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An estimate of the hausdorff dimension of the attractor for homogeneous decaying turbulence
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
AN ESTIMATE OF THE HAUSDORFF DIMENSION OF THE ATFRACTOR FOR HOMOGENEOUS DECAYING TURBULENCE C. FOIAS Indiana University, Bloomington, IN 47405, USA
O.P. MANLEY U.S. Department ofEnergv. Washington, DC 20545, USA
and R. TEMAM Laboratoire d’Analyse Numérique, CNRS and UniversitéParis-Sud, F-91405 Orsay, France Received 15 October 1986; revised manuscript received 10 March 1987; accepted for publication 13 March 1987 Communicated by D.D. Hoim
Previously determined scalings of time and space yielded the representation of decaying turbulence asa statistically stationary 4, a process. Here we find that the dimension of the attractor in the transformed flow regime varies as (Reynolds number)” classical result. Furthermore, we use an argument which establishes a heuristic connection between the deterministic possibly chaotic solutions of the Navier—Stokes equation for decaying turbulence, and a particularfamily of statistical solutions of that equation.
In previous papers [1—3] it was shown how the Kaplan—Yorke formulas [4,5] may be used to establish a bound on the dimension of the attractor for the Navier—Stokes equations. That attractor characterizes the time asymptotic behavior of the deterministic solutions of the Navier—Stokes equations. Parallel work on the statistical description of decaying homogeneous turbulence [6—81has revealed the existence of a transformation of the Navier—Stokes equations to a space—time reference frame in which the family of statistical solutions are stationary with respect to the transformed time. In effect, under this transformation the scales oftime and space are allowed to change in time so as to compensate for the decay, thus preserving the turbulence for all (scaled) time. Hence we see that with respect to that scaled time there is a permanent asymptotic regime in which it is legitimate to study what is the dimension of the attractor, if any, in which the family of statistical solutions resides. In this note we combine the results of the two approaches to the study of turbulent (large Reynolds number B) flows, and estimate an upper bound on the dimension of the attractor for decaying homogeneous turbulence. We show that it is precisely that estimated by intuitive, physical arguments [9], for the number of modes needed to describe such a flow, viz. R We emphasize here that such an attractor does not exist in real space—time, because evidently one cannot speak of non-trivial time-asymptotic behavior of decaying turbulence. Nevertheless, the existence of such an attractor is feasible and the concept is useful in the transformed space—time. We begin by considering the Navier—Stokes equations for the flow field v: ‘~
~+y.Vv=_Vp+VV2v,
140
Vv=0,
~
(1)
0375-960 11871$ 03.50 © Elsevier Science Publishers BY. (North-Holland Physics Publishing Division)
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
where v is the kinematic viscosity, p is the pressure, and the density has been assumed to be constant and equal to unity. It is convenient to change variables in (1) so that in the transformed frame of reference the statistics is independent of the transformed time. Thus on setting t= ‘r[vy2/~(t)] 1/2 v, = u.[VE(t)1y2] 1/4 (2) x, =y1[y2v3k(t)] ~ the evolution of homogeneous decaying turbulence in the transformed space—time is given by ,
,
~=~(u+y.Vu)_u.Vu+V2u_Vq,
,
V u=0,
(3)
