Scaling and singular limits in fluid mechanics
Eduard Feireisl
Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague
joint work with
A. Novotný (Toulon), H. Petzeltová (Prague)
Summer School - Non-Homogeneous Fluids and Flows
Prague, 28 August 2012
Eduard Feireisl
Singular limits in fluid mechanics
Motto
In mathematics you
don’t understand things.
You just get used to
them.
Johann von
Neumann
[1903-1957]
Eduard Feireisl
Singular limits in fluid mechanics
Mathematical model
Viscous, compressible, and heat conducting fluid in motion
mass density % = %(t, x)
absolute temperature ϑ = ϑ(t, x)
velocity field u = u(t, x)
Thermodynamic functions
pressure p = p(%, ϑ)
internal energy e = e(%, ϑ)
entropy s = s(%, ϑ)
Transport
viscous stress S = S(ϑ, ∇x u)
heat flux q = q(ϑ, ∇x ϑ)
Eduard Feireisl
Singular limits in fluid mechanics
Gibbs’ equation and thermodynamic stability
Gibbs’ equation
Willard Gibbs
[1839-1903]
1
ϑDs(%, ϑ) = De(%, ϑ) + p(%, ϑ)D
%
Thermodynamic stability
∂p(%, ϑ)
∂e(%, ϑ)
> 0,
>0
∂%
∂ϑ
Eduard Feireisl
Ludwig Boltzmann
[1844-1906]
Singular limits in fluid mechanics
Constitutive relations
Newton’s rheological law
Isaac Newton
[1643-1727]
S = µ(ϑ) ∇x u +
∇tx u
2
− divx uI + η(ϑ)divx uI
3
Fourier’s law
q = −κ(ϑ)∇x ϑ
Eduard Feireisl
François Marie
Charles Fourier
[1772-1837]
Singular limits in fluid mechanics
Navier-Stokes-Fourier system
Mass conservation
∂t % + divx (%u) = 0
Momentum balance
Claude Louis
Marie Henri Navier
∂t (%u)+divx (%u⊗u)+∇x p(%, ϑ) = divx S(ϑ, ∇x u)
[1785-1836]
Entropy balance
q(ϑ, ∇x ϑ)
∂t (%s(%, ϑ)) + divx (%s(%, ϑ)u) + divx
ϑ
q · ∇x ϑ
1
σ=
S : ∇x u −
ϑ
ϑ
=σ
George
Gabriel
Stokes
[1819-1903]
Scaled Navier-Stokes-Fourier system
Mass conservation
[Sr]∂t % + divx (%u) = 0
Momentum balance
1
1
divx S(ϑ, ∇x u)
[Sr]∂t (%u)+divx (%u⊗u)+
∇x p(%, ϑ) =
2
Re
Ma
Entropy balance
q
1
[Sr]∂t (%s(%, ϑ)) + divx (%s(%, ϑ)u) +
=σ
divx
Pe
ϑ
1
Ma2
1 q · ∇x ϑ
σ=
S : ∇x u −
ϑ
Re
Pe
ϑ
Eduard Feireisl
Singular limits in fluid mechanics
Characteristic numbers - Strouhal number
Strouhal number
[Sr] =
lengthchar
timechar velocitychar
Čeněk Strouhal
[1850-1922]
Scaling by means of Strouhal number is used in the study of the
long-time behavior of the fluid system, where the characteristic
time is large
Eduard Feireisl
Singular limits in fluid mechanics
Mach number
Mach number
velocitychar
[Ma] = p
pressurechar /densitychar
Ernst Mach [1838-1916]
Mach number is the ratio of the characteristic speed
to the speed of sound in the fluid. Low Mach number
limit, where, formally, the speed of sound is becoming
infinite, characterizes incompressibility
Eduard Feireisl
Singular limits in fluid mechanics
Reynolds number
Reynolds number
