On implicitly constituted incompressible fluids
Josef Málek
Mathematical Institute of the Charles University
Sokolovská 83, 186 75 Prague 8, Czech Republic
M. Bulı́ček, P. Gwiazda, J. Málek and A. Swierczewska-Gwiazda
KR Rajagopal
M. Heida, V.Průša
July 6, 2010
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
1 / 54
Formulation of problem
Governing equations
Equations in a bounded domain Ω ⊂ R3 for t ∈ [0, T ]:
div v = 0
v,t + div(v ⊗ v) − div S = −∇p + f
S = ST
• v is the velocity of the fluid
• p is the mean normal stress (pressure) p := − 31 tr T
• f external body forces ( ≡ 0)
• S is the constitutively determined (deviatoric) part of the Cauchy stress
The Cauchy stress: T = −pII + S
Málek (Charles University in Prague)
T = T − 13 (tr T )II + 31 (tr T )II
Implicit fluids & Orlicz spaces
July 6, 2010
2 / 54
Constitutive equations
Point-wisely given constitutive equations
D(v) := ∇v + (∇v)T .
• D (v) - the symmetric part of the velocity gradient: 2D
• Consider merely point-wise relations between D and S (or D and T )
NO integral, differential (rate-type) or stochastic constitutive relations:
S, D ) = 0
G (S
or
G(T
T, D ) = 0
G̃
robust class of fluids
justification to adhoc models
easy incorporation of constraints (as div v = 0)
S) vrs S = H̃
H(D
D)
new class of explicit models D = H (S
Extensions:
S, D , p, x, t, temperature, density, concentration, etc.) = 0
G (S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
3 / 54
Constitutive equations
Point-wisely given constitutive equations
D(v) := ∇v + (∇v)T .
• D (v) - the symmetric part of the velocity gradient: 2D
• Consider merely point-wise relations between D and S (or D and T )
NO integral, differential (rate-type) or stochastic constitutive relations:
S, D ) = 0
G (S
or
G(T
T, D ) = 0
G̃
robust class of fluids
justification to adhoc models
easy incorporation of constraints (as div v = 0)
S) vrs S = H̃
H(D
D)
new class of explicit models D = H (S
Extensions:
S, D , p, x, t, temperature, density, concentration, etc.) = 0
G (S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
3 / 54
Constitutive equations
Navier-Stokes fluids
T = −pII + 2µ∗D ⇐⇒ S = 2µ∗D ⇐⇒ D =
1
S
2µ∗
0 ≤ ξ = S ·D
D|2 = (2µ∗ )−1 |S
S |2
= 2µ∗ |D
1
S |2
D|2 + (2µ∗ )−1 |S
= µ∗ |D
2
µ∗ = 1/2
=⇒ S = D
ξ =S·D =
Málek (Charles University in Prague)
D
S ·D
2
D
D|2 + 21 |S
S |2
+ S2·D
= 12 |D
Implicit fluids & Orlicz spaces
July 6, 2010
4 / 54
Constitutive equations
Navier-Stokes fluids
T = −pII + 2µ∗D ⇐⇒ S = 2µ∗D ⇐⇒ D =
1
S
2µ∗
0 ≤ ξ = S ·D
D|2 = (2µ∗ )−1 |S
S |2
= 2µ∗ |D
1
S |2
D|2 + (2µ∗ )−1 |S
= µ∗ |D
2
µ∗ = 1/2
=⇒ S = D
ξ =S·D =
Málek (Charles University in Prague)
D
S ·D
2
D
D|2 + 21 |S
S |2
+ S2·D
= 12 |D
Implicit fluids & Orlicz spaces
July 6, 2010
4 / 54
Constitutive equations
Navier-Stokes fluids
T = −pII + 2µ∗D ⇐⇒ S = 2µ∗D ⇐⇒ D =
1
S
2µ∗
0 ≤ ξ = S ·D
D|2 = (2µ∗ )−1 |S
S |2
= 2µ∗ |D
1
S |2
D|2 + (2µ∗ )−1 |S
= µ∗ |D
2
µ∗ = 1/2
=⇒ S = D
ξ =S·D =
Málek (Charles University in Prague)
D
S ·D
2
D
D|2 + 21 |S
S |2
+ S2·D
= 12 |D
Implicit fluids & Orlicz spaces
July 6, 2010
4 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Power-law fluids
2−r
1
S| r −1 S
D|r −2D ⇐⇒ S = 2µ∗ |D
D|r −2D ⇐⇒ D = [2µ∗ ]− r −1 |S
T = −pII + 2µ∗ |D
µ∗ = 1/2
r ′ := r /(r − 1)
D|r = |S
S|r /(r −1)
ξ = S · D = |D
=
D
S ·D
r
+
D
S ·D
r′
=
D|r
|D
r
+
S|r
|S
r′
′
Also, for all D , E ∈ R3×3
D) − S̃(E
E) · (D
D − E) ≥ 0 ,
S̃(D
where
for all S 1 , S 2 ∈ R3×3
S1 − S 2 ) · B (S
S1 ) − B (S
S2 ) ≥ 0 ,
(S
B) := 2µ∗ |B
B|r −2B
S̃(B
2−r
S) := 2µ∗ |S
S| r −1 S
where B (S
Generalizations
D|2 )r −2D
S = (1 + |D
D|2 )D
D
S = ν(|D
Málek (Charles University in Prague)
2−r
S|2 ) r −1 S
D = (1 + |S
S|2 )S
S
D = µ(|S
Implicit fluids & Orlicz spaces
July 6, 2010
5 / 54
Constitutive equations
Fluids with shear-rate dependent viscosities
S := S (D
D))
Continuous Explicit Standard Power-Law models (S
D |2 )
µ(|D
D(v)|2 = 1/2|u ′ |2 := κ shear rate
v = (u(y ), 0, 0) =⇒ |D
D|2 ) = 2µ∗ |D
D|r −2
µ(|D
2
2µ∗0
2
2µ∗0 (ǫ
D| ) =
µ(|D
D| ) =
µ(|D
+
D|r −2
µ∗1 |D
1<r <∞
2 r −2
D| )
+ |D
1<r <∞
1∈R
power-law like fluids =⇒ r -coercivity, (r − 1)-growth and strict monotonicity
fluids with shear-rate dependent viscosity
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
6 / 54
Constitutive equations
Classical power-law model for various power-law index
m ∈ (0, +∞)
shear thickening
m=0
Navier–Stokes
m ∈ − 21 , 0
shear thinning
|Tδ |
m = − 12
limit case
2µ
√0
λ
m ∈ −∞, − 21
nonmonotone
0
0
|D|
D|2 )mD
S = (1 + |D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
7 / 54
Constitutive equations
Stress power-law model for various power-law index
D := D (S
S))
Continuous Explicit Stress Power-law models (D
n ∈ −∞, − 21
n = − 21
n ∈ − 21 , 0
n=0
|Tδ |
n ∈ (0, +∞)
0
0
√α
β
|D|
S|2 )nS
D = (1 + |S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
8 / 54
Constitutive equations
Power-law like fluids with activation criteria/discontinuous
stresses
threshold value for the stress to
start flow
Bingham fluid
Herschel-Bingham fluid
S
