(t) ? Pm v (t)j = 0: In [5] an upper bound on the number of determining modes for the 2D NSE was obtained in the form m+1  c Gr(1 + log(Gr))1=2: (3.3) 4  t!1

1

However, in that paper (see also [16, 20]) it is argued, heuristically, that the number of determining modes should be of the order m, where m satis es

m+1  c Gr: 5 1 We will suppose throughout that u(t); v (t) solve Equations (3.1), (3.2) respectively and that jf (t) ? g (t)j ! 0 as t ! 1. Before we show how to eliminate the logarithmic term

in (3.3) we recall a version of the Gronwall lemma that will play a central role. The original version was rst used in [5] to prove (3.3). The more general version stated below is proven in [10].

Lemma 3.1 Let  be a locally integrable real valued function on (0; 1), satisfying for some 0 < T < 1 the following conditions: t+T

1Z lim inf t!1

T

t

( )d = > 0 5

Z t+T 1 lim sup T ?( )d = ? < 1;

t!1

t

= maxf?; 0g. Further, let be a real valued locally integrable function de ned on (0; 1) such that 1 Z t+T + ( )d = 0; lim t!1 where ?

T

t

where = maxf ; 0g. Suppose that  is an absolutely continuous non-negative function on (0; 1) such that d  +   ; a.e. on (0; 1):

+

Then  (t) ! 0 as t ! 1.

dt

Theorem 3.2 Suppose that m satis es

m+1  p3c Gr: 3 1

Then the number of determining modes is not larger than m. That is, if

lim jPm u(t) ? Pm v (t)j = 0;

t!1

then limt!1 ku(t) ? v (t)k = 0.

Proof. The idea of the proof (as in [5]) is to apply Lemma 3.1. Let w(t) = u(t) ? v (t), (t) = Pm w(t) and
(t) = Qm w(t). Then by assumption jj ! 0 as t ! 1. Subtracting Equation (3.2) from (3.1), we obtain

dw + Aw + B(u; w) + B(w; u) ? B(w; w) = f (t) ? g(t): dt Taking the inner product with A
and using (2.5), (2.6), we have 1 d k
k2 +  jA
j2  2 dt j(B(w; w); Au)j + j(B(u; w) + B(w; u) ? B(w; w); A)j + jf (t) ? g(t)jjA
j: For the second term on the right side of the inequality we use (2.3) and jA3=2 j  3m=2j j to obtain

j(B(u; w) + B(w; u) ? B(w; w); A)j = j(B(u; A); w) + B(w; A); u) ? B(w; A); w)j  (2c1ju(t)j1=2ku(t)k1=2jw(t)j1=2kw(t)k1=2 + c1jw(t)jkw(t)k)3m=2j (t)j := M1 (t)j (t)j: Since ku(t)k; kv (t)k remain bounded as t ! 1 (this follows immediately from Equations (3.1), (3.2), using (B (u; u); Au) = 0), M1(t) is bounded as t ! 1. For the rst term we use the equation B (w; w) = B (
;
)+ B (; w)+ B (
;  ) and (2.4) to obtain

j(B(w; w); Au)j  j(B(
;
); Au)j + M2(t)j(t)jjAuj + c3jjjA
jjAuj; 6

where M2 (t) may be chosen to be c3 1m=2(kuk + kv k). Using (2.4), we also have

j(B(
;
); Au)j  c3j
j1=2jA
j1=2k
kjAuj  c3j1A=2
j k
kjAuj: m+1 Applying Young's inequality we conclude

d k
k2 + k
k2  ? 3c23jAuj2   (t); m+1 dt m+1 where (t) = 2M1(t)j j + 2M2(t)jAujj j + 3 c23jAuj2j j2 + 3 jf (t) ? g (t)j2. It follows from a priori estimates on the time average of jAuj (see [5], [9]), Z t+T 2 2 lim sup T1 (3.4) j Auj2d  TF3 + F 2 t!1 t 1 for every T > R0 (here we take T = (1)?1). The assumptions on f; g; j j, and u; v imply that lim t!1 T1 tt+T + ( )d = 0. Set c23jAuj2 :  = m+1 ? 3 m+1 Also from the estimate on the time average of jAuj2 (see (3.4)) we have Z t+T lim sup T1 ? ( )d < 1:

(3.5)

1 Z t+T ( )d > 0 lim inf t!1 T

(3.6)

t!1

Similarly,

p

t

t

holds provided m+1 =1  3c3Gr. Hence, from the above and Lemma 3.1 we conclude that lim t!1 kw(t)k = 0. 2 Let us remark that if we do not require additional regularity of the forcing function f (t), then jAu(t)j2 need not to be uniformly bounded for all t > 0. However, it is always bounded in average (see (3.4)). This means that the function (t) need not to converge to zero , but its average does. For this very reason we use Lemma 3.1, the modi ed version of Gronwall's Lemma.

