Existence of Weak Solutions for a Scale

Existence of Weak Solutions for a Scale

Existence of Weak Solutions for a
Scale-Similarity Model of the Motion of Large
Eddies in Turbulent Flow
Meryem Kaya ∗†
Gazi University
Faculty of Arts and Sciences
Department of Mathematics
Teknikokullar, Ankara, Turkey
May 1, 2003
M athematics Subject Classif ication : 35Q30, 35Q35
Abstract
In turbulent flow the normal procedure has been seek means u of
the fluid velocity u rather than the velocity itself. In large eddy simulation, we use an averaging operator which allows for the separation of
large and small length scales in the flow field. u denotes the eddies of
size O(δ) and larger. Applying local spatial averaging operator with
averaging radius δ to the Navier-Stokes equations gives a new system
of equation governing the large scales. However, it has the well-known
problem of closure. One approach to the closure problem which arises
from averaging the nonlinear term is use of a scale-similarity hypothesis. We consider one such scale similarity model. We prove existence
of weak solutions for the resulting system.
∗
This research of M. Kaya was conducted during a visit to the university of Pittsburgh,
email:[email protected]
†
keywords:Navier -Stokes equations,large eddy simulation, turbulence
1
1
Introduction
The turbulent flow of an incompressible fluid is modelled by solution (u, p)
of the incompressible Navier-Stokes equations:
ut + ∇ · (uu) − Re−1 ∆u + ∇p = f, in Ω, for 0 < t ≤ T,
∇ · u = 0, in Ω, for 0 < t ≤ T,
Z
pdx = 0
u(x, 0) = u0 (x), in Ω, u = 0 on ∂Ω, for 0 < t ≤ T, and
Ω
where Ω ⊂ Rd (d = 2 or d = 3), u : Ω × [0, T ] → Rd is the fluid velocity,
p : Ω → R is the fluid pressure f(x,t) is the (known) body force, u0 (x)
the initial flow field and Re the Reynolds number. There are numerous
approaches to the simulation of turbulent flows in practical settings. One
of the most promising current approaches is large eddy simulation (LES) in
which approximations to local spatial averages of u are calculated. In large
eddy simulation (LES), the filtered quantities and fluctuations are defined as
Z
u(x, t) = gδ ∗ u = gδ (x − x0 )u(x0 , t)dx0 u0 = u − u
R3
where
gδ = δ −3 g(x/δ).
and g is the filter function of characteristic width δ. Applying the filtering
operator to the Navier-Stokes equations gives :
ut + ∇ · (uu) − Re−1 ∆u + ∇p = f ,
∇ · u = 0 in Ω × (0, T ].
(1.1)
The governing equation (1.1) may be rewritten as
ut + ∇ · (u u) − Re−1 ∆u + ∇p + ∇T = f ,
∇ · u = 0 in Ω × (0, T ].
where T denotes the subgrid tensor,
T := uu − u u
(1.2)
which must be modelled. On general approach to closure in LES based on
the scale similarity hypothesis, introduced in 1980 by Bardina, Ferziger and
2
Reynolds [1]. The idea of scale similarity can be thought of as a sort of
extrapolation from the resolved scales to the unresolved scales. The original
Bardina model is given by
uu − u u ∼
= u u − u u.
This model has proven to be highly consistent [15], [3], but stability problems
have been reported in various test of the Bardina model. These have led to
various extensions of Bardina model such as Layton proposed in [12], the
Liu, Maneveau, Katz model [16], Horuitu’s filtered Bardina model [6] and
many ’mixed’ models. In this report we consider a model proposed in [12]
which is another realization of the idea of scale similarity seeking a clear
kinetic energy balance. The model is based on the following three modelling
steps. The nonlinear term is written as [14]
uu = u u + uu0 + u0 u + u0 u0 .
Step1 :The cross terms are modelled by scale similarity:
uu0 + u0 u = u(u − u) + (u − u)u ∼ u(u − u) + (u − u)u.
(1.3)
Step2 :The resolved term u u is modelled with a Boussinesq type assumption
u u ∼ u u + dissipative mechanism on O(δ) scales,
where
∇ · (u u) ∼ ∇ · (u u) − A(δ)u.