drastic changes of the properties
when certain criterion is met
111111111111111111111111111
000000000000000000000000000
K=|u’(r)|
Málek (Charles University in Prague)
formation and dissolution of
blood
chemical reactions/time scale
Implicit fluids & Orlicz spaces
July 6, 2010
9 / 54
Constitutive equations
Power-law like fluids with activation criteria/II
S| > τ ∗
|S
if and only if
S = τ∗
S| ≤ τ ∗
|S
if and only if
D=0
D
D|2 )D
D
+ 2µi (|D
D|
|D
is equivalent to
Similarly:
+
D|2 ) τ ∗ + (|S
S| − τ ∗ )+ D = |S
S| − τ ∗ S
2µi (|D
D|2 )D
D
S = µα (|D
D|2 )D
D
S = µβ (|D
∗
S=µ D
if
if
if
D| < d ∗
|D
D| > d ∗
|D
D| = d ∗ ,
|D
∗
∗ µα (s) and µ
∗ µβ (s)
µ∗ takes any value between µ∗α := lims→d−
β := lims→d+
D| − d ∗ |S
S = M(|D
D|2 )(|D
D| − d ∗ )D
D
| |D
with M(s) := max{µα (s)sgn(s − d ∗ ); µβ (s)sgn(s − d ∗ )}
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
10 / 54
Constitutive equations
Power-law like fluids with activation criteria/II
S| > τ ∗
|S
if and only if
S = τ∗
S| ≤ τ ∗
|S
if and only if
D=0
D
D|2 )D
D
+ 2µi (|D
D|
|D
is equivalent to
Similarly:
+
D|2 ) τ ∗ + (|S
S| − τ ∗ )+ D = |S
S| − τ ∗ S
2µi (|D
D|2 )D
D
S = µα (|D
D|2 )D
D
S = µβ (|D
∗
S=µ D
if
if
if
D| < d ∗
|D
D| > d ∗
|D
D| = d ∗ ,
|D
∗
∗ µα (s) and µ
∗ µβ (s)
µ∗ takes any value between µ∗α := lims→d−
β := lims→d+
D| − d ∗ |S
S = M(|D
D|2 )(|D
D| − d ∗ )D
D
| |D
with M(s) := max{µα (s)sgn(s − d ∗ ); µβ (s)sgn(s − d ∗ )}
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
10 / 54
Constitutive equations
Power-law like fluids with activation criteria/II
S| > τ ∗
|S
if and only if
S = τ∗
S| ≤ τ ∗
|S
if and only if
D=0
D
D|2 )D
D
+ 2µi (|D
D|
|D
is equivalent to
Similarly:
+
D|2 ) τ ∗ + (|S
S| − τ ∗ )+ D = |S
S| − τ ∗ S
2µi (|D
D|2 )D
D
S = µα (|D
D|2 )D
D
S = µβ (|D
∗
S=µ D
if
if
if
D| < d ∗
|D
D| > d ∗
|D
D| = d ∗ ,
|D
∗
∗ µα (s) and µ
∗ µβ (s)
µ∗ takes any value between µ∗α := lims→d−
β := lims→d+
D| − d ∗ |S
S = M(|D
D|2 )(|D
D| − d ∗ )D
D
| |D
with M(s) := max{µα (s)sgn(s − d ∗ ); µβ (s)sgn(s − d ∗ )}
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
10 / 54
Constitutive equations
Power-law like fluids with activation criteria/II
S| > τ ∗
|S
if and only if
S = τ∗
S| ≤ τ ∗
|S
if and only if
D=0
D
D|2 )D
D
+ 2µi (|D
D|
|D
is equivalent to
Similarly:
+
D|2 ) τ ∗ + (|S
S| − τ ∗ )+ D = |S
S| − τ ∗ S
2µi (|D
D|2 )D
D
S = µα (|D
D|2 )D
D
S = µβ (|D
∗
S=µ D
if
if
if
D| < d ∗
|D
D| > d ∗
|D
D| = d ∗ ,
|D
∗
∗ µα (s) and µ
∗ µβ (s)
µ∗ takes any value between µ∗α := lims→d−
β := lims→d+
D| − d ∗ |S
S = M(|D
D|2 )(|D
D| − d ∗ )D
D
| |D
with M(s) := max{µα (s)sgn(s − d ∗ ); µβ (s)sgn(s − d ∗ )}
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
10 / 54
Constitutive equations
Discontinuous response described by a maximal monotone graph
T12
Tmax
shear rate
T1
T2
Málek (Charles University in Prague)
shear rate
T
Implicit fluids & Orlicz spaces
July 6, 2010
11 / 54
Constitutive equations
Perfect plasticity
Ugly “discontinuous” explicit models such as
Perfect plasticity
D| = 0 =⇒ |S
S| ≤ 1
|D
D| > 0 =⇒ S :=
|D
D
D|
|D
can be described by a nice continuous implicit formula
D|S
S − D | + (|S
S| − 1)+ = 0
||D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
12 / 54
Constitutive equations
Implicit theories/I - KR Rajagopal since 2003
Implicit constitutive theory: ability to capture responses of larger set of materials
T, D ) = 0
G (T
Isotropy of the material implies
TD + DT )
α0I + α1T + α2D + α3T 2 + α4D 2 + α5 (T
T2D + DT 2 ) + α7 (T
TD 2 + D 2T ) + α8 (T
T 2D 2 + D 2T 2 ) = 0
+ α6 (T
αi being a functions of
TD ), tr(T
T2D ), tr(D
D2T ), tr(D
D 2T 2 )
tr T , tr D , tr T 2 , tr D 2 , tr T 3 , tr D 3 , tr(T
For incompressible fluids
T=
Málek (Charles University in Prague)
tr T
3
D
I + µ(tr T , tr D 2 )D
Implicit fluids & Orlicz spaces
July 6, 2010
13 / 54
Constitutive equations
Implicit theories/II
Implicit constitutive theory: ability to include constraints in an easy way
If
D)
T = G 1 (D
isotropy of the material implies
T = β0 I + β1 D + β2 D 2
βi = βi (tr D 2 , tr D 3 )
If
T)
D = G 2 (T
isotropy of the material leads to
D = γ 0 I + γ1 T + γ 2 T 2
T2 −
= γ1S + γ2 (T
Málek (Charles University in Prague)
γi = γi (tr T , tr T 2 , tr T 3 )
tr T 2
I)
3
Implicit fluids & Orlicz spaces
July 6, 2010
14 / 54
Constitutive equations
Implicit theories/II
Implicit constitutive theory: ability to include constraints in an easy way
If
D)
T = G 1 (D
isotropy of the material implies
T = β0 I + β1 D + β2 D 2
βi = βi (tr D 2 , tr D 3 )
If
T)
D = G 2 (T
isotropy of the material leads to
D = γ 0 I + γ1 T + γ 2 T 2
T2 −
= γ1S + γ2 (T
Málek (Charles University in Prague)
γi = γi (tr T , tr T 2 , tr T 3 )
tr T 2
I)
3
Implicit fluids & Orlicz spaces
July 6, 2010
14 / 54
Constitutive equations
Implicit formulation - maximal monotone ψ-graph setting
S, D ) ∈ A
D, S ) = 0
(S
⇐⇒
G (D
Assumptions (A is a ψ-maximal monotone graph):
(A1)
(A2)
(0, 0) ∈ A
S1 , D 1 ), (S
S2 , D 2 ) ∈ A
Monotone graph: For any (S
S1 − S 2 ) : (D
D1 − D 2 ) ≥ 0
(S
No strict monotonicity is needed!
