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Published by | profil-zyak-2012 |
Reads | 15 |
Language | English |
∗
λ
λ
λ
∗
es/Lam?
problems
in
from
terface
giv
problems
?
D.
parameter
Mercier,
Stok
S.
sho
where
of
non
the
This
pap
results
er
2,
is
stable
dev
CNRS
oted
FRANCE,
to
oundaries.
some
for
transmission
℄
problems
w
in
passing
v
olving
similar
the
est
Lam?
e
and
pap
Stok
problems
es
pro
systems
in
in
Math?matiques
t
hniques
w
Houy
o-dimensional
1
nonsmo
in
oth
to
domains.
the
W
Here
e
in
rst
and
sho
where
w
b
that
Lam?
these
in
problems
w
ma
?
y
results
b
results
e
T
obtained
h
as
wn
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limit
sc
of
follo
a
ulate
transmission
v
Lam?
ariational
problem
limit
when
vior
the
is
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oth
Stok
de
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t
Univ
results
br?sis,
go
ALEN-
es
to
esp
innit
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y
the
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e
W
?
e
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further
in
transmission
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estigate
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also
eha
the
vior
pap
of
e
the
they
solutions
obtained
of
on
these
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problems
the
near
p
es
singularities,
refer
esp
of
ecially
As
near
regularit
p
regularit
oin
transmission
ts
in
where
our
the
wledge,
in
regularit
terface
not
in
tersects
b
the
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b
systems.
oundaries.
of
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is
particular
In
w
e
e
dieren
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e
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stable
their
ulations
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3
the
b
p
that
erturbation
parameter
oratoire
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ses
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Some
Institut
minimal
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regularit
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results
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:
minimal
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larit
ecially
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the
result
terface
of
tersects
transmission
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Lam?
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problems
refer
obtained
[
in
,
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,
Some
and
n
reference
umerical
there
results
standard
for
problems.
the
w
of
standard
the
systems
singular
tro
exp
in
onen
(see
ts
℄
in
revisit
the
results
asymptotic
that
expansion
er,
are
w
presen
sho
ted
that
and
e
these
from
minimal
results
regularit
transmission
y
problems
results.
y
1
to
In
limit
tro
the
Regularit
erturbation
(for
standard
the
W
Stok
e
system,
in
e
v
to
estigate
6.2
t
[
w
℄
o-dimensional
a
transmission
minimal
problems
y
for
are
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from
Lam?
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and
y
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systems
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o
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p
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olygonal
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studying
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es/Lam?
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line:
er
on
as
ws:
h
sub
w
domain
form
w
the
e
t
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e
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e
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and
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system.
the
Boundary
pro
The
on
the
eha
exterior
is
b
ed
oundary
pieces
and
and
obtained
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y
standard)
wing
transmission
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lo
on
problems
the
Lab
in
de
terface
et
are
Applications
imp
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osed.
FR
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these
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t
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ha
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denis.mercier@univ-v
b
;
eha
viour
near2Ω R
Ω
Ω
Ω
Ω Ω Γ1 2
Ω = Ω ∪Ω , Ω ∩Ω =∅.1 2 1 2
Γ Ω Ω Ω ∩Ω ∂Ω1 2 1 2
Ω n Γ Ω1
1 2u = (u ,u )
1 2Ω u = (u ,u ) Ω,i = 1,2.i ii i
λ,μ,i = 1,2i i
λ μ
Ω Ω λ μi i i
1 2 2u∈H (Ω) p∈L (Ω)
Tσ(u) =μ(∇u+(∇u) )+λdivuId,
Tσ(u,p) =μ(∇u+(∇u) )−pId,
Id
k
k 2 kPH (Ω) ={u∈L (Ω) : u ∈H (Ω ),i = 1,2},i i
kH
1u∈PH (Ω) [u] Γ
1 2[u] =γ u −γ u ,Γ Γ
1 1/2 1γ H (Ω ) H (Γ) u ∈ H (Ω)Γ i
[u] = 0
Ω (P )0
Ωi
∂Ω Γ
a
If
with
sequel.
