Branching Processes in Fluid Mechanics: An application to the - - PowerPoint PPT Presentation

Joint work with Larry Chen, Ron Guenther, Ed Waymire (Oregon State University) and Sun-Chul Kim (Chung Ang University) Branching Processes in Fluid Mechanics: An application to the Navier-Stokes and LANS-alpha equations. Enrique Thomann

Special Semester on Stochastics with Emphasis on Finance

Special Semester on Stochastics with Emphasis on Finance

Outline of the talk

• Navier Stokes and LANSalpha Regularization. What

and why.

• Review of current state of knowledge. A basic

question.

• Problem in Fourier domain.
• Stochastic branching representation of solution. Two

examples and the case of LANSalpha

• Function Spaces. Iteration and contraction mapping.
• Rates of convergence as alpha vanishes.

Special Semester on Stochastics with Emphasis on Finance

Incompressible Navier-Stokes equations in a periodic domain.

Periodic domain D = [−L, L]3, L > 0

− ∂v ∂t + ∇·(v ⊗ v) = ν∆v − ∇p + g ∇·v = 0 ∇· Initial data v(x, 0) = v0(x).

Computational challenges for small Reynolds number. Alternative LANSalpha equation introduces a filtering of high frequencies.

Special Semester on Stochastics with Emphasis on Finance

Naive Regularization (Leray ‘30)

∂v ∂t + ∇·(u ⊗ v) = ν∆v − ∇p + g ∇·v = 0

u = G ∗ v.

Spatial filtering Does not satisfy Kelvin Circulation Theorem

d dt

• γt

v · dr =

• γt

(ν∆v + g) · dr

Special Semester on Stochastics with Emphasis on Finance

Naive Regularization (Leray ‘30)

∂v ∂t + ∇·(u ⊗ v) = ν∆v − ∇p + g ∇·v = 0

u = G ∗ v.

Spatial filtering Does not satisfy Kelvin Circulation Theorem

d dt

• γt

v · dr =

• γt

(ν∆v + g) · dr

Gallavotti challenge - Find a regularization of the Navier- Stokes Equations that satisfy the Kelvin Circulation Theorem. One such regularization is the LANSalpha equation introduced by Foias, Holmes and Titi (2002)

Special Semester on Stochastics with Emphasis on Finance

∂v(α) ∂t + ∇·(u(α) ⊗ v(α)) + (∇u(α))Tv(α) = ν∆v(α) − ∇p + g − ∂v ∂t + ∇·(v ⊗ v) = ν∆v − ∇p + g ∇·v = 0

Navier-Stokes LANS-alpha

∇· ⊗ ∇·v = 0 ∇·v(α) = 0, (1 − α2∆)u(α) = v(α)

Initial data

∇· − data v(α)(x, 0) = v0(x)

Special Semester on Stochastics with Emphasis on Finance

∂v(α) ∂t + ∇·(u(α) ⊗ v(α)) + (∇u(α))Tv(α) = ν∆v(α) − ∇p + g − ∂v ∂t + ∇·(v ⊗ v) = ν∆v − ∇p + g ∇·v = 0

Navier-Stokes LANS-alpha

∇· ⊗ ∇·v = 0 ∇·v(α) = 0, (1 − α2∆)u(α) = v(α)

Initial data

∇· − data v(α)(x, 0) = v0(x)

Spatial Filtering given by the Green function of the Helmholtz operator

u(α) = (1 − α2∆)−1v(α) = G ∗ v(α).

Special Semester on Stochastics with Emphasis on Finance

∂v(α) ∂t + ∇·(u(α) ⊗ v(α)) + (∇u(α))Tv(α) = ν∆v(α) − ∇p + g − ∂v ∂t + ∇·(v ⊗ v) = ν∆v − ∇p + g ∇·v = 0

Navier-Stokes LANS-alpha

∇· ⊗ ∇·v = 0 ∇·v(α) = 0, (1 − α2∆)u(α) = v(α)

Recovering Navier-Stokes. α

α = 0, v(0) = u(0), (∇u(0))Tv(0) = 1 2∇| |v| |2

Initial data

∇· − data v(α)(x, 0) = v0(x)

Spatial Filtering given by the Green function of the Helmholtz operator

u(α) = (1 − α2∆)−1v(α) = G ∗ v(α).

