Approximate Deconvolution Model in a bounded domain with a vertical regularization

0.14 MB·English

by  Hani Ali

Approximate Deconvolution Model in a bounded domain with
a vertical regularization
a
Hani Ali
a
MAP5, CNRS UMR 8145,
Universit´e Paris Descartes,
75006 Paris, France
[email protected]
Abstract
3 We study the global existence issue for a three-dimensional Approximate Deconvolution Model
1 withaverticalfilter. Weconsiderthismodelinaboundedcylindricaldomainwhereweconstruct
0
a unique global weak solution. The proof is based on a refinement of the energy method given
2
by Berselli in [3].
n
a MSC: 76D05; 35Q30; 76F65; 76D03
J

8
1

1 Introduction and notation
]
P
A
The Large eddy simulation model (LES) with anisotropic regularization is introduced in [3] in
.
h order to study flows in bounded domains.
at Instead of choosing the classical filter
m
[ A := I − α21∂12+α22∂22+α23∂32 , (1.1)

1 (cid:0) (cid:1)
the author in [3], considers a horizontal filter
v
2
8 A := I − α2∂2+α2∂2 . (1.2)
h 1 1 2 2
2
4 (cid:0) (cid:1)
This filter is less memory consuming than the classical one [7, 6, 8]. Moreover, there is no need
.
1 to introduce artificial boundary conditions for Helmholtz operator. However, the smoothing
0
created by both of these filters is unnecessary strong. For this reason one can replace the
3
1 classical Laplace operator by the Laplace operator with fractional regularization θ and seek for
:
v the limiting case where we can prove global existence and uniqueness of regular solutions. In
i a series of papers [10, 4, 1, 2], it was shown that the LES models which are derived by using
X
instead of the operator (1.1), the operator A = I+α2θ(−∆)θ for suitable (large enough) θ < 1,
r θ
a are still well-posed.
In this paper, we consider LES models with fractional filter acting only in one variable
A := I +α2θ(−∂ )2θ, 0 ≤ θ ≤ 1. (1.3)
3,θ 3 3
In particular, we study the global existence and uniqueness of solutions to the vertical LES
modelon aboundedproductdomain of thetypeD = Ω×(−π,π), whereΩ is a smoothdomain,
with homogeneous Dirichlet boundary conditions on the lateral boundary ∂Ω ×(−π,π), and
with periodic boundary conditions in the vertical variable. For simplicity, we consider the
domain D = x ∈ R3,x2+x2 < d,−π < x < π with 2π periodicity with respect to x . More
1 2 3 3
precisely, we consider the system of equations
(cid:8) (cid:9)
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∂ w+∇·w⊗w−ν∆w+∇q = f, in D, t > 0,
t
∇·w = 0, in D, t > 0,

w| = 0, t > 0, (1.4)
 ∂Ω×(−π,π)
 w(x ,x ,x ) = w(x ,x ,x +2π), t > 0,
 1 2 3 1 2 3
w(x,0) = w (x) = v .
0 0



We recall that, in theabove equations, the symbol “ ” denotes the vertical filter (1.3), applied

component-by-component to the various tensor fields. Given that the filter is acting only in
the vertical variable, it is possible to require the periodicity only in x . Moreover, we consider
3
the filtered function with homogeneous Dirichlet boundary conditions on the lateral boundary
∂Ω×(−π,π). These boundary conditions of the filtered function are supposed to be the same
as the unfiltered ones, in order to prevent from introducing artificial boundary conditions.
The use of the vertical filter can be extended to a family of Approximate Deconvolution
model. The deconvolution family including (1.4) as the zeroth order case, is given by
∂ w+∇·D (w)⊗D (w)−ν∆w+∇q = f, in D, t > 0,
t N,θ N,θ
∇·w = 0, in D, t > 0,

w| = 0, t > 0, (1.5)
 ∂Ω×(−π,π)

