Content uploaded by Luisa Consiglieri
Author content
All content in this area was uploaded by Luisa Consiglieri on Jul 14, 2023
Content may be subject to copyright.
Content uploaded by Luisa Consiglieri
Author content
All content in this area was uploaded by Luisa Consiglieri on Oct 16, 2015
Content may be subject to copyright.
Related problems
Formulation/Statement of the problem
A (p-q) coupled system in
elliptic nonlinear problems
with nonstandard boundary conditions
Luisa Consiglieri
Department of Mathematics and CMAFUL
Existence of a solution (proof)
The problem
)3,2( nIRboundedOpen n
-
,
inugeeA
inueFf
),(
),(
nn
n
onueeeA
ueG
)()(),(
),(
Find:
n
nii IR
IReu
:)(
:,
,...,1
:0
:
0
oneu
conditionsDirichlet
vectornormal
outwardunit
n
(p-q) coupled system
npp
n
IRIReeF
IRIRFp
,),||1(),(||
:0
:,1
#
#
#
#
1
,
variable, second the onconvex strictly
,continuous
,
ry,Carathéodo
,0)()),,(),,((
||1|),,(|
),,(||:0
:,/12
1#
##
#
eAeA
eA
eA
IRIRIRAnq
q
q
nn
Boundary functions assumptions
0)( 0
0 meas
nql
nq
qn
qn
l
ee
esigne
IRIR
l
if
if
|
:ryCarathéodo
,1
,
)1(
1
),1|(||),(
,0)0,(,0)(]),(),([
:
#
0
nps
np
pn
np
s
uueG
IRIRIRG
s
if
if
:ryCarathéodo
,1
,
)1(
1
,||1),,(0
:
#
0
u= mean value of the fluid velocity
f= mean value of external forces
viscosity eddy viscosity
e mean turbulent kinetic energy (k)
(rate of dissipation of the turbulent energy)
Turbulence modelling
WEAK-RENORMALIZED SOLUTION:
(Navier-Stokes system with Dirichlet conditions)
[Lewandowski, 1994] [Climent & Cara, 1997]
SpaldingLauderofmodel)(
k
Weak solution
);(
)(:))((
0v:)(v);(
0
,1
0
''
0
,1
0
,1
0
:L
:
p'
div
r
pnp
pp
We
LL
onWWu
),,( e
u Find
);(
)()()(),(
0
)1/(,1
0
n
qrr
W
dudxgudedxeeA
);(v,)(v
),(v),(}),(v),({
0
,1
0
p
Wdxuf
dueGeGdxueFeF
xdeFdxueFdxu ),(*),(
deGdueGdu n n ),(*),()(
Main existence result
THEOREM: Under the above assumptions
and additionally
then there exists a weak solution
to the coupled system
, )(),(1' LgLf p
);();( 0
,1
00
,1
0 , L , p'
div rp WeWu
1
)1(
1
n
qn
r
Remark
)()(
11
)1( )1/(,1
LWn
qr
r
n
qn
rqrr
:
Proof’s idea
Existence result: (FIXED POINT ARGUMENT)
);(v,)(v
),(v),(),(v),()(
||||:
0
,1
0
1
1,1
:
p
Wdxuf
duGGdxuFFuu
R
);(,
)(),(:),,(
||||||||||||:),,(
0
)1/(,1
0
3,12,11,1
qrr
Wdhdx
dedxeAhee
RhRRh
DUAL PROBLEM
PRIMAL PROBLEM
First auxiliary existence result
);(v,)(v),(v),(
),(v),()(
0
,1
0
1
:
p
WdxufduGG
dxuFFuu
Estimate
C=Sobolev constant
)1/(1
,'
#
;,1 ||||||||
p
pp f
C
u
0v
Continuous dependence
weakWinuu
uuuuTaking
LinandLin
theoremellichRSobolevFrom
p
m
mm
m
-
:
);(
)()(
)(),(
0
,1
0
11
Lagrange multipliers (2nd result)
:n , L find , For div )();( ''
0
,1
0 spp LWu
dGduGdu
dxFdxuFdxu
),(*),()(
),(*),(
n n
Estimates
)1/()meas(||||)(2||||
)1/()meas(||||)(2||||
,
'#','
,
'#'
'
sus
pup
s
s
sss
s
p
p
ppp
p
n
,
2
R
3
R
[Boccardo and Gallouet, 1989] with Dirichlet conditions ...