for
the
Hence
in
the
is
The
a
y
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oundary
ectorial
through
function
illustrated
dened
2.3
on
this
use
in
,
a
w
n
e
are
denote
p
b
a
y
the
shall
form
e
,
w
w
h
)
whic
es
notations,
and
some
w
that
tro
not
in
of
us
union
Let
of
its
that
.
to
jump
the
b
domain
domain
spaces
where
functional
from
and
4
erators
W
op
Some
functions,
a
The
the
2.2
w
1).
rst
W
Stok
e
e
further
h
in
tro
the
denote
Lam?
w
(or
hitz
Stok
that
es)
e
segmen
b
ts
a
Fig.
is
again
the
(see
whic
in
in
on
F
ector
means
v
b
normal
,
outer
,
unitary
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the
ounded
denotes
us
(see
2.1
℄
ulation
2
2
to
op
b
e
and
p
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ositiv
results
e
regularit
ts.
y
F
w
or
system
systems
v
order
enience
w
e
one
denote
to
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here
that
(resp.
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and
in
)
the
b
piecewise
b
some
t
e
functions
teger
dened
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on
e
of
b
and
e
whose
is
domain,
to
oundary
is
b
assume
is
do
the
w
is
ts.
(resp.
line
usual
er
As
um
).
nite
If
of
.
the
of
of
terior
b
in
space
the
functions
is
h
and
piecewise
een
the
and
.
w
or
et
function
b
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terface
oundary
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olygonal
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a
satisfying
its
,
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set
dened
terface
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in
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an
x
with
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and
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1)
problem
Fig.
of
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oundary
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b
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hitz
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erator
a
5.
with
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p
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rst
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in
analysis.
is
our
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es-Stok
ks
problem,
means
of
w
where
Stok
the
system
is
the
domain
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e
tit
standard
y
oundary
matrix.
on
F
or
with
a
transmission
p
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ositiv
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[4
are
The
supp
osedΓ
Ω Ω1 2
nωS1 ωS2
S
Ω
(P )1
∂Ω Γ
2 2(P ) f ∈L (Ω) (u,p)0
−μ Δu +∇p =f, inΩ, i = 1,2,i i i i i
divu = 0, inΩ, i = 1,2,i i
u = 0on∂Ω,
Γ
[u] = 0,
[σ(u,p)n] = 0.
2 2(P ) f ∈L (Ω) u p p1 1 1
Ω1
−μ Δu +∇p =f , inΩ , 1 1 1 1 1
divu = 0, inΩ ,1 1
−μ Δu −(λ +μ )∇divu =f inΩ ,2 2 2 2 2 2 2
u = 0on∂Ω,
Γ
[u] = 0,
σ(u ,p )n−σ(u )n = 0.1 1 2
oundary
)
in
only
on
dened
in
3
on
(4)-(6)
ws:
(
the
and
one
nd
in
,
with
or
e
F
(1)
ws:
domain
follo
es
as
the
stated
standard
e
with
b
(6)
Similarly
transmission
Figure
1:
form
The
or
satisfying
domain
(1)-(3)
(3)
one
:
and
Stok
on
system
(5)
terface
other
in
with
the
b
with
one
the
(called
terface
)
on
is
:
a
and
some
Lam?-Stok
es
.
(2)
b
problem,
(4)
hereafter:
ulated
as
hence
follo
w
F
e
nd
satisfying
the
hereafter:
Lam?
system
b
eing2 2f ∈L (Ω) u
−μΔu−(λ+μ)∇divu =f inΩ,
u = 0on∂Ω,
Γ
[u] = 0,
[σ(u)n] = 0
1 1
ε = ε =i
λ+μ λ +μi i
−1p =−ε divu
ǫ 2 2 ǫ ǫ(P ) f ∈L (Ω) u p
ǫ ǫ−μΔu +∇p =f, inΩ,
ǫ ǫdivu +εp = 0, inΩ,
ǫu = 0on∂Ω,
Γ
ǫ[u ] = 0
ǫ ǫ ǫ[σ (u ,p ).n] = 0
ǫσ (u,p) =σ(u,p)+μεpId.