 w(x ,x ,x ) = w(x ,x ,x +2π), t > 0,
 1 2 3 1 2 3
w(x,0) = w (x)= v ,
0 0


where D isa deconvolution operator of order N and is given by
N,θ 
N
D = (I −A−1)i, 0 ≤ θ ≤1. (1.6)
N,θ 3,θ
i=0
X
In order to state our main result pertaining to System (1.5), let us introduce our notation.
We set:
π
L2(D) := v ∈ L2(D)3,2π-periodic in x ,and vdx = 0 , (1.7)
3 3
(cid:26) Z−π (cid:27)
endowed with the norm kvk := v·v dx which is the usual norm in L2(D)3.
2
ZD
Then, we define the following spaces that are commonly used in the study of the NSE:
H := v ∈ L2(D), such that ∇·v = 0 and v·n = 0 on ∂D , (1.8)
V := v ∈ H, such that ∇v ∈ L2(D) and v = 0 on ∂D . (1.9)
(cid:8) (cid:9)
Our main result is the foll(cid:8)owing. (cid:9)
Theorem 1.1. Assume f ∈ L2(0,T;H) be a divergence free function and v ∈ H. let 0 ≤
0
N < ∞ be a given and fixed number and let θ > 1. Then problem (1.5), including problem (1.4)
2
when N =0, has a unique regular weak solution.
This result is one of the few results that consider the LES model on a bounded domain. It
holds also true on the whole space R3 and on the torus T . Other α models, with partial filter,
3
will be reported in a forthcoming paper.
The remaining part of the paper is organized as follows. In the second section, we introduce
several notations and spaces. We also prove auxiliary results which will be used in the third
section. In the third section, we prove Theorem 1.1. Finally, a functional inequality is given in
the appendix.
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2 Preliminaries and auxiliary results
Let 1 < p ≤ +∞ and 1 < q ≤ +∞. We denote by LqLp(D) = Lq((−π,+π);Lp(Ω)) the space
v h
+π 1/q
of functions g such that |g(x ,x ,x )|pdx dx )q/pdx <+∞. In what follows, we
1 2 3 1 2 3
Z−π (cid:18)ZΩ (cid:19)
denote by ∇ the gradient operator with respect to the variables x and x . Of course, we have
h 1 2
k∇ vk ≤ k∇vk , for any v ∈V.
h 2 2
Next, we will give some preliminary results which will play a important role in the proof of the
main theorem.
Lemma 2.1. For any u ∈ L2(D)3 satisfying homogeneous Dirichlet boundary conditions on
the boundary ∂Ω×(−π,π) , such that ∇ u ∈ L2(D)3, we have the following estimate
h
1 1
kukL2vL4h(D)3 ≤ Ckuk22k∇huk22 (2.1)
Proof. See in [11].
Lemma 2.2. Let g be a 2π-periodic and mean zero function with g(x) ∈ Hs([−π,π]), and let
s > 1, then we have the following inequality
2
1 1
1−
kgkL∞ ≤ CkgkL22skgkH2ss. (2.2)
Proof. The proof is given in Appendix.
Lemma 2.3. Let s > 1, and suppose that u and ∂su ∈ L2(D)3, then there exists a constant
2
C > 0 such that
1 1
kukL∞v L2h(D)3 ≤ Ckuk21−2sk∂3suk22s.
Proof. We first apply inequality (2.2) in the vertical variable and then use the H¨older
