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

Branching Processes in Fluid Mechanics: An application to the Navier-Stokes and LANS-alpha equations. Enrique Thomann Joint work with Larry Chen, Ron Guenther, Ed Waymire (Oregon State University) and Sun-Chul Kim (Chung Ang University) 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) = v 0 ( 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 Spatial filtering u = G ∗ v . Does not satisfy Kelvin Circulation Theorem d � � v · d r = ( ν ∆ v + g ) · d r dt γ t γ t Special Semester on Stochastics with Emphasis on Finance

Naive Regularization (Leray ‘30) ∂ v ∂ t + ∇ · ( u ⊗ v ) = ν ∆ v − ∇ p + g ∇ · v = 0 Spatial filtering u = G ∗ v . Does not satisfy Kelvin Circulation Theorem d � � v · d r = ( ν ∆ v + g ) · d r dt γ t γ t 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 Navier-Stokes ∂ t + ∇ · ( v ⊗ v ) = ν ∆ v − ∇ p + g ∇ · v = 0 ∇ · v = 0 ∂ v ( α ) + ∇ · ( u ( α ) ⊗ v ( α ) ) + ( ∇ u ( α ) ) Tv ( α ) = ν ∆ v ( α ) − ∇ p + g ∂ t ∇ · v ( α ) = 0 , (1 − α 2 ∆ ) u ( α ) = v ( α ) LANS-alpha ∇ · − Initial data data v ( α ) ( x , 0) = v 0 ( x ) Special Semester on Stochastics with Emphasis on Finance

− ∇ · ⊗ ∂ v Navier-Stokes ∂ t + ∇ · ( v ⊗ v ) = ν ∆ v − ∇ p + g ∇ · v = 0 ∇ · v = 0 ∂ v ( α ) + ∇ · ( u ( α ) ⊗ v ( α ) ) + ( ∇ u ( α ) ) Tv ( α ) = ν ∆ v ( α ) − ∇ p + g ∂ t ∇ · v ( α ) = 0 , (1 − α 2 ∆ ) u ( α ) = v ( α ) LANS-alpha ∇ · − Initial data data v ( α ) ( x , 0) = v 0 ( x ) Spatial Filtering given by the Green function of the Helmholtz operator u ( α ) = (1 − α 2 ∆ ) − 1 v ( α ) = G ∗ v ( α ) . Special Semester on Stochastics with Emphasis on Finance

− ∇ · ⊗ ∂ v Navier-Stokes ∂ t + ∇ · ( v ⊗ v ) = ν ∆ v − ∇ p + g ∇ · v = 0 ∇ · v = 0 ∂ v ( α ) + ∇ · ( u ( α ) ⊗ v ( α ) ) + ( ∇ u ( α ) ) Tv ( α ) = ν ∆ v ( α ) − ∇ p + g ∂ t ∇ · v ( α ) = 0 , (1 − α 2 ∆ ) u ( α ) = v ( α ) LANS-alpha ∇ · − Initial data data v ( α ) ( x , 0) = v 0 ( x ) Spatial Filtering given by the Green function of the Helmholtz operator u ( α ) = (1 − α 2 ∆ ) − 1 v ( α ) = G ∗ v ( α ) . Recovering Navier-Stokes. α ( ∇ u (0) ) Tv (0) = 1 v (0) = u (0) , | 2 α = 0 , 2 ∇ | | v | 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 on 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. ∂ u ∂ ˆ u ∂ t = − | k | 2 ˆ ∂ t = ∆ u + g, u | t =0 = u 0 u + ˆ u | t =0 = ˆ ˆ g, , u 0 � t u 0 ( k ) e − | k | 2 t + 0 e − | k | 2 s ˆ u ( k, t ) = ˆ ˆ g ( k, t − s ) ds Special Semester on Stochastics with Emphasis on Finance