);(
,)(),(
),,(
0
)1/(,1
0
1
theory) data-(L : (SOLA) 1
qrr
W
dhdxdedxeA
hee
Third auxiliary existence result
:/21
||||||||||||,
);()()(,
'
,1,1,
0
,1
0
''
nqn
hCeLenq
WeLhLINM
q
q
qM
q
M
q
M
q
M
For
and For
, 1
1
)1(
1
||||||||||||:
)|(|,1)(
))(/(
,1,1,
n
qn
r
rn
rn
rq
r
hCe
eMee
rqqrr
r
and for
Estimate
))sign(min( functionTest
),( 321 RRCR
Passage to the limit on M
,1,1,1 ||||||||||)(||:
),(,)(
hCe
keke
M
MM
Estimate
)maxmin(
functionTest
[Prignet, 1995] Remarks on existence and uniqueness of solutions ...
[Prignet, 1997] Conditions aux limites non homogènes ...
[dall’Aglio, 1995] SOLA: Solution Obtained as Limit of Approximations
weakLinee
weakWinee
heeheeTaking
M
r
M
MMMM
)()()(
);(
),,(),,(
1
0
,1
0
-
:
etoeofmeasureineconvergencthefrom
ineAeA
M
qr
M
L )(),(),( )1/(
X Hausdorff, locally convex topological vector space
K compact, convex, non empty
L: K P(K) upper semicontinuous
L(x) closed, convex, non empty,
then L admits, at least, one fixed point.
Tychonov-Kakutani-Glicksberg fixed point theorem
),,(
),()(),,(
)()()(:
1
1
111,1
hee
uuuh
LLWX
(sola)
topologies weak the with endowed
))(,,( uue n-
Kx
Fluid-energy coupled system
•Generalized Navier-Stokes-Fourier model
•Generalized Coulomb friction law
•Convective-radiative condition
Governing equations
)3,2( nIRboundedOpen n
INCOMPRESSIBILITY
MOTION EQUATIONS
u in
x
u
n
ii
i
10
fuu
)(
1
ENERGY EQUATION
gDe
)(:
1
uqu
fluid velocity:
stress tensor:
internal energy: e
nii
u,...,1
)(
u
jiij ,
)(
Constitutive law for the stress tensor
ISOTHERMAL (constant coefficients)
[Cioranescu, 1977]
[Naumann & Wulst, 1979]
....
|)](|)()(|),([ 21 u,|u DFeDFeI
tensorstressdeviator
)1()(01
)1()(:,
2
22
#
1#0021
p
pp
ddFpp
ddFdIRIRFF
:convex
pressure
D
DDD
u)uu
uuu
T
2/(
:|| 2
Examples
POWER LAWS (Ostwald & de Waele)
p=2: Newtonian fluid
p>2: dilatant fluid
1<p<2: pseudo-plastic fluid
p=1: Bingham fluid
[Duvaut & Lions, 1972] isothermal Bingham fluid
p
ddF )(
0)(|
)(
u |
D
plasticityofthreshold
GENERALIZED NEWTONIAN FLUIDS
uu DDAeabledifferentiF |)(|),(:||
[Malek, Necas, Rokyta & Ruzicka, 1996] isothermal fluids
Constitutive law for the heat flux
)(),( ee aq
)1|(||)(|1
)(||:
1#
#
q
qnn
q
IRIR
a
a :continuous a
heat capacity
FOURIER LAW (q=2)
EXAMPLE
2
||)(
q
a
e
cp)(
tyconductiviheat
p
ce
q)()(,
Non slip and slip boundary conditions
On
u On
:
0:
0
conditionDirichlet
0)( 0
0 meas
TTT
TT
u ||
u ||
,0),(
0),(
e
e
conditionCoulombfrictionntwisePoi
FRICTIONAL BOUNDARY CONDITION
s
Neu ||),(,0: TT u nu
[Cioranescu, 1977] prescribed normal stress tensor
[Beirão da Veiga, 2005] isothermal Coulomb law
vectornormal
outwardunit
N
n
n
T
:
s=2, linear Navier law
s=3, Chezy-Manning law
s=1:
Convective-radiative boundary condition
TT una
),()(),( eee
On
On
:
0:
0
econditionDirichlet
c
:)4(
:)1(
l
hl
convective heat transfer coefficient
Stefan-Boltzmann constant
EXAMPLE
eeehe l1
||)},({),(
c
1),1|(||),(
0)0,(:
#
0
lee
IRIR
l |
:ryCarathéodo
[J.F. Rodrigues and i, 2003] On stationary flows ...