inequality in the horizontal variable. We get,
1
1
2
2
sup |u(x)|2dx dx ≤ sup |u(x)|2dx dx
1 2 1 2
x3∈[−π,π](cid:18)ZΩ (cid:19) ZΩx3∈[−π,π] !
1
π 1−21s π 21s 2 (2.3)
≤ C |u(x)|2dx |∂su(x)|2dx dx dx
3 3 3 1 2
ZΩ(cid:18)Z−π (cid:19) (cid:18)Z−π (cid:19) !
1 1
≤ Ck∇uk1−2sk∂s∇uk2s.
2 3 2
The following lemma will be useful for the estimate of the nonlinear term.
Lemma 2.4. There exists a positive constant C > 0 such that, for any s > 1 and for any
2
smooth enough divergence-free vector fields u, v, and w, the following estimates hold,
1 1 1 1 1 1
(i) |((u·∇)v,w)|≤ Ckuk2k∇uk2k∇vk1−2sk∂s∇vk2skwk2k∇wk2,
2 2 2 3 2 2 2
and
1 1 1 1 1 1
(ii) |((u·∇)v,w)| ≤ Ckuk2k∇uk2kvk2k∇vk2k∇wk1−2sk∂s∇wk2s.
2 2 2 2 2 3 2
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Proof. (i) By using lemma 2.1 and lemma 2.3 we get
|((u·∇)v,w)|≤ kukL2vL4h(D)3k∇vkL∞v L2h(D)3kwkL2vL4h(D)3
1 1 1 1 1 1
≤ Ckuk2k∇uk2k∇vk1−2sk∂s∇vk2skwk2k∇wk2
2 2 2 3 2 2 2
(ii) We have
|((u·∇)v,w)|≤ v⊗u : ∇w dx ,
(cid:12)ZD (cid:12)
(cid:12) (cid:12)
thus, in order to prove the second inequality i(cid:12)t is sufficient to sw(cid:12)ap the roles of v and w.
(cid:12) (cid:12)
2.1 The vertical filter and the vertical deconvolution operator
Let v beasmooth function of theformv = c (x ,x )eik3x3. Theaction of theverti-
k3∈Z\{0} k3 1 2
calfilteronv(x) = c (x ,x )eik3x3 canbewrittenasA (v) = A (k )c (x ,x )eik3x3,
k3 1 2 P 3,θ θ 3 k3 1 2
k3∈XZ\{0} k3∈XZ\{0}
where the Fourier transform with respect to x of the vertical filter is given by
3
A (k ) = 1+α2θ|k |2θ . (2.4)
θ 3 3
(cid:16) (cid:17)
Therefore, by using the Parseval’s identity with respect to x we get,
3
1
kA2 vk2 = kvk2+α2θk∂θvk2 = (A v,v). (2.5)
3,θ 2 2 3 2 3,θ
Next, we recall some properties of the vertical filter. These properties are proved in [3] for
the horizontal filter and hold true for the vertical filter.
Lemma 2.5. Let f be a smooth function and w a smooth, space-periodic with respect to x , and
3
divergence-free vector field defined on the domain D such that w·n = 0 on ∂D. Let the filtering
be defined by (1.3). Then, the following identities hold true:
f,w = (f,w), (2.6)
w−α2θ∂2θw(cid:0)= w−(cid:1) α2θ∂2θw, (2.7)
3 3
and
∇·(w⊗w),A w = (∇·(w⊗w),w) = 0. (2.8)
3,θ
(cid:16) (cid:17)
Proof. See in [3].
The deconvolution operator D = N (I − A−1)i is constructed by using the vertical
N,θ i=0 3,θ
filter with fractional regularization (1.3). For a fixed N > 0 and for θ = 1, we recover a vertical
P
operator formfrom theVan Cittert deconvolution operator (see [14] and[5]). A straightforward
calculation yields
D c (x ,x )eik3x3
N,θ k3 1 2 
k3∈XZ\{0} (2.9)
  α2θ|k |2θ N+1
= 1+α2θ|k |2θ 1− 3 c (x ,x )eik3x3.
3 1+α2θ|k |2θ k3 1 2
k3∈XZ\{0}(cid:16) (cid:17) (cid:18) 3 (cid:19) !