(A3)
S, D ) there holds
Maximal graph: If for some (S
S − S̃
S) : (D
D − D̃
D) ≥ 0
(S
S, D̃
D) ∈ A
∀ (S̃
then
S, D ) ∈ A
(S
(A4)
S, D ) ∈ A
ψ-graph: There are α ∈ (0, 1] and g > 0 so that for any (S
D) + ψ ∗ (S
S)) − g
S : D ≥ α(ψ(D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
15 / 54
Constitutive equations
What is ψ? An excursion to Orlicz spaces
Assume that ψ : R3×3
sym → R is an N - function (if it depends only on the modulus then
Young function), i.e.,
ψ is convex and continuous
D) = ψ(−D
D)
ψ(D
lim
D|→0+
|D
D)
ψ(D
= 0,
D|
|D
lim
D|→∞
|D
D)
ψ(D
=∞
D|
|D
We define the conjugate function ψ∗:
S) := max (S
S · D − ψ(D
D))
ψ ∗ (S
D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
16 / 54
Constitutive equations
What is ψ? An excursion to Orlicz spaces
Assume that ψ : R3×3
sym → R is an N - function (if it depends only on the modulus then
Young function), i.e.,
ψ is convex and continuous
D) = ψ(−D
D)
ψ(D
lim
D|→0+
|D
D)
ψ(D
= 0,
D|
|D
lim
D|→∞
|D
D)
ψ(D
=∞
D|
|D
We define the conjugate function ψ∗:
S) := max (S
S · D − ψ(D
D))
ψ ∗ (S
D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
16 / 54
Constitutive equations
What is ψ? An excursion to Orlicz spaces/2
• Young inequality:
D) + ψ ∗ (S
S)
S : D ≤ ψ(D
• Orlicz spaces: The Orlicz space Lψ (Ω)d×d is the set of all measurable function
D : Ω → R3×3
sym such that
ˆ
ψ(λ−1D ) dx = 0
lim
λ→∞
Ω
with the norm
DkLψ := inf{λ;
kD
• Hölder inequality
ˆ
Ω
• ∆2 -condition
Málek (Charles University in Prague)
ˆ
Ω
ψ(λ−1D ) dx ≤ 1}
ab dx ≤ 2kakLψ (Ω) kbkLψ∗ (Ω)
D) ≤ C1 ψ(D
D) + C2
ψ(2D
Implicit fluids & Orlicz spaces
July 6, 2010
17 / 54
Constitutive equations
Maximization of entropy production
In order to specify the constitutive relations, the principle of maximal entropy production
(laziness, economy) is used (KR Rajagopal, A Srinivasa):
S ·D = ξ ≥ 0
(*)
D) ≥ 0 and for some fixed S we would like to maximize ξ
Let us assume that ξ := ξ(D
with the constraint (*).
D |2
ξ := 2ν0 |D
D|)|D
D|
ξ = ν(|D
S = 2ν0D
2
Málek (Charles University in Prague)
D|)D
D
S = ν(|D
Implicit fluids & Orlicz spaces
July 6, 2010
18 / 54
Constitutive equations
Maximization of entropy production - dual view
S) ≥ 0 and for some fixed D we would like to maximize ξ
Let us assume that ξ := ξ(S
with the constraint ξ = S · D .
ξ :=
2
S |2
|S
ν0
∗
S|)|S
S|
ξ = ν (|S
D = 2ν0S
2
Málek (Charles University in Prague)
S|)S
S
D = ν ∗ (|S
Implicit fluids & Orlicz spaces
July 6, 2010
19 / 54
Constitutive equations
Maximization of entropy production
D, S) ≥ 0 and
Let us assume that ξ := ξ(D
(i) for some fixed S we would like to maximize ξ with the constraint ξ = S · D
or
(ii) for some fixed D we would like to maximize ξ with the constraint ξ = S · D
ξ :=
ξ=
D|2 +|S
S|2
|D
2
D|r
|D
r
+
′
S|r
|S
r′
Málek (Charles University in Prague)
S=D
D|r −1D
S = |D
Implicit fluids & Orlicz spaces
July 6, 2010
20 / 54
Constitutive equations
Optimality of ψ and ψ ∗
D) + ξ2 (S
S) ≥ 0 - not necessarily conjugate and
Let us assume that ξ := ξ1 (D
(i) for some fixed S we would like to maximize ξ with the constraint ξ = S · D
or
(ii) for some fixed D we would like to maximize ξ with the constraint ξ = S · D
It is the same as maximize ξ1 with the constraint
S) = ξ1 (D
D)
S · D − ξ2 (S
Hence, for D - the point where maximum is reached - we interchange the role of S and
D , so at this point
S · D − ξ2 (S
S)) = ξ1 (D
D)
max (S
S
But it implies
D) := max (S
S · D − ξ2 (S
S)) = ξ1 (D
D)
ξ2∗ (D
S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
21 / 54
Constitutive equations
Optimality of ψ and ψ ∗
D) + ξ2 (S
S) ≥ 0 - not necessarily conjugate and
Let us assume that ξ := ξ1 (D
(i) for some fixed S we would like to maximize ξ with the constraint ξ = S · D
or
(ii) for some fixed D we would like to maximize ξ with the constraint ξ = S · D
It is the same as maximize ξ1 with the constraint
S) = ξ1 (D
D)
S · D − ξ2 (S
Hence, for D - the point where maximum is reached - we interchange the role of S and
D , so at this point
S · D − ξ2 (S
S)) = ξ1 (D
D)
max (S
S
But it implies
D) := max (S
S · D − ξ2 (S
S)) = ξ1 (D
D)
ξ2∗ (D
S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
21 / 54
Constitutive equations
Optimality of ψ and ψ ∗
D) + ξ2 (S
S) ≥ 0 - not necessarily conjugate and
Let us assume that ξ := ξ1 (D
(i) for some fixed S we would like to maximize ξ with the constraint ξ = S · D
or
(ii) for some fixed D we would like to maximize ξ with the constraint ξ = S · D
It is the same as maximize ξ1 with the constraint
S) = ξ1 (D
D)
S · D − ξ2 (S
