BURGERS MODEL OF TURBULENCE:
PROBABILISTIC REPRESENTATIONS
FREE ENERGY AND LARGE DEVIATIONS
C. D. Charalambous and S. Djouadi
11th Mediterranean Conference on Control
and
Automation
June 18-20 , 2003
Rhodes, Greece
0
Outline
1. The 3-Dimensional Navier Stokes Equations
2. Stochastic Representation
3. Free Energy and Relative Entropy
4. Viscous Free and Large Deviations
5. Microscopic Computing via (Max,Plus)
Algebra
1
Navier Stokes Equations in 3D








∂ u(t, x) + u(t, x)·∇u(t, x)
∂t
+∇p(x) = ν∇2u(t, x),
(t, x) ∈ (0, ∞) × 3 (1)






u(0, x) = u0(x), x ∈ 3
where
• u(t, x) =
u1(t, x), u2(t, x), u3(t, x)
;
• p : 3 → is the pressure;
• ν = 2 is the kinematic viscosity.
2
Nonlinear Transformations
• Linear Evolution Equation
∂ S (t, x) = ∇2 S (t, x)
∂t
2
+ 1 p(x)S (t, x),








(t, x) ∈ (0, ∞) × 3,  (2)

S (0, x) = S0 (x),
x ∈ 3 .





• Logarithmic Transformation
V (t, x) = − log S (t, x).
(3)
3
Nonlinear Transformations
• Nonlinear Evolution Equation
∂ V (t, x) = ∇2 V (t, x) − ||∇V (t, x)||2
∂t
2
−p(x),
(t, x) ∈ (0, ∞) × 3,
V (0, x) = V0(x) = − log S0 (x),














3
x∈ . 
(4)
• Relation Between 3D Navier-Stokes Equations and Linear Evolution Equation

V (t, x) = − log S (t, x), 

u(t, x) = ∇ log S (t, x).
(5)


4
Probabilistic Representations
• Let {X (t, ω); t ∈ [0, ∞)} be a Markov
diffusion process defined on the probability space (Ω, A, P ), with filtration
{Ft}t≥0, taking values in the state space
(Σ, B(Σ)).
• Let M(Σ) denote the set of probability
measures on (Σ, B(Σ)).
• For T > 0 define
2
LF ((0, T ); ) = {φ : [0, T ] × Ω → ; φ(·)
T
is Ft − adapted, and E{
φ(t, ω)2dt} < ∞}.
0
5
Probabilistic Representations
• Let µ ∈ M(Σ) and define the relative
entropy of µ with respect to P by
I(µ|P )
dµ
dµ
log ( dP )dµ if µ << P , dP ∈ L1(µ)
Σ
=
∞ otherwise
• First Probabilistic Representation:
S (t, x) = Ex{S0 (X (t))
× exp ( 1 0t p(X (s))ds)};
dX (s) =
√
dW (s),
X0 = x,








(6)







where {W (t, ω); t ∈ (0, ∞)} is a standard
Brownian motion process, and Ex[·] denotes
expectation with respect to measure P .
6
Probabilistic Representations
• Second Probabilistic Representation
Define the Free Energy of p with respect
to measure P by









F (p) = log S (t, x)
= log Ex{S0 (X (t)) exp ( 1 0t p(X (s))ds)}; (7)





√


dX (s) = dW (s), X0 = x,
Then
V (t, x) = −F (p)








µ
= − supµ∈M(Σ) {Ex [ − V0(X (t))  (8)





t

+ 0 p(X (s))ds] − I(µ |P )}.
7
Probabilistic Representations
• Stochastic Optimization
µ
V (t, x) = − supγ∈L2 (0,t;) Ex { − V0(Xt)
F
t
+ 0 [p(X (s)) − |γ(s)|2]ds};
dX (s) = γ(s)ds +
√
dW (s),
X0 = x.

















Theorem 1 The 3-Dimensional velocity field
u(t, x) is the gradient of the value function
V (t, x), specifically,

V (t, x) = − log S (t, x), 

u(t, x) = ∇ log S (t, x).