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Thus
D c (x ,x )eik3x3 = D (k )c (x ,x )eik3x3, (2.10)
N,θ k3 1 2  N,θ 3 k3 1 2
k3∈XZ\{0} k∈XZ\{0}
where we have fork ∈ Z\{0}, and θ ≥ 0, 
3
D (k ) = 1, (2.11)
0,θ 3
1 ≤ D (k ) ≤ N +1 for each N > 0, (2.12)
N,θ 3
and D (k ) ≤ A for a fixed α > 0. (2.13)
N,θ 3 3,θ
From the previous hypothesis, one can prove the following Lemma by adapting the results
summarized in the isotropic case in [4]:
Lemma 2.6. For θ ≥ 0, k ∈ Z\{0} and for each N > 0, there exists a constant C > 0 such
3
that for all v sufficiently smooth we have
kvk ≤ kD (v)k ≤ (N +1)kvk , (2.14)
2 N,θ 2 2
kvk2 ≤ CkDN,θ(v)k2 ≤ CkA3,θ12DN21,θ(v)k2, (2.15)
kA3,θ21DN21,θ(v)k2 ≤kvk2, (2.16)
kvk22+α2θk∂3θvk22 ≤ kA3,θ12DN12,θ(v)k22. (2.17)
3 Existence and uniqueness results
In this section, we give a definition of what is called a regular weak solution to problem (1.5).
Then, we give the proof of theorem 1.1.
Definition 3.1. Let f ∈ L2(0,T;H) be a divergence free function and v ∈ H. For any
0
0 ≤ θ ≤ 1 and 0 ≤ N < ∞, w is called a “regular” weak solution to problem (1.5) if the
following properties are satisfied:
w ∈ C(0,T;H)∩L2(0,T;V), (3.1)
∂θw ∈ C(0,T;H)∩L2(0,T;V), (3.2)
3
∂ w ∈ L2(0,T;V∗), (3.3)
t
and the velocity w fulfills
T
(∂ w,ϕ)+ ∇·(D (w)⊗D (w)),ϕ +ν(∇w,∇ϕ) dt
t N,θ N,θ
Z0 (cid:16) (cid:17) (3.4)
T
= f,ϕ dt for all ϕ ∈ L2(0,T;V).
Z0
(cid:0) (cid:1)
Moreover,
w(0) = w . (3.5)
0
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3.1 Proof of Theorem 1.1
The proof of Theorem 1.1 follows the classical scheme. We start by constructing approximated
solution wn via Galerkin method. Then, we seek for a priori estimates that are uniform with
respect to n. Next, we take the limit in theequations after having usedcompactness properties.
Finally, we show that the constructed solution is unique thanks to Gronwall’s lemma [15].
Step 1(Galerkinapproximation). Wedenoteby ϕk3 +∞ theeigenfunctionsoftheStokes
|k3|=1
operatoronD,withDirichletboundaryconditionson∂D andwithperiodicityonlywithrespect
(cid:8) (cid:9)
to x . Theexplicit expressionof these eigenfunctions can befoundin [12]. Theseeigenfunctions
3
are linear combinations of W (x ,x )eik3x3, where k ∈ Z\{0}, for certain families of smooth
k3 1 2 3
functions W : Ω → R vanishing at ∂Ω.
k3
We set
wn(t,x) = cn (x ,x ,t)ϕk3(x).
k3 1 2 (3.6)
k3∈Z\{X0},|k3|≤n
Since we are dealing with real functions, one has to take suitable conjugation and alternatively
use linear combinations of sines and cosines in the variable x . We look for wn(t,x) which is
3
determined by the following system of equations
∂ wn,ϕk3 +(∇·(D (wn)⊗D (wn)),ϕk3)+ν(∇wn,∇ϕk3)
t N,θ N,θ
 (cid:16) = hf,ϕ(cid:17)k3i , |k | = 1,2,...,n, (3.7)
 3

