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Regularity results of Stokes Lamé interfa e problems

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Niveau: Supérieur, Doctorat, Bac+8
Regularity results of Stokes/Lamé interfa e problems D. Mer ier, S. Ni aise ? Abstra t This paper is devoted to some transmission problems involving the Lamé and Stokes systems in two-dimensional nonsmooth domains. We rst show that these problems may be obtained as a limit of a transmission Lamé problem when the Lamé oe ient ? goes to innity. We further investigate the behavior of the solutions of these problems near geometri al singularities, espe ially near orner points where the interfa e interse ts the boundaries. In parti ular we show stable de ompositions with respe t to the perturbation parameter ?. Some minimal regularity results are dedu ed from similar minimal regu- larity result of transmission Lamé problems obtained in [3?. Some numeri al results for the al ulation of the singular exponents in the asymptoti expansion are presented and onrmed these minimal regularity results. 1 Introdu tion We investigate two-dimensional transmission problems for the Lamé and Stokes systems of the following types: Two polygonal domains are onne ted via an interfa e line: on ea h subdomain we onsider either the Stokes system or the Lamé system. Boundary onditions on the exterior boundary pie es and (non standard) transmission onditions on the interfa e are imposed. It is well known that the solutions of these boundary-transmission problems have singular behaviour near orners, espe ially where the interfa e interse ts the boundaries.

  • iennes

  • transmission lamé

  • boundary onditions

  • pie ewise onstant

  • mer ieruniv-valen

  • onstant given

  • dimensional transmission problems

  • ?1 µ1?u1


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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