Hence, for D - the point where maximum is reached - we interchange the role of S and
D , so at this point
S · D − ξ2 (S
S)) = ξ1 (D
D)
max (S
S
But it implies
D) := max (S
S · D − ξ2 (S
S)) = ξ1 (D
D)
ξ2∗ (D
S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
21 / 54
Constitutive equations
Optimality of ψ and ψ ∗
D) + ξ2 (S
S) ≥ 0 - not necessarily conjugate and
Let us assume that ξ := ξ1 (D
(i) for some fixed S we would like to maximize ξ with the constraint ξ = S · D
or
(ii) for some fixed D we would like to maximize ξ with the constraint ξ = S · D
It is the same as maximize ξ1 with the constraint
S) = ξ1 (D
D)
S · D − ξ2 (S
Hence, for D - the point where maximum is reached - we interchange the role of S and
D , so at this point
S · D − ξ2 (S
S)) = ξ1 (D
D)
max (S
S
But it implies
D) := max (S
S · D − ξ2 (S
S)) = ξ1 (D
D)
ξ2∗ (D
S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
21 / 54
Constitutive equations
Optimality of ψ and ψ ∗ - more general models
Non-polynomial growth
S ∼ (1 + |D
D |2 )
r −2
2
D|)D
D =⇒ ψ(D
D) ∼ |D
D|r ln(1 + |D
D|)
ln(1 + |D
Anisotropic case - different growth
D|rij −2D =⇒ ψ(D
D) ∼
S ij ∼ |D
X
D|rij
|D
Different upper and lower growth in principle - ψ has different polynomial upper
D) := ψ(|D
D|): for certain 1 < r ≤ q < ∞ there are
and lower growth, for ψ(D
positive constants c1 , c2 , c3 and c4 so that
c1 s r − c2 ≤ ψ(s) ≤ c3 s q + c4
=⇒
′
′
c1∗ s q − c2∗ ≤ ψ ∗ (s) ≤ c3∗ s r + c4∗
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
22 / 54
Constitutive equations
Optimality of ψ and ψ ∗ - more general models
Non-polynomial growth
S ∼ (1 + |D
D |2 )
r −2
2
D|)D
D =⇒ ψ(D
D) ∼ |D
D|r ln(1 + |D
D|)
ln(1 + |D
Anisotropic case - different growth
D|rij −2D =⇒ ψ(D
D) ∼
S ij ∼ |D
X
D|rij
|D
Different upper and lower growth in principle - ψ has different polynomial upper
D) := ψ(|D
D|): for certain 1 < r ≤ q < ∞ there are
and lower growth, for ψ(D
positive constants c1 , c2 , c3 and c4 so that
c1 s r − c2 ≤ ψ(s) ≤ c3 s q + c4
=⇒
′
′
c1∗ s q − c2∗ ≤ ψ ∗ (s) ≤ c3∗ s r + c4∗
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
22 / 54
Constitutive equations
Optimality of ψ and ψ ∗ - more general models
Non-polynomial growth
S ∼ (1 + |D
D |2 )
r −2
2
D|)D
D =⇒ ψ(D
D) ∼ |D
D|r ln(1 + |D
D|)
ln(1 + |D
Anisotropic case - different growth
D|rij −2D =⇒ ψ(D
D) ∼
S ij ∼ |D
X
D|rij
|D
Different upper and lower growth in principle - ψ has different polynomial upper
D) := ψ(|D
D|): for certain 1 < r ≤ q < ∞ there are
and lower growth, for ψ(D
positive constants c1 , c2 , c3 and c4 so that
c1 s r − c2 ≤ ψ(s) ≤ c3 s q + c4
=⇒
′
′
c1∗ s q − c2∗ ≤ ψ ∗ (s) ≤ c3∗ s r + c4∗
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
22 / 54
Constitutive equations
Optimality of ψ and ψ ∗ - more general models
Non-polynomial growth
S ∼ (1 + |D
D |2 )
r −2
2
D|)D
D =⇒ ψ(D
D) ∼ |D
D|r ln(1 + |D
D|)
ln(1 + |D
Anisotropic case - different growth
D|rij −2D =⇒ ψ(D
D) ∼
S ij ∼ |D
X
D|rij
|D
Different upper and lower growth in principle - ψ has different polynomial upper
D) := ψ(|D
D|): for certain 1 < r ≤ q < ∞ there are
and lower growth, for ψ(D
positive constants c1 , c2 , c3 and c4 so that
c1 s r − c2 ≤ ψ(s) ≤ c3 s q + c4
=⇒
′
′
c1∗ s q − c2∗ ≤ ψ ∗ (s) ≤ c3∗ s r + c4∗
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
22 / 54
Constitutive equations
What is the goal?
• Goal = existence result for as general constitutive relationships as possible
• Using large data apriori estimates (Ω bounded and nice, nice b.c.)
Steady case
ˆ
Ω
D) + ψ ∗ (S
S) dx ≤ C
ψ(D
Unsteady case
sup kvk22 +
t
ˆ
0
T
ˆ
Ω
D) + ψ ∗ (S
S) dx dt ≤ C
ψ(D
• If the function spaces that are under control are “slightly better” than just to
guarantee that all terms in weak formulation are meaningful, does there exist a weak
solution?
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
23 / 54
Constitutive equations
What is the goal?
• Goal = existence result for as general constitutive relationships as possible
• Using large data apriori estimates (Ω bounded and nice, nice b.c.)
Steady case
ˆ
Ω
D) + ψ ∗ (S
S) dx ≤ C
ψ(D
Unsteady case
sup kvk22 +
t
ˆ
0
T
ˆ
Ω
D) + ψ ∗ (S
S) dx dt ≤ C
ψ(D
• If the function spaces that are under control are “slightly better” than just to
guarantee that all terms in weak formulation are meaningful, does there exist a weak
solution?
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
23 / 54
Constitutive equations
The goal more precisely
If the function spaces generated by the apriori large data energy estimates are
compactly embedded into L2 , does there exist a weak solution?