where s= —ln(l r), ~(t) = ~~(l + ~/2t/yv 1/2)_2, ~(t) is the mean energy dissipation, and y =\/i~~. Further detailed discussion of (3) may be found in ref. [81. Note however the presence of two additional linear terms in (3) a modification of the Navier—Stokes equations introduced by the change of variables (2). They act as a forcing function; hence, unlike the case of the unforced Navier—Stokes equation (1), it is possible for some of the Lyapunov exponents associated to (3) to. be positive. In the transformed frame, length is measured in units of the Taylor microlength, and energy density is equal to y2, as is the mean dissipation. We remark that the time evolution of homogeneous decaying turbulence as given by the appropriate statistical solutions of (1) can be obtained by applying the inverse of the transformation (2) to a stationary homogeneous statistical solution, ~u,of (3). The linearized form of (3) around some solution u ( I), i.e. the equation of variation, is —
—
—uVz—z.Vu+ ~(z+yVz)+vV2z—
~‘~z~A[u(x,t)Iz,
V~
(4)
In what follows we do not need the explicit expression for the functional derivative of the pressure gradient. Eq. (4) helps us estimate the Hausdorff dimension dH(X) of the attractor X of eq. (4) supplemented with some suitable boundary conditions on a convenient volume Q to be specified later on. Given the Lyapunov exponents A,,A 2,... ordered in a decreasing sequence, an estimate of that dimension is [1,2] (5)
~
where N is an integer such that for a well-chosen initial flow velocity field u ( x,t = 0) = u0(x)
(~~ N+ I
(6)
We show now how with the use of (4) an estimate of this sum, and hence of the dimension of the attractor of (3) may be related to the statistical properties of decaying turbulence. Thus the sum of the first N+ 1 Lyapunov exponents is /N+ \ 1 ~N±, (\~)~i)(uo)=lim-J ~ (A[u(t)]ØJ(’r),Øf(’r))Qd’r, (7) i=1
0
where Ø~(t)is the orthonormal basis of the space spanned by w1(t), w2(t),...,wN+I(t), the w~being a suitably chosen set of the independent solutions of (4) in the domain defined by Q. Here (a,b)o=~J (ab) dx, i.e. an average over a volume
Q,
taken to be a cube of as yet unspecified linear extent; çb~(t) depends on time 141
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
as well as on initial conditions. Clearly it is impossible to get an explicit expression for the integrand in (7). Fortunately, as can be shown [1], we can obtain a useful result by weakening (7) to an inequality: fN+l \ N+1 ~ A, )(uo)~lim sup ~ (A[u(i)]Oj,01)0d’r, (7’)
—J
\
/
/=1
~
~
I
where now {Ø~}~1+lare all possible systems oforthonormal smooth solenoidal vector fields such that the surface integral
J (~)
dS=0,
with ÔQ the surface of Q, and ô/ôn signifying the derivative normal to that surface. Physically the need for this boundary condition on Ø~ is implied by mass conservation. The supremum is taken over all sets {Ø~}~’. As will become apparent, no further conditions on the Ø~are needed to complete the argument. Now, the integrand of (7’) depends only on the initial conditions, u0. It is known that in a similar case, namely that of Navier—Stokes equations with time independent forces, the time average analogous to (7’) has an ergodic property [10]. We conjecture that the same is true in the present instance with a feedback like force. Namely, we believe that associated with some realistic initial conditions, u0, there exists at least one invariant measure (or probability distribution) ~ such that l~ N+I f N+I lim_J sup ~ (A[u(t)]Oj,0j)Qth=J sup ~ (A(u)Of,Of)Qd~,(u), ~
0
~
I
(01)
(8)
I
where, technically speaking the limit is taken in the generalized sense of ref. [10]. Evidently, given that the are solenoidal vectors, the inner product of 0 with the variation ofthe pressure gradient vanishes identically. Associated with (3) there is a stationary homogeneous statistical solution u. We conjecture that ~t may be used instead of y,,,~.Then it follows from (8) that N is such that N+I r N+1 sup ~ (A(u)OJ.,OJ)Qd~u(u) (9)
~ A1~J
i=I
{O~}
I
and N+I
N+I
N+I
N+I
1
1
I
I
~ (A(u)0~,0~)~~— ~ (VO~VØ~,)c~—~ ~ (0~,0~)~+ ~ (01, IVUI)Q,