Related coupled systems
[Duvaut & Lions, 1972] Transfert de chaleur dans un fluide de Bingham ...
(constant plasticity threshold, without convective terms,
and DIRICHLET condition for fluid motion)
To precise the structure of the variational inequality
it will envolve Lagrange multipliers
0|)(|
|)(|
)(
)()( u if
u
u
u D
D
D
DI
[Baranger & Mikelic, 1994] ... écoulement quasi-Newtonian ...
(without convective term, and DIRICHLET conditions)
The weak formulation
),,( e
u Find
)1/(
,):(
)()()(
qrr
WddxgD
dedxeedxe
u u
a u
TT
pTT Vvuvfuv
uv vuu
,)(||||)(
}),(),({:
dxde
dxDeFDeFdxD
ss
),( u DeF
0
,1
''
0
,1
0:)(
))((,:))((
0v,0,0:))((
onWWe
LL
ononinW
r
np
jiij
nnp
N
np
:
:Y
vvv :Vu
r
p'
p
Assumptions
ENERGY-DEPENDENT PARAMETERS
1
),(0::
eIRIR
:yield friction ryCarathéodo
10
10
),(
),(::,
e
eIRIR
:esviscositi ryCarathéodo
10 ),(::
eIRIR
:tyconductivi ryCarathéodo
otherwise and
if ,
,1
)1(
1
s
np
pn
np
s
(p,q,l) coercive- and growth- ness for
nql
nq
nqn
nq
l
IReesigne
n
n
pF
F
if ,
if
convex, strictly
a
1
2
1,
)1)(1(
1
,,0)()),(),(( 2
3
),,(
1
Strict monotonicity for a:
0)())()((
aa
np
n
q
np
n
n
nnp
pn
q
,
2
1
2
3
,
)1(
)12(
Existence result
THEOREM: Under the above assumptions
and additionally
then there exists a weak solution
to the coupled system
,Lf )()( 1' Lg
p
r
We , Y , Vu p'p
1
)1(
1
n
qn
r
Proof’s idea
Existence result: (FIXED POINT ARGUMENT)
pTT
Vp
Vvuvfuv
uv vuw :wuu
ww
,)(||||)(
),(),(:),(
PROBLEMPRIMAL
||||||||:),(
1
2,11
dxd
dxDFDFdxD
RR
ss
)1/(
4,13,12,11
,
)()()(:),,,(
||||||||||||||||:),,,(
qrr
Wdhdx
dedxedxehee
RhRRRh
a w w
ww Vp
DUAL PROBLEM
First auxiliary existence result
pTT Vvuvfuv
uv vuw :w,uu
,)(||||)(
),(),(:)(
1
dxd
dxDFDFdxD
ss
Estimate
C=Sobolev constant
1
:R
)1/(1
,'
##
||||||||
p
p
Cfu Vp
p
V0v
Continuous dependence
weakinTaking
LinandLin
n
n
p
pn
pn
tin
mmmmm
m
t
m
-V uu :wuu ,wuu
, L ww
p
),()(
)(),(
2
3
),(
11
Lagrange multipliers (2nd result)
: , Y - find ,VuFor Tp'p
21
I
ddsCds
onanddds
dxFdxDFdxD
dxFdxDFdxD
s
sTT
TT
uu
uu
u u
u u
'
2
222
1
111
)(
)()(||)(:1
)(||||)(:1
)(
)(|)(|)(:
)(
)(|)(|)(:
Estimates
41,1
311
:||||
:||:||
RCR
RRCD p
TT
,
u
u
)1/(
1
,
)()()(
),,,(
qrr
Wdhdx
dedxedxe
hee
a w
theory) data-(L :w (SOLA) 1
Third auxiliary existence result
),,,(
),(),(),,,(
)()()(:
1
1
111,1
hee
h
LLWX
w (sola)
wu, wuu w
topologies weak the with endowed
Vp
),:,,( TT u- u u
De
Conclusion
Open problems
landqofceindependen
sandpofceindependen
[...]
regularity
Acknowledgement:
Université de Pau et des Pays de L’Adour