4 Determining Nodes Let

E = fx1; x2; : : :; xN g be a collection of points in . The set E is called a set of determining nodes if for any two solutions u and v solving Equations (3.1), (3.2) respectively, and satisfying lim (u(xj ; t) ? v (xj ; t)) = 0; j = 1; : : :; N; t!1 7

we have

lim ju(t) ? v (t)j = 0: The existence of a set of determining nodes was rst proven in [7]. Later in [9] an upper bound for the number of determining nodes for the NSE described in Section 2 was found to be proportional to Gr2(1 + log Gr). Rather than nding an estimate on the maximum distance the nodes can be separated (as in [7], [9]), we simply estimate the number of nodes needed to be p determining directly. For this we divide the domain into N equal squares of side l = L= N . Further we place one of the points xj ; j = 1; : : :; N in each subsquare. The key to our estimates is the following lemma (cf. [9], [10]) which is proven in the appendix. t!1

Lemma 4.1 For every w 2 D(A) set (w) = 1max jw(xj )j: j N Then

jwj2  4L22(w) + cN8L2 jAwj2; 2 kwk2  c9N2(w) + c10NL jAwj2; 2 kwk21 = sup jw(x)j2  c9N2(w) + c10NL jAwj2: 4

x2

(4.1) (4.2) (4.3)

Let u(t); v (t) solve (3.1), (3.2) respectively, and suppose jf (t) ? g (t)j ! 0 as t ! 1 as before.

Theorem 4.2 Let be divided into N equal squares with the points E = fx1; x2; : : :; xN g distributed one in each square. Then E is a set of determining nodes provided N  c11Gr; p 2 where c11 = 4 2c3c10 .

Proof. With the exception of the use of Lemma 4.1 and a minor modi cation described below, the proof is the same as in [9]. We therefore only provide a sketch of the proof here. Again set w(t) = u(t) ? v (t). Then  (w) ! 0 as t ! 1. Subtracting Equation (3.2) from (3.1) and using (2.5), (2.6), (2.4), we have 1d 2 2 2 dt kwk +  jAwj  j(B (w; w); Au)j + jf ? g jjAwj  c3kwk1kwkjAuj + jf ? gjjAwj: Now we use (4.3) to estimate kwk1 . Further, instead of using the estimate jAwj  as in [9], we use (4.2) to obtain 2 2 jAwj2  N kwk2 ? c9N  (w) :

11=2kwk,

c10L2

c10L2

8

After using Young's inequality and some algebra, we nd d kwk2 + kwk2  N ? 2c23c10L2 jAuj2  (t); where

dt

c10L2

N

2 2(w) p p 2 (t) = c9N + 2 c 3 c9 N (w)kwkjAuj + jf (t) ? g (t)j: 2 c10L  R t+T 1 As in the proof of Theorem 3.2, we have limt!1 T t + ( )d = 0. Set 2 c L2 10 jAuj2:  = cNL2 ? 2c3N 10 Again from the estimate on the time average of jAuj2, inequality (3.4), we have Z lim sup 1

t!1

Similarly,

p

T

t+T

t

?( )d < 1:

1 Z t+T ( )d > 0 lim inf t!1 T t

holds provided N  2c3c101L2 Gr. Hence, from the above and Lemma 3.1 we conclude that lim t!1 kw(t)k = 0. 2

5 Determining Finite Volume Elements p

We divide into N equal squares of side l = L= N , and label the squares by Q1 ; : : :; QN . Here we suppose that the average values of solutions on each of the Qj 's is known. For this we set Z huiQ = N2 u(x)dx

L

j

Qj

for every 1  j  N . A set of volume elements is said to be determining if for any two solutions u and v solving Equations (3.1), (3.2) respectively, and satisfying lim (huiQ ? hv iQ ) = 0; t!1 we have lim ju(t) ? v (t)j = 0: t!1 The notion of determining volume elements was introduced in [8]. The existence of determining volume elements for the NSE was shown in [10]. Moreover, it was found there that the number of volume elements need to be determine the solutions was of the order Gr2. Theorem 5.1 Let be divided into N equal squares. Suppose lim (huiQ ? hv iQ ) = 0; t!1 for 1  j  N . Then the volume elements are determining, that is, lim ku(t) ? v (t)k = 0; t!1 provided N  c12Gr. j

j

j

j

9

Proof. Set w(t) = u(t) ? v (t) and (w) = max1j N jhwiQ j. With the exception of replacing the estimate jAwj  11=2kwk, with the analog of (4.2), j

p

p

jAwj  2 LN kwk ? 2 L6N (w); (see Lemma 3.1 of [10]), the proof is exactly the same as in [10] (and Theorem 4.2 above). We omit further details. 2

