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M
M PP EE JJ
Mathematical Physics Electronic Journal
ISSN 1086-6655
Volume 12, 2006
Paper 2
Received: Nov 15, 2005, Revised: Mar 19, 2006, Accepted: Apr 3, 2006
Editor: R. de la Llave
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
OF THE NAVIER-STOKES EQUATIONS
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
Abstract. Two constructions of homogeneous and isotropic statistical solutions of the
3D Navier-Stokes system are presented. First, homogeneous and isotropic probability
measures supported by weak solutions of the Navier-Stokes system are produced by averaging over rotations the known homogeneous probability measures, supported by such
solutions, of [VF1], [VF2]. It is then shown how to approximate (in the sense of convergence of characteristic functionals) any isotropic measure on a certain space of vector
fields by isotropic measures supported by periodic vector fields and their rotations. This is
achieved without loss of uniqueness for the Galerkin system, allowing for the Galerkin approximations of homogeneous statistical Navier-Stokes solutions to be adopted to isotropic
approximations. The construction of homogeneous measures in [VF1], [VF2] then applies
to produce homogeneous and isotropic probability measures, supported by weak solutions
of the Navier-Stokes equations. In both constructions, the restriction of the measures at
t = 0 is well defined and coincides with the initial measure.
The second author was partially supported by RFBI grant 04-01-00066.
1
2
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
1. Introduction
The Kolmogorov theory of turbulence describes the behavior of homogeneous and isotropic
fluid flows, i.e. flows with statistical properties independent of translations, reflections, and
rotations in 3D-space. The mathematical equivalent of such flows are translation, reflection,
and rotation invariant probability measures P supported by Navier-Stokes solutions. Given an
appropriate initial measure µ on the space of initial conditions and any finite time interval [0, T ],
the measure P should yield µ at t = 0 is some sense. It should also yield measures µ t for each
time t in [0, T ], each invariant under space translations, reflections and rotations, that represent
the flow of µ under the Navier-Stokes equations as in [H].
The existence of such measures P that satisfy all the assumptions above for translations in
space, called homogeneous, was proved by Vishik and Fursikov in [VF1] and is included in
detail in [VF2]. See also [FT]. This note adopts the construction from [VF1], [VF2] to produce
measures P that, in addition to being invariant under translations, are also invariant under
rotations and reflections. Such measures will be called homogeneous and isotropic. (To
emphasize the new elements of the construction, the convention here will be that “isotropic”
does not imply “homogeneous.”) One of the main results of this note is:
Theorem 1.1. Given µ
b homogeneous and isotropic measure on a space H 0 (r) of vector fields on
R3 with the finite energy density, there exists measure Pb on L2 (0, T, H0 (r)), homogeneous and
isotropic with respect to the space variables, supported by weak solutions of the Navier-Stokes
equations, and with finite energy density that satisfies the standard energy inequality. On the
support of Pb right limits with respect to time are well-defined in an appropriate norm and the
right limit at t = 0 yields µ
b.
(For the definitions of all notions used in the formulation of Theorem 1.1, see sections 2 and 3
below.)
To prove Theorem 1.1 the homogeneous statistical solution P constructed in [VF1], [VF2]
with initial measure µ
b is averaged over all rotations and reflections. The resulting measure
Pb satisfies all conditions of Theorem 1.1 above and is therefore the desired homogeneous and
isotropic statistical solution. This plan is realized in sections 2 and 3.
It has to be emphasized, however, that existence theorems obtained via convergent sequences
of “simpler” approximations of the constructed solution are as a rule much more useful in
mathematical physics than the so called “pure existence theorems.” For this reason, sections 4,
5, and 6 below are devoted to constructing such approximations of isotropic statistical solutions.
The construction of homogeneous statistical solutions in [VF1], [VF2] is based on Galerkin
approximations of measures that are supported by divergence free periodic vector fields with
trigonometric polynomials as components. The main difficulty in extending this construction to
isotropic measures is that the space of such vector fields, whereas invariant under translations,
is not invariant under rotations.
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
3
The construction in sections 4, 5, and 6 is based on the observation that the space of such
vector fields AND all their rotations and reflections should suffice for invariance under rotations
and reflections.
It is then necessary to construct Galerkin approximations of isotropic measures on this class
of vector fields. In this sense, the crux of the matter is Sections 4.2, 4.3, and 6.1. Note that
the constructions of sections 4, 5, and 6 not only offer approximations of the isotropic statistical
solution constructed in section 3 but they also allow for a construction of isotropic statistical
solutions that is formally independent on the results of section 3.
This paper considers the case of 3D Navier-Stokes equations, although the arguments here
are applicable in 2D case as well.
2. Isotropic measures
2.1. Definitions. Non-trivial measures invariant under translations exist on weighted Sobolev
spaces of vector fields defined over R 3 , p. 208, [VF2]:
Definition 2.1. For k non negative integer and r < −3/2, define H k (r) to be the space of
solenoidal vector fields
u(x) = (u1 (x), u2 (x), u3 (x)),
(2.1)
divu =
x = (x1 , x2 , x3 ) ∈ R3 ,
3
X
∂uj
j=1
∂xj
= 0,
with finite (k, r)-norm:
(2.2)
kuk2k,r =
Z
R3
¡
¯
¯2
X ¯¯ ∂ |α| u(x) ¯¯
¢
r
1 + |x|2
¯ α1 α2 α3 ¯ dx,
¯ ∂x1 ∂x2 ∂x3 ¯
|α|≤k
3
r2 =
R
< u, ∂φ >2 + < u, ∆φ >2 +
∂t
3
X
< uj u,
j=1
∂φ
>2 dt = 0,
∂xj
)
¡
¢
for all φ ∈ C0∞ (0, T ) × R3 ∩ C((0, T ); H0 (r)) ,
R3
u(x) · v(x) dx.
To define restrictions at any time t ∈ [0, T ] one works with the following norms: For B N =
{|x| < N } the ball of radius N in R3 , and for k.ks the standard Sobolev norm in W s,2 (R3 ) =
L2s (R3 ), define the dual norm
kv|BN k−s =
(3.2)
sup
w∈C0∞ (BN )
< v, w >2
.
kwks
Using this and following [VF2], p.245, define
(3.3)
kukBV −s = kukL2 (0,T ;H0 (r)) +
∞
X
N =1
1
22N C(N )
|u|N .
Here C(N ) are constants from (5.3) and (5.4) (see below) and |u| N is defined as follows:
(3.4)
|u|N = vrai sup ku(t, ·)|BN k−s + sup
t∈[0,T ]
l
X
{tj } j=1
vrai
sup
t,τ ∈[tj−1 ,tj )
k(u(t, ·) − u(τ, ·))|BN k−s ,
8
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
where sup{tj } is the supremum over all partitions t 0 < · · · < tl , l ∈ N of the segment [0, T ].
Define
BV −s = {u ∈ L2 (0, T ; H0 (r)) : kukBV −s < ∞}.
(3.5)
The merit of the BV −s norm is that for u in BV −s the limits
γt0 (u) := lim u(t, .)
(3.6)
t→t+
0
exist for all t0 ∈ [0, T ], if taken with respect to the norm
!1/2
à ∞
X
1
2
(3.7)
ku(t, .)kΦ−s =
ku(t, .)|BN k−s
,
22N C(N )
N =1
see [VF2], Chapter VII, Lemma 8.2.
Note also that for a homogeneous measure µ the pointwise averages
Z
Z
2
(3.8)
|u| (x) µ(du),
|∇u|2 (x) µ(du)
can be defined by the equalities
Z Z
Z
Z
2
2
(3.9)
|u(x)| φ(x) dx µ(du) = |u(x)| µ(du) φ(x) dx
(3.10)
Z Z
2
|∇u(x)| φ(x) dx µ(du) =
Z
2
|∇u(x)| µ(du)
Z
∀ φ ∈ L1 (R3 )
φ(x) dx
and they are independent of x ∈ R3 , see Chapter VII, section 1 of [VF2]. The expressions in
(3.8) are well defined since the left hand sides in (3.9), (3.10) are finite for each φ ∈ C 0∞ (R3 ).
The first expression in (3.8) is the energy density and the second one is the density of
the energy dissipation.
Since the translation operator Th (along x) is well defined on the space L 2 (0, T ; H0 (r)) of
vector fields u(t, x) dependent not only on x but on t as well, one can introduce the notion of
homogeneity in x:
Definition 3.2. A measure P (A), A ∈ B(L2 (0, T ; H0 (r))) is called homogeneous in x if for
each h ∈ R3 :
(3.11)
Th∗ P
= P ⇐⇒
Z
f (u)
L2 (0,T ;H0 (r))
Th∗ P (du)
=
Z
f (u) P (du),
L2 (0,T ;H0 (r))
for any P -integrable f on H 0 (r).
The following definition summarizes the properties of homogeneous statistical solutions of the
Navier-Stokes equations as they were produced in Chapter VII of [VF2]:
Definition 3.3. Given homogeneous probability measure µ on B(H 0 (r)) possessing finite energy
density, a homogeneous statistical solution of the Navier-Stokes equations with initial
condition µ is a probability measure P on B(L 2 (0, T ; H0 (r))) such that:
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
(1) P is homogeneous in x.
c ) = 1, where W
c = L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S , s >
(2) P (W
9
11
2 .
(3) For all A ∈ B(H0 (r)),
P (γ0−1 A) = µ(A),
(3.12)
where
c : γ0 u ∈ A}.
γ0−1 A = {u ∈ W
(4) For each t in [0, T ],
¶
Z
Z µ
Z t
2
2
|∇u| (τ, x) dτ P (du) ≤ C |u(x)|2 µ(du),
(3.13)
|u(t, x)| +
0
where the expression in the left hand side of (3.13) is defined similarly to (3.8).
The main result of [VF2], Chapter VII, then reads as follows:
Theorem 3.4. Given µ homogeneous measure on H 0 (r) with finite energy density,
Z
|u|2 (x) µ(du) < ∞,
(3.14)
H0 (r)
there exists homogeneous statistical solution of the Navier-Stokes equations P with initial condition µ.
Remark 3.5. The definition above is a rephrasing of Definition 11.1 of [VF2], with one minor
change: It asks that P is supported by L 2 (0, T ; H1 (r))∩BV −s ∩GN S rather than some subset of it.
Since [VF2] produces some subset supporting a homogeneous statistical solution, it automatically
produces a homogeneous statistical solution according to the definition here.
Remark 3.6. In addition, the family of homogeneous measures µ t := P ◦ γt−1 on H0 (r) satisfies
the Hopf equation, [VF2], Chapter VIII.
Define now isotropic and homogeneous statistical solutions. First define isotropic in x
measures P on B(L2 (0, T ; H0 (r))). (This can be done since for each ω ∈ O(3) operator
Rω u(t, x) ≡ ωu(t, ω −1 x) is well defined on L2 (0, T ; H0 (r)).)
Definition 3.7. A measure P (A), A ∈ B(L 2 (0, T ; H0 (r))) is called isotropic in x if for each
ω ∈ O(3):
(3.15)
Rω∗ P
= P ⇐⇒
Z
f (u)
L2 (0,T ;H0 (r))
Rω∗ P (du)
=
Z
f (u) P (du),
L2 (0,T ;H0 (r))
for any P -integrable f on L2 (0, T ; H0 (r)).
For the definition of homogeneous and isotropic statistical solutions one has only to add in
Definition 3.3 the property of rotation and reflection invariance:
Definition 3.8. Given homogeneous and isotropic probability measure µ
b on B(H 0 (r)), a homogeneous and isotropic statistical solution of the Navier-Stokes equations with initial
condition µ
b is a probability measure Pb on B(L2 (0, T ; H0 (r))) such that:
(1) Pb is homogeneous and isotropic in x.
10
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
c ) = 1, where W
c = L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S , s >
(2) Pb(W
(3) Pb(γ −1 A) = µ
b(A) for every A ∈ B(H 0 (r)).
11
2 ,.
0
(4) For each t in [0, T ],
¶
µ
Z
Z
Z t
2
2
b
|∇u| (τ, x) dτ P (du) ≤ C
(3.16)
|u(t, x)| +
L2 (0,T ;H0 (r))
0
H0 (r)
|u(x)|2 µ
b(du).
3.2. Construction of homogeneous and isotropic statistical solutions. To construct
homogeneous and isotropic statistical solutions several preliminary assertions need to be proved
first. For these, use the definition of the norm k · k s of Sobolev space W s,2 (R3 ) through Fourier
transform:
(3.17)
kφk2s
=
Z
R3
b 2 (ξ) dξ,
(1 + |ξ| ) |φ|
2 s
where
b =
φ(ξ)
1
(2π)3/2
Z
e−ix·ξ φ(x) dx.
R3
Lemma 3.9. For any matrix ω ∈ O(3) the following equalities hold:
kRω φks = kφks ,
kRω φkBV −s = kφkBV −s ,
(3.18)
kRω φkΦ−s = kφkΦ−s .
Proof. By the definition of Fourier transform and by virtue of (2.16)
Z
1
−iω −1 x·ω −1 ξ
d
R
φ(ξ)
=
e
ωφ(ω −1 x) dx
ω
(2π)3/2 R3
Z
(3.19)
1
−1
b
=
e−iy·ω ξ ωφ(y) dy = Rω φ(ξ).
3/2
(2π)
R3
By (3.17), (3.19), and (2.16)
kRω φk2s
=
Z
R3
(3.20)
=
Z
R3
b −1 ξ)|2 dξ
(1 + |ω −1 ξ|2 )s |ω φ(ω
2
b
(1 + |η|2 )s |ω φ(η)|
dη = kφk2s ,
which proves the first equality in (3.18).