(10)
8
(9)
Viscous Free NS Equations and Large Deviations
• Existence and uniqueness of solutions,
in the limit, as → 0, are established using the theory of viscosity solutions [YongZhou 99].
Proposition 1 Suppose ϕ(x) = p(x), V0(x),
S0 (x), satisfy the condition
(11)
|ϕ(x) − ϕ(z)| ≤ K||x − z||, ∀x, z ∈ 3, K > 0;
Then S , V ∈ C([0, T ] × 3) are unique viscosity solutions of (2), (4).
Suppose (11) holds and
− lim log S0 (x) = V00(x),
→0
uniformly on compact set in 3.
Then
lim V (t, x) = V 0(t, x)
→0
9
uniformly on compact sets in [0, T ] × 3,
and V 0 ∈ C([0, T ]×3) is a unique viscosity
solution of
∂ V 0(t, x) = −||∇V 0(t, x)||2
∂t
−p(x),








(t, x) ∈ (0, ∞) × 3, 

V 0(0, x) = V00(x),
x ∈ 3 .
(12)





Proof. This follows from [Yong-Zhou 99].
Viscous Free NS Equations and Large Deviations
• Weak Representation of the Solution
of the Navier Stokes Equation, as ν →
0.
Let φ ∈ Cb(3) such that lim→0 ∇φ(x) =
∇φ0(x) uniformly on compact sets in 3.
Introduce
3
=
u 2 (x, t)φ(x)dx
j
3
log S (x, t)
=−
3
V (x, t)
∂ φ (x)dx
∂xj
∂ φ (x)dx
∂xj
Then by the Laplace-Varadhan Theorem
Lemma of Large Deviations
lim
→0 3
2
u (x, t)φ(x)dx
j
= sup { − V 0(t, x) +
x∈3
∂ 0
φ (x)}
∂xj
10
Viscous Free Burgers Equation
Remark 1 The above observations lead to the
conclusion that in the limit, as, ν → 0, integration is replaced by maximization. We shall
demenstrate this observation shortly.
• Special Case: u Scalar, p = 0.
Then










V 0(t, x)
∞
ξ 1
= − lim→0 log −∞ exp ( − 0 u0(σ)dσ ) (13)





2


(ξ−x)
1

√
exp ( − 2t )dξ
×

2πt
x Assume f0(x) = 0 u0(σ)dσ is bounded and
continuous, and lim→0 f (x) = f00(x) =
x 0
0 u0(σ)dσ, uniformly on compact subsets
of .
11
Viscous Free Burgers Equation
Then
0
V (t, x) = − sup { −
ξ∈
ξ
0
(ξ − x)2
0
}.
(14)
u0(σ)dσ −
2t
Define
ξ
(ξ − x)2
0
ξ ∈ arg sup { −
}. (15)
u0(σ)dσ −
2t
0
ξ∈
If V 0 ∈ Cx1() then ∀(t, x) ∈ (0, ∞)×, we have
x
−
ξ
V 0(t, x) =
.
u0(t, x) =
∂x
t
∂
(16)
Moreover, if u0
0 has no discontinuities, then
(ξ − x)
).
= u0
(ξ
0
t
12
Viscous Free Burgers Equation
Consequently,
0 u0(t, x) = u0
0(x − u0(ξ )t)
(17)
which the well-known solution of the viscosity
free Burgers equation








∂ u0(t, x)
∂t
∂ u0(t, x) = 0,
+u0(t, x) ∂x
(18)
(t, x) ∈ (0, ∞) × , 
u0(0, x) = u0
0(x),






x ∈ .
• Generalizations to 3D Navier-Stokes Equations. The above Exposition is Applicable
to the 3D Navier-Stokes equations

V (t, x) = − log S (t, x), 


u(t, x) = ∇ log S (t, x). 
(19)
13
The (Max,Plus) Algebra and Large Deviations
• (Max,Plus) Measures and Expectation.
Let “+” and “.” denote the usual addition
and multiplication operations, over .
¯ = {−∞} ∪ = [−∞, ∞].
Let The algebras of interest are the (log,plus)
¯.
and the (max,plus), defined over • (log,plus) Algebra: Addition and multiplication are defined, for ∈ (0, ∞), by
b
a
a ⊕ b = log { exp( ) + exp( )};
b
a
¯.
a ⊗ b = log { exp( ). exp( )}, a, b ∈ 14
The (Max,Plus) Algebra and Large Deviations
• (max,plus) Algebra: Addition and multiplication are defined by
a ⊕ b = max{a, b};
a ⊗ b = a + b,
¯
a, b ∈ ¯
• For any sequence {ai}n
i=1 ⊂ , independent
of ∈ (0, ∞)
n n
ai
lim
ai = lim log
exp ( )
→0
→0
i=1
i=1
=
n
ai = max ai,
i
i=1
n n
ai
lim
ai = lim log
exp { }
→0
→0
i=1
i=1
=
n
i=1
ai =
n
ai.
i=1
15
The (Max,Plus) Algebra and Large Deviations
Consequently, the (log,plus) algebra converges
to the (max,plus) algebra as → 0. This convergence is a property of Large Deviations theory, and it applies to general complete separable metric spaces on which probability measures are defined.
Definition 1 Let (Ω, F ) be a measurable space.
µ is a measure with respect to the (max,plus)
algebra if the following conditions hold.
1. µ(A) ∈ [0, −∞],
∀A ∈ F ;
2. µ(Ω) = 0;
3. µ( i Ai) =
∀{Ai} ∈ F .
µ(A
),
A
i
i Aj = ∅, ∀i = j,
i
16
The (Max,Plus) Algebra and Large Deviations
• (max,plus) Measures are Obtained from
the Theory of Rare Events
Let {(Ω, F , P )}> be a family of probability measures indexed by > 0. Define
µ(A) = log P (A),
A ∈ F.
(20)
Suppose the following limit exists
µ(A) = lim µ(A) = lim log P (A)
→0
→0
= ess sup {I(x); x ∈ A},
A∈F
• where I : Ω → [−∞, 0] is an upper semicontinuous (u.s.c) function; it is the socalled action function (or functional)
17
The (Max,Plus) Algebra and Large Deviations
Then µ is a (Max,Plus) finite-additive measure.
Moreover,
µ(
∞