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
24 / 54
Constitutive equations
Results - power-law like fluid - Explicit
Compact embedding is available if r >
Lebesgue and Sobolev spaces
6
5
r = 2 Lerray (1934)
r≥
r≥
r≥
11
for unsteady, r ≥ 95 steady; Ladyzhenskaya, JL Lions 60’s
5
9
unsteady; Bellout, Bloom, Nečas, Málek, Růžička 90’s
5
8
unsteady; Frehse, Málek, Steinhauer (2000)
5
6
steady; Frehse, Málek, Steinhauer (2003) Diening, Málek,
5
r>
(2008)
r>
6
5
Steinhauer
unsteady; Diening, Růžička, Wolf (2009)
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
25 / 54
Constitutive equations
Results - power-law like fluid - implicit (discontinuous)
Lebesgue and Sobolev spaces
r≥
r>
11
- strictly monotone operators - Gwiazda, Málek, S̀wierczewska
5
9
Herschel-Bulkley model - Málek, Růžička, Shelukhin(2005)
5
6
steady - strictly monotone graph - Bulı́ček, Gwiazda, Málek,
5
(2007)
r>
S̀wierczewska (2009)
r>
6
5
unsteady; Bulı́ček, Gwiazda, Málek, S̀wierczewska-Gwiazda (2010)
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
26 / 54
Constitutive equations
Answers - implicit
Orlicz and Orlicz-Sobolev spaces
subcritical - Gwiazda, S̀wierczevska-Gwiazda et al (2009)
strict monotone operators
the energy equality holds
the term (v ⊗ v) · D (v) ∈ L1 - the solution is an admissible test
function in the weak formulation
supercritical - Bulı́ček, Gwiazda, Málek, S̀wierczewska-Gwiazda (2010)
(A1)-(A4) - maximal monotone ψ-graph
D) = ψ(|D
D|);
ψ(D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
27 / 54
Constitutive equations
Methods
subcritical case
energy equality - v is an addmissible test function
Minty’s method
difficulties if ψ does not satisfy ∆2 condition
supercritical case
generalized Minty’s method
Lipschitz approximation in Orlicz-Sobolev spaces
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
28 / 54
Constitutive equations
Methods
subcritical case
energy equality - v is an addmissible test function
Minty’s method
difficulties if ψ does not satisfy ∆2 condition
supercritical case
generalized Minty’s method
Lipschitz approximation in Orlicz-Sobolev spaces
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
28 / 54
Constitutive equations
Generalized Minty’s method - Convergence lemma
Assume that
A is a maximal monotone ψ-graph satisfying (A1)–(A4)
′
Sn }∞
Dn }∞
{S
n=1 and {D
n=1 satisfy for some Q ⊂ Q
for a.a. (t, x) ∈ Q ′ ,
Sn , D n ) ∈ A
(S
weakly in Lψ (Q ′ ),
Dn ⇀ D
lim sup
n→∞
ˆ
Q′
Sn ⇀ S
ˆ
S n · D n dx dt ≤
S · D dx dt.
∗
weakly in Lψ (Q ′ ),
Q′
Then for almost all (t, x) ∈ Q ′ we have
S, D ) ∈ A
(S
Lemma - Local version
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
29 / 54
Constitutive equations
Application of Convergence lemma to Stokes-like problems
To find (v, p, S ) such that
v ∈ W01,r (Ω)
div v = 0
′
p ∈ Lr (Ω)
D ∈ Lψ (Ω)
− div S = −∇p + f in D′ (Ω),
∗
S ∈ Lψ (Ω)
S(x), D (v(x))) ∈ A for a.a. x ∈ Ω
(S
Theorem. For any f ∈ W01,r
∗
and Ω there is a weak solution to the Problem P.
Proof is based on the following steps:
D take one S ∗ := SD∗ so that (S
S∗ , D ) ∈ A) and its
Take any selection (∀D
η-regularizations leads to η-approximations Pη
Galerkin N-approximations of Pη give finitedimensional problems PN,η
For fix N ∈ N : letting η → 0 we obtain PN
Uniform estimates follow from (A4); Letting N → ∞ and applying Convergence
S, D ) ∈ A a.e. in Ω
lemma one concludes (S
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
30 / 54
Constitutive equations
Extension to subcritical problems
If
compactness in L2 is available (which follows if r >
for unsteady problems)
r > 11
5
9
5
for steady problems and
v is an admissible test function in the weak formulation of the problem
Then the results holds for
div(v ⊗ v) − div S = −∇p + f in Ω
v,t + div(v ⊗ v) − div S = −∇p + f in (0, T ) × Ω and v(0, ·) = v0 ∈ L2div
Supercritical problems: vn − v is not admissible test function =⇒ need for its
appropriate truncation
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
31 / 54
Constitutive equations
Lipschitz approximations of Sobolev function/1
Calderon, Ziemer, Acerbi and Fusco, ...
Theorem. (Diening, Málek, Steinhauer ’08, Frehse, Málek, Steinhauer ’03)
Let 1 < q < ∞ and Ω ∈ C 0,1 . Let
un ∈ W01,q (Ω)d
and un ⇀ 0 weakly in W01,q (Ω)d .
Set
K := sup kun k1,q < ∞,
n
γn := kun kq → 0
√
Let θn > 0 be such that (e.g. θn := γn )
θn → 0
and
(n → ∞).
γn
→0
θn
(n → ∞).
j
Let µj := 22 .
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
32 / 54
Constitutive equations
Lipschitz approximations of Sobolev function/2
Then there exists a sequence λn,j > 0 with
µj ≤ λn,j ≤ µj+1 ,
and a sequence un,j ∈ W01,∞ (Ω)d such that for all j, n ∈ N
kun,j k∞ ≤ θn → 0
n,j
(n → ∞),
k∇u k∞ ≤ c λn,j ≤ c µj+1
and
{un,j 6= un } ⊂ Ω ∩ {Mun > θn } ∪ {M(∇un ) > 2 λn,j } ,
and for all j ∈ N and n → ∞
un,j → 0
un,j ⇀ 0
∗
∇un,j ⇀ 0
Málek (Charles University in Prague)
strongly in Ls (Ω)d for all s ∈ [1, ∞],
weakly in W01,s (Ω)d for all s ∈ [1, ∞),
weakly- ∗ in L∞ (Ω)d×d .
Implicit fluids & Orlicz spaces
July 6, 2010
33 / 54
Constitutive equations
Lipschitz approximations of Sobolev function/3
Furthermore, for all n, j ∈ N
|{un,j 6= un }|d ≤
ckun kq1,q
+c
λqn,j
γn
θn
q
and
k∇un,j χ{un,j 6=un } kq ≤ c kλn,j χ{un,j 6=un } kq ≤ c
γn
µj+1 + c ǫj ,
θn
where ǫj := K 2−j/q vanishes as j → ∞. The constant c depends on Ω.
• based on the continuity of the Hardy-Littlewood maximal function in Lp - In Orlicz
space setting it requires ∆2 -condition and log -continuity w.r.t. x or (t, x)
• Goal is to avoid using continuity of Hardy-Littelwood maximal function; apply weak
(1, 1)-estimates
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
34 / 54
Constitutive equations
Lipschitz approximations of ”Orlicz-Sobolev” functions
Lemma
1
Sn }∞
{un }∞
n=1 such that
n=1 tends strongly to 0 in L and {S
ˆ
Sn |) + ψ(|∇un |) dx ≤ C ∗ (C ∗ > 1).