Equalities (3.20), (2.16), and (3.2) give:
kRω v|BN k−s =
(3.21)
=
< Rω v, φ >2
kφks
φ∈C0∞ (BN )
sup
< v, (Rω )−1 φ >2
= kv|BN k−s ,
k(Rω )−1 φks
φ∈C0∞ (BN )
sup
since Rω : C0∞ (BN ) → C0∞ (BN ) is an isomorphism.
Equality (3.21) and definition (3.4) of | · | N imply the equality
(3.22)
|Rω φ|N = |φ|N .
This identity, (3.21), and the definitions (3.3), (3.7) of the norms k·k BV −s , k·kΦ−s , imply directly
the second and third equalities in (3.18).
¤
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
11
Lemma 3.10. For every ω ∈ O(3) and for each homogeneous measure µ(A), A ∈ B(H 0 (r))
Z
Z
2
∗
|u(x)| Rω µ(du) = |u(x)|2 µ(du),
Z
Z
(3.23)
2
∗
|∇u(x)| Rω µ(du) = |∇u(x)|2 µ(du).
Proof. By (2.6), (2.7), (3.9), and (2.16), for each ω ∈ O(3)
Z
Z Z
Z
2
∗
|ωu(ω −1 x)|2 φ(ωω −1 x) dx dµ(u)
|u(x)| Rω µ(du) φ(x) dx =
R3
Z Z
|u(y)|2 φ(ωy) dy dµ(u)
=
3
Z R
Z
(3.24)
2
φ(ωy) dy
= |u(y)| dµ(u)
R3
Z
Z
φ(x) dx,
= |u(x)|2 dµ(u)
R3
which proves the first equality of (3.23). If y = ω −1 x, i.e. yl = ωkl xk , then by (2.16), (2.18),
obtain
Z X
j
Z X
∂u(ω −1 x) 2
|
| φ(ωω −1 x) dx
∂xj
j
Z X
∂up (y)
∂up (y)
ωjl
=
ωjm
φ(ωy) dy
∂yl
∂ym
j,p
Z
= |∇y u(y)|2 φ(ωy) dy.
∂u(ω −1 x) 2
|ω
| φ(x) dx =
∂xj
(3.25)
Using these identities, the second equality of (3.23) can be proved similarly to (3.24).
¤
Recall now that the set GN S has been introduced in Definition 3.1.
Lemma 3.11. For each ω ∈ O(3) the equality R ω GN S = GN S holds.
Proof. Prove first that if u satisfies (3.1) then R ω u satisfies (3.1) as well, for every ω ∈ O(3).
For this note that, by Lemma 2.3,
(3.26)
u ∈ L2 (0, T ; H0 (r)) ⇒ Rω u ∈ L2 (0, T ; H0 (r)),
φ ∈ C((0, T ); H0 (r)) ∩ C0∞ ((0, T ) × R3 ) ⇒ Rω φ ∈ C((0, T ); H0 (r)) ∩ C0∞ ((0, T ) × R3 ).
So let u satisfy (3.1). Then (2.16) and the well-known fact that Laplace operator is invariant
under orthogonal change of variables yield:
(3.27)
< Rω u,
∂(Rω−1 )φ
∂φ
+ ∆φ >=< u,
+ ∆(Rω−1 )φ > .
∂t
∂t
12
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
If y = ω −1 x = ω ∗ x, i.e. yl = ωkl xk , then taking into account (2.18) and (2.16) calculate:
Z
Z
∂φl (x)
∂φ
dx = ωjk uk (ω −1 x)ωlm um (ω −1 x)
dx
(Rω u)j Rω u ·
∂xj
∂xj
Z
∂φl (ωy)
(3.28)
= ωjk uk (y)ωlm um (y)ωjp
dy
∂yp
Z
∂ωlm φl (ωy)
= uk (y)um (y)
dy.
∂yk
Then
(3.29)
3
X
j=1
3
X
∂(Rω−1 )φ
∂φ
< uj u,
>2 =
>2 .
< (Rω u)j Rω u,
∂xj
∂xj
j=1
Adding (3.27) and (3.29), integrating the resulting equality with respect to t over [0, T ], and
taking into account that u satisfies (3.1), shows that R ω u satisfies (3.1) as well.
¤
Lemma 3.12. γ0 commutes with Rω for any rotation ω.
Proof. By Lemma 3.9, ku(t, .)kΦ−s = kRω u(t, .)kΦ−s . Therefore if limt→0+ u(t, .) = γ0 (u), then
limt→0+ Rω u(t, .) = Rω γ0 (u), and limt→0+ Rω u(t, .) = γ0 (Rω u), i.e. γ0 (Rω u) = Rω γ0 (u).
Recall that for each B ∈ B(H 0 (r))
(3.30)
c : γ0 u ∈ B},
γ0−1 B = {u(t, x) ∈ W
c is the set of Definition 3.3 or (equivalently) of Definition 3.8.
where W
Lemma 3.13. For B ∈ B(H 0 (r)),
(3.31)
Rω γ0−1 (B) = γ0−1 (Rω B),
∀ ω ∈ O(3).
Proof. Using (2.18), one can prove similarly to Lemma 2.3 that
(3.32)
Rω H1 (r) = H1 (r)
∀ ω ∈ O(3),
for H1 (r) as in Definition 2.1. This, together with lemmas 3.9 and 3.11, imply that
(3.33)
Therefore
c=W
c
Rω W
∀ ω ∈ O(3).
u ∈ Rω γ0−1 (B) ⇒ u = Rω v, v ∈ γ0−1 (B)
⇒ γ0 u = γ0 (Rω v), v ∈ γ0−1 (B)
(3.34)
⇒ γ0 u = Rω γ0 (v), v ∈ γ0−1 (B), by Lemma 3.12,
⇒ γ0 u = Rω b, b ∈ B
⇒ u = γ0−1 Rω b, b ∈ B
⇒ u ∈ γ0−1 (Rω B).
¤
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
13
Conversely,
u ∈ γ0−1 (Rω B) ⇒ γ0 (u) = Rω b, b ∈ B
⇒ Rω−1 γ0 (u) = b, b ∈ B
⇒ γ0 (Rω−1 u) = b, b ∈ B, by Lemma 3.12,
(3.35)
⇒ Rω−1 u = γ0−1 (b), b ∈ B
⇒ Rω−1 u ∈ γ0−1 (B),
⇒ u ∈ Rω γ0−1 (B).
¤
Theorem 3.14. Given µ
b homogeneous and isotropic measure on H 0 (r) with finite energy density,
Z
(3.36)
H0 (r)
|u|2 (x) µ
b(du) < ∞,
there exists homogeneous and isotropic statistical solution Pb of the Navier-Stokes equations with
initial condition µ
b.
Proof. Ignoring for the moment that µ
b is also isotropic, let P be the homogeneous statistical
c =
solution with initial condition the homogeneous µ
b guaranteed by Theorem 3.4. The set W
L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S is invariant under rotations by (3.33). Applying the analogue of
operation (2.22) on the homogeneous measure P obtain:
(3.37)
Pb (A) =
Z
O(3)
Rω∗ P (A) dω =
Z
O(3)
P (Rω−1 A) dω,
c ). Repeating the proof of Proposition 2.7 for the measure Pb shows that Pb is
for any A ∈ (B)(W
c ) = 1 by Theorem 3.4, equality (3.33) implies that
homogeneous and isotropic in x. Since P (W
c ) = P (R−1 W
c ) = P (W
c ) = 1 for each ω ∈ O(3). Hence, Pb(W
c ) = 1 by definition (3.37).
Rω∗ P (W
ω
That Pb has initial condition µ
b follows from
Pb(γ0−1 B) =
=
(3.38)
Z
O(3)
Z
O(3)
=
Z
O(3)
for any B ∈ B(H0 (r)).
P (Rω−1 γ0−1 (B)) dω
P (γ0−1 (Rω−1 B)) dω, by Lemma 3.13
µ
b(Rω−1 B) dω, by (3.12)
=µ
b(B), since µ
b is also isotropic,
14
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
For the energy inequality, use (3.37), Lemma 3.10, and (3.13) to get:
¶
µ
Z
Z t
2
2
|∇u| (τ, x) dτ Pb(du)
|u(t, x)| +
L2 (0,T ;H0 (r))
=
(3.39)
=
Z
Z
O(3)
L2 (0,T ;H0 (r))
L2 (0,T ;H0 (r))
≤C
Z
H0 (r)
which proves (3.16).
0
Z
µ
µ
2
|u(t, x)| +
|u(t, x)|2 +
Z
Z
t
2
|∇u| (τ, x) dτ
0
t
|∇u|2 (τ, x) dτ
0
¶
¶
Rω ∗ P (du) dω
P (du)
|u(x)|2 µ
b(du),
¤
4. Galerkin approximation of isotropic statistical solutions
4.1. Isotropic measures on periodic vector fields. Let M l be as in [VF2]:
½ X
¾
ik·x
(4.1)
Ml =
ak e
: ak · k = 0, ak = a−k ∀ k ,
k∈ πl Z3 ,
|k|≤l
the finite-dimensional space of divergence-free, 3D, real, vector valued trigonometric polynomials
of degree l and period 2l. Then the inclusion
Ml ⊂ H0 (r)
(4.2)
holds for all l. [VF2], Appendix II, shows explicitly how, starting from any homogeneous probability measure on H0 (r), one can construct homogeneous probability measures µ l on H0 (r),
supported solely by Ml for each l, and approximating µ in the sense of characteristic functionals.
The trouble, of course, is that Ml is not invariant under rotations. The following definitions
address this point.
dl be the union of all rotations of elements of M l :
Definition 4.1. Let M
[
dl =
(4.3)
M
Rω M (l)
ω∈O(3)
in H0 (r).
dl the topology τ generated by sets of the form
Consider on M
(4.4)
{Rω m : ω ∈ ρ, m ∈ σ, where ρ ⊂ O(3), σ ⊂ Ml are open sets}.
Since O(3) and Ml are finite-dimensional sets, the topology τ coincides with the topology
dl ) is generated by
generated by the enveloping space H 0 (r). Therefore, the Borel σ-algebra B(M
sets of the form (4.4). Moreover, it is clear that
(4.5)
dl ) = B(H0 (r)) ∩ M
dl ≡ {A ∩ M
dl : A ∈ B(H0 (r))}
B(M
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
15
Note that for each fixed ω the elements of R ω Ml are of the form
X
(4.6)
bk eiωk·x , bk · ωk = 0 for all k.
k∈ πl Z3 ,
|k|≤l
Using definition (2.6) of the push forward measure, proceed to:
dl ) be the push-forward of the product of the Haar measure
Definition 4.2. Let µbl (A), A ∈ B(M
on O(3) and the measure µl on Ml via the map (ω, u) 7→ Rω u:
µbl (A) = (H × µl ){(ω, u) ∈ O(3) × Ml : Rω u ∈ A}.
(4.7)
As for any push-forward measure, by (2.7),
Z
(4.8)
for any µbl -integrable f .
cl
M
f (v) µbl (dv) =
Z
Ml
Z
f (Rω u) dωµl (du),
O(3)
dl ⊂ H0 (r), the domains of
Since µl is supported on Ml ⊂ H0 (r) and µbl is supported on M
dl , Ml in (4.8) can change to H 0 (r). Comparing then (4.7), (4.8) to the definitions
integration M
of averaging (2.22), (2.23) it follows that the measure µbl is the averaging of µl over O(3):
µbl (A) =
(4.9)
Z
O(3)
Rω∗ µl (A) dω
∀ A ∈ B(H0 (r))
Proposition 4.3. µbl is homogeneous and isotropic.
dl = M
dl and the invariance of the Haar measure, obtain for
Proof. Using the equality Rω−1 M
each µbl -integrable f :
Z
dl
M
f (w)Rω∗ 0 µbl (dw)
=
=
Z
cl
M
Z
Ml
(4.10)
=
Z
Ml
=
Z
dl
M
f (Rω0 v) µbl (dv)
Z
f (Rω0 Rω u) dω µl (du)
O(3)
Z
f (Rω u) dω µl (du)
O(3)
f (v) µbl (dv),
i.e. µ
b is invariant with respect of rotations: R ω∗ 0 µ
b=µ
b for each ω0 ∈ O(3).
16
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
dl = M
dl , T −1 Ml = Ml imply:
Similarly, the equalities Th−1 M
ωh
Z
Z
f (Th v) µbl (dv)
f (w)Th∗ µbl (dw) =
dl
M
=
=
(4.11)
=
=
=
Z
Z
Z
Z
Z
dl
M
Ml
Z
O(3)
O(3)
Ml
dl
M
f (Th Rω u) dω µl (du)
O(3)
Z
Ml
Z
Z
f (Rω Tω−1 h u) µl (du) dω
f (Rω u) µl (du) dω, by the homogeneity of µl ,
Ml
f (Rω u) dω µl (du)
O(3)
f (v) µbl (dv).
¤
4.2. Galerkin equations for Fourier coefficients on R ω Ml . Let H s (Πl ) be the space of
periodic vector fields
(4.12)
½
¾
X
X
ik·x
2
2 s
2
H (Πl ) = u(x) =
ak e , ak = (ak1 , ak2 , ak3 ), kuks =
(1 + |k| ) |ak | < ∞ .
s
k∈ πl Z3
k∈ πl Z3
Here
(4.13)
Πl = {x = (x1 , x2 , x3 ) : |xj | ≤ l, j = 1, 2, 3}
is the cube of periods for these vector fields.