the last term resulting from the use of the Schwarz inequality. Note that as in (6), (9) does not depend explicitly on the pressure gradient because of the divergence-free nature of Ø~. But (Ø~,0~)~= 1 for all j, thus we can apply the following generalization [121 of the Lieb—Thirring inequality [11]
J(
3 Q_S/
~
N-I-I
01)
dx~c
2”3(N+l),
N+I
(10)
0 ~ (VØ1,VØ~)0+c0Q
where c
0 (as c1, c2 in the sequel) is an absolute constant of order of unity. This yields 213(N+l)513—Q~’3(N+l)~ ~ (VØ N±I c~’Q and (seeref. [1]) 142
3, VØJ)Q
(lOa)
Volume 122, number 3,4
PHYSICS LETTERS A
N+I
8 June 1987
N+I
~I (0j,IVuI)Q~ ~ (V0f,V0j)Q+c,$IVuI5/2dx. Q
(lOb)
To proceed further we need an estimate ofthe moment of IV ui5”2 with respect to the probability distribution ~s.We get it by noting that under the scaling transformation (o~,~v)(x) =~v(Ax) the homogeneous statistical solutions ofthe Navier—Stokes equations (i.e. the probability distribution y~within the self-similar family determined by ~t) transform as ~ ~ [8]. Physically that signifies a transformation from a flow characterized by a mean dissipation ~in a fluid with a viscosity v, to another fluid flow with a mean dissipation = 1~and viscosity v’ = v~/A.With that in mind we find that the “5/2 moment” of V v scales as follows: ~‘
(v~I2)~
$~J IV
UI 5/2 dx d~0~~3(u) = () 3)514v5”4
$ ~J
IV ~15~2 dx d
4u”~(v)
where u = a~v.Hence on using the scaling transformation implied by the change of variables (2) there results: 514v514 Vv~5”2dxd~u’~(v)=y5”2 IVuI5”~dxd~t(u) ~—
J.~f
$~J
If intermittency effects are taken into account, the left-hand side may depend on the averaging volume Q, and therefore the ratio on the right hand side would depend on y. However on neglecting intermittency we can assume that this ratio is independent of y, i.e. IVuI5”2dxdi~(u)=c
5”2.
(lOc)
2y This result can be confirmed by applying the Holder inequality. Thus 3/4
Vv~5”2dxdu~’~<(J~JIVvI2 dxd/1~)
($~J
1/4
IVvI~dxd/1~)
=$ ~ J E(x)(x+y) dxdp~I~~,o(e/v)3”4.
When intermittency effects are ignored, the autocorrelation ofthe dissipation is simply ~2 for all y fore in the transformed frame we recover in part (1Oc). Hence on taking into account (1Oa )— (1 Oc) there follows
[9]. There-
N+ I
~ ~
[c,j’(N+ 1
)5”3Q2”3
512] (11) 1c2Qy For large Reynolds numbers we can ignore the middle term in the bracket. Now since the length of the coherent structures in the turbulent flow is of the order of1~,we take the cube Q to have an edge of that size. Then necessarily Q~ 3 y since in our notation all lengths are expressed in units of the Taylor microlength. We note now that the bracketed expression in (11) changes sign when (N+ 1)~”~ ~c 5”2l~,i.e. 0c1 c2y N+l~y312l~=y912~R914 (12) —
—(N+ l)Q2’3 —~c
.
~
.
143
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
Thus we find that the Hausdorif dimension of the attractor is no larger than the well-known estimate of the number of degrees of freedom for flows with large Reynolds numbers. We see that this estimate is somewhat smaller than that obtained previously [3],namely that dH(X) R We attribute that to the restriction of the flows to those for which the statistical properties are represented by the homogeneous measures ~~=j~~t) associated to ~z.The estimate (12) suggests that ~zis concentrated on a finite dimensional attractor with dimension R 9/4; probably this attractor is only a part of the largest attractor which may be of dimension R The preceding arguments establish a heuristic connection between the deterministic, but possibly chaotic solutions of Navier—Stokes equations governing decaying turbulence, and the stationary statistical solution p~ of (3). However we believe that ultimately our heuristic arguments can be confirmed in a rigorous manner. In summary, then, we have shown the plausible existence of an attractor for decaying turbulence. Further, we have found that its Hausdorff dimension does not exceed B consistent with the classical estimate of the number of degrees of freedom in turbulent flow at large Reynolds numbers.