μ λ λ ε→ (0,0)1 2
ǫ(P ) (P )0
μ λ λ ε → 02 1 1
ǫ(P ) (P )1
ǫ 1 2 2 2(P ) V = (H (Ω)) f ∈ L (Ω) .0
ǫ ǫ(P ) u ∈V
Z Z
ǫ ǫ(μ∇u ·∇v +(λ+μ)divu divv)dx = f·vdx,∀v∈V.
Ω Ω
ǫu ∈VZ
a a(u,v) = (μ∇u·∇v+(λ+μ)divudivv)dx
ΩZ
l l(v) = fvdx
Ω
(7)-(9)
x
the
e
and
set
let
on
w
)
if
uit
and
go
es
terface
to
in
innit
dened
y
(or
alen
equiv
b
alen
Lam?-Lam?
tly
e
,
,
Similarly
with
.
(9)
of
and
)
(13)
w
the
e
e
see
w
that
wn
limit
b
formal
and
the
linear
is
,
is
a
the
Indeed,
formal
v
limit
where
of
problem:
that
solution
see
(7)
e
natural
w
on
.
:
The
The
next
of
with
)
from
.
y
e
form
a
y
rigorous
(i.
pro
if
of
in
of
auxiliary
these
,
formal
problem
argumen
written
ts.
as
2.4
V
of
ariational
ws:
form
F
ulations
and
of
hereafter:
the
problems
problem.
The
let
w
e
eak
ha
form
w
ulation
us
of
the
problem
for
tly
nd
alen
of
equiv
hereafter:
(or
(8)
is
the
v
in
ery
standard.
:
Let
(12)
y
Then
innit
to
existence
go
uniqueness
and
terface
let
the
4
satisfying
F
results
x
the
e
w
of
and
bilinear
if
if
,
b
tly
w
Consequen
set
ormally
e.
,
and
these
(11)
t
e
w
tro
Then
the
the
unkno
w
function
eak
then
form
previous
ulation
of
e
problem
equiv
o
tly
problems
follo
the
b
tin
is
y
to
the
nd
form
e
dened
view
y
as
or
the
nd
satisfying
satisfying
(13)
(10)-(12)
limit
(10)
of
will
giv
and(P )0 Z
2 2L (Ω) ={p∈L (Ω) : p(x)dx = 0} f,u p0
Ω
(P )0
2v∈V q∈L (Ω))0
a (u,v)+b (v,p) =l(v),∀v∈V,0 0
2b (u,q) = 0,∀q∈L (Ω),0 0
Z Z
a (u,v) = μ∇u·∇vdx, b (u,q) =− qdivudx.0 0
Ω Ω
2(P ) (u,p)∈V ×L (Ω)0 0
(P ) (u,p ) ∈1 1
2V ×L (Ω )1
a (u,v)+b (v,p) =l(v),∀v∈V,1 1
2b (u,q) = 0,∀q∈L (Ω ),1 1
a1
Z
a (u,v) = μ ∇u .∇v dx1 1 1 1
Ω1
Z
+ (μ ∇u ·∇v +(λ +μ )divu divv )dx,2 2 2 2 2 2 2
Ω2
Z
b (u,q) =− qdivudx.1
Ω1
a a V b b0 1 0 1
b (v,q)0 2sup ≥β kqk ,∀q∈L (Ω),0 0,Ω 0kvk1,Ωv∈V\{0}
b (v,q)1 2sup ≥β kqk ,∀q∈L (Ω ),1 0,Ω 11kvkv∈V\{0} 1,Ω
β ,β0 1
2q ∈L (Ω ) q˜1
Ω
q Ω ,1
q˜=
c Ω ,2
c
Z
1
c =− q(x)dx,
|Ω |2 Ω1
e
of
in
a
wing
unique
to
solution
Corollary
to
b
(14)
en
and
inner
(15)
.
follo
the
w
w
from
as
the
dened
(14)
the
y
for
of
functions
Let
follo
t.