On the space C 1 (0, T ; H 2 (Πl )) the Navier-Stokes system can be written in the form:
∂t u − ∆u + π(u, ∇)u = 0, div u=0,
(4.14)
where π : L2 (Πl ) → {u ∈ L2 (Πl ) : divu = 0} is the projection on solenoidal vector fields. It is
P
standard that substitution of the Fourier series u(x) = k ak eik·x into (4.14) yields the following
system for the Fourier coefficients a k (t):
∂t ak + |k|2 ak +
(4.15)
X
i((ak′ ·k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3
k∈
(ak′ · k′′ )(ak′′ · k)
k) = 0,
|k|2
π 3
Z .
l
Let pl : H 2 (Πl ) → Ml be projection on trigonometric polynomials:
X
X
(4.16)
H 2 (Πl ) ∋ u(x) =
ak eikx
ak eikx 7→ pl u(x) =
k∈ πl Z3
k∈ πl Z3 ,|k|≤l
ak · k = 0,
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
17
As is well-known, to get Galerkin approximations of Navier-Stokes system one restricts (4.14)
to C 1 (0, T ; Ml ) and applies the operator pl to (4.14) to obtain:
∂t u − ∆u + pl π(u, ∇)u = 0, div u=0,
(4.17)
where u ∈ C 1 (0, T ; Ml ). In terms of the Fourier coefficients of the Galerkin approximations this
will have the form:
∂t ak + |k|2 ak +
X
i((ak′ ·k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′ |≤l
(4.18)
k∈
(ak′ · k′′ )(ak′′ · k)
k) = 0,
|k|2
ak · k = 0,
π 3
Z , |k| ≤ l.
l
Proposition 4.4. For each ω ∈ O(3) the following holds:
½
¾
X
imx
(4.19)
Rω Ml = v(x) =
bm e
.
m∈ πl ωZ3 ,
|m|≤l
Moreover,
P
u(x) =
k∈ πl Z3
|k|≤l
(4.20)
Rω u(x) =
ak eikx
P
bm eimx
m∈ πl ωZ3
⇒ bm = ωaω−1 m .
|m|≤l
Proof. Let u(x) =
P
ikx
|k|≤l ak e
∈ Ml . Then using the definition Rω u(x) = ωu(ω −1 x) and
applying the change of variables ωk = m get:
X
(4.21)
Rω u(x) =
ωak eiωkx =
k∈ πl Z3 ,
|k|≤l
X
ωaω−1 m eimx
m∈ πl ωZ3 ,
|m|≤l
This proves (4.19) and (4.20).
¤
Proposition 4.5. For each ω ∈ O(3) the Galerkin approximations for the Navier-Stokes equations on the space C 1 (0, T ; Rω Ml ) are of the following form:
(4.22)
∂t bm (t) + |m|2 bm +
X
m′ +m′′ =m,
m′ ,m′′ ∈ πl ωZ3 ,
|m′ |≤l, |m′′ |≤l
¶
µ
(bm′ · m′′ )(bm′′ · m)
m
= 0,
i (bm′ ·m′′ )bm′′ −
|m|2
m∈
bm · m = 0,
π 3
ωZ , |m| ≤ l.
l
Proof. To obtain the Galerkin approximations on C 1 (0, T ; Rω Ml ) for the Navier-Stokes equations, repeat the procedure above that leads to the Galerkin approximations (4.18): Re-write
18
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
(4.14) on the space of periodic fields C 1 (0, T ; Rω H 2 (Πl )) in terms of Fourier coefficients to get
the following analog of (4.15):
(4.23)
X
∂t bm + |m|2 bm +
m′ +m′′ =m,
m′ ,m′′ ∈ πl ωZ3
µ
¶
(bm′ · m′′ )(bm′′ · m)
m
= 0,
i (bm′ ·m′′ )bm′′ −
|m|2
bm · m = 0,
π 3
ωZ ,
l
and then repeat the derivation of (4.17), (4.18) from (4.14), (4.15) to finally get (4.22) from
m∈
(4.23).
¤
Now supplement (4.17) and (4.18) with the initial condition
X
X
(4.24)
u(t, x)|t=0 =
ak (t)eikx |t=0 = u0 (x) =
ak0 eikx
k∈ πl Z3 ,
|k|≤l
k∈ πl Z3 ,
|k|≤l
and
π 3
Z , |k| ≤ l
l
Moreover, supplement (4.22) with the initial condition
π
(4.26)
bm (t)|t=0 = bm0 , m ∈ ωZ3 , |k| ≤ l
l
Then, as is well-known, the Cauchy problem (4.17), (4.24), (equivalently, the Cauchy problem
(4.25)
ak (t)|t=0 = ak0 ,
k∈
for the ordinary differential equations (4.18), (4.25)) has a unique solution in C 1 (0, T ; Ml ). Call
this solution Sl (u0 ). Analogously, the Cauchy Problem (4.22),(4.26) possesses a unique solution.
Write this solution as the Fourier polynomial
X
(4.27)
Sl (v0 ) =
bm (t)eimx , where v0 =
m∈ πl ωZ3 ,
|k|≤l
X
bm0 eimx
m∈ πl ωZ3 ,
|k|≤l
Lemma 4.6. Let u0 ∈ Ml . Then Rω Sl (u0 ) solves (4.22) with initial condition v 0 = Rω u0 .
Moreover, if Sl (u0 ) admits the Fourier decomposition
X
ak (t)eikx ,
(4.28)
Sl (u0 ) =
k∈ πl Z3 ,
|k|≤l
then {ωak } satisfies
(4.29)
2
∂t ωak + |k| ωak +
X
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′|≤l
¶
µ
(ωak′ · ωk′′ )(ωak′′ · ωk)
′′
ωk = 0,
i (ωak′ · ωk )ωak′′ −
|k|2
ωak · ωk = 0,
k∈
π 3
Z , |k| ≤ l.
l
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
19
Proof. Let
(4.30)
X
Rω Sl (u0 ) =
bm (t)eimx
m∈ πl ωZ3 ,
|m|≤l
where, by (4.20),
(4.31)
bm = aω−1 m .
The assertion of the Lemma will be proved once it shown that the {b m (t)} satisfy (4.22). Substitution of (4.31) into the left hand side of (4.22), the change of variables m = ωk, and the fact
that ω ∈ O(3) yield:
∀k∈
π 3
Z , |k| ≤ l;
l
2
∂t ωak + |ωk| ωak +
X
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′|≤l
(4.32)
½
= ω ∂t ak + |k|2 ak +
X
¶
µ
(ωak′ · ωk′′ )(ωak′′ · ωk)
′′
ωk
i (ωak′ · ωk )ωak′′ −
|ωk|2
i((ak′ · k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′ |≤l
(ak′ · k′′ )(ak′′ · k)
k
|k|2
¾
= 0,
since {ak (t)} satisfies (4.18).
¤
Remark 4.7. As will be shown, more is true: The Galerkin PDE has a unique solution for any
dl , see (4.47) below.
initial condition v in M
The task now is to show that inclusions u 1 ∈ Ml and Rω u1 = u2 ∈ Ml , for some ω, implies
that for the same ω, Rω Sl u1 stays in Ml for all t, still solving the Galerkin system on M l . This
will then show that Rω Sl u1 = Sl u2 .
Definition 4.8. Given an isometry ω of R 3 , let Kω be the set of all elements k in the lattice
π 3
lZ
such that ωk also belongs to the lattice.
Lemma 4.9. Let u1 , u2 be vector fields of the form
(4.33)
u1 (x) =
X
ak eik·x ,
u2 (x) =
|k|≤l
X
|k|≤l
bk eik·x ,
where all
k∈
π 3
Z ,
l
not necessarily divergence free. Then for some ω isometry of R 3 , Rω u1 = u2 if and only if all
k’s in the representation of u1 are in Kω .
20
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
Proof. Since Rω u1 = u2 , Rω u1 is periodic of period 2l. Therefore
X
ωak eiωk·x =
|k|≤l
(4.34)
X
ωak eiωk·(x+(2l)j )
|k|≤l
=
X
ωak ei2l(ωk)j eiωk·x , j = 1, 2, 3,
|k|≤l
for (2l)j denoting the vector with 2l as j coordinate and zeroes on the rest. Now since e iωk·x
are orthonormal on the image of the cube Πl = [−l, l]3 under ω, this implies that
1 = ei2l(ωk)j ,
(4.35)
therefore
2l(ωk)j = 2πNkj , Nkj ∈ Z.
(4.36)
Therefore ωk ∈
π 3
Z . The converse is clear.
l
¤
Lemma 4.10. Let u1 in Ml of the form ,
(4.37)
u1 =
X
ak eik·x ,
k ∈ Kω .
|k|≤l
Then the Galerkin solution in Ml with initial condition u1 is of the form
(4.38)
u(t) =
X
ak (t)eik·x ,
ak (t) = 0 f or all t f or k ∈
/ Kω .
|k|≤l
Proof. In Ml , first solve the system
∂t ak = −|k|2 ak ,
k∈
/ Kω |k| ≤ l
¶
µ
X X
(ak′ )j ikj′′ ak′′ · k
′′
k = −|k|2 ak , k ∈ Kω , |k| ≤ l
(ak′ )j ikj ak′′ −
(4.39) ∂t ak +
2
|k|
′
′′
j
k +k =k
|k ′ |≤l
|k ′′ |≤l
with initial conditions
(4.40)
ak (0) = 0,
k∈
/ Kω
ak (0) = ak ,
k ∈ Kω ,
using the ak ’s from (4.37).
In particular
(4.41)
ak (t) = 0, for all t, for k not in Kω .
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
21
Now if k ∈
/ Kω , and k ′ + k ′′ = k, then either k ′ ∈
/ Kω or k ′′ ∈
/ Kω . Then the unique solution
of (4.39) (4.40) also solves the system
¶
X X µ
(ak′ )j ikj′′ ak′′ · k
′′
k = −|k|2 ak , k ∈
/ Kω , |k| ≤ l
(ak′ )j ikj ak′′ −
∂t ak +
2
|k|
′
′′
(4.42)
∂t ak +
j
k +k =k
|k ′ |≤l
|k ′′ |≤l
X
X
j
k ′ +k ′′ =k
|k ′ |≤l
|k ′′ |≤l
µ
(ak′ )j ikj′′ ak′′ −
(ak′ )j ikj′′ ak′′ · k
k
|k|2
¶
= −|k|2 ak , k ∈ Kω , |k| ≤ l,
i.e. solves the Galerkin system with initial condition u 1 .
¤
Proposition 4.11. Let u1 , u2 be in Ml of the form
X
X
(4.43)
u1 (x) =
ak eik·x , u2 (x) =
bκ eiκ·x ,
|k|≤l
|κ|≤l
Then u2 = Rω u1 implies Sl u2 = Rω Sl u1 .
Proof. Let
(4.44)
S l u1 =
X
aκ (t) eiκ·x ,
aκ (0) = aκ ,
S l u2 =
|κ|≤l
X
bκ (t) eiκ·x ,
bκ (0) = bκ .
|κ|≤l
By Lemma 4.6, Rω Sl u1 is the unique solution of the Galerkin system on R ω Ml with initial
condition Rω u1 , i.e. by (4.29) it solves
(4.45)
∂t ωak +
X
j
X
k ′ +k ′′ =k
|k ′ |≤l
|k ′′ |≤l
µ
(ωak′ )j i(ωk ′′ )j ωak′′ −
(ωak′ )j i(ωk ′′ )j ωak′′ · ωk
ωk
|ωk|2
Observe that in this system if ωk 6= κ for some κ in the lattice
π 3
lZ
¶
= −|ωk|2 ωak .
then ωak (t) = 0 for all t, by
Lemma 4.10. Now rename ωk = κ, ωak (t) = cκ (t), ωak (0) = cκ (0) = bκ , ωk ′ = κ′ , ωk ′′ = κ′′ to
get
(4.46)
∂t cκ +
X
j
X
κ′ +κ′′ =κ
|κ′ |≤l
µ
′′
(cκ′ )j i(κ )j cκ′′
¶
(cκ′ )j i(κ′′ )j cκ′′ · κ
κ = −|κ|2 cκ ,
−
|κ|2
|κ′′ |≤l
with initial conditions Rak . This gives a permutation of the Galerkin system on M l with initial
condition u2 . Therefore cκ (t) = bκ (t) for all κ ∈ πl Z3 .
¤
dl , there
4.3. Isotropic Galerkin approximations of statistical solution. Now given v in M
dl as
are ω in O(3) and u in Ml such that v = Rω u. Extend Sl from Ml to M
(4.47)
Sbl v = Rω Sl u.
22
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
This is well defined by Proposition 4.11: If R ω1 u1 = Rω2 u2 then a straightforward calculation
shows that Rω−1 ω1 u1 = u2 , therefore Rω−1 ω1 Sl u1 = Sl u2 , or Rω1 Sl u1 = Rω2 Sl u2 . In particular,
2
2
Sbl satisfies
Sbl Rω u = Rω Sl u
(4.48)
∀ u ∈ Ml .
Let v = Rω u and w = Rω1 v = Rωω1 u where u ∈ Ml . Applying Rω1 to both parts of (4.47)
gives Rω1 Sbl v = Rωω1 Sl u. On the other hand, (4.47) for w can be written as Sbl w = Rωω1 Sl u.
Comparing these two equalities gives
(4.49)
[VF2], p. 219 shows that
(4.50)
Rω1 Sbl v = Sbl Rω1 v,
dl .
∀v∈M
Th Sl = Sl Th .