~ ~.
~.
~
This research was partially supported by the US Department of Energy Grant DE-FGO2-86ER25020 and by the Institute for Nonlinear Science of University of California at San Diego. We thank Professor H. Abarbanel and the referee for useful comments on previous versions of this Note.
References [1] [2] [3] [4]
P. Constantin, C. Foias and R. Temam, Attractors representing turbulent flows, Mem. Am. Math. Soc. 53 (1985) No. 314. P. Constantin, C. Foias, O.P. Manley and R. Temam, CR. Acad. Sci. 297 (1984) 599. P. Constantin, C. Foias, O.P. Manleyand R. Temam, J. Fluid Mech. 150 (1985) 427. J. Kaplan and J.A. Yorke, Chaotic behavior of multidimensional difference equations, in: Lecture notes in mathematics, Vol. 730. Functional differential equations of approximation of fixed points, eds. H.O. Peitgen and H.O. Walther (Springer, Berlin, 1979). [5] P. Constantin and C. Foias, Comm. Pure AppI. Math. 38 (1985) 1. [6] C. Foias and R. Temam, Commun. Math. Phys. 90 (1983) 187. [7] C. Foias, O.P. Manley and R. Temam, Phys. Rev. Lett. 51(1983) 617. [8] C. Foias, OP. Manley and R. Temam, Self-similar invariant families ofturbulent flows, submitted to Phys. Fluids. [9] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Addison-Wesley, Reading, 1959). [101 C. Foias, Rend. Sem. Mat. Univ. Padova 49 (1973) 9. [11] E.H. Lieb, Commun. Math. Phys. 92 (1984) 473. [12] J.M. Ghidaglia, M. Marion and R. Temam, Private communication (1986).
144
PHYSICS LETTERS A
8 June 1987
AN ESTIMATE OF THE HAUSDORFF DIMENSION OF THE ATFRACTOR FOR HOMOGENEOUS DECAYING TURBULENCE C. FOIAS Indiana University, Bloomington, IN 47405, USA
O.P. MANLEY U.S. Department ofEnergv. Washington, DC 20545, USA
and R. TEMAM Laboratoire d’Analyse Numérique, CNRS and UniversitéParis-Sud, F-91405 Orsay, France Received 15 October 1986; revised manuscript received 10 March 1987; accepted for publication 13 March 1987 Communicated by D.D. Hoim
Previously determined scalings of time and space yielded the representation of decaying turbulence asa statistically stationary 4, a process. Here we find that the dimension of the attractor in the transformed flow regime varies as (Reynolds number)” classical result. Furthermore, we use an argument which establishes a heuristic connection between the deterministic possibly chaotic solutions of the Navier—Stokes equation for decaying turbulence, and a particularfamily of statistical solutions of that equation.