Special Semester on Stochastics with Emphasis on Finance

Current Theory - Brief survey

• Foias, Holmes and Titi (2002), Existence, regularity and

convergence of subsequences as alpha vanishes.

• Marsden and Shkoller (2003) Introduced the

Langrangian averaged Euler equations.

• Linshiz and Titi (2006) MHD-alpha models. Rate of

convergence as alpha vanishes ?

Special Semester on Stochastics with Emphasis on Finance

Probabilistic representation of solutions - An answer to the rate of convergence question for LANS-alpha.

• LeJan-Sznitman (1997) Probabilistic

representation of NS in Fourier Domain.

• Bhattacharya et al (2003) Notion of

Majorizing kernels.

• Ramirez (2006) Numerical methods based
• n stochastic representations applied to

Burgers equation.

• Chen et al (2008) Rate of convergence.

Special Semester on Stochastics with Emphasis on Finance

An illustrative example. Forced Heat equation.

ˆ u(k, t) = ˆ u0(k)e−|k|2t +

t

0 e−|k|2sˆ

g(k, t − s)ds ∂u ∂t = ∆u + g, u|t=0 = u0 ∂ˆ u ∂t = −|k|2ˆ u + ˆ g, , ˆ u|t=0 = ˆ u0

Special Semester on Stochastics with Emphasis on Finance

An illustrative example. Forced Heat equation.

ˆ u(k, t) = ˆ u0(k)e−|k|2t +

t

0 e−|k|2sˆ

g(k, t − s)ds ∂u ∂t = ∆u + g, u|t=0 = u0 ∂ˆ u ∂t = −|k|2ˆ u + ˆ g, , ˆ u|t=0 = ˆ u0

P(S > t) = e−|k|2t

Exponential random variable ˆ u(k, t) = ˆ u0(k)e−|k|2t + t |k|2e−|k|2s ˆ g(k, t − s) |k|2 ds

Special Semester on Stochastics with Emphasis on Finance

An illustrative example. Forced Heat equation.

ˆ u(k, t) = ˆ u0(k)e−|k|2t +

t

0 e−|k|2sˆ

g(k, t − s)ds ∂u ∂t = ∆u + g, u|t=0 = u0 ∂ˆ u ∂t = −|k|2ˆ u + ˆ g, , ˆ u|t=0 = ˆ u0

P(S > t) = e−|k|2t

Exponential random variable

X(k, t) =

  

ˆ u0(k) if S > t

ˆ g(k,t−S) |k|2

if S ≤ t.

ˆ u(k, t) = ˆ u0(k)e−|k|2t + t |k|2e−|k|2s ˆ g(k, t − s) |k|2 ds

ˆ u(k, t) = E[X(k, t)]

Special Semester on Stochastics with Emphasis on Finance

Probabilistic representation for LANS-alpha

− ∂v(α) ∂t + ∇·(u(α) ⊗ v(α)) + (∇u(α))Tv(α) = ν∆v(α) − ∇p + g ∇·v(α) = 0, (1 − α2∆)u(α) = v(α)

( )

Incompressibility Leray Projection

time, β = 2π/L.