Applying to both parts of (4.47) the translation operator T h and using (2.25), (4.50) gives
(4.51)
Th Sbl v = Rω Tω−1 h Sl u = Rω Sl Tω−1 h u.
On the other hand, applying Sbl as defined in (4.47) to Th v = Th Rω u = Rω Tω−1 h u, obtain
Sbl Th v = Rω Sl Tω−1 h u. Comparing this equality with (4.51) obtain
(4.52)
Define
(4.53)
Th Sbl v = Sbl Th v
dl .
∀v∈M
−1
Pbl (A) = µbl (Sbl A),
for any Borel subset A of L2 (0, T, H0 (r)) where, recall,
(4.54)
−1
dl ).
Sbl A = γ0 (A ∩ Sbl M
dl it is enough to consider Borel sets A satisfying A ∩
Since the measure µbl is supported on M
dl 6= ∅.
Sbl M
Definition (4.53) is the isotropic version of the measure P l defined in [VF2]:
(4.55)
Pl (A) = µl (Sl −1 A)
for any Borel subset A of L2 (0, T, H0 (r)).
Lemma 4.12. Pbl is homogeneous and isotropic.
Proof. It suffices to show that for all A ∈ B(L 2 (0, T ; H0 (r))):
(4.56)
and
(4.57)
−1
−1
Sbl Rω A = Rω Sbl A
−1
−1
Sbl Th A = Th Sbl A,
−1
since Pbl (A) = µbl (Sbl A) and µbl is homogeneous and isotropic.
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
23
Using (4.54), (4.49),
−1
dl )
Sbl Rω A = γ0 (Rω A ∩ Rω Rω−1 Sbl M
dl )
= γ0 Rω (A ∩ Sbl Rω−1 M
(4.58)
dl )
= Rω γ0 (A ∩ Sbl M
−1
= Rω Sbl A.
Also, by (4.54), (4.52)
(4.59)
−1
dl )
Sbl Th A = γ0 (Th A ∩ Th T−h Sbl M
dl )
= γ0 Th (A ∩ Sbl T−h M
dl )
= Th γ0 (A ∩ Sbl M
= Th Sbl
−1
A.
¤
The following relation between Pl and Pbl allows the known estimates on for Pl to be carried
over to Pbl . Once again, let a to be the action map of rotations on vector fields:
(4.60)
a(ω, u) = Rω u
Lemma 4.13. The equality holds:
Pbl (A) = (Pl × H)(a−1 A).
where H is the Haar measure on O(3), normalized.
Proof.
−1
Pbl (A) = µbl (Sbl A)
= (µl × H)(a−1 Sbl
(4.61)
−1
A)
−1
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Rω u0 ∈ Sbl A}
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Sbl Rω u0 ∈ A}
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Rω Sl u0 ∈ A}, by (4.48)
= (Pl × H){(Sl u0 , ω) ∈ C 1 (0, T ; Ml ) × O(3) : Rω Sl u0 ∈ A}
= (Pl × H)(a−1 A).
¤
24
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
5. Convergence of isotropic Galerkin approximations
5.1. Galerkin Approximation of Homogeneous Statistical Solutions. This section summarizes the Galerkin approximation in [VF2]. Recall that the following are shown in Chapter
VII there:
Given any µ homogeneous probability measure on H 0 (r), there exist for each l homogeneous
probability measures µl on H0 (r), supported on Ml , converging to µ in characteristic i.e.:
Z
Z
i
ei µ(du), l → ∞,
e
µl (du) →
(5.1)
H0 (r)
H0 (r)
for any test function ν.
5.1.1. Convergence. The probability measures P l , defined by (4.55) are homogeneous in x if
the initial µ is homogeneous, and converge weakly to a homogeneous probability measure P on
L2 (0, T ; H0 (r)). Weak convergence relies on the following three uniform estimates on P l : There
are constants C, C(N ) independent of l such that for each t in [0, T ],
¶
Z
Z µ
Z t
2
2
|∇u| (τ, x) dτ Pl (du) ≤ C |u0 (x)|2 µl (du0 ),
(5.2)
|u(t, x)| +
0
(5.3)
Z
ku|BN k−s Pl (du) ≤ C(N ),
(5.4)
Z
k∂t u|BN k−s Pl (du) ≤ C(N )
for kv|BN k−s the dual norm (3.2), and with s > 11/2.
5.1.2. P is a homogeneous statistical solution of the Navier-Stokes system. As already remarked,
the weak limit P is supported on L2 (0, T ; H1 (r))∩ BV −s ∩ GN S so that each u(., .) in the support
of P satisfies the weak form (3.1) of Navier-Stokes equations, and the right limit in time γ t u
exists for each t with respect to the Φ −s norm. This extra regularity of P relies on the estimate
(5.5)
Z
kukBV −s P (du) ≤ ∞,
s > 11/2
for kukBV −s the norm (3.3). In addition, P satisfies energy estimate (3.13).
5.1.3. P solves the initial value problem. The measure defined by P (γ 0−1 A), for A Borel of H0 (r),
is the initial measure µ. The proof of this relies on the convergence (5.1).
5.2. Homogeneous and Isotropic Statistical Solutions via isotropic Galerkin Approximations. In the setting of isotropic measures, the construction of the previous subsection can
be repeated as follows:
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
25
5.2.1. Approximation of initial measure. Given initial homogeneous and isotropic µ
b on H 0 (r),
construct its (merely) homogeneous approximation µ l , with µl → µ
b in characteristic as in the
previous section. Then use Definitions 4.1 and 4.2 to obtain from the µ l ’s probability measures
dl . Convergence in characteristic
µbl that are now homogeneous and isotropic and supported by M
still holds, thanks to the following:
Lemma 5.1. µbl converges to µ
b in characteristic as l → ∞.
Proof. For µbl defined as above,
χµbl (ν) =
=
(5.6)
=
=
Convergence Theorem,
χµbl (ν) =
Z
Z
Z
Z
R
Since |χµl ((Rω−1 ν)| ≤ 1 and
(5.7)
Z
O(3)
H0 (r)
H0 (r)
O(3)
O(3)
χµbl (ν) → χµb (ν) as l → ∞.
Z
Z
eiRω u·ν dω µl (du)
O(3)
H0 (r)
eiu·Rω−1 ν µl (du) dω
χµl (Rω−1 ν) dω.
O(3) 1
dω = 1, (5.1) implies, via the Lebesgue Dominated
χµl (Rω−1 ν) dω →
as l → ∞. Therefore, since
Z
Z
χµb (Rω−1 ν) dω =
(5.8)
O(3)
eiu·ν µbl (du)
O(3)
Z
H0 (r)
Z
O(3)
χµb (Rω−1 ν) dω
eiRω u·ν µ
b(du) dω = χµb (ν),
¤
5.2.2. Convergence to a homogeneous and isotropic measure. Now construct the homogeneous
and isotropic measures Pbl according to (4.53). To prove weak convergence the following is needed
Lemma 5.2. The following estimates hold:
Z
(5.9)
ku|BN k−s Pbl (du) ≤ C(N ),
(5.10)
Z
k∂t u|BN k−s Pbl (du) ≤ C(N ).
Proof. By Lemma 4.13 and relations (3.21),
Z
Z Z
b
kRω ∂t u|BN k−s dω Pl (du)
k∂t u|BN k−s Pl (du) =
O(3)
(5.11)
Z
= k∂t u|BN k−s Pl (du).
26
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
[VF2] Chapter VII, Theorem 3.1 proves that the right hand side of (5.11) is bounded by a
constant C(N ) that depends on N but does not depend on l. This proves (5.10). The bound
(5.9) is proved similarly.
¤
The following is also needed to repeat the proof of the weak convergence of the Pbl ’s.
Lemma 5.3. With pointwise averages defined as in (3.8),(3.9),(3.10), for any t in [0, T ],
¶
Z µ
Z t
¡ 2
¢
2
2
|u| (τ, x) + |∇u| (τ, x) dτ Pbl (du)
|u| (t, x) +
0
(5.12)
Z
≤ C |u|2 (x) µbl (du).
Proof. Using an equality similar to (3.9), (3.10) one can show that
Z
Z
Z
2
(5.13)
|u| (t, x) µbl (du) =
|u|2 (t, x) Rω∗ µl (du) dω,
O(3)
(5.14)
Z µ
2
|u| (t, x) +
=
Z
Z
t¡
2
2
¢
|u| (t, x) + |∇u| (t, x) dτ
0
O(3)
Z µ
2
|u| (t, x) +
Z tµ
2
¶
Pbl (du)
2
|u| (t, x) + |∇u| (t, x)
0
¶
dτ
¶
Rω∗ Pl (du) dω.
Applying to (5.13), (5.14) Lemma 3.10 and taking into account the inequality for Galerkin
approximations of the homogeneous statistical solution
¶
Z µ
Z t
¡ 2
¢
2
2
|u| (t, x) + |∇u| (t, x) dτ Pl (du)
|u| (t, x) +
0
(5.15)
Z
≤ C |u|2 (x) µl (du)
that was proved in [VF2] Chapter VII, Lemma 2.4, yields (5.12).
¤
Lemma 5.4. The measures µbl satisfy
Z
Z
2
(5.16)
|u| (x) µbl (du) ≤ |u|2 (x) µl (du)
¤
Proof. This follows from (5.13) and Lemma 3.10.
With Lemmas 5.2, 5.3, 5.4 established, there are no further obstacles in repeating the arguments in [VF2] to show that the family Pbl is weakly compact. The argument for this in [VF2]
is that the measures Pl are supported on the space Ω of elements in L 2 (0, T ; H0 (r)) with the
following norm finite:
(5.17)
∞
X
N =1
1
2N C(N )
(kukL2 (0,T ;H−s (BN )) + k
∂u
k 1
+ kukL2 (0,T ;H1 (r1 )) ) < ∞,
−s
∂t L (0,T ;H (BN ))
s > 11/2,
3
r < r1 < − .
2
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
27
Then Ω is compactly embedded in L 2 (0, T ; H0 (r)), [VF2], Chapter VII, Lemma 5.2, and
Z
(5.18)
sup kukΩ Pl (du) < ∞,
l
[VF2], proof of Theorem 6.1 in Chapter VII. From the estimates of Lemmas 5.2, 5.3 for the
measures Pbl , the energy conservation (5.12), and the uniform estimate (5.4), it becomes clear
that the all measures Pbl are supported on Ω and the uniform estimate (5.18), with P l changed
to Pbl , still holds.
b be the limit of some weakly convergent subsequence of Pbl ’s, as l → ∞.
Let Q
b weakly on L2 (0, T ; H0 (r)), then
Lemma 5.5. If Pbl are homogeneous and isotropic and Pbl ⇒ Q
b is homogeneous and isotropic.
Q
b weakly then by definition
Proof. If Pbl ⇒ Q
Z
Z
b
b
(5.19)
f (u) Pl (du) → f (u) Q(du).
for any f continuous and bounded on L 2 (0, T ; H0 (r)). Since the measures Pbl are homogeneous
and isotropic, by definitions 2.2, 2.4
Z
Z
b
f (u) Pl (du) = f (Th u) Pbl (du)
Z
(5.20)
= f (Rω u) Pbl (du),
for any h ∈ R3 and ω ∈ O(3). These equalities and
Z
Z
b
f (Th u) Pbl (du) → f (Th u) Q(du),
Z
Z
(5.21)
b
b
f (Rω u) Pl (du) → f (Rω u) Q(du),
imply
(5.22)
Z
b
f (u) Q(du)
=
=
Z
Z
b
f (Th u) Q(du)
b
f (Rω u) Q(du).
¤
b is in
5.2.3. Extra regularity for right t-limits and the initial condition. That the support of Q
addition in L2 (0, T ; H1 (r)) ∩ BV −s (where right limits with respect to time are well defined by
[VF2], Chapter VII, Lemma 8.2) uses only the estimate
Z
b
(5.23)
kukBV −s Q(du)
≤∞
for kukBV −s the norm (3.3), which follows as in the proof of Lemma 5.2. Define therefore γ 0 by
b as the initial value of Q.
b
(3.6) and think of γ0∗ Q
28
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
That the initial value γ0∗ P of the the homogeneous statistical solution P of Theorem 3.4 is the
initial measure µ is shown in [VF2] as Theorem 10.1, Lemma 10.1, Lemma 10.2, and Theorem
10.2 of Chapter VII there. Of these, Lemma 10.1, Lemma 10.2, and Theorem 10.2 of Chapter
b Theorem 10.1 uses only the convergence in characteristic of the
VII are valid verbatim for Q.
µl ’s to µ. The analogous convergence of the µbl ’s to µ
b was established here as Lemma 5.1.
b
6. The support of the measure Q.
This section uses the approach of [VF2] Chapter VII, Section 7, to show that the homogeneous
b is supported by generalized solutions of the Navier-Stokes equations. To
and isotropic measure Q
realize this approach, some subtle points regarding the definition of the equations for isotropic
Galerkin approximations in the x-representation need to be addressed first.