(1.4)
The operator A(δ)w takes the general form A(δ)w = R∗ ∇ · TF (Rw), where
R is a restriction operator to the finest resolved scales. It is defined by the
use of its variational representation
−(A(δ)w, v) = (νF (δ)D(w − w), D(v − v)),
where νF (δ) is the fine scale fluctuation coefficient. This simplifies to
A(δ)w ∼ ∇ · (νF (δ)D(w − w) − (w − w)) where D(w) := 12 (∇w + ∇wt ).
Step3:The u0 u0 term are modelled by a Boussinesq hypothesis that
u0 u0 ∼ −νT (δ, u)(∇u + ∇ut )
3
(1.5)
where νT (δ, u) is called turbulent viscosity coefficient. Using (1.3), (1.4)
and (1.5) in (1.1), the model written below, (w, q) denotes as usual the
approximations to (u, p),
wt + ∇ · (w w) + ∇ · (w(w − w) + (w − w)w) − ∇ · (νT (δ, w)(∇w + ∇wt ))
− ∇q − Re−1 ∆w − A(δ)w = f ,
∇ · w = 0 in Ω × (0, T ]
(1.6)
where w, f : Ω×[0, T ] → Rd , q : Ω → R. Boundary and zero mean conditions
must be imposed on (1.6). There are several possibilities for the turbulent
viscosity coefficient. The most common ones used in computational practice
are a bulk viscosity νT = νT (δ), the viscosity of [7], νT = (0.17)δ |w − w|
and the Smagorinsky model, see [17, 9, 2, 11]
νT (δ, w) = (cs δ)2 ∇w + ∇wt .
(1.7)
We shall assume νT = 0 namely, there is no extra viscosity terms. With
(1.7) or νT = νT (δ) our results can be easily extended. Before starting to
prove the existence of weak solution for the model, we will give a proof that
the model, is given by (1.6), is Galilean invariant. It has been shown that
the filtered form of Navier-Stokes equation are Galilean invariant [18]. Thus
it is enough to show
e + W )) = ∇ · T(w)
e
∇ · (T(w
for any constant vector W. To this end we will give the following Lemma.
Lemma 1.1. Let consider the model of the subgrid tensor
T =uu − u u ∼ w w + w(w − w) + (w − w)w − (cs δ)2 ∇w + ∇wt (∇w + ∇wt )
e
− (νF (δ)D(w − w) − (w − w) − ww = T(w),
e + W ) = ∇ · T(w)
e
∇ · T(w
for any constant vector W.
Proof.
e + W ) = (w + W )(w + W ) + (w + W )(w + W − (w + W ))
T(w
+ (w + W − (w + W ))(w + W )
− (cs δ)2 ∇(w + W ) + ∇(w + W )t (∇(w + W ) + ∇(w + W )t )
− (νF (δ)D(w + W − (w + W )) − (w + W − w + W )) − (w + W )(w + W )
4
Since W is a constant vector, W = W, W = W, W w = W w, wW = wW .
Thus,
e + W ) = w w + w(w − w) + (w − w)w − (cs δ)2 ∇w + ∇wt (∇w + ∇wt )
T(w
− (νF (δ)D(w − w) − (w − w) − ww + (w − w)W + W (w − w)
+ W (w − w) + (w − w)W .
Hence we have
e
∇·T(w+W
) = ∇·Te(w)+∇·(w−w)W +∇·(W (w−w))+∇·((w−w)W )+∇·(W (w−w)).
Since the averaging preserves incompressibility [18], that is ∇·w = ∇·w = 0,
so we have
e + W ) = ∇ · T(w).
e
∇ · T(w
This complete the proof.
2
Existence of Solutions
In this section we consider the question of existence of weak solutions to the
following systems. Thus, we seek (w, q) satisfying
wt + ∇ · (w w) + ∇ · (w(w − w) + (w − w)w) − ∇q − Re−1 ∆w
− A(δ)w = f , ∇ · w = 0, in Ω × (0, T ]
(2.1)
w(x, 0) = gδ ∗ u0 (x), in Ω,
Z
Z
Z
w(xj + L, t) = w(xj , t) and u0 dx = 0, f dx = 0, qdx = 0.
Ω
Ω
(2.2)
(2.3)
Ω
We shall begin by giving the definition of weak solution. Let D(Ω) =
{ψ ∈ C0∞ (Ω) : ∇.ψ = 0 in Ω}, H(Ω) be the completion of D(Ω) in L2 (Ω)
H 1 (Ω) be the completion of D(Ω) in W 1,2 (Ω) and ψ ∈ D(Ω).