?
and
e
of
,
on
in
y
eak
and
If
the
the
the
that
the
viscosit
(resp.
uous
and
taking
e
p
easily
ts
satises
smo
the
rst
so-called
inf-sup
of
while
adapt
one,
the
the
tin
k:
metho
As
d
oin
w
saddle-p
its
the
a
(2)-(3),
ws:
us
ulation
dev
are
elop
y
ed
form
in
t
y
I.5.1
.
of
is
[
is
?
nd
℄
rst
(see
of
also
pro
℄
the
for
then
our
some
transmission
ositiv
problem
,
satisfying
W
oth
and
solution
is
The
simply
assertion
a
ws
Stok
from
es
I.2.4
system
[
(15)
℄
that
for
h
w
with
use
a
follo
discon
tric
using
for
and
(14).
parts
previously
y
e
b
sho
tegration
let
in
that
after
e
in
extension
obtain,
to
and
dened
e
follo
w
w
,
form
(resp.
of
function
problem
a
and
with
in
(1)
b
Hence
where
the
is
w
eak
giv
form
b
ulation
bilinear
of
is
Existence
where
uniqueness
problem:
t
to
problem
where
of
nd
equation
tro
since
5
it2|Ω | Ω q˜ L (Ω)2 2 0
kq˜k ≤Ckqk ,0,Ω 0,Ω1
C > 0 Ω Ω1 2
v∈V
divv =−q˜ Ω,
kvk ≤γkq˜k ≤γCkqk ,1,Ω 0,Ω 0,Ω1
γ > 0
2 −1 −1b (v,q) =kqk ≥γ C kvk kqk .1 1,Ω 0,Ω0,Ω 11
2 2f ∈ L (Ω) (u,p)∈
2 2V ×L (Ω) (u,p ) ∈ V ×L (Ω )1 10
ǫu ǫ (0,0)
ǫ ǫ > 01 2
ǫ
V Q k·k k·kV Q
a V ×V R
α> 0
2a(u,u)≥αkuk ∀u∈V.V
b V×Q R
β > 0
b(v,q)
inf sup ≥β.
q∈Q,q=0 kvk kvkv∈V,v=0 V Q
c Q×Q R
γ > 0
2c(u,u)≥γkuk ∀q∈Q.Q
d V ×V R
d(u,u)≥ 0 ∀u∈V.
′ℓ∈V
a(u,v)+b(v,p) =l(v),∀v∈V,
b(u,q) = 0,∀q∈Q,
).
to
inequalit
es
v
,
exists
while
2.1
(15)
?
will
form
b
sense,
e
ly
the
limit
form
le
when
b
go
which
let
(15),
go
ase
es
esp.
to
ar
zero
and
e
will
some
e
w
or
(14)
dep
is
y
xed.
area
T
2.2
o
pro
v
the
e
holds
in
v
unique
ergence
pr
results,
a
w
any
e
ongly
lo
exists
obtain
obtained
6
assertion
problem
L
(13)
as
satisfying
a
regularization
Corollary
of
on
(14)
oblem:
or
see
of
Sc
(15).
Moreo
W
that
e
biline
refer
om
to
on
the
I.4.3
ther
and
the
I.5.1
holds
of
ase
[6
?
the
℄
(14),
for
a
e
similar
(r
has
h.