wn(0) = P (w ) =P (v ),
n 0 n 0
wherePn denotes the projection operator over Hn := Spanhϕ1,...,ϕni. Therefore, the classical
Caratheodory theory [16] implies the short-time existence of solutions to (3.7). Next, we derive
estimates on cn that is uniform w.r.t. n. These estimates imply that the solution of (3.7),
constructed on a short time interval [0,Tn[, exists for all t ∈[0,T].
Step 2 (A priori estimates) We need to derive an energy inequality for wn. This can be
obtained by using A D (wn) as a test function in (3.7). Thanks to the explicit expression
3,θ N,θ
of the eigenfunctions, and the properties of A and D , the quantity A D (wn) is a
3,θ N,θ 3,θ N,θ
legitimate test function since it still belongs to H . Standard manipulations and the use of
n
Lemma 2.5 combined with the following identities
1 d 1 1
(∂ wn,A D (wn)) = kD2 (wn)k2 +α2θk∂θD2 (wn)k2
t 3,θ N,θ 2dt N,θ 2 3 N,θ 2
(cid:18) (cid:19) (3.8)
1 d 1 1
= kA2 D2 (wn)k2,
2dt 3,θ N,θ 2
1 1
−∆wn,A D (wn) = k∇D2 (wn)k2+k∂θ∇D2 (wn)k2
3,θ N,θ N,θ 2 3 N,θ 2
(cid:18) (cid:19) (3.9)
(cid:0) (cid:1) 1 1
= k∇A2 D2 (wn)k2,
3,θ N,θ 2
and
1 1
f,A D (wn) = D2 (f),D2 (wn) , (3.10)
3,θ N,θ N,θ N,θ
(cid:18) (cid:19)
(cid:0) (cid:1)
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lead to the a priori estimate
1 1 1 t 1 1
kA2 D2 (wn)k2 +ν k∇A2 D2 (wn)k2 ds
2 3,θ N,θ 2 3,θ N,θ 2
t 1 1 Z0 1 1 1 (3.11)
= D2 (f),D2 (wn) ds+ kA2 D2 (vn)k2.
N,θ N,θ 2 3,θ N,θ 0 2
Z0 (cid:18) (cid:19)
By using the Cauchy-Schwartz inequality, the Poincar´e inequality combined with the Young
inequality and inequality (2.16), we conclude from (3.11) the following inequality
1 1 t 1 1
sup kA2 D2 (wn)k2 +ν k∇A2 D2 (wn)k2 ds
3,θ N,θ 2 3,θ N,θ 2
t∈[0,Tn[ Z0 (3.12)
C(N +1) T
≤ kvnk2+ kfk2 ds
0 2 ν 2
Z0
which immediately implies that the existence time is independent of n and it is possible to take
T = Tn.
We deduce from (3.12) and (2.17) that
t
sup kwnk2+k∂θwnk2 +ν k∇wnk2+k∂θ∇wnk2 ds
2 3 2 2 3 2
t∈[0,T](cid:16) (cid:17) Z0 (cid:16) (cid:17) (3.13)
C(N +1) T
≤ kv k2+ kfk2 ds.
0 2 ν 2
Z0
It follows from the above inequality that
wn ∈ L∞(0,T;H)∩L2(0,T;V ), (3.14)
∂θwn ∈ L∞(0,T;H)∩L2(0,T;V ). (3.15)
3
Thus, in one hand we get,
∆wn ∈ L2(0,T;V∗), (3.16)
∂θ∆wn ∈ L2(0,T;V∗). (3.17)
3
In the other hand, it follows from (2.14) that
D (wn) ∈ L∞(0,T;H)∩L2(0,T;V), (3.18)
N,θ
∂θD (wn) ∈ L∞(0,T;H)∩L2(0,T;V). (3.19)
3 N,θ
Theinequality(3.18)allowsustofindestimatesonthenonlinearterm∇·(D (wn)⊗D (wn)).
N,θ N,θ
For that we use lemma 2.4. For all ϕ ∈ V we have
∇·(D (wn)⊗D (wn)),ϕ ≤ |(∇·(D (wn)⊗D (wn)),ϕ)|
N,θ N,θ N,θ N,θ
(3.20)
(cid:12)(cid:12)(cid:16) ≤ CkD (wn)(cid:17)k(cid:12)(cid:12)k∇D (wn)k k∇ϕk1−21θk∇∂θϕk21θ.
(cid:12) N,θ 2(cid:12) N,θ 2 2 3 2
We also have
1 1
A2 ∇·(D (wn)⊗D (wn)),ϕ ≤ ∇·(D (wn)⊗D (wn)),A2 ϕ
3,θ N,θ N,θ N,θ N,θ 3,θ
(cid:12)(cid:18) (cid:19)(cid:12) (cid:12)(cid:18) (cid:19)(cid:12) (3.21)
(cid:12) (cid:12) (cid:12) 1 1 1 1(cid:12)