In previous papers [1—3] it was shown how the Kaplan—Yorke formulas [4,5] may be used to establish a bound on the dimension of the attractor for the Navier—Stokes equations. That attractor characterizes the time asymptotic behavior of the deterministic solutions of the Navier—Stokes equations. Parallel work on the statistical description of decaying homogeneous turbulence [6—81has revealed the existence of a transformation of the Navier—Stokes equations to a space—time reference frame in which the family of statistical solutions are stationary with respect to the transformed time. In effect, under this transformation the scales oftime and space are allowed to change in time so as to compensate for the decay, thus preserving the turbulence for all (scaled) time. Hence we see that with respect to that scaled time there is a permanent asymptotic regime in which it is legitimate to study what is the dimension of the attractor, if any, in which the family of statistical solutions resides. In this note we combine the results of the two approaches to the study of turbulent (large Reynolds number B) flows, and estimate an upper bound on the dimension of the attractor for decaying homogeneous turbulence. We show that it is precisely that estimated by intuitive, physical arguments [9], for the number of modes needed to describe such a flow, viz. R We emphasize here that such an attractor does not exist in real space—time, because evidently one cannot speak of non-trivial time-asymptotic behavior of decaying turbulence. Nevertheless, the existence of such an attractor is feasible and the concept is useful in the transformed space—time. We begin by considering the Navier—Stokes equations for the flow field v: ‘~
~+y.Vv=_Vp+VV2v,
140
Vv=0,
~
(1)
0375-960 11871$ 03.50 © Elsevier Science Publishers BY. (North-Holland Physics Publishing Division)
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
where v is the kinematic viscosity, p is the pressure, and the density has been assumed to be constant and equal to unity. It is convenient to change variables in (1) so that in the transformed frame of reference the statistics is independent of the transformed time. Thus on setting t= ‘r[vy2/~(t)] 1/2 v, = u.[VE(t)1y2] 1/4 (2) x, =y1[y2v3k(t)] ~ the evolution of homogeneous decaying turbulence in the transformed space—time is given by ,
,
~=~(u+y.Vu)_u.Vu+V2u_Vq,
,
V u=0,
(3)
where s= —ln(l r), ~(t) = ~~(l + ~/2t/yv 1/2)_2, ~(t) is the mean energy dissipation, and y =\/i~~. Further detailed discussion of (3) may be found in ref. [81. Note however the presence of two additional linear terms in (3) a modification of the Navier—Stokes equations introduced by the change of variables (2). They act as a forcing function; hence, unlike the case of the unforced Navier—Stokes equation (1), it is possible for some of the Lyapunov exponents associated to (3) to. be positive. In the transformed frame, length is measured in units of the Taylor microlength, and energy density is equal to y2, as is the mean dissipation. We remark that the time evolution of homogeneous decaying turbulence as given by the appropriate statistical solutions of (1) can be obtained by applying the inverse of the transformation (2) to a stationary homogeneous statistical solution, ~u,of (3). The linearized form of (3) around some solution u ( I), i.e. the equation of variation, is —
—
—uVz—z.Vu+ ~(z+yVz)+vV2z—
~‘~z~A[u(x,t)Iz,
V~
(4)
In what follows we do not need the explicit expression for the functional derivative of the pressure gradient. Eq. (4) helps us estimate the Hausdorff dimension dH(X) of the attractor X of eq. (4) supplemented with some suitable boundary conditions on a convenient volume Q to be specified later on. Given the Lyapunov exponents A,,A 2,... ordered in a decreasing sequence, an estimate of that dimension is [1,2] (5)
~
where N is an integer such that for a well-chosen initial flow velocity field u ( x,t = 0) = u0(x)
(~~ N+ I
(6)
We show now how with the use of (4) an estimate of this sum, and hence of the dimension of the attractor of (3) may be related to the statistical properties of decaying turbulence. Thus the sum of the first N+ 1 Lyapunov exponents is /N+ \ 1 ~N±, (\~)~i)(uo)=lim-J ~ (A[u(t)]ØJ(’r),Øf(’r))Qd’r, (7) i=1