Definition 2.1. Let u0 ∈ H(Ω), f ∈ L2 (ΩT ). A measurable function
w : ΩT −→ Rn is a weak solution of the problem (2.1)- (2.2) in ΩT if
5
a)w ∈ VT = L2 (0, T ; H 1 ) ∩ L∞ (0, T ; H) ;
b)w verifies
Z t
[−Re−1 (∇w, ∇ψ) + (w w, ∇ψ) + (w(w − w) + (w − w)w, ∇ψ)
0
Z t
− νF (δ)(D(w − w), D(ψ − ψ))]ds = − (f , ψ)ds + (w(t), ψ) − (w0 , ψ)
0
where for T ∈ (0, ∞), ΩT = Ω × [0, T ].
Before we prove of the existence of weak solutions of (2.1)- (2.3) we give
the following Lemma. It is proved [12]. Here we shall give this proof briefly.
This Lemma gives a useful result about the following (nonstandard) trilinear
form.
Lemma 2.1. Let b(u,v,w) denote the (nonstandard) trilinear form:
Z
b(u, v, w) := u v : ∇w + [u(v − v) + (u − u)v] : ∇wdx.
Ω
Suppose the averaging used in L2 (Ω) self-adjoint and commutes with differentiation,
w ∈ L2 (Ω) and ∇w ∈ L2 (Ω) are periodic with zero mean. Then
Z
I = ∇.[w w + w(w − w) + (w − w)w].wdx = 0.
Ω
Proof. Integration by parts and using the properties of the averaging operator
yields
Z
I = [w w + w(w − w) + (w − w)w] : ∇wdx
Ω
Z
[w w : ∇w + ww : ∇w − w w : ∇w + ww : ∇w − w w : ∇w]dx.
=
Ω
An easy index calculation shows that
Z
Z
uv : ∇wdx = u.(∇w)vdx
Ω
Ω
6
which is the more familiar trilinear form. Making this change gives
Z
I = [w.(∇w)w + w.(∇w)w + w.(∇w)w − 2w.(∇w)w]dx.
Ω
Since ∇·w = 0, the third term vanishes. By the assumption on the averaging
process, ∇·w = 0, so the last term vanishes. We use the usual skew symmetry
property we obtain
Z
w.(∇w)w + w.(∇w)w = 0.
Ω
Thus I=0.
Theorem 2.1. Let T > 0, and Ω be any domain in Rd . Then for any given
u0 ∈ L2 (Ω) f ∈ L2 (ΩT ) there exist at least one weak solution to (2.1)- (2.3)
in ΩT .
Proof. We shall use the Faedo-Galerkin method following the presentation of
Galdi in the Navier-Stokes case [4]. Let D(Ω) =: {ψ ∈ C0∞ : ∇.ψ = 0 inΩ},
H(Ω) be the completion of D(Ω) in L2 (Ω) H 1 (Ω) be the completion of D(Ω)
in W 1,2 (Ω) {ψr } ⊂ D(Ω) be the orthonormal basis of H(Ω). We shall look
for approximating solutions v k of the problem (2.1)- (2.3) which have the
form
k
X
k
ckr (t)ψr (x), k ∈ N.
(2.4)
v (x, t) =
r=1
k
In (2.1) we set w = v , multiply by ψr and integrate over Ω we obtain
d k
(v , ψr ) − (v k v k , ∇ψr ) + Re−1 (∇v k , ∇ψr ) + νF (δ)(D(v k − v k ), D(ψr − ψ r ))
dt
− ((v k (v k − v k ) + (v k − v k )v k , ∇ψ r ) = (f , ψr ).
Note that since ∇ · u = 0, it follows ∆u = 2∇ · D(u). The symmetry of
deformation tensor yields
1
(∇u, ∇v) = (D(u), D(v)).
2
Thus, we obtain the following equality.
d k
νF (δ)
(v , ψr ) − (v k v k , ∇ψr ) + Re−1 (∇v k , ∇ψr ) +
(∇(v k − v k ), ∇(ψr − ψ r ))
dt
2
− (v k (v k − v k ) + (v k − v k )v k , ∇ψ r ) = (f , ψr ).