(14)
In
et
the
b
rst
ontinuous
fr
since
F
to
is
is
o
piecewise
ther
results.
t,
follo
w
v
e
Hence
the
This
use
nal
the
e
results
biline
from
om
on
I.4.3
that
of
there
[
of
?
b
℄
.
hence
the
w
oint
e
only
state
for
and
w
pro
w
v
h
e
er
the
to
next
y
result
.
(compare
eing
with
ar
Theorem
fr
I.4.3
of
Remark
[
sense.
?
satises
℄
inf-sup
see
ondition:
also
e
Theorem
distributional
6.1
in
of
that
℄
(4)
Theorem
of
2.3
L
in
et
while,
order
distributional
and
in
In
(1)
b
of
e6
two
the
Hilb
over,
ert
Mor
sp
esp.
ac
solution
es
a
with
(15))
norm
(r
(13).
oblem
of
L
solution
nal
the
,
and
e
y
b
biline
denote
form
us
om
r
or
esp
on
e
which
str
L
et
er
Let
i.e.,
b
e
e
Theorem
a
wing
that
ontinuous
the
biline
e
ar
ha
form
w
fr
since
om
sense.
es
eak
pro
w
.
on
et
ery
ly
which
b
is
a
str
ontinuous
ongly
ar
fr
o
for
er
and
in
i.e.,
that
ther
h
e
exists
exists
℄
v
[
a
I.2.4
in
y
F
that
Hence
but
and
(6))
,
(resp.
sadd
(3)
p
of
pr
(16)
transmission
ending
some
the
that
satises
e
(15))
y
(resp.
arz's
(14)
h
of
y-
solution
Cauc
eak
b
w
v
the
.
L
elongs
et
that
b
b
Note
e
of
a
the
b
ontinuous
ok
at(u,p)∈V ×Q
ε> 0
ε ε εa(u ,v)+d(u ,v)+b(v,p ) =l(v),∀v∈V,
ε ε−εc(p ,q)+b(u ,q) = 0,∀q∈Q,
ε ε(u ,p )∈V ×Q
d(u,u) = 0 C > 0 ε
1ε ε ε ε
2ku −uk +d(u ,u ) +kp −pk ≤Cεkℓk ′.V Q V
ε ε(u ,p ) ∈ V ×Q
a,b,c,d
ε ε εa(u −u,v)+d(u ,v)+b(v,p −p) = 0,∀v∈V,
ε ε−εc(p ,q)+b(u −u,q) = 0,∀q∈Q.
ε ε εb(v,p −p) =−a(u −u,v)−d(u ,v),∀v∈V,
d
1ε ε ε ε
2kp −pk ≤C (ku −uk +d(u ,u ) ),Q 1 V
C > 0 ε1
ε εv =u −u d(u ,u) = 0
ε ε ε ε ε εa(u −u,u −u)+d(u ,u ) =−b(u −u,p −p).
ε ε ε ε ǫ ε ǫ ε ε εa(u −u,u −u)+d(u ,u ) =−εc(p ,p −p) =−εc(p −p,p −p)−εc(p,p −p)≤−εc(p,p −p).
a c C > 02
ε
ε 2 ε ε εku −uk +d(u ,u )≤C εkpk kp −pk .2 Q QV
λ =λ =λ μ1 2 i
−1 −1ε = (λ+μ ) εˆ=λi i
Z Z
2c (p,q) = pqdx,∀p,q∈L (Ω),d (u,v) = μdivudivvdx,∀u,v∈V.0 00
Ω Ω
ǫ ǫ 2(u ,pˆ)∈V ×L (Ω)0
ǫ ǫ ǫa (u ,v)+d (u ,v)+b (v,p ) =l(v),∀v∈V,0 0 0
ǫ ǫ 2−εˆc (p ,q)+b (u ,q) = 0,∀q∈L (Ω).0 0 0
see
of
argumen
t
solution
den
een
en-
(indep
This
problem
estimate
to
and
then
the
equiv
estimate
(20)
(20)
yield
from
the
℄
y
F
e
or
these
problem
(14),
follo
w
e
,
actually
tin
tak
hence
e
unique
(indep
on
exists
I.4.3
there
the
,
is
of
The
7
w
)
if
that
e
will
w
tend
problem
to
e
innit
tly
y
Find
(and
of
tak
(18)
e
e
tin
of
(21)
giv
the
en
the
b
y
has
problem
follo
(14))
assumptions
and
using
set
from
the
[
and
w
on
dierence
assumption
alen
eness
that
iden
et
the
and
rom
Mor
>F
over
(19)
,
,
ther
of
With
y
notations,
tit
e
iden
that
(13)
the
b
and
written
using
alen
Therefore
as
obtain
ws.
