2 Description of the model
2.1 GOY and Sabra shell models
Let H be the set of all sequences u = u
1
, u
2
, . . . of complex numbers such that P
n
|u
n
|
2
∞. We consider H as a real Hilbert space endowed with the inner product
·, · and the norm |·| of the form u, v = Re
X
n ≥1
u
n
v
∗ n
, |u|
2
= X
n ≥1
|u
n
|
2
, 2.1
where v
∗ n
denotes the complex conjugate of v
n
. Let k 0, µ 1 and for every n ≥ 1, set k
n
= k µ
n
. Let A : DomA
⊂ H → H be the non-bounded linear operator defined by Au
n
= k
2 n
u
n
, n = 1, 2, . . . ,
DomA = n
u ∈ H :
X
n ≥1
k
4 n
|u
n
|
2
∞ o
. The operator A is clearly self-adjoint, strictly positive definite since Au, u
≥ k
2
|u|
2
for u ∈ DomA.
For any α 0, set
H
α
= DomA
α
= {u ∈ H : X
n ≥1
k
4 α
n
|u
n
|
2
+∞}, kuk
2 α
= X
n ≥1
k
4 α
n
|u
n
|
2
for u ∈ H
α
. 2.2
Let H
= H, V := DomA
1 2
= n
u ∈ H :
X
n ≥1
k
2 n
|u
n
|
2
+∞ o
; also set H = H
1 4
, kuk
H
= kuk
1 4
. Then V as each of the spaces
H
α
is a Hilbert space for the scalar product u, v
V
= Re P
n
k
2 n
u
n
v
∗ n
, u, v
∈ V and the associated norm is denoted by kuk
2
= X
n ≥1
k
2 n
|u
n
|
2
. 2.3
The adjoint of V with respect to the H scalar product is V
′
= {u
n
∈ C
N
: P
n ≥1
k
−2 n
|u
n
|
2
+∞} and V
⊂ H ⊂ V
′
is a Gelfand triple. Let 〈u , v〉 = Re
P
n ≥1
u
n
v
∗ n
denote the duality between u
∈ V and v
∈ V
′
. Clearly for 0 ≤ α β, u ∈ H
β
and v ∈ V we have
kuk
2 α
≤ k
4 α−β
kuk
2 β
, and kvk
2 H
≤ |v| kvk, 2.4
where the last inequality is proved by the Cauchy-Schwarz inequality. Set u
−1
= u = 0, let a, b be real numbers and B : H × V → H or B : V × H → H denote the bilinear
operator defined by [Bu, v]
n
= −i
ak
n+1
u
∗ n+1
v
∗ n+2
+ bk
n
u
∗ n
−1
v
∗ n+1
− ak
n −1
u
∗ n
−1
v
∗ n
−2
− bk
n −1
u
∗ n
−2
v
∗ n
−1
2.5
for n = 1, 2, . . . in the GOY shell-model see, e.g., [25] or [Bu, v]
n
= −i
ak
n+1
u
∗ n+1
v
n+2
+ bk
n
u
∗ n
−1
v
n+1
+ ak
n −1
u
n −1
v
n −2
+ bk
n −1
u
n −2
v
n −1
,
2.6 in the Sabra shell model introduced in [21].
2554
Note that B can be extended as a bilinear operator from H × H to V
′
and that there exists a constant C
0 such that given u, v ∈ H and w ∈ V we have |〈Bu, v , w〉| + | Bu, w , v
| + | Bw, u , v| ≤ C |u| |v| kwk. 2.7
An easy computation proves that for u, v ∈ H and w ∈ V resp. v, w ∈ H and u ∈ V ,
〈Bu, v , w〉 = − Bu, w , v resp. Bu, v , w = − Bu, w , v .
2.8 Furthermore, B : V
× V → V and B : H × H → H; indeed, for u, v ∈ V resp. u, v ∈ H we have kBu, vk
2
= X
n ≥1
k
2 n
|Bu, v
n
|
2
≤ C kuk
2
sup
n
k
2 n
|v
n
|
2
≤ C kuk
2
kvk
2
, 2.9
|Bu, v| ≤ C kuk
H
kvk
H
. For u, v in either H,
H or V , let Bu := Bu, u. The anti-symmetry property 2.8 implies that |〈Bu
1
− Bu
2
, u
1
− u
2
〉
V
| = |〈Bu
1
− u
2
, u
2
〉
V
| for u
1
, u
2
∈ V and |〈Bu
1
− Bu
2
, u
1
− u
2
〉| = |〈Bu
1
− u
2
, u
2
〉| for u
1
∈ H and u
2
∈ V . Hence there exist positive constants ¯ C
1
and ¯ C
2
such that |〈Bu
1
− Bu
2
, u
1
− u
2
〉
V
| ≤ ¯ C
1
ku
1
− u
2
k
2
ku
2
k, ∀u
1
, u
2
∈ V, 2.10
|〈Bu
1
− Bu
2
, u
1
− u
2
〉| ≤ ¯ C
2
|u
1
− u
2
|
2
ku
2
k, ∀u
1
∈ H, ∀u
2
∈ V . 2.11
Finally, since B is bilinear, Cauchy-Schwarz’s inequality yields for any α ∈ [0,
1 2
], u, v ∈ V : A
α
Bu − A
α
Bv , A
α
u − v ≤
A
α
Bu − v, u + A
α
Bv, u − v , A
α
u − v ≤ Cku − vk
2 α
kuk + kvk. 2.12
In the GOY shell model, B is defined by 2.5; for any u ∈ V , Au ∈ V
′
we have 〈Bu, u, Au〉 = Re
− i X
n ≥1
u
∗ n
u
∗ n+1
u
∗ n+2
µ
3n+1
k
3
a + bµ
2
− aµ
4
− bµ
4
. Since
µ 6= 1, a1 +
µ
2
+ bµ
2
= 0 if and only if
〈Bu, u , Au〉 = 0, ∀u ∈ V. 2.13
On the other hand, in the Sabra shell model, B is defined by 2.6 and one has for u ∈ V ,
〈Bu, u , Au〉 = k
3
Re − i
X
n ≥1
µ
3n+1
h a + b µ
2
u
∗ n
u
∗ n+1
u
n+2
+ a + bµ
4
u
n
u
n+1
u
∗ n+2
i .
Thus Bu, u, Au = 0 for every u ∈ V if and only if a + bµ
2
= a + bµ
4
and again µ 6= 1 shows that
2.13 holds true.
2.2 Stochastic driving force
Kategori