[Re] =
densitychar velocitychar lengthchar
viscositychar
Osborne Reynolds
[1842-1912]
High Reynolds number is attributed to turbulent flows, where the
viscosity of the fluid is negligible
Eduard Feireisl
Singular limits in fluid mechanics
Péclet number
Péclet number
[Pe] =
Jean Claude
Eugène Péclet
[1793-1857]
pressurechar velocitychar lengthchar
heat conductivitychar temperaturechar
High Péclet number corresponds to low heat conductivity of the
fluid that may be attributed to turbulent flows
Eduard Feireisl
Singular limits in fluid mechanics
Inviscid incompressible limit
Mass conservation
∂t % + divx (%u) = 0
Momentum balance
∂t (%u) + divx (%u ⊗ u) +
1
∇x p(%, ϑ) = εa divx S(ϑ, ∇x u)
ε2
Entropy production
q(ϑ, ∇x ϑ)
∂t (%s(%, ϑ)) + divx (%s(%, ϑ)u) + ε divx
ϑ
1
2+a
b q(ϑ, ∇x ϑ) · ∇x ϑ
=
ε
S(ϑ, ∇x u) : ∇x u − ε
ϑ
ϑ
b
Eduard Feireisl
Singular limits in fluid mechanics
Boundary conditions and total energy conservation
Navier’s slip
u · n|∂Ω = 0, εc [S(ϑ, ∇x u)n]tan + β(ϑ)u|∂Ω = 0, c, β > 0
Energy insulation
q(ϑ, ∇x ϑ) · n|∂Ω = −β(ϑ)εd |u|2 |∂Ω , d = 2 + a − c − b
Total mass and energy conservation
Z
Z
d
d
% dx = 0,
ε2 %|u|2 + %e(%, ϑ) dx = 0
dt Ω
dt Ω
Eduard Feireisl
Singular limits in fluid mechanics
Target (limit) system
Incompressible Euler system
divx v = 0
∂t v + divx (v ⊗ v) + ∇x Π = 0
v · n|∂Ω = 0
Leonhard Paul
Euler [1707-1783]
Transport equation for temperature deviation
∂t T + v · ∇x T = 0
Eduard Feireisl
Singular limits in fluid mechanics
Ballistic free energy
Ballistic free energy
HΘ (%, ϑ) = % e(%, ϑ) − Θs(%, ϑ)
Jerald LaVerne
Ericksen [*1924]
Coercivity
2
∂%,%
HΘ (%, Θ) =
1
∂% p(%, Θ)
%
1
∂ϑ HΘ (%, ϑ) = (ϑ − Θ) ∂ϑ e(%, ϑ)
ϑ
Eduard Feireisl
Singular limits in fluid mechanics
Coercivity
Coercivity of the ballistic free energy
∂HΘ (r , Θ)
(% − r ) − HΘ (r , Θ)
∂%
≥ c(K ) |% − r |2 + |ϑ − Θ|2 for all (%, ϑ) ∈ K
HΘ (%, ϑ) −
HΘ (%, ϑ) −
∂HΘ (r , Θ)
(% − r ) − HΘ (r , Θ)
∂%
≥ c(K ) (1 + %e(%, ϑ) + %s(%, ϑ)) whenever (%, ϑ) ∈ [0, ∞)2 \ K .
K ⊂ (0, ∞)2 a compact containing r , Θ
Eduard Feireisl
Singular limits in fluid mechanics
Relative entropy (scaled)
Relative entropy
Eε %, ϑ, ur , Θ, U =
Z 1
1
∂HΘ (r , Θ)
2
%|u − U| + 2 HΘ (%, ϑ) −
(% − r ) − HΘ (r , Θ)
ε
∂%
Ω 2
Eduard Feireisl
Singular limits in fluid mechanics
Dissipative solutions
Relative entropy inequality
h iτ
Eε %, ϑ, ur , Θ, U
t=0
Z τZ
Θ a
b−2 q(ϑ, ∇x ϑ) · ∇x ϑ
+
ε S(ϑ, ∇x u) : ∇x u − ε
dx dt
ϑ
0
Ω ϑ
Z τZ
Θβ 2
a−c
+ε
|u| dSx dt
0
∂Ω ϑ
Z τ
≤
Rε (%, ϑ, u, r , Θ, U) dt
0
for any r > 0, Θ > 0, U satisfying relevant boundary conditions
Eduard Feireisl
Singular limits in fluid mechanics
Remainder
Rε (%, ϑ, u, r , Θ, U)
Z
τ
=
0
Z % ∂t U + u · ∇x U · (U − u) + εa S(ϑ, ∇x u) : ∇x U dx dt