6 Appendix We now give the proof of Lemma 4.1. We begin with an auxiliary lemma.

Lemma 6.1 Let 1 = [0; ]  [0; d] and u(x; y) 2 H 1( 1). Then 

Z

0

ju(x; 0)j2dx  2d?1kuk2L ( ) + dk@u=@yk2L ( ): 2

2

1

(6.1)

1

Proof. Without loss of generality we can assume that u is a smooth function. Then from by the fundamental theorem of calculus we have

u2(x; 0) = u2(x; y) ?

y

Z

0

@ 2 @y u (x; s)ds:

Integrating over 1, with respect to x and y , and applying the Cauchy-Schwarz inequality, we obtain Z  d ju(x; 0)j2dx  kuk2L2( 1 ) + 2dkukL2( 1) k@u=@ykL2( 1) : 0

The result follows after an application of Young's inequality. 2 Now consider a square Q = [0; l]  [0; l]. For any two points in the square (x1; y ); (x2; y ) we have Z 2

2 x2 @u(s; y )

@u(
; y ) 2

ju(x1; y) ? u(x2; y)j = : @x ds  l @x x1

L2 (0;l)

We apply (6.1) to @u=@x with d replaced with the maximal distance of the y coordinate of the points (x1; y ); (x2; y ) from the horizontal walls. That is, d = maxfy; l ? y g  l=2. We arrive at

2

2

@u

@ 2u 2 2

ju(x1; y) ? u(x2; y)j  4 @x + l @y@x

: L2 (Q)

L2 (Q)

By symmetry a similar inequality holds for points of the form (x; y1); (x; y2). Now for arbitrary points in Q, (x1; y1); (x2; y2), we have ju(x1; y1) ? u(x2; y2)j  ju(x1; y1) ? u(x2; y1)j + ju(x2; y1) ? u(x2; y2)j. Applying the above inequality to each part, we obtain

ju(x1; y1) ? u(x2; y2)j  2

4kruk2 2

L (Q)

10

@ 2 u

2

@y@x L2(Q)

+ l2

!

1=2

:

(6.2)

p

Now divide the domain into N equal squares of side l = L= N . Again label the squares Qj ; j = 1; : : :; N , and set  (w) = max1j N jw(xj )j, where now xj is any point in Qj . It follows from integrating (6.2) over Qj that

@ 2w kwk2L (Q )  2l22(w) + 32l2krwk2L (Q ) + 8l4 @y@x 2

2

j

j







2

L2 (Qj )

:

@ 2w j  c jAwj (again, j
j means the Summing over j and using krwk2  jwjjAwj; j @y@x 13 2 L ( ) norm), and an application of Young's inequality, we obtain

jwj2  4Nl22(w) + c8l4jAwj2 which is Equation (4.1). The other equations in Lemma 4.1 follow from kwk2  jwjjAwj, Agmon's inequality kwk21  c22 jwjjAwj and a repeated use of Young's inequality (see [10]).

Acknowledgments This work was supported in part by the NSF and the Joint Univer-

sity of California { Los Alamos National Laboratory Institute for Cooperative Research (INCOR) Program for Climate Modeling. Partial support has also come from the Army Research Oce contract number DAAL03-89-C-0038 while E.S.T. was visiting the Army High Performance Computing Research Center at the University of Minnesota.

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[10] D.A. Jones, E.S. Titi, Determination of the solutions of the Navier-Stokes equations by nite volume elements, Physica, D60, (1992), 165-174. [11] M. Kwak, Finite dimensional inertial form for the 2D Navier-Stokes equations, Indiana Univ. Math. Jour., 41, (1992), 927-982. [12] M. Kwak, Fourier spanning dimension of the 2D Navier-Stokes equations, to appear in Nonlinear Anal., TMA. [13] I. Kukavica, On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity, 5, (1992), 997-1006. [14] J.L. Lions, Quelques Method de Resolution de Problem aux Limites Non Lineaire, Dunod, Paris, 1969. [15] V.X. Liu, On locally determining integrals for the Navier-Stokes Equations, preprint. [16] O.P. Manley, Y.M. Treve, Minimum number of modes in approximate solutions to equations of hydrodynamics, Phys. Lett. A, 82, (1981), 88-90. [17] C. Marchioro, An example of absence of turbulence for any Reynolds number, Comm. Math. Phys., 105, (1986), 99-106. [18] F. Takens, Detecting strange attractors in turbulence, Lecture Notes in Math., 898, Springer-Verlag, New York, (1981), 366-381. [19] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983. [20] Y.M. Treve, O.P. Manley, Energy conserving Galerkin approximations for 2D hydrodynamics and MHD Benard convection, Physica, D4, (1982), p.319. [21] Z. Shao, E.S. Titi, Parameterizing the global attractor of the Navier-Stokes equations by nodal values, preprint.

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