Aspect Ratio

ˆ u(k, t) = ˆ v(k, t) 1 + α2|βk|2

Take Fourier Transform, integrate in time and eliminate the pressure using the projection

k · ˆ v(k, t) = 0

πk ∇p = 0

Special Semester on Stochastics with Emphasis on Finance

First and last term can be interpreted using an exponential random

• variable. To provide a probabilistic representation to the quadratic terms,

use a branching process with the aid of Majorizing kernels. ˆ v(k, t) = exp[−ν|βk|2t]ˆ v0(k) −i t exp[−ν|βk|2s]

• j

βk · ˆ v(j, t − s) 1 + α2|βj|2 πk(ˆ v(k − j, t − s))ds −i t exp[−ν|βk|2s]

• j

βπk(j)ˆ v(j, t − s) · ˆ v(k − j, t − s) 1 + α2|βj|2 ds + t exp[−ν|βk|2s]ˆ g(k, t − s)ds Mild Formulation

Special Semester on Stochastics with Emphasis on Finance

(−i) t exp[−ν|βk|2s]

• j

βk · ˆ v(j, t − s) 1 + α2|βj|2 πk(ˆ v(k − j, t − s))ds = t ν|βk|2 exp[−ν|βk|2s]

• j

|βk| (1 + α2|βj|2) 1 ν|βk|2 (−i) [ek · ˆ v(j, t − s)] πk(ˆ v(k − j, t − s))ds t

• Dealing with the quadratic terms - a particular term

Special Semester on Stochastics with Emphasis on Finance

(−i) t exp[−ν|βk|2s]

• j

βk · ˆ v(j, t − s) 1 + α2|βj|2 πk(ˆ v(k − j, t − s))ds = t ν|βk|2 exp[−ν|βk|2s]

• j

|βk| (1 + α2|βj|2) 1 ν|βk|2 (−i) [ek · ˆ v(j, t − s)] πk(ˆ v(k − j, t − s))ds t

• Dealing with the quadratic terms - a particular term

− · − − − = t ν|βk|2 exp[−ν|βk|2s]

• j

|βk|h ∗ h(k) (1 + α2|βj|2)ν|βk|2

• h(k)q0m(α)

(j,k−j)

h(k − j)h(j) h ∗ h(k)

• W (j,n;k)

(−i)

• ek · ˆ

v(j, t − s) h(j)

• πk(ˆ

v(k − j, t − s) h(k − j) )

• Q0(a,b;j,n)

ds

Special Semester on Stochastics with Emphasis on Finance

χ(k, t) = ˆ v(k, t) h(k) , ϕ(k, t) = ˆ g(k, t) ν|βk|2h(k)q3 Fourier transform rescaled by h(k). Equivalent LANS-alpha formulation χ(k, t) = exp[−ν|βk|2t]χ0(k) +

2

• l=0

ql t ν|βk|2 exp[−ν|βk|2s]

• j,n

m(α)

l

(j, n)

• multipliers

Ql(χ(j, t − s), χ(n, t − s); j, n)

• branching

W (j, n; k) ds + q3 t ν|βk|2 exp[−ν|βk|2s] ϕ(k, t − s)ds

Special Semester on Stochastics with Emphasis on Finance

|, W (j, n; k) = h(j)h(n) h ∗ h(k) δk(j, n), 0 < qi < 1,

3

• i=0

qi = 1, |Ql(a, b; j, n)| ≤ |a||b| Branching distribution Offspring Type Probabilities kv1 + kv2 = kv W (j, n : k) = h(j)h(n) (h ∗ h)(k) δk(j, n) Sv, kv, κv Indexed by v ∈ V = ∪∞

n=0{1, 2}n.

Sv

L

= Exp(ν|kv|2), P(κv = i) = qi, i = 0, 1, 2, 3. Random Variables and distribution

Special Semester on Stochastics with Emphasis on Finance

• (α)(kv, t) =

                       χ0(kv) if Sv ≥ t ϕ(kv, t − Sv) if Sv < t, and κv = 3 m(α)

l

(kv1, kv2) Ql

• (α)(kv1, t − Sv),

(α)(kv2, t − Sv); kv1, kv2

• if Sv < t, and κv = l = 3.

Multiplicative functional - Common Probability Space for all α!