(2.5)
7
If we write (2.4), in (2.5) this represent a system of ordinary differential
equations of the form
k
k
X
X
d
−1
cki (∇ψi , ∇ψr )
ckr (t) −
cki ckj ((gδ ∗ ψi )(gδ ∗ ψj ), ∇ψr ) + Re
dt
i=1
i,j=1
k
νF (δ) X
+
ckj (∇(ψj − gδ ∗ ψj ), ∇(ψr − gδ ∗ ψr )
2 j=1
−
k
X
cki ckj [(gδ ∗ ψi )(ψj − gδ ∗ ψj ) + (ψj − gδ ∗ ψj )(gδ ∗ ψi ), ∇(gδ ∗ ψr )]
i=1
= (f , ψr ) = f r r = 1, · · · , k
(2.6)
with the initial condition
ckr (0) = c0r = (v0 , ψr ).
(2.7)
Since f r ∈ L2 (0, T ) for all r = 1, · · · , k, from the elementary theory of
ordinary differential equations we know the problem admits a unique solution
ckr ∈ W 1,2 (0, Tk ) where Tk ≤ T .
Multiplying (2.6) by ckr and summing over r from 1 to k we get:
1 d
vtk 2 − (v k v k , ∇v k ) + νF (δ) ∇(v k − v k )2 − (v k (v k − v k )+
2
2
2 dt
2
2
(v k − v k )v k , ∇v k ) + Re−1 ∇v k 2 = (f , v k ).
We integrate this equality, we obtain
Z t
Z t
k 2
k 2
v + 2Re−1
∇v ds − 2 (v k v k , ∇v k )ds
2
2
0
Z t
Z0 t
∇(v k − v k )2 ds − 2 (v k (v k − v k ) + (v k − v k )v k , ∇v k )ds
+ νF (δ)
2
0
Z t 0
= 2 (f , v k )ds + kv0k k22
0
with v0k = vk (0). We consider third and last term in the left hand side of the
above equality. Let us write these two terms nonstandard trilinear form:
Z
k
k
k
b(v , v , v ) = −2 [v k ∇v k v k +v k ∇v k v k −v k ∇v k v k +v k ∇v k v k −v k ∇v k v k ]dx.
Ω
(2.8)
8
From Lemma 2.1 I = b(v k , v k , v k ) = 0. In the last equality, we use I=0,
Schwarz inequality, Poincaré-Friedrichs inequality, and since kv0k k ≤ kv0 k,
we obtain,
Z t
Z t
k 2
2
−1
∇(v k − v k )2 ds
∇v 2 ds + νF (δ)
kvk k2 + Re
2
0
0
Z t
2
f ds + kv0 k2 .
≤ CRe
2
2
0
where C is a constant. Then we easily deduce the following bound
Z t
2
−1
∇vk 2 2 ds ≤ M for all t ∈ [0, T ]
kvk k2 + Re
2
(2.9)
0
with M independent of t and k. We shall now investigate the properties of
convergence of the sequence {vk } when k → ∞. To this end we begin to show
that, for any fixed r ∈ N the sequence of functions
Grk (t) ≡ (vk (x, t), ψr )
is uniformly bounded and uniformly continuous in t ∈ [0, T ]. The uniform
boundness follows at once from (2.9). To show the uniform continuity,
integrating (2.5) with respect to t from s to t and using Schwarz inequality
we obtain
Z t
k k
k
r
r
k
b(v , v , ψr ) dτ
|Gk (t) − Gk (s)| = (v (x, t) − v (x, s), ψr ) ≤
s
Z t
Z t
k
νF (δ)
k
k −1
∇v k∇ψr k dτ
+
∇(v − v ) ∇(ψr − ψ r ) dτ + Re
2
s
s
Z t
f kψr k dτ.
+
(2.10)
s
On the other hand an easy index calculation shows that
Z
Z
uv : ∇wdx = u.(∇w)vdx
Ω
Ω
which is more familiar trilinear form. Making this change in the following
formula
Z
k
k
b(v , v , ψr ) := (v k v k : ∇ψr + (v k (v k − v k ) + (v k − v k )v k ) : ∇ψ r )dx.