Ai) = ess sup I(ω) : ω ∈
i=1

= sup {ess sup {I(ω) : ω ∈ Ai} }
i
= sup µ(Ai),
i
∞
i=1


Ai

Ai ∩ Aj = ∅, ∀i = j, {Ai}n
i=1 ∈ F .
Unfortunately, µ does not have a density because µ({ω}) = −∞, ∀ω ∈ Ω. However, if we
replace the measure by µ(A) = sup {I(ω); ω ∈
A}, A ∈ F , then µ has I as its density.
18
The (Max,Plus) Algebra and Large Deviations
• Expectation of Indicator Function. Define the indicator function with respect to
the (max,plus) measure by
χA(ω) =
0 if ω ∈ A
−∞ if ω ∈ A
Then
Eµ[χA] = µ(A) = sup {χA(ω) + I(ω); ω ∈ Ω}
= sup {I(ω); ω ∈ A},
A ∈ F.
19
The (Max,Plus) Algebra and Large Deviations
• Expectation of Simple Functions. For
{Ai} ∈ F , Ai ∩Aj = ∅, ∀i = j, define the simple function with respect to the (max,plus)
measure by
n
f (ω) =
i=1
ai ⊗ χAi (ω).
Then
Eµ[f ] =
n
αi ⊗ µ(Ai).
i=1
20
The (Max,Plus) Algebra and Large Deviations
• Large Deviations Principle .
Let {(X , BX , P )}>0 be a family of complete probability spaces indexed by and
for A ∈ BX , let
µX (A) = log P (A), µX (A) = lim µX (A),
→0
provided the limit exists.
We say that this probability space satisfies
the Large Deviations Principle (LDP) with
real-valued rate function IX (·), denoted by
{(X , BX , P )}>0 ∼ IX (x) if there exists a
function IX : X → [−∞, 0] called the action functional which satisfies the following
properties.
21
The (Max,Plus) Algebra and Large Deviations
1. −∞ ≤ IX (x) ≤ 0, ∀x ∈ X
2. IX (·) is Upper Semicontinuous (u.s.c.)
3. For each m > −∞ the set {x; m ≤ IX (x)}
is a compact set in X .
4. For each C ∈ BX
lim sup log P (C) ≤ sup IX (x)
→0
(21)
x∈C¯
where C¯ is the closure of the set C ∈ BX .
5. For each C ∈ BX
lim inf log P () ≥ sup IX (x)
→0
(22)
x∈0
22
where C 0 is the interior of the set C ∈ BX .
6. If C ∈ BX is such that
sup IX (x) = sup IX (x) = sup IX (x) (23)
x∈C 0
x∈C¯
x∈C
then
µX (C) = sup IX (x)
(24)
x∈C
and C ∈ X is called a continuity set of IX (·).
Current Work
• Microscopic Computation via
the (Max,Plus) Measure
• Entropy and Information Theory via
the (Max,Plus) Measure
23
References
1. M.L. Landahl and E.M.-Christensen, Turbulence and Random Processes in Fluid Mechanics, Cambridge University Press, 2nd Edition (1992).
2. J. Yong and X.Y. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equatoions,
Springer-Verlag, (1999).
3. J.-D. Deuschel and D.W. Stroock, Large
Deviations, Academic Press Inc., (1989).
4. S.R.S. Varadhan, Large Deviations and Applications, Society for Industrial and Applied
Mathematics, (1984).
5. D.W. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, (1984).
24
6. W.M. McEneaney and C.D. Charalambous,
“Large Deviations Theory, Induced Log-Plus
and Max-Plus Measures and their Applications,”
in Mathematical Theory of Network and Systems 2000, Perpignan, France, June 19-23,
2000.