Special Semester on Stochastics with Emphasis on Finance

• (α)(kv, t) =

                       χ0(kv) if Sv ≥ t ϕ(kv, t − Sv) if Sv < t, and κv = 3 m(α)

l

(kv1, kv2) Ql

• (α)(kv1, t − Sv),

(α)(kv2, t − Sv); kv1, kv2

• if Sv < t, and κv = l = 3.

Multiplicative functional - Common Probability Space for all

Theorem: Assume that ˆ v0(k), ˆ g(k, s) and h(k) are such that E(| (α)(k, t)|) is finite for all k ∈ Z3, 0 ≤ t ≤ T. Then ˆ v(α)(k, t) = h(k)E( (α)(k, t)) is a mild solution of the LANSα equation.

α!

Special Semester on Stochastics with Emphasis on Finance

Conditions on Finite expectation: Sledge hammer approach Require multiplieres and rescaled initial data and forcing to be bounded by one. m(k) = h ∗ h(k) h(k)ν|βk| Characterizes majorizing kernels

m(α)

l

(j, k) = m(k) α2|βj|l|βk/2|2−l (1 + α2|βj|2)(1 + α2|β|k − j||2)ql ≤ m(k) ql , m(α)

0 (j, k) = m(k)

1 q0(1 + α2|βj|2 ≤ m(k) q0

Special Semester on Stochastics with Emphasis on Finance

Conditions on Finite expectation: Sledge hammer approach Require multiplieres and rescaled initial data and forcing to be bounded by one. m(k) = h ∗ h(k) h(k)ν|βk| Characterizes majorizing kernels

m(α)

l

(j, k) = m(k) α2|βj|l|βk/2|2−l (1 + α2|βj|2)(1 + α2|β|k − j||2)ql ≤ m(k) ql , m(α)

0 (j, k) = m(k)

1 q0(1 + α2|βj|2 ≤ m(k) q0

Definition: A Majorizing Kernel is function h : Z3 → (0, ∞) such that h ∗ h(k) ≤ C|k|h(k) Standardize iff C = 1.

Special Semester on Stochastics with Emphasis on Finance

Functional Spaces Fh = {v ∈ D′ : ||v||h ≡ sup

0≤t≤T,k

|ˆ v(k, t)| h(k) < ∞}. Note that for any c > 0 that, Fh = Fch, ||v||ch = 1 c ||v||h.

Theorem: Global existence for small initial data Let h be a standardized majorizing kernel. Take q3 = 1/2 and q0 = q1 = q2 = 1/6. Let BR ⊂ Fh denote the ball of radius R centered at 0, where R = νβ/6. If the initial data v0 ∈ BR and ∆−1g ∈ BνR/2, then the solution of each LANSα v(α)exists and is unique for all t > 0. Furthermore for each k ∈ Z3, one has lim

α→0 v(α)(k, t) = v(0)(k, t).

Special Semester on Stochastics with Emphasis on Finance

Finding majorizing kernels

Proposition:For measurable h : R3 → [0, ∞), define h ∗c h(k) :=

• Rd h(k − η)h(η)dη,

k ∈ R3, h ∗d h(k) :=

• k∈Z3

h(k − j)h(j), k ∈ Z3. Suppose h ∗c h(k) ≤ c|k|h(k), k ∈ R3. Let Qk(1) denote the unit cube centered at k ∈ Z3. If there are constants c1, c2 such that c2h(k) ≤ h(η) ≤ c1h(k), ∀η ∈ Qk(1), then c2

2h ∗d h(k) ≤ h ∗c h(k) ≤ c2 1h ∗d h(k),

k ∈ Z3.

Examples h(k) = 1 |k|2 , h(k) = e−|k| |k|

Special Semester on Stochastics with Emphasis on Finance

Rate of Convergence

Theorem: Let h ∈ l1(Z3) be a standardized majorizing kernel satisfying the following further moment conditions:

• j

|j|h(j) < ∞,

• j

|j|lh(j)h(k − j) < ∞, k ∈ Z3, l = 2, 3. Take q0 = q1 = q2 = 1

6 and q3 = 1

• 2. Let γ = νβ2

2 . Let R = νβ 6 and

suppose v0 ∈ BR, ∆−1g ∈ B νR

2 . Then LANSα has a unique global

solution for all α ≥ 0. Moreover, there is a positive constant A(T), not depending on α, such that T ||v(α)(·, t) − v(0)(·, t)||L2(T 3)dt ≤ A(T)α.