6.1. Equations for isotropic Galerkin approximations in the x-representation. Section 4 defined isotropic Galerkin approximations by introducing and investigating the Galerkin
equations in terms of Fourier coefficients. Here, a complete description of the Galerkin equations
in the x-representation is given, beginning with a more precise determination of the domain of
their definition.
dl defined by (4.1), (4.3), introduce the set of periodic vector
In addition to the sets Ml and M
fields Nl with the cube of periods Πl define
M PP EE JJ
Mathematical Physics Electronic Journal
ISSN 1086-6655
Volume 12, 2006
Paper 2
Received: Nov 15, 2005, Revised: Mar 19, 2006, Accepted: Apr 3, 2006
Editor: R. de la Llave
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
OF THE NAVIER-STOKES EQUATIONS
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
Abstract. Two constructions of homogeneous and isotropic statistical solutions of the
3D Navier-Stokes system are presented. First, homogeneous and isotropic probability
measures supported by weak solutions of the Navier-Stokes system are produced by averaging over rotations the known homogeneous probability measures, supported by such
solutions, of [VF1], [VF2]. It is then shown how to approximate (in the sense of convergence of characteristic functionals) any isotropic measure on a certain space of vector
fields by isotropic measures supported by periodic vector fields and their rotations. This is
achieved without loss of uniqueness for the Galerkin system, allowing for the Galerkin approximations of homogeneous statistical Navier-Stokes solutions to be adopted to isotropic
approximations. The construction of homogeneous measures in [VF1], [VF2] then applies
to produce homogeneous and isotropic probability measures, supported by weak solutions
of the Navier-Stokes equations. In both constructions, the restriction of the measures at
t = 0 is well defined and coincides with the initial measure.
The second author was partially supported by RFBI grant 04-01-00066.
1
2
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
1. Introduction
The Kolmogorov theory of turbulence describes the behavior of homogeneous and isotropic
fluid flows, i.e. flows with statistical properties independent of translations, reflections, and
rotations in 3D-space. The mathematical equivalent of such flows are translation, reflection,
and rotation invariant probability measures P supported by Navier-Stokes solutions. Given an
appropriate initial measure µ on the space of initial conditions and any finite time interval [0, T ],
the measure P should yield µ at t = 0 is some sense. It should also yield measures µ t for each
time t in [0, T ], each invariant under space translations, reflections and rotations, that represent
the flow of µ under the Navier-Stokes equations as in [H].
The existence of such measures P that satisfy all the assumptions above for translations in
space, called homogeneous, was proved by Vishik and Fursikov in [VF1] and is included in
detail in [VF2]. See also [FT]. This note adopts the construction from [VF1], [VF2] to produce
measures P that, in addition to being invariant under translations, are also invariant under
rotations and reflections. Such measures will be called homogeneous and isotropic. (To
emphasize the new elements of the construction, the convention here will be that “isotropic”
does not imply “homogeneous.”) One of the main results of this note is:
Theorem 1.1. Given µ
b homogeneous and isotropic measure on a space H 0 (r) of vector fields on
R3 with the finite energy density, there exists measure Pb on L2 (0, T, H0 (r)), homogeneous and
isotropic with respect to the space variables, supported by weak solutions of the Navier-Stokes
equations, and with finite energy density that satisfies the standard energy inequality. On the
support of Pb right limits with respect to time are well-defined in an appropriate norm and the
right limit at t = 0 yields µ
b.
(For the definitions of all notions used in the formulation of Theorem 1.1, see sections 2 and 3
below.)
To prove Theorem 1.1 the homogeneous statistical solution P constructed in [VF1], [VF2]
with initial measure µ
b is averaged over all rotations and reflections. The resulting measure
Pb satisfies all conditions of Theorem 1.1 above and is therefore the desired homogeneous and
isotropic statistical solution. This plan is realized in sections 2 and 3.
It has to be emphasized, however, that existence theorems obtained via convergent sequences
of “simpler” approximations of the constructed solution are as a rule much more useful in
mathematical physics than the so called “pure existence theorems.” For this reason, sections 4,
5, and 6 below are devoted to constructing such approximations of isotropic statistical solutions.
The construction of homogeneous statistical solutions in [VF1], [VF2] is based on Galerkin
approximations of measures that are supported by divergence free periodic vector fields with
trigonometric polynomials as components. The main difficulty in extending this construction to
isotropic measures is that the space of such vector fields, whereas invariant under translations,
is not invariant under rotations.
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
3
The construction in sections 4, 5, and 6 is based on the observation that the space of such
vector fields AND all their rotations and reflections should suffice for invariance under rotations
and reflections.
It is then necessary to construct Galerkin approximations of isotropic measures on this class
of vector fields. In this sense, the crux of the matter is Sections 4.2, 4.3, and 6.1. Note that
the constructions of sections 4, 5, and 6 not only offer approximations of the isotropic statistical
solution constructed in section 3 but they also allow for a construction of isotropic statistical
solutions that is formally independent on the results of section 3.
This paper considers the case of 3D Navier-Stokes equations, although the arguments here
are applicable in 2D case as well.
2. Isotropic measures
2.1. Definitions. Non-trivial measures invariant under translations exist on weighted Sobolev
spaces of vector fields defined over R 3 , p. 208, [VF2]:
Definition 2.1. For k non negative integer and r < −3/2, define H k (r) to be the space of
solenoidal vector fields
u(x) = (u1 (x), u2 (x), u3 (x)),
(2.1)
divu =
x = (x1 , x2 , x3 ) ∈ R3 ,
3
X
∂uj
j=1
∂xj
= 0,
with finite (k, r)-norm:
(2.2)
kuk2k,r =
Z
R3
¡
¯
¯2
X ¯¯ ∂ |α| u(x) ¯¯
¢
r
1 + |x|2
¯ α1 α2 α3 ¯ dx,
¯ ∂x1 ∂x2 ∂x3 ¯
|α|≤k
3
r2 =
R
< u, ∂φ >2 + < u, ∆φ >2 +
∂t
3
X
< uj u,
j=1
∂φ
>2 dt = 0,
∂xj
)
¡
¢
for all φ ∈ C0∞ (0, T ) × R3 ∩ C((0, T ); H0 (r)) ,
R3
u(x) · v(x) dx.
To define restrictions at any time t ∈ [0, T ] one works with the following norms: For B N =
{|x| < N } the ball of radius N in R3 , and for k.ks the standard Sobolev norm in W s,2 (R3 ) =
L2s (R3 ), define the dual norm
kv|BN k−s =
(3.2)
sup
w∈C0∞ (BN )
< v, w >2
.
kwks
Using this and following [VF2], p.245, define
(3.3)
kukBV −s = kukL2 (0,T ;H0 (r)) +
∞
X
N =1
1
22N C(N )
|u|N .
Here C(N ) are constants from (5.3) and (5.4) (see below) and |u| N is defined as follows:
(3.4)
|u|N = vrai sup ku(t, ·)|BN k−s + sup
t∈[0,T ]
l
X
{tj } j=1
vrai
sup
t,τ ∈[tj−1 ,tj )
k(u(t, ·) − u(τ, ·))|BN k−s ,
8
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
where sup{tj } is the supremum over all partitions t 0 < · · · < tl , l ∈ N of the segment [0, T ].
Define
BV −s = {u ∈ L2 (0, T ; H0 (r)) : kukBV −s < ∞}.
(3.5)
The merit of the BV −s norm is that for u in BV −s the limits
γt0 (u) := lim u(t, .)
(3.6)
t→t+
0
exist for all t0 ∈ [0, T ], if taken with respect to the norm
!1/2
à ∞
X
1
2
(3.7)
ku(t, .)kΦ−s =
ku(t, .)|BN k−s
,
22N C(N )
N =1
see [VF2], Chapter VII, Lemma 8.2.
Note also that for a homogeneous measure µ the pointwise averages
Z
Z
2
(3.8)
|u| (x) µ(du),
|∇u|2 (x) µ(du)
can be defined by the equalities
Z Z
Z
Z
2
2
(3.9)
|u(x)| φ(x) dx µ(du) = |u(x)| µ(du) φ(x) dx
(3.10)
Z Z
2
|∇u(x)| φ(x) dx µ(du) =
Z
2
|∇u(x)| µ(du)
Z
∀ φ ∈ L1 (R3 )
φ(x) dx
and they are independent of x ∈ R3 , see Chapter VII, section 1 of [VF2]. The expressions in
(3.8) are well defined since the left hand sides in (3.9), (3.10) are finite for each φ ∈ C 0∞ (R3 ).
The first expression in (3.8) is the energy density and the second one is the density of
the energy dissipation.
Since the translation operator Th (along x) is well defined on the space L 2 (0, T ; H0 (r)) of
vector fields u(t, x) dependent not only on x but on t as well, one can introduce the notion of
homogeneity in x:
Definition 3.2. A measure P (A), A ∈ B(L2 (0, T ; H0 (r))) is called homogeneous in x if for
each h ∈ R3 :
(3.11)
Th∗ P
= P ⇐⇒
Z
f (u)
L2 (0,T ;H0 (r))
Th∗ P (du)
=
Z
f (u) P (du),
L2 (0,T ;H0 (r))
for any P -integrable f on H 0 (r).
The following definition summarizes the properties of homogeneous statistical solutions of the
Navier-Stokes equations as they were produced in Chapter VII of [VF2]:
Definition 3.3. Given homogeneous probability measure µ on B(H 0 (r)) possessing finite energy
density, a homogeneous statistical solution of the Navier-Stokes equations with initial
condition µ is a probability measure P on B(L 2 (0, T ; H0 (r))) such that:
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
(1) P is homogeneous in x.
c ) = 1, where W
c = L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S , s >
(2) P (W
9
11
2 .
(3) For all A ∈ B(H0 (r)),
P (γ0−1 A) = µ(A),
(3.12)
where
c : γ0 u ∈ A}.
γ0−1 A = {u ∈ W
(4) For each t in [0, T ],
¶
Z
Z µ
Z t
2
2
|∇u| (τ, x) dτ P (du) ≤ C |u(x)|2 µ(du),
(3.13)
|u(t, x)| +
0
where the expression in the left hand side of (3.13) is defined similarly to (3.8).
The main result of [VF2], Chapter VII, then reads as follows:
Theorem 3.4. Given µ homogeneous measure on H 0 (r) with finite energy density,
Z
|u|2 (x) µ(du) < ∞,
(3.14)
H0 (r)
there exists homogeneous statistical solution of the Navier-Stokes equations P with initial condition µ.
Remark 3.5. The definition above is a rephrasing of Definition 11.1 of [VF2], with one minor
change: It asks that P is supported by L 2 (0, T ; H1 (r))∩BV −s ∩GN S rather than some subset of it.
Since [VF2] produces some subset supporting a homogeneous statistical solution, it automatically
produces a homogeneous statistical solution according to the definition here.
Remark 3.6. In addition, the family of homogeneous measures µ t := P ◦ γt−1 on H0 (r) satisfies
the Hopf equation, [VF2], Chapter VIII.
Define now isotropic and homogeneous statistical solutions. First define isotropic in x
measures P on B(L2 (0, T ; H0 (r))). (This can be done since for each ω ∈ O(3) operator
Rω u(t, x) ≡ ωu(t, ω −1 x) is well defined on L2 (0, T ; H0 (r)).)
Definition 3.7. A measure P (A), A ∈ B(L 2 (0, T ; H0 (r))) is called isotropic in x if for each
ω ∈ O(3):
(3.15)
Rω∗ P
= P ⇐⇒
Z
f (u)
L2 (0,T ;H0 (r))
Rω∗ P (du)
=
Z
f (u) P (du),
L2 (0,T ;H0 (r))
for any P -integrable f on L2 (0, T ; H0 (r)).
For the definition of homogeneous and isotropic statistical solutions one has only to add in
Definition 3.3 the property of rotation and reflection invariance:
Definition 3.8. Given homogeneous and isotropic probability measure µ
b on B(H 0 (r)), a homogeneous and isotropic statistical solution of the Navier-Stokes equations with initial
condition µ
b is a probability measure Pb on B(L2 (0, T ; H0 (r))) such that:
(1) Pb is homogeneous and isotropic in x.
10
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
c ) = 1, where W
c = L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S , s >
(2) Pb(W
(3) Pb(γ −1 A) = µ
b(A) for every A ∈ B(H 0 (r)).
11
2 ,.
0
(4) For each t in [0, T ],
¶
µ
Z
Z
Z t
2
2
b
|∇u| (τ, x) dτ P (du) ≤ C
(3.16)
|u(t, x)| +
L2 (0,T ;H0 (r))
0
H0 (r)
|u(x)|2 µ
b(du).
3.2. Construction of homogeneous and isotropic statistical solutions. To construct
homogeneous and isotropic statistical solutions several preliminary assertions need to be proved
first. For these, use the definition of the norm k · k s of Sobolev space W s,2 (R3 ) through Fourier
transform:
(3.17)
kφk2s
=
Z
R3
b 2 (ξ) dξ,
(1 + |ξ| ) |φ|
2 s
where
b =
φ(ξ)
1
(2π)3/2
Z
e−ix·ξ φ(x) dx.
R3
Lemma 3.9. For any matrix ω ∈ O(3) the following equalities hold:
kRω φks = kφks ,
kRω φkBV −s = kφkBV −s ,
(3.18)
kRω φkΦ−s = kφkΦ−s .
Proof. By the definition of Fourier transform and by virtue of (2.16)
Z
1
−iω −1 x·ω −1 ξ
d
R
φ(ξ)
=
e
ωφ(ω −1 x) dx
ω
(2π)3/2 R3
Z
(3.19)
1
−1
b
=
e−iy·ω ξ ωφ(y) dy = Rω φ(ξ).
3/2
(2π)
R3
By (3.17), (3.19), and (2.16)
kRω φk2s
=
Z
R3
(3.20)
=
Z
R3
b −1 ξ)|2 dξ
(1 + |ω −1 ξ|2 )s |ω φ(ω
2
b
(1 + |η|2 )s |ω φ(η)|
dη = kφk2s ,
which proves the first equality in (3.18).