0
where Ø~(t)is the orthonormal basis of the space spanned by w1(t), w2(t),...,wN+I(t), the w~being a suitably chosen set of the independent solutions of (4) in the domain defined by Q. Here (a,b)o=~J (ab) dx, i.e. an average over a volume
Q,
taken to be a cube of as yet unspecified linear extent; çb~(t) depends on time 141
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
as well as on initial conditions. Clearly it is impossible to get an explicit expression for the integrand in (7). Fortunately, as can be shown [1], we can obtain a useful result by weakening (7) to an inequality: fN+l \ N+1 ~ A, )(uo)~lim sup ~ (A[u(i)]Oj,01)0d’r, (7’)
—J
\
/
/=1
~
~
I
where now {Ø~}~1+lare all possible systems oforthonormal smooth solenoidal vector fields such that the surface integral
J (~)
dS=0,
with ÔQ the surface of Q, and ô/ôn signifying the derivative normal to that surface. Physically the need for this boundary condition on Ø~ is implied by mass conservation. The supremum is taken over all sets {Ø~}~’. As will become apparent, no further conditions on the Ø~are needed to complete the argument. Now, the integrand of (7’) depends only on the initial conditions, u0. It is known that in a similar case, namely that of Navier—Stokes equations with time independent forces, the time average analogous to (7’) has an ergodic property [10]. We conjecture that the same is true in the present instance with a feedback like force. Namely, we believe that associated with some realistic initial conditions, u0, there exists at least one invariant measure (or probability distribution) ~ such that l~ N+I f N+I lim_J sup ~ (A[u(t)]Oj,0j)Qth=J sup ~ (A(u)Of,Of)Qd~,(u), ~
0
~
I
(01)
(8)
I
where, technically speaking the limit is taken in the generalized sense of ref. [10]. Evidently, given that the are solenoidal vectors, the inner product of 0 with the variation ofthe pressure gradient vanishes identically. Associated with (3) there is a stationary homogeneous statistical solution u. We conjecture that ~t may be used instead of y,,,~.Then it follows from (8) that N is such that N+I r N+1 sup ~ (A(u)OJ.,OJ)Qd~u(u) (9)
~ A1~J
i=I
{O~}
I
and N+I
N+I
N+I
N+I
1
1
I
I
~ (A(u)0~,0~)~~— ~ (VO~VØ~,)c~—~ ~ (0~,0~)~+ ~ (01, IVUI)Q,
the last term resulting from the use of the Schwarz inequality. Note that as in (6), (9) does not depend explicitly on the pressure gradient because of the divergence-free nature of Ø~. But (Ø~,0~)~= 1 for all j, thus we can apply the following generalization [121 of the Lieb—Thirring inequality [11]
J(
3 Q_S/
~
N-I-I
01)
dx~c
2”3(N+l),
N+I
(10)
0 ~ (VØ1,VØ~)0+c0Q
where c
0 (as c1, c2 in the sequel) is an absolute constant of order of unity. This yields 213(N+l)513—Q~’3(N+l)~ ~ (VØ N±I c~’Q and (seeref. [1]) 142
3, VØJ)Q
(lOa)
Volume 122, number 3,4
PHYSICS LETTERS A
N+I
8 June 1987
N+I
~I (0j,IVuI)Q~ ~ (V0f,V0j)Q+c,$IVuI5/2dx. Q
(lOb)
To proceed further we need an estimate ofthe moment of IV ui5”2 with respect to the probability distribution ~s.We get it by noting that under the scaling transformation (o~,~v)(x) =~v(Ax) the homogeneous statistical solutions ofthe Navier—Stokes equations (i.e. the probability distribution y~within the self-similar family determined by ~t) transform as ~ ~ [8]. Physically that signifies a transformation from a flow characterized by a mean dissipation ~in a fluid with a viscosity v, to another fluid flow with a mean dissipation = 1~and viscosity v’ = v~/A.With that in mind we find that the “5/2 moment” of V v scales as follows: ~‘
(v~I2)~
$~J IV
UI 5/2 dx d~0~~3(u) = () 3)514v5”4
$ ~J
IV ~15~2 dx d
4u”~(v)
where u = a~v.Hence on using the scaling transformation implied by the change of variables (2) there results: 514v514 Vv~5”2dxd~u’~(v)=y5”2 IVuI5”~dxd~t(u) ~—
J.~f
$~J
If intermittency effects are taken into account, the left-hand side may depend on the averaging volume Q, and therefore the ratio on the right hand side would depend on y. However on neglecting intermittency we can assume that this ratio is independent of y, i.e. IVuI5”2dxdi~(u)=c
5”2.