Special Semester on Stochastics with Emphasis on Finance

δ(k, t) = v(α)(k, t)−v(0)(k, t), k ∈ Z3, ∆(t) := sup

k

|δ(k, t)|, t ≥ 0. Common probability space. ∆(t) ≤ sup

k

{2M[

• j

h(j) 1 + α2|βj|2 ] t |βk|e−ν|βk|2s∆(t − s)ds +M2α2

j

|βj|2h(j)h(k − j) 1 + α2|βj|2 t |βk|e−ν|βk|2sds +1 22M

• j
• D(α)(j, k)
• h(j)

t e−ν|βk|2s∆(t − s)ds +1 2M2

j

• D(α)(j, k)
• h(j)h(k − j)

t e−ν|βk|2sds}. ≤ sup

k

{H1 + F1 + H2 + F2}.

Main points of proof

Special Semester on Stochastics with Emphasis on Finance

D(α)(j, k) = πk(βj)

• 1

1 + α2|βj|2 − 1 1 + α2|β(k − j)|2

• ≤

2α2|πk(βj)||βj|2 (1 + α2|βj|2)(1 + α2|β(k − j)|2) | | | − | Moments conditions m0 =

• j

h(j) 1 + α2|βj|2 , m1 =

• j

|j|h(j), and mℓ =

• j

|j|ℓ h(j)h(k − j) h ∗ h(k) , l = 2, 3.

Special Semester on Stochastics with Emphasis on Finance

Forcing term estimates F1 + F2 ≤ α2 M2β ν (m2 + m3). Homogeneous term estimates γ = νβ2/2 H1 ≤ Mm0 t e−γs 1 √νs ∆(t − s)ds. Integral inequality of Abel type ∆(t) ≤ M∗

• α2 +

t e−γ(t−s)

• ν(t − s)

∆(s)ds

• ,

Let ˜ ∆(t) = eγt∆(t). ˜ ∆(t) ≤ M∗

• α2eγt +

t 1

• ν(t − s)

˜ ∆(s)ds

• ,

Special Semester on Stochastics with Emphasis on Finance

Invert Abel inequality ˜ ∆(t) ≤ M∗α2

• eγt + 1

√ν c(t)

• + M∗2π

ν t ˜ ∆(s)ds Gronwall inequality t ˜ ∆(s)ds ≤ M∗α2 t

• eγs + 1

√ν c(s)

• e

−M∗2π ν

(t−s)ds = α2C(s)

Special Semester on Stochastics with Emphasis on Finance

T

• v(α)(·, s) − v(0)(·, s)
• L2(T 3) ds

=

T

• e− γ s

2 e

γ s

2

ˆ

v(α)(·, s) − ˆ v(0)(·, s)

• l2(Z3) ds
• T
• e−γ s

1

2

T

• eγ s sup

k

• ˆ

v(α)(k, s) − ˆ v(0)(k, s)

• k
• ˆ

v(α)(k, s) − ˆ v(0)(k, s)

• ds

1

2

• 1 − e−γ T

γ

• 2R

T

• eγ s(s)ds

1

2

α

• 2RC(T)1 − e−γ T

γ

= α A(T),

Cauchy-Schwartz and Plancherel

Special Semester on Stochastics with Emphasis on Finance

• Replace pointwise estimates on initial data and forcing

by asymptotic estimates.

• Obtain the finiteness of the expected value of the

multiplicative functional without using simple bounds

• n the factors.
• Role of incompressibility.
• Obtain representation in physical space.

Special Semester on Stochastics with Emphasis on Finance