Ω
9
it gives
k
Z
k
v k .∇ψr v k + v k .∇ψ r v k + v k .∇ψ r v k − 2v k .∇ψ r v k .
b(v , v , ψr ) =
Ω
By the usual skew symmetry property of this trilinear form, we obtain
Z
k
k
b(v , v , ψr ) = −v k .∇v k ψr − v k .∇v k ψ r − v k .∇v k ψ r + 2v k .∇v k ψ r
Ω
Using Cauchy-Schwarz inequality and Young inequality for convolutions we
get
Z t
Z t
k k
k
√
b(v , v , ψr ) ≤ s1 max v (x, t) t − s( ∇v k 2 ) 12
t
s
s
Z t
k
√
k 2 1
+ s2 max v (x, t) t − s( ∇v ) 2
t
s
where s1 = maxx∈Ω |ψr (x)| and s2 = 4maxx∈Ω ψ r (x). Now we use this
inequality and triangle inequality in (2.10) we obtain
Z t
Z t
k 2 1
√
2 1
r
r
2
2
|Gk (t) − Gk (s)| ≤ max kvk (x, t)k t − s s1 ( k∇vk k ) + s2 (
∇v ) )
t
s
s
Z t
Z t
k 2 1
2 1
√
νF (δ) √
−1
∇v ) 2 + Re k∇ψr k t − s( ∇v k ) 2
s3 t − s(
2
s
s
Z t
√
1
+ max kψr k t − s( kf k2 ) 2
x∈Ω
s
where s3 = 2 k∇ψr k . Because of (2.9), the right hand side of this inequality converges to zero uniformly as t → s. Grk (t) is equicontinuity. By the
Ascoli-Arzela theorem, from the sequence {Grk (t)}k∈N we may then select a
subsequence which we continue to denote by {Grk (t)}k∈N uniformly converging to a continuous function Gr (t). The selected sequence {Grk (t)}k∈N may
depend on r. However using Cantor diagonalization method, we end up by
with a sequence again denoted by {Grk (t)}k∈N converging to Gr for all r ∈ N
uniformly in t ∈ [0, T ]. This information together with (2.9) and the weak
compactness of the space H, allows us to infer the existence of v(t) ∈ H(Ω)
such that
lim (vk (t)−v(t), ψr ) = 0 uniformly in t ∈ [0, T ] and f or all r ∈ N. (2.11)
k→∞
10
vk (t) converges weakly in L2 to v(t), uniformly in t ∈ [0, T ] that is
lim (vk (t) − v(t), u) = 0 uniformly in t ∈ [0, T ] f or all u ∈ L2 (Ω). (2.12)
k→∞
In view of (2.9) v ∈ L∞ (0, T ; H(Ω)) . Again from (2.9) by the weak of
compactness of the space L2 (ΩT )
Z t
lim
(∂m (vk − v), w)ds = 0 for all w ∈ L2 (ΩT ) m = 1, · · · , n (2.13)
k→∞
0
(with ∂m = ∂x∂m ) v ∈ L2 (0, T ; H 1 (Ω)) [4]. It is shown that (2.11) imply the
strong convergence of {vk } to v in L2 (w × [0, T ]) for all w ⊂ Ω, that is
Z
T
kvk (t) − v(t)k22,Q dt = 0
lim
k→∞
(2.14)
0
in [4] where Q is a cube in Rn . Now with the help (2.12)- (2.14), we shall
now show that v is a weak solution to (2.1)- (2.2). Since we already proved
that v ∈ VT , it remains to show v satisfy (2.3). Integrating (2.5) from 0 to
t < T we find
−1
Z
−Re
t
Z
t
(∇v , ∇ψr )ds +
(v k (v k − v k ) + (v k − v k )v k , ∇ψ r )ds
0
0
Z t
Z
νF (δ) t
k k
(∇(v k − v k ), ∇(ψr − ψ r ))ds
+ (v v , ∇ψr ) −
2
0
Z t 0
= − (f , ψr )ds + (v k (t), ψr ) − (vo , ψr )
k
(2.15)
0
Now we consider second and third terms of the left hand side of the equation
(2.15) by the usual skew symmetry property we write
Z tZ
k
k
[−v k ∇v k ψr − v k ∇v k ψ r − v k ∇v k ψ r + 2v k ∇v k ψ r ]dx.
b(v , v , ψr ) =
0
Ω
From (2.12) and (2.13) we get
Z
k
lim (v (t) − v(t), ψr ) = 0, lim
k→∞
k→∞
t
(∇v k (s) − ∇v(s), ∇ψr )ds = 0.