(cid:12) ≤CkD (wn)k k(cid:12)∇D(cid:12) (wn)k k∇A2 ϕk1−2θk∇∂θA2 ϕk2θ(cid:12)
(cid:12) N,θ 2 (cid:12) (cid:12)N,θ 2 3,θ 2 3 3,θ 2(cid:12)
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1 1 1 1 1 1 1 1
Since k∇∂θϕk2θ ≤ k∇ϕk2θ, k∇A2 ϕk1−2θ ≤ k∇ϕk1−2θ and k∇∂θA2 ϕk2θ ≤ k∇ϕk2θ we get
3 2 2 3,θ 2 2 3 3,θ 2 2
that
∇·(D (wn)⊗D (wn)),ϕ ≤ CkD (wn)k k∇D (wn)k k∇ϕk , (3.22)
N,θ N,θ N,θ 2 N,θ 2 2
(cid:12)(cid:16) (cid:17)(cid:12)
and (cid:12) (cid:12)
(cid:12) (cid:12)
1
A2 ∇·(D (wn)⊗D (wn)),ϕ ≤ CkD (wn)k k∇D (wn)k k∇ϕk . (3.23)
3,θ N,θ N,θ N,θ 2 N,θ 2 2
(cid:12)(cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
T(cid:12)hus we obtain (cid:12)
(cid:12) (cid:12)
∇·(D (wn)⊗D (wn))∈ L2(0,T;V∗), (3.24)
N,θ N,θ
1
A2 ∇·(D (wn)⊗D (wn))∈ L2(0,T;V∗). (3.25)
3,θ N,θ N,θ
From eqs. (3.7), (3.14), (3.25), we also obtain that
∂ wn ∈ L2(0,T;V∗), (3.26)
t
and
1
∂ A2 wn ∈L2(0,T;V∗). (3.27)
t 3,θ
Step 3 (Limit n → ∞) It follows from the estimates (3.14)-(3.27) and the Aubin-Lions
compactness lemma (see [13] for example) that there exists a not relabeled subsequence of wn
and w such that
wn ⇀∗ w weakly∗ in L∞(0,T;H), (3.28)
D (wn)⇀∗ D (w) weakly∗ in L∞(0,T;H), (3.29)
N,θ N,θ
wn ⇀ w weakly in L2(0,T;V ), (3.30)
∂θwn ⇀ ∂θw weakly in L2(0,T;V ), (3.31)
3 3
D (wn)⇀ D (w) weakly in L2(0,T;V ), (3.32)
N,θ N,θ
∂θD (wn)⇀ ∂θD (w) weakly in L2(0,T;V ), (3.33)
3 N,θ 3 N,θ
∂ wn ⇀ ∂ w weakly in L2(0,T;V ∗), (3.34)
t t
1 1
∂ A2 wn ⇀ ∂ A2 w weakly in L2(0,T;V ∗), (3.35)
t 3,θ t 3,θ
wn → w strongly in L2(0,T;H), (3.36)
D (wn)→ D (w) strongly in L2(0,T;H), (3.37)
N,θ N,θ
From (3.32) and (3.37), it follows that
∇·(D (wn)⊗D (wn)) → ∇·(D (w)⊗D (w)) strongly in L1(0,T;L1(D)3),
N,θ N,θ N,θ N,θ
(3.38)
Finally, since the sequence ∇·(D (wn)⊗D (wn)) is bounded in L2(0,T;V∗), it converges
N,θ N,θ
weakly, upto a subsequence, to some χ in L2(0,T;V∗). The previous result and the uniqueness
of the limit allow us to claim that χ = ∇·(D (w)⊗D (w)). Consequently,
N,θ N,θ
∇·(D (wn)⊗D (wn)) ⇀ ∇·(D (w)⊗D (w)) weakly in L2(0,T;V∗). (3.39)
N,θ N,θ N,θ N,θ
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The above established convergences are clearly sufficient to take the limit in (3.7) and conclude
that the velocity w satisfies (3.4). Moreover, from (3.30) and (3.34), one can deduce by a
classical interpolation argument [9] that
w ∈ C(0,T;H). (3.40)
Furthermore, from the strong continuity of w with respect to the time and with values in H,
we deduce that w(0) = w .
0
Step 4 (Uniqueness) Next, we will show the continuous dependence of the solutions on the
initial data and in particular the uniqueness.
Let θ > 1 and let (w ,q ) and (w ,q ) be any two solutions of (1.5) on the interval [0,T], with
2 1 1 2 2
initial values w (0) and w (0). Let us denote by δw = w −w . We subtract the equations
1 2 2 1
for w from the equations for w . Then, we obtain
1 2
∂ δw−ν∆δw+∇·(D (w )⊗D (w ))−∇·(D (w )⊗D (w )) = 0, (3.41)
t N,θ 2 N,θ 2 N,θ 1 N,θ 1
and δw = 0 at the initial time.
1