ψ ∗ (|S
Ω
∗
Then for arbitrary λ ∈ R+ and k ∈ N there exists λmax < ∞ and there exists sequence
n
k
n
n
of {λkn }∞
n=1 and the sequence uk (going to zero) and open sets En := {uk 6= u } such
that λkn ∈ [λ∗ , λmax ] and for any sequence αkn
unk ∈ W 1,p ,
D(unk )k∞ ≤ C λkn ,
kD
C∗
,
ψ(λkn )
k
αnk ψ(λkn /αnk )
αn
n
n
∗
S · D (uk )| dx ≤ CC
|S
+
k
ψ(λkn )
|Ω ∩ Enk | ≤ C
ˆ
Ω∩Enk
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
35 / 54
Constitutive equations
Application of Lipschitz approximations to steady flows
• We have suitable approximations (vn , S n ) and their weak limits (v, S ), we need to show
S, D (v)) ∈ A
that (S
• Test the approximative n- problem by Lipschitz approximation of vn − v, i.e.,
unk := (vn − v)k
S, D ) ∈ A
• One gets (here S is such that (S
k
ˆ
αk ψ(λkn /αnk )
αn
Sn − S ) : D (unk ) ≤ CC ∗
(S
lim
+ n
n→∞ un =un
k
ψ(λkn )
k
• Hölder inequality gives
ˆ
ˆ
Sn − S ) · D (vn − v)|ε ≤
|(S
lim
n→∞
Ω
Málek (Charles University in Prague)
un =unk
+
ˆ
un 6=unk
Implicit fluids & Orlicz spaces
≤ small terms → 0
July 6, 2010
36 / 54
Constitutive equations
Application of Lipschitz approximations to steady flows
• We have suitable approximations (vn , S n ) and their weak limits (v, S ), we need to show
S, D (v)) ∈ A
that (S
• Test the approximative n- problem by Lipschitz approximation of vn − v, i.e.,
unk := (vn − v)k
S, D ) ∈ A
• One gets (here S is such that (S
k
ˆ
αk ψ(λkn /αnk )
αn
Sn − S ) : D (unk ) ≤ CC ∗
(S
lim
+ n
n→∞ un =un
k
ψ(λkn )
k
• Hölder inequality gives
ˆ
ˆ
Sn − S ) · D (vn − v)|ε ≤
|(S
lim
n→∞
Ω
Málek (Charles University in Prague)
un =unk
+
ˆ
un 6=unk
Implicit fluids & Orlicz spaces
≤ small terms → 0
July 6, 2010
36 / 54
Constitutive equations
Application of Lipschitz approximations to steady flows
• We have suitable approximations (vn , S n ) and their weak limits (v, S ), we need to show
S, D (v)) ∈ A
that (S
• Test the approximative n- problem by Lipschitz approximation of vn − v, i.e.,
unk := (vn − v)k
S, D ) ∈ A
• One gets (here S is such that (S
k
ˆ
αk ψ(λkn /αnk )
αn
Sn − S ) : D (unk ) ≤ CC ∗
(S
lim
+ n
n→∞ un =un
k
ψ(λkn )
k
• Hölder inequality gives
ˆ
ˆ
Sn − S ) · D (vn − v)|ε ≤
|(S
lim
n→∞
Ω
Málek (Charles University in Prague)
un =unk
+
ˆ
un 6=unk
Implicit fluids & Orlicz spaces
≤ small terms → 0
July 6, 2010
36 / 54
Constitutive equations
Application of Generalized Minty’s method/Convergence
lemma
Sn − S ) · D (vn − v) to 0; if the strictly monotone
point-wise convergence of (S
property available the proof is finished
if only monotone property is available: apply bf biting lemma; Since
Sn − S ) · D (vn − v) is bounded in L1 there is sequence of non-increasing sets
(S
Ak+1 ⊂ Ak , limk→∞ |Ak | = 0 such that
Sn − S ) · D (vn − v) converges to 0 weakly in L1 (Ω \ Ak )
(S
nonnegativity & point-wise & weak implies strong in L1 (Ω \ Ak )
strong & weak implies for any bounded ϕ
ˆ
ˆ
S n · D (vn )ϕ =
lim
n→∞
Ω\Ak
Ω\Ak
S · D (v)ϕ
apply Convergence lemma
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
37 / 54
Constitutive equations
Application of Generalized Minty’s method/Convergence
lemma
Sn − S ) · D (vn − v) to 0; if the strictly monotone
point-wise convergence of (S
property available the proof is finished
if only monotone property is available: apply bf biting lemma; Since
Sn − S ) · D (vn − v) is bounded in L1 there is sequence of non-increasing sets
(S
Ak+1 ⊂ Ak , limk→∞ |Ak | = 0 such that
Sn − S ) · D (vn − v) converges to 0 weakly in L1 (Ω \ Ak )
(S
nonnegativity & point-wise & weak implies strong in L1 (Ω \ Ak )
strong & weak implies for any bounded ϕ
ˆ
ˆ
S n · D (vn )ϕ =
lim
n→∞
Ω\Ak
Ω\Ak
S · D (v)ϕ
apply Convergence lemma
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
37 / 54
Constitutive equations
Concluding Remarks
Extension of (homogeneous, incompressible) fluids of power-law type to fully
implicit constitutive theory characterized by maximal monotone ψ-graphs (not
necessarilly of power-law type)
Thermodynamically consistent
Ability to capture shear thinning/thickening, activation criteria,
pressure thickening
”Complete” large data existence theory both for steady and unsteady flows
From explicit to implicit, from (v, p) formulation to (v, p, S ) setting
D|) to ψ(D
D)
Extension 1: from ψ(|D
Extension 2: unsteady flows, full thermodynamic setting
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
38 / 54
Constitutive equations
1
2
3
4
5
6
J. Frehse, J. Málek and M. Steinhauer: On analysis of steady flows of fluids with
shear-dependent viscosity based on the Lipschitz truncation method, SIAM J.