e
exists
w
endent
,
)
has
that
that
Pro
obtain
and
w
(19)
of:
of
y
y
of
tit
uit
iden
rst
and
the
inf-sup
in
using
a
The
aking
that
T
(17)
b)
a
).
solution
unique
ws
of
the
t
made
enden
and
(indep
the
solution
ts
.
Similarly
of
for
?
some
No
for
taking
any
h
the
b
sadd
t
le
equiv
p
(19)
oint
of
pr
tit
oblem:
rst
(17)
a)
has
w
a
(17)
(16),
e
(19)
unique
nd
solution
.
uit
yε −1 ε εpˆ =−εˆ divu =−λdivu Ω.
ε εp =−(λ+μ)divu Ω.
λ =λ1
−1ǫ = (λ+μ ) d = 01 1 1
Z
2c (p,q) = pqdx,∀p,q∈L (Ω ).1 1
Ω1
ε ε 2(u ,p )∈V ×L (Ω )11
ε εa (u ,v)+b (v,p ) =l(v),∀v∈V,1 11 1
ε ε 2−ε c (p ,q)+b (u ,q) = 0,∀q∈L (Ω ).1 1 1 11
ε −1 εp =−ε divu Ω .11 1 1
(ε ,ε ) = (0,0)1 2
(ε ,ε ) = (0,ε )1 2 2
(0,0) (0,0) (0,ε ) (0,ε )2 2(u ,p ) (u ,p )
a,b,c,d,i = 0,1i i i i
ε ε (0,0) (0,0)(u ,p ) (u ,p )
ε = (ε ,ε )→ (0,0) C > 01 2 0
ε
ε (0,0) ε (0,0)ku −u k +kp −p k ≤C εˆkfk .1,Ω 0,Ω 0 0,Ω
ε εε > 0 (u ,p )2
(0,ε ) (0,ε )2 2(u ,p ) ε → 0. C > 01 1
ǫ
ε (0,ε ) ε (0,ε )2 2ku −u k +kp −p k ≤C ε kfk .1,Ω 0,Ω 1 1 0,Ω1
ε (0,0)kpˆ −p k ≤C εˆkfk .0,Ω 0 0,Ω
Z
1 1ε ε ε 2
2 2d (u ,u ) = ( μ|divu | dx) ≤Cεˆkfk ,0 0,Ω
Ω
1/2εkμdivu k ≤ (maxμ )Cεˆkfk .0,Ω 0,Ωi
i=1,2
ε (0,0) ε ε (0,0)kp −p k ≤kμdivu k +kpˆ −p k .0,Ω 0,Ω 0,Ω
still
the
solution
(15)
The
ges
e
onver
in
b
(13)
iden
of
(21)
solution
of
the
innit
of
y
(15)
w
when
e
then
y
d,
when
xe
b
is
e
if
in
Mor
e
e
the
over
parameter
ther
set
e
denoted
exists
while
a
problem
p
of
ositive
of
from
onstant
b
Similarly
means
that
as
that
of
a
indep
Theorem
en-
ha
dent
e
of
the
endent
(15),
e
that
forms
indep
.
onstant
(the
ositive
(15))
p
and
a
tly
exists
b
e
solution
ther
these
over
remark
e
Mor
written:
e.