(lOc)
2y This result can be confirmed by applying the Holder inequality. Thus 3/4
Vv~5”2dxdu~’~<(J~JIVvI2 dxd/1~)
($~J
1/4
IVvI~dxd/1~)
=$ ~ J E(x)(x+y) dxdp~I~~,o(e/v)3”4.
When intermittency effects are ignored, the autocorrelation ofthe dissipation is simply ~2 for all y fore in the transformed frame we recover in part (1Oc). Hence on taking into account (1Oa )— (1 Oc) there follows
[9]. There-
N+ I
~ ~
[c,j’(N+ 1
)5”3Q2”3
512] (11) 1c2Qy For large Reynolds numbers we can ignore the middle term in the bracket. Now since the length of the coherent structures in the turbulent flow is of the order of1~,we take the cube Q to have an edge of that size. Then necessarily Q~ 3 y since in our notation all lengths are expressed in units of the Taylor microlength. We note now that the bracketed expression in (11) changes sign when (N+ 1)~”~ ~c 5”2l~,i.e. 0c1 c2y N+l~y312l~=y912~R914 (12) —
—(N+ l)Q2’3 —~c
.
~
.
143
Volume 122, number 3,4
PHYSICS LETTERS A
8 June 1987
Thus we find that the Hausdorif dimension of the attractor is no larger than the well-known estimate of the number of degrees of freedom for flows with large Reynolds numbers. We see that this estimate is somewhat smaller than that obtained previously [3],namely that dH(X) R We attribute that to the restriction of the flows to those for which the statistical properties are represented by the homogeneous measures ~~=j~~t) associated to ~z.The estimate (12) suggests that ~zis concentrated on a finite dimensional attractor with dimension R 9/4; probably this attractor is only a part of the largest attractor which may be of dimension R The preceding arguments establish a heuristic connection between the deterministic, but possibly chaotic solutions of Navier—Stokes equations governing decaying turbulence, and the stationary statistical solution p~ of (3). However we believe that ultimately our heuristic arguments can be confirmed in a rigorous manner. In summary, then, we have shown the plausible existence of an attractor for decaying turbulence. Further, we have found that its Hausdorff dimension does not exceed B consistent with the classical estimate of the number of degrees of freedom in turbulent flow at large Reynolds numbers.
~ ~.
~.
~
This research was partially supported by the US Department of Energy Grant DE-FGO2-86ER25020 and by the Institute for Nonlinear Science of University of California at San Diego. We thank Professor H. Abarbanel and the referee for useful comments on previous versions of this Note.
References [1] [2] [3] [4]
P. Constantin, C. Foias and R. Temam, Attractors representing turbulent flows, Mem. Am. Math. Soc. 53 (1985) No. 314. P. Constantin, C. Foias, O.P. Manley and R. Temam, CR. Acad. Sci. 297 (1984) 599. P. Constantin, C. Foias, O.P. Manleyand R. Temam, J. Fluid Mech. 150 (1985) 427. J. Kaplan and J.A. Yorke, Chaotic behavior of multidimensional difference equations, in: Lecture notes in mathematics, Vol. 730. Functional differential equations of approximation of fixed points, eds. H.O. Peitgen and H.O. Walther (Springer, Berlin, 1979). [5] P. Constantin and C. Foias, Comm. Pure AppI. Math. 38 (1985) 1. [6] C. Foias and R. Temam, Commun. Math. Phys. 90 (1983) 187. [7] C. Foias, O.P. Manley and R. Temam, Phys. Rev. Lett. 51(1983) 617. [8] C. Foias, OP. Manley and R. Temam, Self-similar invariant families ofturbulent flows, submitted to Phys. Fluids. [9] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Addison-Wesley, Reading, 1959). [101 C. Foias, Rend. Sem. Mat. Univ. Padova 49 (1973) 9. [11] E.H. Lieb, Commun. Math. Phys. 92 (1984) 473. [12] J.M. Ghidaglia, M. Marion and R. Temam, Private communication (1986).
144