Applying A2 to (3.41) we obtain
3,θ
1 1 1
A2 ∂ δw−νA2 ∆δw+A2 ∇·(D (w )⊗D (w ))
3,θ t 3,θ 3,θ N,θ 2 N,θ 2
(3.42)
1
−A2 ∇(·D (w )⊗D (w )) = 0
3,θ N,θ 1 N,θ 1
1
One can take A2 D (δw) ∈ L2(0,T;V) as test function in (3.42). Let us mention that,
3,θ N,θ
1 1
∂ A2 δw ∈ L2(0,T;V∗) and A2 D (δw) ∈ L2(0,T;V). Thus, by using Lions-Magenes
t 3,θ 3,θ N,θ
Lemma [9] we have
1 1 1d 1 1
h∂tA32,θδw,A32,θDN,θ(δw)iV∗,V = 2dkA32,θDN2,θ(δw)k22.
Using lemma 2.5 and the divergence free condition we get:
1 d 1 1 1 1
kA2 D2 (δw)k2+νk∇A2 D2 (δw)k2 ≤ (D (δw)·∇)D (w ),D (δw) . (3.43)
2dt 3,θ N,θ 2 3,θ N,θ 2 N,θ N,θ 2 N,θ
(cid:0) (cid:1)
We estimate now the right-hand side by using lemma 2.4 as follows,
(D (δw)·∇)D (w ),D (δw)
N,θ N,θ 2 N,θ
(3.44)
1 1
(cid:12)(cid:0) ≤ kD (δw)k k∇D (δw)k(cid:1)k(cid:12)∇D (w )k1−2θk∂θ∇D (w )k2θ
(cid:12) N,θ 2 N,θ 2(cid:12) N,θ 2 2 3 N,θ 2 2
Hence, by using the Young inequality combined with lemma 2.6, we obtain that there exists a
constant C > 0 that depends on N and ν such that
(D (δw)∇)D (w ),D (δw)
N,θ N,θ 2 N,θ
1 1 ν (3.45)
(cid:12)(cid:0) ≤ Ckδwk2k∇w k2−(cid:1)θ(cid:12)k∂θ∇w kθ + k∇δwk2
(cid:12) 2 2 2 (cid:12) 3 2 2 2 2
By using (2.17) we have
1 d
kδwk2+α2θk∂θδwk2 +ν k∇δwk2+α2θk∇δwk2
2dt 2 3 2 2 2
(3.46)
(cid:16) 1 d 1 1(cid:17) (cid:16) 1 1 (cid:17)
≤ kA2 D2 (δw)k2+νk∇A2 D2 (δw)k2
2dt 3,θ N,θ 2 3,θ N,θ 2
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From (3.44), (3.46) we get
d
kδwk2+α2θk∂θδwk2 +ν k∇δwk2+α2θk∇δwk2
dt 2 3 2 2 2
(3.47)
(cid:16) (cid:17) (cid:16) 1 1(cid:17)
≤ Ckδwk2k∇w k2−θk∂θ∇w kθ.
2 2 2 3 2 2
1 1
Since k∇w k2−θk∂θ∇w kθ ∈ L1([0,T]), we conclude by using Gronwall’s inequality the con-
2 2 3 2 2
tinuousdependenceof thesolutions ontheinitialdataintheL∞([0,T],H)norm. Inparticular,
if δw = 0 then δw = 0 and the solutions are unique for all t ∈ [0,T]. In addition, since T > 0
0
is arbitrary chosen, this solution may be uniquely extended for all time.
This finishes the proof of Theorem 1.1.
Acknowledgement I would like to thank Taoufik Hmidi for his valuable discussions, help
and advices.
Appendix: Proof of the functional estimate (2.2)
Let g bea 2π-periodic and mean zero function with g(x) ∈ Hs([−π,π]), and let s > 1. If we
2
write g(x) as the sum of its Fourier series g(x) = gkeikx, then, we can estimate kgkL∞
k∈Z\{0}
X
by
kgkL∞ ≤ |gk|.
k∈Z\{0}
X
We then break up the sum into low and high wave-number components,
kgkL∞ ≤ |gk|+ |gk|.
0<|k|≤κ |k|>κ
X X
We now use the Cauchy-Schwarz inequality on each part,
1 1 1 1
2 2 2 2
kgkL∞ ≤ |gk|2 1 + |k|2s|gk|2 |k|−2s .
       
0<|k|≤κ 0<|k|≤κ |k|>κ |k|>κ
X X X X
       
Since
1≤ Cκ and |k|−2s ≤ Cκ−2s+1,
0<|k|≤κ |k|>κ
X X
the above inequality becomes
1 1
kgkL∞ ≤ C κ2kgkL2 +κ−s+2kgkHs .
(cid:16) (cid:17)
To make both terms on the right-hand side the same, we choose
1
kgks
κ = Hs.
1
kgks
L2
This yields the following estimate
1 1
1−
kgkL∞ ≤ CkgkL22skgkH2ss,
which is (2.2).
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