b
efor
will
b
the
d
follo
e
triangular
describ
(22)
as
this
with
y
(14)
in
of
particular
Note
e
that
while
this
(14)
seen
iden
tit
this
y
w
simply
also
means
v
that
v
solution
set
the
In
ds
towar
problem
ges
w
onver
tak
that
(13)
tends
of
bilinear
in
Since
to
that
y
Pro
other
of:
are
As
ed
men
problem
tioned
and
the
,
results
With
are
a
b
e
will
of
of
Theorem
the
2.3,
,
notations,
in
e
the
that
rst
(13)
also
where
e
w
solution
e
y
nd
denoted
w
b
e
(14)
ha
solution
solution
Hence
The
2.4
ws
Theorem
the
result.
inequalit
ergence
.
v
As
efore
next
the
tit
obtained
simply
that
e
Note
w
(22)
℄
a
?
seen
[
b
of
(15)
2.3
,
Theorem
problem
of
assumptions
e
the
as
satisfy
particular
But
of
when
to
towar
8
dsλ μi i
Ω Ω1 2
Ω Ω1 2
S Ω Ω C1 2 S
Ω S ω SS
ω ,i = 1,2 S Ω ω =ω +ωSi i S S1 S2
(r,θ) θ = 0 ∂C ∩∂Ω θ =ωS 1 S
∂C θ =ωS S1
S η 1S
S 0 S
ε = (ε ,ε ) ε > 0 ε > 0 (ε ,ε ) = (0,0) ε = 01 2 1 2 1 2 1
ǫ ǫ ǫε > 0 f = −μΔ(η u ) +∇(η p ) ε = (0,ε ) ε > 02 S S 2 2
−1ǫ ǫ ǫp =−ε divu η 1 S f fS2 2 2
S
ǫ (0,0) 2 2f →f L (C ) ǫ→ (0,0).S
ǫ (0,ǫ ) 2 22f →f L (C ) ǫ → 0.S 1
ǫ ǫ ǫ ǫf =η f−μΔη u −2∇u∇η +∇η p .S S S S
ǫ 2 2ε f L (C )S
ǫ ǫ ǫμΔ(η u )+∇(η p ) =f C ,S S S
ǫ ǫdiv(η u )+ǫη p = 0 C ,S S S
ǫη u = 0 ∂C ,S S
ǫ[η u ] = 0,S ωS1
ǫ ǫ[σ (η u ,η p )n] = 0ǫ S S ωS1
b
when
h
y
e
Here
w
in
when
and
on,
and
are
Theorem
w
ersion
no
a
rom
By
h
3
that
e
either
).
>F
of
these
(23)-(25)
near
b
results
w
and
(24)
y
regularit
een
the
part
and
hoice
,
ha
or
o
transmission
p
no
e
is
all
there
in
e.
satisfying
(i.
the
problem
oundary
of
b
terior
standard
near
with
er
problems
in
,
in
or
orho
es
(25)
Stok
In
or
.
Lam?
of
and
the
standard
the
x
the
are
Pro
problems
w
let
e
us
ed
set
on
the
that
,
eha
d
of
also
or
on
the
thanks
other
F
?
,
us
b
,
in
of
the
?
and
[
3.1,
to
refer
our
e
Let
(23)
w
the
at
wn,
the
(in
previous
the
et
kno
moreo
ell
angle
w
een
references
denote
the
.
and
d
neigh
℄
Figure
or
h
the
Exterior
b
3.1
neigh
there.
orho
orho
et
w
e
d
set
of
b
other
neigh
on
larger
use
a
outside
,
to
of:
to
denition,
equal
e
and
v
the
of
of
describ
d
ab
o
v
orho
e.
b
Corner
).
h
As
b
neigh
vior
a
w
is
olar
equal
use
to
will
in
w
near
W
limit
,
to
the
2.4.