Equalities (3.20), (2.16), and (3.2) give:
kRω v|BN k−s =
(3.21)
=
< Rω v, φ >2
kφks
φ∈C0∞ (BN )
sup
< v, (Rω )−1 φ >2
= kv|BN k−s ,
k(Rω )−1 φks
φ∈C0∞ (BN )
sup
since Rω : C0∞ (BN ) → C0∞ (BN ) is an isomorphism.
Equality (3.21) and definition (3.4) of | · | N imply the equality
(3.22)
|Rω φ|N = |φ|N .
This identity, (3.21), and the definitions (3.3), (3.7) of the norms k·k BV −s , k·kΦ−s , imply directly
the second and third equalities in (3.18).
¤
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
11
Lemma 3.10. For every ω ∈ O(3) and for each homogeneous measure µ(A), A ∈ B(H 0 (r))
Z
Z
2
∗
|u(x)| Rω µ(du) = |u(x)|2 µ(du),
Z
Z
(3.23)
2
∗
|∇u(x)| Rω µ(du) = |∇u(x)|2 µ(du).
Proof. By (2.6), (2.7), (3.9), and (2.16), for each ω ∈ O(3)
Z
Z Z
Z
2
∗
|ωu(ω −1 x)|2 φ(ωω −1 x) dx dµ(u)
|u(x)| Rω µ(du) φ(x) dx =
R3
Z Z
|u(y)|2 φ(ωy) dy dµ(u)
=
3
Z R
Z
(3.24)
2
φ(ωy) dy
= |u(y)| dµ(u)
R3
Z
Z
φ(x) dx,
= |u(x)|2 dµ(u)
R3
which proves the first equality of (3.23). If y = ω −1 x, i.e. yl = ωkl xk , then by (2.16), (2.18),
obtain
Z X
j
Z X
∂u(ω −1 x) 2
|
| φ(ωω −1 x) dx
∂xj
j
Z X
∂up (y)
∂up (y)
ωjl
=
ωjm
φ(ωy) dy
∂yl
∂ym
j,p
Z
= |∇y u(y)|2 φ(ωy) dy.
∂u(ω −1 x) 2
|ω
| φ(x) dx =
∂xj
(3.25)
Using these identities, the second equality of (3.23) can be proved similarly to (3.24).
¤
Recall now that the set GN S has been introduced in Definition 3.1.
Lemma 3.11. For each ω ∈ O(3) the equality R ω GN S = GN S holds.
Proof. Prove first that if u satisfies (3.1) then R ω u satisfies (3.1) as well, for every ω ∈ O(3).
For this note that, by Lemma 2.3,
(3.26)
u ∈ L2 (0, T ; H0 (r)) ⇒ Rω u ∈ L2 (0, T ; H0 (r)),
φ ∈ C((0, T ); H0 (r)) ∩ C0∞ ((0, T ) × R3 ) ⇒ Rω φ ∈ C((0, T ); H0 (r)) ∩ C0∞ ((0, T ) × R3 ).
So let u satisfy (3.1). Then (2.16) and the well-known fact that Laplace operator is invariant
under orthogonal change of variables yield:
(3.27)
< Rω u,
∂(Rω−1 )φ
∂φ
+ ∆φ >=< u,
+ ∆(Rω−1 )φ > .
∂t
∂t
12
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
If y = ω −1 x = ω ∗ x, i.e. yl = ωkl xk , then taking into account (2.18) and (2.16) calculate:
Z
Z
∂φl (x)
∂φ
dx = ωjk uk (ω −1 x)ωlm um (ω −1 x)
dx
(Rω u)j Rω u ·
∂xj
∂xj
Z
∂φl (ωy)
(3.28)
= ωjk uk (y)ωlm um (y)ωjp
dy
∂yp
Z
∂ωlm φl (ωy)
= uk (y)um (y)
dy.
∂yk
Then
(3.29)
3
X
j=1
3
X
∂(Rω−1 )φ
∂φ
< uj u,
>2 =
>2 .
< (Rω u)j Rω u,
∂xj
∂xj
j=1
Adding (3.27) and (3.29), integrating the resulting equality with respect to t over [0, T ], and
taking into account that u satisfies (3.1), shows that R ω u satisfies (3.1) as well.
¤
Lemma 3.12. γ0 commutes with Rω for any rotation ω.
Proof. By Lemma 3.9, ku(t, .)kΦ−s = kRω u(t, .)kΦ−s . Therefore if limt→0+ u(t, .) = γ0 (u), then
limt→0+ Rω u(t, .) = Rω γ0 (u), and limt→0+ Rω u(t, .) = γ0 (Rω u), i.e. γ0 (Rω u) = Rω γ0 (u).
Recall that for each B ∈ B(H 0 (r))
(3.30)
c : γ0 u ∈ B},
γ0−1 B = {u(t, x) ∈ W
c is the set of Definition 3.3 or (equivalently) of Definition 3.8.
where W
Lemma 3.13. For B ∈ B(H 0 (r)),
(3.31)
Rω γ0−1 (B) = γ0−1 (Rω B),
∀ ω ∈ O(3).
Proof. Using (2.18), one can prove similarly to Lemma 2.3 that
(3.32)
Rω H1 (r) = H1 (r)
∀ ω ∈ O(3),
for H1 (r) as in Definition 2.1. This, together with lemmas 3.9 and 3.11, imply that
(3.33)
Therefore
c=W
c
Rω W
∀ ω ∈ O(3).
u ∈ Rω γ0−1 (B) ⇒ u = Rω v, v ∈ γ0−1 (B)
⇒ γ0 u = γ0 (Rω v), v ∈ γ0−1 (B)
(3.34)
⇒ γ0 u = Rω γ0 (v), v ∈ γ0−1 (B), by Lemma 3.12,
⇒ γ0 u = Rω b, b ∈ B
⇒ u = γ0−1 Rω b, b ∈ B
⇒ u ∈ γ0−1 (Rω B).
¤
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
13
Conversely,
u ∈ γ0−1 (Rω B) ⇒ γ0 (u) = Rω b, b ∈ B
⇒ Rω−1 γ0 (u) = b, b ∈ B
⇒ γ0 (Rω−1 u) = b, b ∈ B, by Lemma 3.12,
(3.35)
⇒ Rω−1 u = γ0−1 (b), b ∈ B
⇒ Rω−1 u ∈ γ0−1 (B),
⇒ u ∈ Rω γ0−1 (B).
¤
Theorem 3.14. Given µ
b homogeneous and isotropic measure on H 0 (r) with finite energy density,
Z
(3.36)
H0 (r)
|u|2 (x) µ
b(du) < ∞,
there exists homogeneous and isotropic statistical solution Pb of the Navier-Stokes equations with
initial condition µ
b.
Proof. Ignoring for the moment that µ
b is also isotropic, let P be the homogeneous statistical
c =
solution with initial condition the homogeneous µ
b guaranteed by Theorem 3.4. The set W
L2 (0, T ; H1 (r)) ∩ BV −s ∩ GN S is invariant under rotations by (3.33). Applying the analogue of
operation (2.22) on the homogeneous measure P obtain:
(3.37)
Pb (A) =
Z
O(3)
Rω∗ P (A) dω =
Z
O(3)
P (Rω−1 A) dω,
c ). Repeating the proof of Proposition 2.7 for the measure Pb shows that Pb is
for any A ∈ (B)(W
c ) = 1 by Theorem 3.4, equality (3.33) implies that
homogeneous and isotropic in x. Since P (W
c ) = P (R−1 W
c ) = P (W
c ) = 1 for each ω ∈ O(3). Hence, Pb(W
c ) = 1 by definition (3.37).
Rω∗ P (W
ω
That Pb has initial condition µ
b follows from
Pb(γ0−1 B) =
=
(3.38)
Z
O(3)
Z
O(3)
=
Z
O(3)
for any B ∈ B(H0 (r)).
P (Rω−1 γ0−1 (B)) dω
P (γ0−1 (Rω−1 B)) dω, by Lemma 3.13
µ
b(Rω−1 B) dω, by (3.12)
=µ
b(B), since µ
b is also isotropic,
14
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
For the energy inequality, use (3.37), Lemma 3.10, and (3.13) to get:
¶
µ
Z
Z t
2
2
|∇u| (τ, x) dτ Pb(du)
|u(t, x)| +
L2 (0,T ;H0 (r))
=
(3.39)
=
Z
Z
O(3)
L2 (0,T ;H0 (r))
L2 (0,T ;H0 (r))
≤C
Z
H0 (r)
which proves (3.16).
0
Z
µ
µ
2
|u(t, x)| +
|u(t, x)|2 +
Z
Z
t
2
|∇u| (τ, x) dτ
0
t
|∇u|2 (τ, x) dτ
0
¶
¶
Rω ∗ P (du) dω
P (du)
|u(x)|2 µ
b(du),
¤
4. Galerkin approximation of isotropic statistical solutions
4.1. Isotropic measures on periodic vector fields. Let M l be as in [VF2]:
½ X
¾
ik·x
(4.1)
Ml =
ak e
: ak · k = 0, ak = a−k ∀ k ,
k∈ πl Z3 ,
|k|≤l
the finite-dimensional space of divergence-free, 3D, real, vector valued trigonometric polynomials
of degree l and period 2l. Then the inclusion
Ml ⊂ H0 (r)
(4.2)
holds for all l. [VF2], Appendix II, shows explicitly how, starting from any homogeneous probability measure on H0 (r), one can construct homogeneous probability measures µ l on H0 (r),
supported solely by Ml for each l, and approximating µ in the sense of characteristic functionals.
The trouble, of course, is that Ml is not invariant under rotations. The following definitions
address this point.
dl be the union of all rotations of elements of M l :
Definition 4.1. Let M
[
dl =
(4.3)
M
Rω M (l)
ω∈O(3)
in H0 (r).
dl the topology τ generated by sets of the form
Consider on M
(4.4)
{Rω m : ω ∈ ρ, m ∈ σ, where ρ ⊂ O(3), σ ⊂ Ml are open sets}.
Since O(3) and Ml are finite-dimensional sets, the topology τ coincides with the topology
dl ) is generated by
generated by the enveloping space H 0 (r). Therefore, the Borel σ-algebra B(M
sets of the form (4.4). Moreover, it is clear that
(4.5)
dl ) = B(H0 (r)) ∩ M
dl ≡ {A ∩ M
dl : A ∈ B(H0 (r))}
B(M
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
15
Note that for each fixed ω the elements of R ω Ml are of the form
X
(4.6)
bk eiωk·x , bk · ωk = 0 for all k.
k∈ πl Z3 ,
|k|≤l
Using definition (2.6) of the push forward measure, proceed to:
dl ) be the push-forward of the product of the Haar measure
Definition 4.2. Let µbl (A), A ∈ B(M
on O(3) and the measure µl on Ml via the map (ω, u) 7→ Rω u:
µbl (A) = (H × µl ){(ω, u) ∈ O(3) × Ml : Rω u ∈ A}.
(4.7)
As for any push-forward measure, by (2.7),
Z
(4.8)
for any µbl -integrable f .
cl
M
f (v) µbl (dv) =
Z
Ml
Z
f (Rω u) dωµl (du),
O(3)
dl ⊂ H0 (r), the domains of
Since µl is supported on Ml ⊂ H0 (r) and µbl is supported on M
dl , Ml in (4.8) can change to H 0 (r). Comparing then (4.7), (4.8) to the definitions
integration M
of averaging (2.22), (2.23) it follows that the measure µbl is the averaging of µl over O(3):
µbl (A) =
(4.9)
Z
O(3)
Rω∗ µl (A) dω
∀ A ∈ B(H0 (r))
Proposition 4.3. µbl is homogeneous and isotropic.
dl = M
dl and the invariance of the Haar measure, obtain for
Proof. Using the equality Rω−1 M
each µbl -integrable f :
Z
dl
M
f (w)Rω∗ 0 µbl (dw)
=
=
Z
cl
M
Z
Ml
(4.10)
=
Z
Ml
=
Z
dl
M
f (Rω0 v) µbl (dv)
Z
f (Rω0 Rω u) dω µl (du)
O(3)
Z
f (Rω u) dω µl (du)
O(3)
f (v) µbl (dv),
i.e. µ
b is invariant with respect of rotations: R ω∗ 0 µ
b=µ
b for each ω0 ∈ O(3).
16
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
dl = M
dl , T −1 Ml = Ml imply:
Similarly, the equalities Th−1 M
ωh
Z
Z
f (Th v) µbl (dv)
f (w)Th∗ µbl (dw) =
dl
M
=
=
(4.11)
=
=
=
Z
Z
Z
Z
Z
dl
M
Ml
Z
O(3)
O(3)
Ml
dl
M
f (Th Rω u) dω µl (du)
O(3)
Z
Ml
Z
Z
f (Rω Tω−1 h u) µl (du) dω
f (Rω u) µl (du) dω, by the homogeneity of µl ,
Ml
f (Rω u) dω µl (du)
O(3)
f (v) µbl (dv).
¤
4.2. Galerkin equations for Fourier coefficients on R ω Ml . Let H s (Πl ) be the space of
periodic vector fields
(4.12)
½
¾
X
X
ik·x
2
2 s
2
H (Πl ) = u(x) =
ak e , ak = (ak1 , ak2 , ak3 ), kuks =
(1 + |k| ) |ak | < ∞ .
s
k∈ πl Z3
k∈ πl Z3
Here
(4.13)
Πl = {x = (x1 , x2 , x3 ) : |xj | ≤ l, j = 1, 2, 3}
is the cube of periods for these vector fields.