0
11
(2.16)
Furthermore let Q be a cube containing the support of ψr , then we have
Z t
Z t
[(v k ∇v k , ψr ) − (v∇v, ψr )]ds ≤ ((v k − v)∇v k , ψr )Q ds
0
Z 0t
+ (v∇(v k − v), ψr )Q ds
(2.17)
0
We consider first term of the right hand side and using Cauchy-Schwarz
inequality we obtain
Z t
Z t
k
v − v ∇v k max |ψr (x)| .
((v k − v)∇v k , ψr )Q ≤
0
x∈Q
0
Setting s1 : maxx∈Q |ψr (x)| and using (2.9) and Young inequality for convolution, we have
Z t
Z t
((v k − v)∇v k , ψr )Q ≤ Cs1 M 12 ( v k − v 2 ) 21
2,Q
0
o
where C is a constant. Thus using (2.14) we get:
Z t
k
k
lim (v − v)∇v , ψr )Q ds = 0
k→∞
(2.18)
0
We also have:
Z t
X
n Z t
k
k
(v∇(v − v), ψr )Q ds ≤
(∂m (v − v), v i mψr )Q ds
0
0
m=1
n Z t
X
k
≤
(∂m (v − v), gδ ∗ ((gδ ∗ vi )ψr ))Q ds
m=1
0
and since gδ ∗ ((gδ ∗ vi )ψr ) ∈ L2 (ΩT ), (2.13) implies
Z t
k
lim (v∇(v − v), ψr )Q ds = 0.
k→∞
Relation (2.18)- (2.19) yield:
Z t
k
k
lim (v ∇v , ψr ) − (v∇v, ψr ))ds = 0.
k→∞
(2.19)
0
0
12
(2.20)
Now we consider the second term of b(v k , v k , ψr ). Again let Q be a cube
containing the support of ψr ,then we have
Z t
Z t
((v k ∇v k , ψ r ) − (v∇v, ψ r ))ds ≤ ((v k − v)∇v k , ψ r )Q ds
0
Z 0t
+ (v∇(v k − v), ψ r )Q ds .
(2.21)
0
We use Cauchy-Schwarz, the first term of the right hand side of (2.21) we
obtain
Z t
Z t
Z t
(v k − v)∇v k , ψ r )Q ds ≤ s2 ( v k − v 2 ds) 12 ( ∇v k 2 ds) 21
2,Q
2,Q
0
0
0
Using (2.9) and Young inequality, we get
Z t
Z t
k
k
(v − v)∇v , ψ r )Q ds ≤ Cs2 M 12 ( v k − v 2 ds) 21
2,Q
0
0
Thus using (2.14), we obtain:
Z t
k
k
lim ((v − v)∇v , ψ r )Q ds = 0
k→∞
(2.22)
0
Now we consider the second term of the right hand side of the equation
(2.21) we write
Z t
n Z t
X
(∂m (v k − v), v m ψ r )Q ds
(v k ∇(v k − v), ψ r )Q ≤
0
m=1
0
and since v m ψ r ∈ L2 (ΩT ) (2.13) implies
Z t
k
k
lim (v ∇(v − v), ψ r )Q ds = 0
k→∞
(2.23)
0
Relation (2.22)- (2.23) yield:
Z t
k
k
lim ((v ∇v , ψ r ) − (v∇v, ψ r ))ds = 0
k→∞
0
13
(2.24)
Similarly we consider third term of b(v k , v k , ψr ) we write
Z t
Z t
[(vk ∇v k , ψ r ) − (v∇v, ψ r )]ds ≤ ((v k − v)∇v k , ψ r )Q ds
0
Z 0t
+ (v∇(v k − v), ψ r )Q .
(2.25)
0
Again using Cauchy-Schwarz, Young inequality, and (2.9) in the first term
of the right hand side of the equation (2.25) we get:
Z t
Z t
1
k
k
((v − v)∇v , ψ r )Q ds ≤ Cs2 M 2 ( v k − v 2 ds) 21 .