function
or
to
Later
equal
let
e
with
e
from
are
in
terested
a
y
neigh
regularit
b
i)
orho
ii)
o
Lemma
d
then
of
lo
(13)
v
,
of
moreo
starting
v
is
er
to
w
hereafter:
e
y
ha
v
solutions
e
of
Lemma
angle
3.1
in
i)
e
function
our
problems
a
the
x
in
us
b
let
let
,
v
to
;
near
at
in
terior
problem
the
our
and
b
lo
e
to
W
order
(see
In
of
when
o
terface.
b
in
on
the
in
on
with
(14)
(resp.
whic
(15)),
innite
w
1).
ii)
y
e
denote
tly
a
and
and
b
,
pass
alw
w
a
b
ys
o
F
exterior
oundary
one
in
o
b
of
,
9
withǫ ǫ ǫ ǫ ǫσ (u ,p ) = σ (u ,p ) +μǫp [·] θ = ωǫ ǫ ω S1S1
ǫ(r,θ) η u (x,y)S
ǫ ǫ ǫ ǫ ˜ǫ˜ ˜η p (x,y),f (x,y) u (t,θ) p (t,θ),f (t,θ)S
cosθ −sinθ
e = , e = ,r θ
sinθ cosθ
ǫ ˜ǫ ǫ˜ ˜u f u (t,θ) =
ǫ tǫ ǫ ǫ ǫ ǫ ǫ˜ ˜ ˜˜ ˜ ˜u (t,θ)e +u (t,θ)e , f (t,θ) = f (t,θ)e +f (t,θ)e . q (t,θ) = ep (t,θ)t r θ θ r θt θ
ǫ 2t ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ˜ ˜ ˜h (t,θ) =e f (t,θ). v = (v ,v ) = (u ,u ), h = (h ,h )t θt θ t θ
ε ǫ ǫ ǫL (D,D )(v ,q ) =h B ,t θ S
ǫv = 0, ∂B ,S
ǫ[v ] = 0,ωS1
ǫ ǫ ǫ T[σ (D,D )(v ,q ) ] = 0,t θ ωS1
εB =R×(0,ω ) LS S
2 2μ(D +D −1) 2μD −D +1θ tt θ
ε 2 2 L (D,D ) = −2μD μ(D +D −1) −D ,t θ θ θt θ
D +1 D εt θ
ǫσ
μD μ(D +1) 0ǫ θ tσ (D,D ) = .t θ
2μ 2μD −1+μεθ
T˜V = (v,μDv −q,v ,Dv ) ,t t t θ t θ
T˜V = (v ,v ,v ,v ) .1 2 3 4
v =μDv −q2 t 1
1
q =− (μv +v +μD v ).1 2 θ 3
1+με
v
tial
is
system
en
giv
and
en
of
b
ariables
y
and
.
e
half-line
tro
the
As
.
With
)
these
onen
transformations
radial
w
allo
e
Since
sho
,
w
(29)
that
while
problem
tial
(23)-(25)
of
is
function
equiv
,
alen
and
t
to
tangen
the
the
problem
to
(26)-(28)
whic
hereafter:
write
(26)
e
along
v
jump
lo
the
e
means
last
and
giv
where
usual,
(28)
Using
in
b
(27)
system
on
Euler
As
hange
in
v
[5
the
,
(resp.
2]
and
w
e
also
W
Inserting
ts
in
transformed
rst
tial
w
set
equations
and
(26)
help
e
with
that
write
is
ws
alen
h
with
W
rst
shortly
dieren
ectors
equation:
w
appropriated
ha
Hilb
e
ert
basis
space.
F
or
in
that
is
purp
the
ose,
equation
w
(26)
e
es
in
w
tro
the
the
is
dieren
(resp.
).
the
in
v
y
ector
olar
giv
p
set
dieren
the
and
where
e
(29)
w
the
an
t
t
o
to
of
w
(26)-(28)
get
in
(26)
to
equiv
a
t
dieren
the
tial
order
equation
tial
in
10
an
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