On the space C 1 (0, T ; H 2 (Πl )) the Navier-Stokes system can be written in the form:
∂t u − ∆u + π(u, ∇)u = 0, div u=0,
(4.14)
where π : L2 (Πl ) → {u ∈ L2 (Πl ) : divu = 0} is the projection on solenoidal vector fields. It is
P
standard that substitution of the Fourier series u(x) = k ak eik·x into (4.14) yields the following
system for the Fourier coefficients a k (t):
∂t ak + |k|2 ak +
(4.15)
X
i((ak′ ·k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3
k∈
(ak′ · k′′ )(ak′′ · k)
k) = 0,
|k|2
π 3
Z .
l
Let pl : H 2 (Πl ) → Ml be projection on trigonometric polynomials:
X
X
(4.16)
H 2 (Πl ) ∋ u(x) =
ak eikx
ak eikx 7→ pl u(x) =
k∈ πl Z3
k∈ πl Z3 ,|k|≤l
ak · k = 0,
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
17
As is well-known, to get Galerkin approximations of Navier-Stokes system one restricts (4.14)
to C 1 (0, T ; Ml ) and applies the operator pl to (4.14) to obtain:
∂t u − ∆u + pl π(u, ∇)u = 0, div u=0,
(4.17)
where u ∈ C 1 (0, T ; Ml ). In terms of the Fourier coefficients of the Galerkin approximations this
will have the form:
∂t ak + |k|2 ak +
X
i((ak′ ·k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′ |≤l
(4.18)
k∈
(ak′ · k′′ )(ak′′ · k)
k) = 0,
|k|2
ak · k = 0,
π 3
Z , |k| ≤ l.
l
Proposition 4.4. For each ω ∈ O(3) the following holds:
½
¾
X
imx
(4.19)
Rω Ml = v(x) =
bm e
.
m∈ πl ωZ3 ,
|m|≤l
Moreover,
P
u(x) =
k∈ πl Z3
|k|≤l
(4.20)
Rω u(x) =
ak eikx
P
bm eimx
m∈ πl ωZ3
⇒ bm = ωaω−1 m .
|m|≤l
Proof. Let u(x) =
P
ikx
|k|≤l ak e
∈ Ml . Then using the definition Rω u(x) = ωu(ω −1 x) and
applying the change of variables ωk = m get:
X
(4.21)
Rω u(x) =
ωak eiωkx =
k∈ πl Z3 ,
|k|≤l
X
ωaω−1 m eimx
m∈ πl ωZ3 ,
|m|≤l
This proves (4.19) and (4.20).
¤
Proposition 4.5. For each ω ∈ O(3) the Galerkin approximations for the Navier-Stokes equations on the space C 1 (0, T ; Rω Ml ) are of the following form:
(4.22)
∂t bm (t) + |m|2 bm +
X
m′ +m′′ =m,
m′ ,m′′ ∈ πl ωZ3 ,
|m′ |≤l, |m′′ |≤l
¶
µ
(bm′ · m′′ )(bm′′ · m)
m
= 0,
i (bm′ ·m′′ )bm′′ −
|m|2
m∈
bm · m = 0,
π 3
ωZ , |m| ≤ l.
l
Proof. To obtain the Galerkin approximations on C 1 (0, T ; Rω Ml ) for the Navier-Stokes equations, repeat the procedure above that leads to the Galerkin approximations (4.18): Re-write
18
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
(4.14) on the space of periodic fields C 1 (0, T ; Rω H 2 (Πl )) in terms of Fourier coefficients to get
the following analog of (4.15):
(4.23)
X
∂t bm + |m|2 bm +
m′ +m′′ =m,
m′ ,m′′ ∈ πl ωZ3
µ
¶
(bm′ · m′′ )(bm′′ · m)
m
= 0,
i (bm′ ·m′′ )bm′′ −
|m|2
bm · m = 0,
π 3
ωZ ,
l
and then repeat the derivation of (4.17), (4.18) from (4.14), (4.15) to finally get (4.22) from
m∈
(4.23).
¤
Now supplement (4.17) and (4.18) with the initial condition
X
X
(4.24)
u(t, x)|t=0 =
ak (t)eikx |t=0 = u0 (x) =
ak0 eikx
k∈ πl Z3 ,
|k|≤l
k∈ πl Z3 ,
|k|≤l
and
π 3
Z , |k| ≤ l
l
Moreover, supplement (4.22) with the initial condition
π
(4.26)
bm (t)|t=0 = bm0 , m ∈ ωZ3 , |k| ≤ l
l
Then, as is well-known, the Cauchy problem (4.17), (4.24), (equivalently, the Cauchy problem
(4.25)
ak (t)|t=0 = ak0 ,
k∈
for the ordinary differential equations (4.18), (4.25)) has a unique solution in C 1 (0, T ; Ml ). Call
this solution Sl (u0 ). Analogously, the Cauchy Problem (4.22),(4.26) possesses a unique solution.
Write this solution as the Fourier polynomial
X
(4.27)
Sl (v0 ) =
bm (t)eimx , where v0 =
m∈ πl ωZ3 ,
|k|≤l
X
bm0 eimx
m∈ πl ωZ3 ,
|k|≤l
Lemma 4.6. Let u0 ∈ Ml . Then Rω Sl (u0 ) solves (4.22) with initial condition v 0 = Rω u0 .
Moreover, if Sl (u0 ) admits the Fourier decomposition
X
ak (t)eikx ,
(4.28)
Sl (u0 ) =
k∈ πl Z3 ,
|k|≤l
then {ωak } satisfies
(4.29)
2
∂t ωak + |k| ωak +
X
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′|≤l
¶
µ
(ωak′ · ωk′′ )(ωak′′ · ωk)
′′
ωk = 0,
i (ωak′ · ωk )ωak′′ −
|k|2
ωak · ωk = 0,
k∈
π 3
Z , |k| ≤ l.
l
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
19
Proof. Let
(4.30)
X
Rω Sl (u0 ) =
bm (t)eimx
m∈ πl ωZ3 ,
|m|≤l
where, by (4.20),
(4.31)
bm = aω−1 m .
The assertion of the Lemma will be proved once it shown that the {b m (t)} satisfy (4.22). Substitution of (4.31) into the left hand side of (4.22), the change of variables m = ωk, and the fact
that ω ∈ O(3) yield:
∀k∈
π 3
Z , |k| ≤ l;
l
2
∂t ωak + |ωk| ωak +
X
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′|≤l
(4.32)
½
= ω ∂t ak + |k|2 ak +
X
¶
µ
(ωak′ · ωk′′ )(ωak′′ · ωk)
′′
ωk
i (ωak′ · ωk )ωak′′ −
|ωk|2
i((ak′ · k′′ )ak′′ −
k ′ +k ′′ =k,
k ′ ,k ′′ ∈ πl Z3 ,
|k ′ |≤l,|k ′′ |≤l
(ak′ · k′′ )(ak′′ · k)
k
|k|2
¾
= 0,
since {ak (t)} satisfies (4.18).
¤
Remark 4.7. As will be shown, more is true: The Galerkin PDE has a unique solution for any
dl , see (4.47) below.
initial condition v in M
The task now is to show that inclusions u 1 ∈ Ml and Rω u1 = u2 ∈ Ml , for some ω, implies
that for the same ω, Rω Sl u1 stays in Ml for all t, still solving the Galerkin system on M l . This
will then show that Rω Sl u1 = Sl u2 .
Definition 4.8. Given an isometry ω of R 3 , let Kω be the set of all elements k in the lattice
π 3
lZ
such that ωk also belongs to the lattice.
Lemma 4.9. Let u1 , u2 be vector fields of the form
(4.33)
u1 (x) =
X
ak eik·x ,
u2 (x) =
|k|≤l
X
|k|≤l
bk eik·x ,
where all
k∈
π 3
Z ,
l
not necessarily divergence free. Then for some ω isometry of R 3 , Rω u1 = u2 if and only if all
k’s in the representation of u1 are in Kω .
20
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
Proof. Since Rω u1 = u2 , Rω u1 is periodic of period 2l. Therefore
X
ωak eiωk·x =
|k|≤l
(4.34)
X
ωak eiωk·(x+(2l)j )
|k|≤l
=
X
ωak ei2l(ωk)j eiωk·x , j = 1, 2, 3,
|k|≤l
for (2l)j denoting the vector with 2l as j coordinate and zeroes on the rest. Now since e iωk·x
are orthonormal on the image of the cube Πl = [−l, l]3 under ω, this implies that
1 = ei2l(ωk)j ,
(4.35)
therefore
2l(ωk)j = 2πNkj , Nkj ∈ Z.
(4.36)
Therefore ωk ∈
π 3
Z . The converse is clear.
l
¤
Lemma 4.10. Let u1 in Ml of the form ,
(4.37)
u1 =
X
ak eik·x ,
k ∈ Kω .
|k|≤l
Then the Galerkin solution in Ml with initial condition u1 is of the form
(4.38)
u(t) =
X
ak (t)eik·x ,
ak (t) = 0 f or all t f or k ∈
/ Kω .
|k|≤l
Proof. In Ml , first solve the system
∂t ak = −|k|2 ak ,
k∈
/ Kω |k| ≤ l
¶
µ
X X
(ak′ )j ikj′′ ak′′ · k
′′
k = −|k|2 ak , k ∈ Kω , |k| ≤ l
(ak′ )j ikj ak′′ −
(4.39) ∂t ak +
2
|k|
′
′′
j
k +k =k
|k ′ |≤l
|k ′′ |≤l
with initial conditions
(4.40)
ak (0) = 0,
k∈
/ Kω
ak (0) = ak ,
k ∈ Kω ,
using the ak ’s from (4.37).
In particular
(4.41)
ak (t) = 0, for all t, for k not in Kω .
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
21
Now if k ∈
/ Kω , and k ′ + k ′′ = k, then either k ′ ∈
/ Kω or k ′′ ∈
/ Kω . Then the unique solution
of (4.39) (4.40) also solves the system
¶
X X µ
(ak′ )j ikj′′ ak′′ · k
′′
k = −|k|2 ak , k ∈
/ Kω , |k| ≤ l
(ak′ )j ikj ak′′ −
∂t ak +
2
|k|
′
′′
(4.42)
∂t ak +
j
k +k =k
|k ′ |≤l
|k ′′ |≤l
X
X
j
k ′ +k ′′ =k
|k ′ |≤l
|k ′′ |≤l
µ
(ak′ )j ikj′′ ak′′ −
(ak′ )j ikj′′ ak′′ · k
k
|k|2
¶
= −|k|2 ak , k ∈ Kω , |k| ≤ l,
i.e. solves the Galerkin system with initial condition u 1 .
¤
Proposition 4.11. Let u1 , u2 be in Ml of the form
X
X
(4.43)
u1 (x) =
ak eik·x , u2 (x) =
bκ eiκ·x ,
|k|≤l
|κ|≤l
Then u2 = Rω u1 implies Sl u2 = Rω Sl u1 .
Proof. Let
(4.44)
S l u1 =
X
aκ (t) eiκ·x ,
aκ (0) = aκ ,
S l u2 =
|κ|≤l
X
bκ (t) eiκ·x ,
bκ (0) = bκ .
|κ|≤l
By Lemma 4.6, Rω Sl u1 is the unique solution of the Galerkin system on R ω Ml with initial
condition Rω u1 , i.e. by (4.29) it solves
(4.45)
∂t ωak +
X
j
X
k ′ +k ′′ =k
|k ′ |≤l
|k ′′ |≤l
µ
(ωak′ )j i(ωk ′′ )j ωak′′ −
(ωak′ )j i(ωk ′′ )j ωak′′ · ωk
ωk
|ωk|2
Observe that in this system if ωk 6= κ for some κ in the lattice
π 3
lZ
¶
= −|ωk|2 ωak .
then ωak (t) = 0 for all t, by
Lemma 4.10. Now rename ωk = κ, ωak (t) = cκ (t), ωak (0) = cκ (0) = bκ , ωk ′ = κ′ , ωk ′′ = κ′′ to
get
(4.46)
∂t cκ +
X
j
X
κ′ +κ′′ =κ
|κ′ |≤l
µ
′′
(cκ′ )j i(κ )j cκ′′
¶
(cκ′ )j i(κ′′ )j cκ′′ · κ
κ = −|κ|2 cκ ,
−
|κ|2
|κ′′ |≤l
with initial conditions Rak . This gives a permutation of the Galerkin system on M l with initial
condition u2 . Therefore cκ (t) = bκ (t) for all κ ∈ πl Z3 .
¤
dl , there
4.3. Isotropic Galerkin approximations of statistical solution. Now given v in M
dl as
are ω in O(3) and u in Ml such that v = Rω u. Extend Sl from Ml to M
(4.47)
Sbl v = Rω Sl u.
22
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
This is well defined by Proposition 4.11: If R ω1 u1 = Rω2 u2 then a straightforward calculation
shows that Rω−1 ω1 u1 = u2 , therefore Rω−1 ω1 Sl u1 = Sl u2 , or Rω1 Sl u1 = Rω2 Sl u2 . In particular,
2
2
Sbl satisfies
Sbl Rω u = Rω Sl u
(4.48)
∀ u ∈ Ml .
Let v = Rω u and w = Rω1 v = Rωω1 u where u ∈ Ml . Applying Rω1 to both parts of (4.47)
gives Rω1 Sbl v = Rωω1 Sl u. On the other hand, (4.47) for w can be written as Sbl w = Rωω1 Sl u.
Comparing these two equalities gives
(4.49)
[VF2], p. 219 shows that
(4.50)
Rω1 Sbl v = Sbl Rω1 v,
dl .
∀v∈M
Th Sl = Sl Th .