Math. Anal. 34, 1064-1083, 2003
L. Diening, J. Málek and M. Steinhauer: On Lipschitz truncations of Sobolev
functions (with variable exponent) and their selected applications, ESAIM:
Control, Optimisation & Calc. Var., 14, 211-232,2008
J. Málek, M. Růžička and V.V. Shelukhin: Herschel-Bulkley Fluids: Existence
and regularity of steady flows, Mathematical Models and Methods in Applied
Sciences, 15, 1845–1861, 2005
P. Gwiazda, J. Málek and A. Świerczewska: On flows of an incompressible fluid
with a discontinuous power-law-like rheology, Computers & Mathematics with
Applications, 53, 531–546, 2007
M. Bulı́ček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda: On steady
flows of an incompressible fluids with implicit power-law-like theology, Advances in
Calculus of Variations, 2, 109–136 2009
M. Bulı́ček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda: On unsteady
flows of implicitly contituted incompressible fluids, to appear in the Preprint series
of the Nečas Center, 2010
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
39 / 54
Newtonian incompressible heat-conducting fluids
Newtonian and Fourier fluids
Newtonian homogeneous incompressible fluid
D(v)
S = 2ν(e)D
D(v)
or T = −pII + 2ν(e)D
Fourier fluid
q = −κ(e)∇e
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
40 / 54
Newtonian incompressible heat-conducting fluids
”Equivalent” formulation of the balance of energy/1
div v = 0
v,t + div(v ⊗ v) − div S = −∇p
Sv)
(e + |v|2 /2),t + div((e + |v|2 /2 + p)v) + div q = div (S
is equivalent (if v is admissible test function in BM) to
div v = 0
v,t + div(v ⊗ v) − div S = −∇p
e,t + div(ev) + div q = S · D (v)
Helmholtz decomposition u = udiv + ∇g v
Leray’s projector P : u 7→ udiv
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
41 / 54
Newtonian incompressible heat-conducting fluids
”Equivalent” formulation of the balance of energy/2
div v = 0
v,t + div(v ⊗ v) − div S = −∇p
Sv)
(e + |v|2 /2),t + div((e + |v|2 /2 + p)v) + div q = div (S
is equivalent (if v is admissible test function in BM) to
div v = 0
v,t + P div(v ⊗ v) − P div S = 0
e,t + div(ev) + div q = S · D (v)
Advantages/Disadvantages
+ pressure is not included into the 2nd formulation
+ minimum principle for e if S · D (v) ≥ 0
− S · D(v) ∈ L1 while Sv ∈ Lq with q > 1, 2nd form is derived form of BE
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
42 / 54
Newtonian incompressible heat-conducting fluids
D(v) and q = −κ(e)∇e Newtonian case - S = ν(e)D
bounded ν, κ
Theorem 1. (M. Bulı́ček, E. Feireisl, J. Málek ’06 and ’08) Assume that
ν ∗ ≥ ν(s) ≥ ν∗ > 0 and κ∗ ≥ κ(s) ≥ κ∗ > 0
∂Ω ∈ C 1,1 , v0 ∈ L2n,div , e0 ∈ L1 , e0 ≥ C ∗ > 0 in Ω, h ∈ L1 (0, T ).
Then for all T > 0, 0 ≤ λ < 1 there is suitable weak solution {v, p, e}
1,2
v ∈ C (0, T ; L2weak ) ∩ L2 (0, T ; Wn,div
)
tr v ∈ L2 (0, T ; L2 (∂Ω))
5
5
p ∈ L 3 (0, T ; L 3 )
e∈
L∞ (0, T ; L1 )
(p +
∩
´
Lm (Q),
Ω p(t, x)dx = g (t)
∇e ∈ Ln (Q) with m
10
10
|v|2
)v ∈ L 9 (0, T ; L 9 )
2
Málek (Charles University in Prague)
∈ h1, 53 ), n ∈ h1, 54 )
5
5
D (v)v ∈ L 4 ([0, T ]; L 4 )
Implicit fluids & Orlicz spaces
July 6, 2010
43 / 54
Newtonian incompressible heat-conducting fluids
Weak vrs Suitable weak solutions
Governing equations
D(v)
div v = 0
v,t + div(v ⊗ v) = −∇p + div ν(. . . )D
|v|2 |v|2 D(v) v
e+
+ div e + p +
v − div(κ(. . . )∇e) = div ν(. . . )D
2 ,t
2
Formulation of the second law of thermodynamics
D(v)|2
e,t + div(ev) − div(κ(e)∇e) ≥ ν(e)|D
equivalent to
D(v)|2 ≤ div(ν(e)D
D(v)v − (p + 12 |v|2 )v)
( 12 |v|2 ),t + ν|D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
44 / 54
Newtonian incompressible heat-conducting fluids
Energy estimates and their consequences
D(v)
div v = 0
v,t + div(v ⊗ v) = −∇p + div ν(. . . )D
|v|2 |v|2 D(v) v
e+
+ div e + p +
v − div(κ(. . . )∇e) = div ν(. . . )D
2 ,t
2
D(v)|2
e,t + div(ev) − div(κ(. . . )∇e)(≥) = ν(. . . )|D
´
Ω
´T
0
e+
|v|2 2 (t, x)dx
≤
´
Ω
e0 +
|v0 |2 dx
2
=⇒ e ∈ L∞ (L1 ) v ∈ L∞ (L2 )
D(v)|2 dx ≤ C
ν(. . . )|D
=⇒
∇v ∈ L2 (L2 )
D(v)|2 ≥ 0, =⇒ e > C ∗ a.e. , e ∈ Lm (Lm ) , ∇(e)(1−s)/2 ∈ L2 (L2 )
ν(. . . )|D
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
45 / 54
Newtonian incompressible heat-conducting fluids
Estimates for the pressure
Equation for the pressure:
D(v))
p = (−∆)−1 div div(v ⊗ v − ν(. . . )D
v ∈ L∞ (L2 ) and ∇v ∈ L2 (L2 ) =⇒ v ∈ L10/3 (L10/3 ) and p ∈ L5/3 (L5/3 )
No-slip BCs for NSEs: Lp maximal regularity for the evolutionary Stokes system
(Solonnikov ’77, Giga, Giga, Sohr ’85)
f ∈ Lp (Lq ) =⇒ v,t , ∇(2) v, ∇p ∈ Lp (Lq )
No-slip BC for generalized NSEs with v (e) does not hold.
Navier’s slip: v · n = 0 solutions of homogeneous Neumann problem for Laplace
equations are admissible
(p, div ϕ) = −(p, −∆h)
− ∆h = |p|α p
Integrable pressure exists for domains with Lipschitz boundary, etc.
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
46 / 54
Newtonian incompressible heat-conducting fluids
Further consequences of energy estimates
D(v)
div v = 0
v,t + div(v ⊗ v) = −∇p + div ν(. . . )D
|v|2 |v|2 D(v) v
e+
+ div e + p +
v − div(κ(. . . )∇e) = div ν(. . . )D
2 ,t
2
D(v)|2
e,t + div(ev) − div(κ(. . . )∇e)(≥) = ν(. . . )|D
∗
v,t ∈ L5/2 (W 1,5/2 ) = L−5/3 (W 1,−5/3 )
′
e,t ∈ L1 (W −1,q ) with q > 10
Aubin-Lions lemma and its generalization: v and e precompact in Lm (Lm )
for m ∈ [1, 53 )
Trace theorem and Aubin-Lions lemma: pre-compactness of v on ∂Ω
Two steps in the proof of existence
Stability of the system w.r.t. weakly converging sequences
Constructions of approximations (several levels), derivation of uniform
estimates, weak limits - candidates for the solutions, taking limits in
nonlinearities
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
47 / 54
Newtonian incompressible heat-conducting fluids
Newtonian and Fourier fluids - unbounded ν and κ
α, β > 0:
ν(e) ∼ (1 + e α )
κ(e) ∼ (1 + e β )
NSF: ν decreases with increasing e
TKE: ν increases with increasing k
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
a
)
ν(e) = ν0 exp( b+e
√
ν(k, ℓ) = ν0 + ν1 ℓ k
July 6, 2010
48 / 54
Newtonian incompressible heat-conducting fluids
Conjecture - Bulı́ček, Lewandowski, Málek
ν(e) := ν0 e α
and
κ(e) := µ0 e α .