Ω
+ε
a−c
Z
Z
τ
Z h
Z
βu · U dSx dt
0
1
+ 2
ε
τ
∂Ω
i
%
p(r , Θ)−p(%, ϑ) divU+ (U−u)·∇x p(r , Θ) dxdt
r
0
Ω
Z τZ 1
− 2
% s(%, ϑ) − s(r , Θ) ∂t Θ + u · ∇x Θ dx dt
ε 0 Ω
Z τZ
q(ϑ, ∇x ϑ)
b−2
+ε
· ∇x Θ dxdt
ϑ
0
Ω
Z Z
1 τ
r − %
+ 2
∂t p(r , Θ) + U · ∇x p(r , Θ) dx dt
ε 0 Ω r
Initial data and far field behavior
Initial data
(1)
(1)
(1)
(1)
(1)
(1)
%(0, ·) = % + ε %0,ε , %0,ε → %0 in L2 (Ω) and weakly-(*) in L∞ (Ω)
ϑ(0, ·) = ϑ+ε ϑ0,ε , ϑ0,ε → ϑ0 in L2 (Ω) and weakly-(*) in L∞ (Ω)
u(0, ·) = u0,ε → v0 in L2 (Ω; R 3 ), v0 ∈ W k,2 (Ω; R 3 ), k >
Far field
% → %, ϑ → ϑ, u → 0 as |x| → ∞
Eduard Feireisl
Singular limits in fluid mechanics
5
2
Convergence
b > 0, 0 < c < a <
10
,
3
Asymptotic incompressibility
ess sup k %ε (t, ·) − % kL2 +L5/3 (Ω) ≤ εc
t∈(0,T )
√
%ε u ε →
p
2
3
% v in L∞
loc ((0, T ]; Lloc (Ω; R ))
and weakly-(*) in L∞ (0, T ; L2 (Ω; R 3 ))
Temperature deviation
ϑε − ϑ
q
3
→ T in L∞
loc ((0, T ]; Lloc (Ω; R )), 1 ≤ q < 2 ,
ε
and weakly-(*) in L∞ (0, T ; L2 (Ω))
Eduard Feireisl
Singular limits in fluid mechanics
Uniform bounds
The uniform bounds independent of ε are obtained by taking
r = %, Θ = ϑ, U = 0
in the relative entropy inequality
Energy bounds
%ε − % ess sup ≤ c,
ε L2 +L5/3 (Ω)
t∈(0,T )
ϑε − ϑ ≤ c,
ess sup 2
ε
t∈(0,T )
L (Ω)
√
ess sup k %ε uε kL2 (Ω;R 3 ) ≤ c
t∈(0,T )
Eduard Feireisl
Singular limits in fluid mechanics
Dissipation
Integral bounds
Z TZ
a
ε
0
2
∇x uε + ∇tx uε − 2 divx uε I dx dt ≤ c
3
Ω
Z TZ
a−c
ε
|uε |2 dSx dt ≤ c
0
εb−2
Z
0
T
∂Ω
Z
|∇x ϑε |2 dx dt ≤ c
Ω
Eduard Feireisl
Singular limits in fluid mechanics
First order approximation
Linearization
ε∂t
ε∂t (%ε uε ) + ∇x
%ε − %
ε
+ divx (%ε uε ) = 0
%ε − %
ϑε − ϑ
+ ∂ϑ p(%, ϑ)
∂% p(%, ϑ)
ε
ε
!
= f1,ε
!
ϑε − ϑ
%ε − %
+ %∂% s(%, ϑ)
∂t %∂ϑ s(%, ϑ)
ε
ε
! #
"
ϑε − ϑ
%ε − %
+divx
%∂ϑ s(%, ϑ)
+ %∂% s(%, ϑ)
uε = f2,ε
ε
ε
Eduard Feireisl
Singular limits in fluid mechanics
Stability
Another application of the relative entropy inequality
Take
rε = % + εRε , Θε = ϑ + εTε , Uε = v + ∇x Φε
as test functions in the relative entropy inequality
Acoustic equation
ε∂t (αRε + βTε ) + ω∆Φε = 0
ε∂t ∇x Φε + ∇x (αRε + βTε ) = 0
Transport equation
∂t (δTε − βRε ) + Uε · ∇x (δTε − βRε ) + (δTε − βRε )divx Uε = 0
Eduard Feireisl
Singular limits in fluid mechanics
Lighthill’s acoustic equation
Wave equation
ε∂t Z + ∆Φ = 0, ε∂t Φ + Z = 0,
Neumann boundary condition
∇x Φ · n|∂Ω = 0,
Initial conditions
Michael James
Lighthill
[1924-1998]
Φ(0·) = Φ0 , Z (0, ·) = Z0 ,
Eduard Feireisl
Singular limits in fluid mechanics
Neumann Laplacean
Domain of definition
Z
Z
n
D(−∆N ) = w ∈ W 1,2 (Ω) ∇x w · ∇x φ dx =
g φ dx
Ω
for a certain g ∈ L2 (Ω) and all φ ∈ Cc∞ (Ω)
Ω
o
−∆N w = g .