2,Q
0
0
Using (2.14) we get:
Z t
lim ((v k − v)∇v k , ψ r )Q ds = 0.
k→∞
(2.26)
0
Now we consider the second term of the right hand side of (2.25)
Z t
X
n Z t
k
k
(v∇(v − v, ψ r )Q ds ≤
(∂m (v − v), vm ψ r )Q ds
0
0
m=1
We use the properties of convolutions, we obtain
Z t
n Z t
X
∂m (v k − v), gδ ∗ (vm ψ r )Q ds
(v∇(v k − v, ψ r )Q ds ≤
0
m=1
0
since gδ ∗ (vm ψ r ) ∈ L2 (ΩT ), (2.13) implies
Z t
k
lim (v∇(v − v, ψ r )Q ds = 0.
k→∞
(2.27)
0
Relation (2.26)- (2.27) yield:
Z t
lim [(vk ∇v k , ψ r ) − (v∇v, ψ r )]ds = 0
k→∞
(2.28)
0
Now we consider last term of b(v k , v k , ψr ) again we can write
Z t
Z t
k
k
k
[(v k ∇v , ψ r ) − (v∇v, ψ r )]ds ≤ ((v − v)∇v , ψ r )Q ds
0
Z 0t
+ (v∇(v k − v), ψ r )Q ds .
0
14
(2.29)
Similarly, using Cauchy-Schwarz, Young inequality, (2.9) and (2.14) in the
first term of the right hand side of (2.29) we get:
Z t
k
k
(2.30)
lim ((v − v)∇v , ψ r )Q ds = 0,
k→∞
0
Besides this, we get the following inequality for the second term of the equation (2.29)
Z t
X
n Z t
k
k
(v∇(v − v), ψ r )Q ds ≤
(∂m (v − v), v m ψ r )Q .
0
0
m=1
From the properties of convolution we write
Z t
n Z t
X
k
k
(v∇(v − v), ψ r )Q ds ≤
(∂m (v − v), gδ ∗ (v m ψ r ))Q 0
m=1
0
and since gδ ∗ (v m ψ r ) ∈ L2 (ΩT ). From (2.13) we obtain
Z t
k
lim (v∇(v − v), ψ r )Q ds = 0.
k→∞
Thus relation (2.30)- (2.31) yield:
Z t
k
lim [(v k ∇v , ψ r ) − (v∇v, ψ r )]ds = 0
k→∞
(2.31)
0
(2.32)
0
Finally, we consider fourth term the left hand side of (2.15). Again let Q be
a cube containing the support of ψr , then we have
Z t
k
k
[(∇(v − v), ∇(ψr − ψ r )) − (∇(v − v), ∇(ψr − ψ r ))]ds
0
Z t
Z t
≤ (∇(v k − v), ∇(ψr − ψ r ))Q ds + (∇(v k − v), ∇(ψr − ψ r ))Q ds .
0
0
Since ∇(ψr − ψ r ) ∈ L2 (ΩT ) and using (2.13) we get
Z t
k
lim (∇(v − v), ∇(ψr − ψ r ))Q = 0.
k→∞
0
15
(2.33)
Similarly since gδ ∗ ∇(ψr − ψ r ) ∈ L2 (ΩT ) and using (2.13) it gives
Z t
k
lim (∇(v − v), ∇(ψr − ψ r ))Q ds = 0.
k→∞
Using (2.33) and (2.34) we get
Z t
k
k
lim (∇(v − v ) − ∇(v − v), ∇(ψr − ψ r ))ds = 0
k→∞
(2.34)
0
(2.35)
0
Therefore taking the limit over k → ∞ in (2.15) and using (2.16), (2.20),
(2.24), (2.28), (2.32), (2.35), we get
−1
−Re
Z
t
Z
t
Z
t
(∇v, ∇ψr ) +
(v(v − v) + (v − v)v, ∇ψ r )ds +
(v v, ∇ψr )ds
0
0
0
Z t
Z
νF (δ) t
(∇(v − v), ∇(ψr − ψ r )) = − (f , ψr )ds + (v(t), ψr ) − (v0 , ψr )
−
2
0
0
(2.36)
However, from Lemma 2.3 in [4] we know that every function ψ ∈ D(Ω) can
be uniformly approximated in C 2 (Ω) by functions of the form
ψN (x) =
N
X
γr ψr (x) N ∈ N, γr ∈ R
r=1
so writing (2.36) with ψN in place of ψr and we may pass to the limit N → ∞
in this new relation and use the fact that v ∈ L2 (0, T ; H 1 ) ∩ L∞ (0, T ; H) to
show v is a weak solution of (2.1)- (2.2).
Acknowledgement : I would like to thank Prof. Dr. W.J.Layton for
this problem and his valuable comments and several helpful discussions.
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