(1)
Conjecture. Let α ∈ R, ν and µ are of the form (1). Then there exists a
δ > 0 and C ∗ > 0 such that for any suitable weak solution (v, p, e) to NSF
with unbounded material coefficients the following implication holds:
If
ˆ ˆ
0
−1
B1 (0)
D(v)|2 dx dt ≤ δ
ν(e)|D
then
|v(t, x)| ≤ C ∗
1
in (− , 0) × B 1 (0).
2
2
For α ≡ 0: NSEs - Conjecture holds (CKN ’82, Vasseur ’07). Statement:
If Conjecture holds for α ≥ 61 then the corresponding suitable weak
solution has bounded velocity
Málek (Charles University in Prague)
Implicit fluids & Orlicz spaces
July 6, 2010
49 / 54
Newtonian incompressible heat-conducting fluids
Scaling property of NSF
(v, p, e) solve NSF on some neighborhood of (0, 0): (−ℓA0 , 0) × Bℓ0 (0) with some
A > 0 and ℓ0 > 0. Then
vℓ (t, x) := ℓB v(ℓA t, x)
with
A :=
pℓ (t, x)
2 − 2α
,
1 − 2α
:= ℓ2B p(ℓA t, x)
B :=
1
1 − 2α
eℓ (t, x) := ℓ2B e(ℓA t, ℓx)
α 6=
1
2
solves NSF in (−1, 0) × B1 (0). Conjecture applied on (vℓ , eℓ ) leads to
δ≥
=
ˆ
0
−1
ˆ 1
−1
ˆ
D(vℓ )|2 dx dt
ν(kℓ )|D
ˆ
D(v(ℓA t, ℓx))|2 dx dt
ℓ2Bα+2B+2 (k(ℓA t, ℓx))α |D
B1 (0)
6α−1
= ℓ 1−2α
B1 (0)
ˆ 0
−ℓA
Málek (Charles University in Prague)
ˆ
Bℓ (0)
D(v)|2 dx dt.
k α |D
Implicit fluids & Orlicz spaces
July 6, 2010
50 / 54
Newtonian incompressible heat-conducting fluids
Existence result - unbounded ν and κ
Theorem 2. (M. Bulı́ček, R. Lewandowski, J. Málek, 2010)
ν, κ fulfil the growth condition
β≥0
0≤α<
Assume that
2β
2
+
5
3
Then for any set of data there exists (suitable) weak solution (v, p, e) to the
system in consideration, completed by Navier’s slip boundary conditions, such that
1,2
v ∈ C (0, T ; L2weak ) ∩ L2 (0, T ; Wn,div
)
p ∈ Lq (0, T ; Lq )
tr v ∈ L8/5 (0, T ; L8/5 (∂Ω))
q < min{5/3, 2 − 2α/(α + β + 5/3)}
β+1
2
′
′
v,t ∈ Lq (0, T ; Wn−1,q ) e,t ∈ M(0, T ; W −1,10/9 ) E,t ∈ L1 (0, T ; W −1,10/9 )
e ∈ L∞ (0, T ; L1 ) ,
e≥0
and
(1 + e)s − 1 ∈ L2 (0, T ; W 1,2 ) s <
lim (kv(t) − v0 k22 + ke(t) − e0 k1 ) = 0
t→0+
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Implicit fluids & Orlicz spaces
July 6, 2010
51 / 54
Newtonian incompressible heat-conducting fluids
Maximal L2 -regularity for Stokes-Fourier system
Navier-Stokes sytem
div v = 0
v,t + div(v ⊗ v) − div(2ν0D(v)) + ∇p = F
Maximal Lq -regularity for the evolutionary (linear) Stokes system
div v = 0
v,t − div(2ν0D (v)) + ∇p = F
F ∈ Lr (0, T ; Lr (Ω)d ) =⇒ v,t , ∇p, ∇2 v ∈ Lr (0, T ; Lr (Ω))
Q: Maximal Lq -regularity for the evolutionary (non-linear) Stokes-Fourier
div v = 0
D(v)) + ∇p = F
v,t − div(ν(e)D
D(v)|2
e,t − div(κ(e)∇e) = ν(e)|D
Simplifications: periodic problem, κ(e) = 1, ν0 ≤ v (e) ≤ ν1 , r = 2
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Implicit fluids & Orlicz spaces
July 6, 2010
52 / 54
Newtonian incompressible heat-conducting fluids
L2 -maximal regularity like result for Stokes-Fourier Eqs
Theorem 3. (M. Bulı́ček, P. Kaplický, J. Málek Applicable Analysis 2010)
Let d ≥ 2. Let
√
e0 ∈ W 1,2 (Ω) e0 ≥ emin
F ∈ L2 (0, T ; L2 (Ω)d )
ˆ
1,2
d
1,2
d
u = 0}
v0 ∈ Wdiv (Ω) := {u ∈ W (Ω) ; div u = 0,
Ω
C 0,1 (R+ ) fulfills
ν ′ (s)
Assume that ν ∈
(ε > 0)
2
1
−
≤
≤
15(s − emin + ε)
ν(s)
40(s − emin + ε)
for all s ∈ (emin , ∞)
Then there exists a triple (v, e, p) that solves (SF) such that
1,2
v ∈ L∞ (0, T ; Wdiv
(Ω)d ) ∩ L2 (0, T ; W 2,2 (Ω)d ) ∩ W 1,2 (0, T ; L2 (Ω)d )
√
e∈L
d+2
d
∞
(0, T ; W 2,
e ∈ L (0, T ; W
1,2
Málek (Charles University in Prague)
d+2
d
(Ω)) ∩ W 1,
(Ω))
d+2
d
(0, T ; L
2
d+2
d
p ∈ L (0, T ; W
Implicit fluids & Orlicz spaces
(Ω))
1,2
(Ω))
July 6, 2010
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Newtonian incompressible heat-conducting fluids
Example
−
2
ν ′ (s)
1
≤
≤
15(s − emin + ε)
ν(s)
40(s − emin + ε)
If ν(e) = ν0 exp(
for all s ∈ (emin , ∞)
a
) that the above conditions holds if emin > 2a − b
b+e
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