Neumann Laplacean
The Neumann Laplacean −∆N is a non-negative self-adjoint
operator on the Hilbert space L2 (Ω)
Eduard Feireisl
Singular limits in fluid mechanics
Duhamel’s formula
Acoustic potential
p
1
t
i
Z0
exp i −∆N
Φ0 − √
2
ε
−∆N
p
1
t
i
+ exp −i −∆N
Z0
Φ0 + √
2
ε
−∆N
Φ(t, ·) =
Time derivative
p
i
1
th p
exp i −∆N
i −∆N [Φ0 ] + Z0
2
ε
h
i
p
p
1
t
+ exp −i −∆N
−i −∆N [Φ0 ] + Z0
2
ε
Z (t, ·) =
Jean-Marie
Constant Duhamel
[1797-1872]
Strichartz estimates for the flat Laplacean
Strichartz estimates
Z ∞
√
p
exp ±i −∆t [h] q
L (R 3 )
−∞
dt ≤ khkpH 1,2 (R 3 )
1
1 3
= + , q<∞
2
p q
Local energy decay
Z
∞
−∞
√
2
χ exp ±i −∆t [h] α,2
H
(R 3 )
dt ≤ c(χ)khk2H α,2 (R 3 )
3
α ≤ , χ ∈ Cc∞ (R 3 )
2
Eduard Feireisl
Singular limits in fluid mechanics
Limiting absorption principle
Limiting absorption principle
The cut-off resolvent operator
(1 + |x|2 )−s/2 ◦ [−∆N − µ ± iδ]−1 ◦ (1 + |x|2 )−s/2 , δ > 0, s > 1
can be extended as a bounded linear operator on L2 (Ω) for δ → 0
and µ belonging to compact subintervals of (0, ∞).
Tosio Kato [1917-1999]
Robert S. Strichartz
Kato’s theorem
Theorem
Let A be a closed densely defined linear operator and H a
self-adjoint densely defined linear operator in a Hilbert space
X . For λ 6∈ R, let RH [λ] = (H − λId)−1 denote the resolvent
of H. Suppose that
Γ=
sup
λ6∈R, v ∈D(A∗ ), kv kX =1
kA ◦ RH [λ] ◦ A∗ [v ]kX < ∞.
Then
π
sup
w ∈X , kw kX =1 2
Z
∞
−∞
kA exp(−itH)[w ]k2X dt ≤ Γ2 .
Eduard Feireisl
Singular limits in fluid mechanics
Frequency localized local decay estimates
Decay estimates
∞
Z
0
p
2
χG (−∆N ) exp ±i −∆N t [h] 2
L (Ω)
dt ≤ ckhk2L2 (Ω)
Scaled estimates
Z
0
T
p
t 2
[h]
dt ≤ εc(T )khk2L2 (Ω)
χG (−∆N ) exp ±i −∆N
ε
L2 (Ω)
G ∈ Cc∞ (0, ∞), χ ∈ Cc∞ (Ω)
Eduard Feireisl
Singular limits in fluid mechanics
Admissible domains
Limiting absorption principle
The operator ∆N satisfies the limiting absorption principle in Ω.
Strichartz estimates on “larger” domain
There is a domain such that D ∩ {|x| > R} = Ω ∩ {|x| > R} and
∆N satisfies the Strichartz estimates in D.
Local decay on “larger” domain
The operator ∆N satisfies the local energy decay estimates in D.
Frequency localized Strichartz estimates
Z
∞
−∞
p
p
G (−∆N ) exp ±i −∆N t [h] q
L (Ω)
≤ c(G )khkpL2 (Ω)
1
1 3
= + , q < ∞, G ∈ Cc∞ (0, ∞)
2
p q