Applying to both parts of (4.47) the translation operator T h and using (2.25), (4.50) gives
(4.51)
Th Sbl v = Rω Tω−1 h Sl u = Rω Sl Tω−1 h u.
On the other hand, applying Sbl as defined in (4.47) to Th v = Th Rω u = Rω Tω−1 h u, obtain
Sbl Th v = Rω Sl Tω−1 h u. Comparing this equality with (4.51) obtain
(4.52)
Define
(4.53)
Th Sbl v = Sbl Th v
dl .
∀v∈M
−1
Pbl (A) = µbl (Sbl A),
for any Borel subset A of L2 (0, T, H0 (r)) where, recall,
(4.54)
−1
dl ).
Sbl A = γ0 (A ∩ Sbl M
dl it is enough to consider Borel sets A satisfying A ∩
Since the measure µbl is supported on M
dl 6= ∅.
Sbl M
Definition (4.53) is the isotropic version of the measure P l defined in [VF2]:
(4.55)
Pl (A) = µl (Sl −1 A)
for any Borel subset A of L2 (0, T, H0 (r)).
Lemma 4.12. Pbl is homogeneous and isotropic.
Proof. It suffices to show that for all A ∈ B(L 2 (0, T ; H0 (r))):
(4.56)
and
(4.57)
−1
−1
Sbl Rω A = Rω Sbl A
−1
−1
Sbl Th A = Th Sbl A,
−1
since Pbl (A) = µbl (Sbl A) and µbl is homogeneous and isotropic.
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
23
Using (4.54), (4.49),
−1
dl )
Sbl Rω A = γ0 (Rω A ∩ Rω Rω−1 Sbl M
dl )
= γ0 Rω (A ∩ Sbl Rω−1 M
(4.58)
dl )
= Rω γ0 (A ∩ Sbl M
−1
= Rω Sbl A.
Also, by (4.54), (4.52)
(4.59)
−1
dl )
Sbl Th A = γ0 (Th A ∩ Th T−h Sbl M
dl )
= γ0 Th (A ∩ Sbl T−h M
dl )
= Th γ0 (A ∩ Sbl M
= Th Sbl
−1
A.
¤
The following relation between Pl and Pbl allows the known estimates on for Pl to be carried
over to Pbl . Once again, let a to be the action map of rotations on vector fields:
(4.60)
a(ω, u) = Rω u
Lemma 4.13. The equality holds:
Pbl (A) = (Pl × H)(a−1 A).
where H is the Haar measure on O(3), normalized.
Proof.
−1
Pbl (A) = µbl (Sbl A)
= (µl × H)(a−1 Sbl
(4.61)
−1
A)
−1
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Rω u0 ∈ Sbl A}
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Sbl Rω u0 ∈ A}
= (µl × H){(u0 , ω) ∈ Ml × O(3) : Rω Sl u0 ∈ A}, by (4.48)
= (Pl × H){(Sl u0 , ω) ∈ C 1 (0, T ; Ml ) × O(3) : Rω Sl u0 ∈ A}
= (Pl × H)(a−1 A).
¤
24
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
5. Convergence of isotropic Galerkin approximations
5.1. Galerkin Approximation of Homogeneous Statistical Solutions. This section summarizes the Galerkin approximation in [VF2]. Recall that the following are shown in Chapter
VII there:
Given any µ homogeneous probability measure on H 0 (r), there exist for each l homogeneous
probability measures µl on H0 (r), supported on Ml , converging to µ in characteristic i.e.:
Z
Z
i
ei µ(du), l → ∞,
e
µl (du) →
(5.1)
H0 (r)
H0 (r)
for any test function ν.
5.1.1. Convergence. The probability measures P l , defined by (4.55) are homogeneous in x if
the initial µ is homogeneous, and converge weakly to a homogeneous probability measure P on
L2 (0, T ; H0 (r)). Weak convergence relies on the following three uniform estimates on P l : There
are constants C, C(N ) independent of l such that for each t in [0, T ],
¶
Z
Z µ
Z t
2
2
|∇u| (τ, x) dτ Pl (du) ≤ C |u0 (x)|2 µl (du0 ),
(5.2)
|u(t, x)| +
0
(5.3)
Z
ku|BN k−s Pl (du) ≤ C(N ),
(5.4)
Z
k∂t u|BN k−s Pl (du) ≤ C(N )
for kv|BN k−s the dual norm (3.2), and with s > 11/2.
5.1.2. P is a homogeneous statistical solution of the Navier-Stokes system. As already remarked,
the weak limit P is supported on L2 (0, T ; H1 (r))∩ BV −s ∩ GN S so that each u(., .) in the support
of P satisfies the weak form (3.1) of Navier-Stokes equations, and the right limit in time γ t u
exists for each t with respect to the Φ −s norm. This extra regularity of P relies on the estimate
(5.5)
Z
kukBV −s P (du) ≤ ∞,
s > 11/2
for kukBV −s the norm (3.3). In addition, P satisfies energy estimate (3.13).
5.1.3. P solves the initial value problem. The measure defined by P (γ 0−1 A), for A Borel of H0 (r),
is the initial measure µ. The proof of this relies on the convergence (5.1).
5.2. Homogeneous and Isotropic Statistical Solutions via isotropic Galerkin Approximations. In the setting of isotropic measures, the construction of the previous subsection can
be repeated as follows:
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
25
5.2.1. Approximation of initial measure. Given initial homogeneous and isotropic µ
b on H 0 (r),
construct its (merely) homogeneous approximation µ l , with µl → µ
b in characteristic as in the
previous section. Then use Definitions 4.1 and 4.2 to obtain from the µ l ’s probability measures
dl . Convergence in characteristic
µbl that are now homogeneous and isotropic and supported by M
still holds, thanks to the following:
Lemma 5.1. µbl converges to µ
b in characteristic as l → ∞.
Proof. For µbl defined as above,
χµbl (ν) =
=
(5.6)
=
=
Convergence Theorem,
χµbl (ν) =
Z
Z
Z
Z
R
Since |χµl ((Rω−1 ν)| ≤ 1 and
(5.7)
Z
O(3)
H0 (r)
H0 (r)
O(3)
O(3)
χµbl (ν) → χµb (ν) as l → ∞.
Z
Z
eiRω u·ν dω µl (du)
O(3)
H0 (r)
eiu·Rω−1 ν µl (du) dω
χµl (Rω−1 ν) dω.
O(3) 1
dω = 1, (5.1) implies, via the Lebesgue Dominated
χµl (Rω−1 ν) dω →
as l → ∞. Therefore, since
Z
Z
χµb (Rω−1 ν) dω =
(5.8)
O(3)
eiu·ν µbl (du)
O(3)
Z
H0 (r)
Z
O(3)
χµb (Rω−1 ν) dω
eiRω u·ν µ
b(du) dω = χµb (ν),
¤
5.2.2. Convergence to a homogeneous and isotropic measure. Now construct the homogeneous
and isotropic measures Pbl according to (4.53). To prove weak convergence the following is needed
Lemma 5.2. The following estimates hold:
Z
(5.9)
ku|BN k−s Pbl (du) ≤ C(N ),
(5.10)
Z
k∂t u|BN k−s Pbl (du) ≤ C(N ).
Proof. By Lemma 4.13 and relations (3.21),
Z
Z Z
b
kRω ∂t u|BN k−s dω Pl (du)
k∂t u|BN k−s Pl (du) =
O(3)
(5.11)
Z
= k∂t u|BN k−s Pl (du).
26
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
[VF2] Chapter VII, Theorem 3.1 proves that the right hand side of (5.11) is bounded by a
constant C(N ) that depends on N but does not depend on l. This proves (5.10). The bound
(5.9) is proved similarly.
¤
The following is also needed to repeat the proof of the weak convergence of the Pbl ’s.
Lemma 5.3. With pointwise averages defined as in (3.8),(3.9),(3.10), for any t in [0, T ],
¶
Z µ
Z t
¡ 2
¢
2
2
|u| (τ, x) + |∇u| (τ, x) dτ Pbl (du)
|u| (t, x) +
0
(5.12)
Z
≤ C |u|2 (x) µbl (du).
Proof. Using an equality similar to (3.9), (3.10) one can show that
Z
Z
Z
2
(5.13)
|u| (t, x) µbl (du) =
|u|2 (t, x) Rω∗ µl (du) dω,
O(3)
(5.14)
Z µ
2
|u| (t, x) +
=
Z
Z
t¡
2
2
¢
|u| (t, x) + |∇u| (t, x) dτ
0
O(3)
Z µ
2
|u| (t, x) +
Z tµ
2
¶
Pbl (du)
2
|u| (t, x) + |∇u| (t, x)
0
¶
dτ
¶
Rω∗ Pl (du) dω.
Applying to (5.13), (5.14) Lemma 3.10 and taking into account the inequality for Galerkin
approximations of the homogeneous statistical solution
¶
Z µ
Z t
¡ 2
¢
2
2
|u| (t, x) + |∇u| (t, x) dτ Pl (du)
|u| (t, x) +
0
(5.15)
Z
≤ C |u|2 (x) µl (du)
that was proved in [VF2] Chapter VII, Lemma 2.4, yields (5.12).
¤
Lemma 5.4. The measures µbl satisfy
Z
Z
2
(5.16)
|u| (x) µbl (du) ≤ |u|2 (x) µl (du)
¤
Proof. This follows from (5.13) and Lemma 3.10.
With Lemmas 5.2, 5.3, 5.4 established, there are no further obstacles in repeating the arguments in [VF2] to show that the family Pbl is weakly compact. The argument for this in [VF2]
is that the measures Pl are supported on the space Ω of elements in L 2 (0, T ; H0 (r)) with the
following norm finite:
(5.17)
∞
X
N =1
1
2N C(N )
(kukL2 (0,T ;H−s (BN )) + k
∂u
k 1
+ kukL2 (0,T ;H1 (r1 )) ) < ∞,
−s
∂t L (0,T ;H (BN ))
s > 11/2,
3
r < r1 < − .
2
HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS
27
Then Ω is compactly embedded in L 2 (0, T ; H0 (r)), [VF2], Chapter VII, Lemma 5.2, and
Z
(5.18)
sup kukΩ Pl (du) < ∞,
l
[VF2], proof of Theorem 6.1 in Chapter VII. From the estimates of Lemmas 5.2, 5.3 for the
measures Pbl , the energy conservation (5.12), and the uniform estimate (5.4), it becomes clear
that the all measures Pbl are supported on Ω and the uniform estimate (5.18), with P l changed
to Pbl , still holds.
b be the limit of some weakly convergent subsequence of Pbl ’s, as l → ∞.
Let Q
b weakly on L2 (0, T ; H0 (r)), then
Lemma 5.5. If Pbl are homogeneous and isotropic and Pbl ⇒ Q
b is homogeneous and isotropic.
Q
b weakly then by definition
Proof. If Pbl ⇒ Q
Z
Z
b
b
(5.19)
f (u) Pl (du) → f (u) Q(du).
for any f continuous and bounded on L 2 (0, T ; H0 (r)). Since the measures Pbl are homogeneous
and isotropic, by definitions 2.2, 2.4
Z
Z
b
f (u) Pl (du) = f (Th u) Pbl (du)
Z
(5.20)
= f (Rω u) Pbl (du),
for any h ∈ R3 and ω ∈ O(3). These equalities and
Z
Z
b
f (Th u) Pbl (du) → f (Th u) Q(du),
Z
Z
(5.21)
b
b
f (Rω u) Pl (du) → f (Rω u) Q(du),
imply
(5.22)
Z
b
f (u) Q(du)
=
=
Z
Z
b
f (Th u) Q(du)
b
f (Rω u) Q(du).
¤
b is in
5.2.3. Extra regularity for right t-limits and the initial condition. That the support of Q
addition in L2 (0, T ; H1 (r)) ∩ BV −s (where right limits with respect to time are well defined by
[VF2], Chapter VII, Lemma 8.2) uses only the estimate
Z
b
(5.23)
kukBV −s Q(du)
≤∞
for kukBV −s the norm (3.3), which follows as in the proof of Lemma 5.2. Define therefore γ 0 by
b as the initial value of Q.
b
(3.6) and think of γ0∗ Q
28
S. DOSTOGLOU, A. V. FURSIKOV, AND J. D. KAHL
That the initial value γ0∗ P of the the homogeneous statistical solution P of Theorem 3.4 is the
initial measure µ is shown in [VF2] as Theorem 10.1, Lemma 10.1, Lemma 10.2, and Theorem
10.2 of Chapter VII there. Of these, Lemma 10.1, Lemma 10.2, and Theorem 10.2 of Chapter
b Theorem 10.1 uses only the convergence in characteristic of the
VII are valid verbatim for Q.
µl ’s to µ. The analogous convergence of the µbl ’s to µ
b was established here as Lemma 5.1.
b
6. The support of the measure Q.
This section uses the approach of [VF2] Chapter VII, Section 7, to show that the homogeneous
b is supported by generalized solutions of the Navier-Stokes equations. To
and isotropic measure Q
realize this approach, some subtle points regarding the definition of the equations for isotropic
Galerkin approximations in the x-representation need to be addressed first.
6.1. Equations for isotropic Galerkin approximations in the x-representation. Section 4 defined isotropic Galerkin approximations by introducing and investigating the Galerkin
equations in terms of Fourier coefficients. Here, a complete description of the Galerkin equations
in the x-representation is given, beginning with a more precise determination of the domain of
their definition.
dl defined by (4.1), (4.3), introduce the set of periodic vector
In addition to the sets Ml and M
fields Nl with the cube of periods Πl define