An illustrative example. Forced Heat equation. ∂ u ∂ ˆ u ∂ t = − | k | 2 ˆ ∂ t = ∆ u + g, u | t =0 = u 0 u + ˆ u | t =0 = ˆ ˆ g, , u 0 � t u 0 ( k ) e − | k | 2 t + 0 e − | k | 2 s ˆ u ( k, t ) = ˆ ˆ g ( k, t − s ) ds Exponential random variable P ( S > t ) = e − | k | 2 t � t | k | 2 e − | k | 2 s ˆ g ( k , t − s ) u 0 ( k ) e − | k | 2 t + u ( k , t ) = ˆ ˆ ds | k | 2 0 Special Semester on Stochastics with Emphasis on Finance

An illustrative example. Forced Heat equation. ∂ u ∂ ˆ u ∂ t = − | k | 2 ˆ ∂ t = ∆ u + g, u | t =0 = u 0 u + ˆ u | t =0 = ˆ ˆ g, , u 0 � t u 0 ( k ) e − | k | 2 t + 0 e − | k | 2 s ˆ u ( k, t ) = ˆ ˆ g ( k, t − s ) ds Exponential random variable P ( S > t ) = e − | k | 2 t � t | k | 2 e − | k | 2 s ˆ g ( k , t − s ) u 0 ( k ) e − | k | 2 t + u ( k , t ) = ˆ ˆ ds | k | 2 0  u 0 ( k ) ˆ if S > t  u ( k , t ) = E [ X ( k , t )] ˆ X ( k, t ) = g ( k,t − S ) ˆ if S ≤ t. | k | 2  Special Semester on Stochastics with Emphasis on Finance

Probabilistic representation for LANS-alpha − ∂ v ( α ) + ∇ · ( u ( α ) ⊗ v ( α ) ) + ( ∇ u ( α ) ) Tv ( α ) ν ∆ v ( α ) − ∇ p + g = ∂ t ∇ · v ( α ) = 0 , (1 − α 2 ∆ ) u ( α ) v ( α ) = ( ) k · ˆ v (k , t ) = 0 Incompressibility π k � ∇ p = 0 Leray Projection Aspect Ratio time, β = 2 π /L . ˆ v (k , t ) u (k , t ) = ˆ 1 + α 2 | β k | 2 Take Fourier Transform, integrate in time and eliminate the pressure using the projection Special Semester on Stochastics with Emphasis on Finance

Mild Formulation exp[ − ν | β k | 2 t ]ˆ ˆ v (k , t ) = v 0 (k) � t β k · ˆ v (j , t − s ) � exp[ − ν | β k | 2 s ] 1 + α 2 | β j | 2 π k (ˆ v (k − j , t − s )) ds − i 0 j � t βπ k (j)ˆ v (j , t − s ) · ˆ v (k − j , t − s ) � exp[ − ν | β k | 2 s ] − i ds 1 + α 2 | β j | 2 0 j � t exp[ − ν | β k | 2 s ]ˆ + g (k , t − s ) ds 0 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. Special Semester on Stochastics with Emphasis on Finance

Dealing with the quadratic terms - a particular term � t β k · ˆ v (j , t − s ) � exp[ − ν | β k | 2 s ] ( − i ) 1 + α 2 | β j | 2 π k (ˆ v (k − j , t − s )) ds 0 j � t | β k | 1 � ν | β k | 2 exp[ − ν | β k | 2 s ] = (1 + α 2 | β j | 2 ) ν | β k | 2 0 j ( − i ) [e k · ˆ v (j , t − s )] π k (ˆ v (k − j , t − s )) ds � t � Special Semester on Stochastics with Emphasis on Finance

Dealing with the quadratic terms - a particular term � t β k · ˆ v (j , t − s ) � exp[ − ν | β k | 2 s ] ( − i ) 1 + α 2 | β j | 2 π k (ˆ v (k − j , t − s )) ds 0 j � t | β k | 1 � ν | β k | 2 exp[ − ν | β k | 2 s ] = (1 + α 2 | β j | 2 ) ν | β k | 2 0 j ( − i ) [e k · ˆ v (j , t − s )] π k (ˆ v (k − j , t − s )) ds · − − − − � t � t � | β k | h ∗ h (k) h (k − j) h (j) � ν | β k | 2 exp[ − ν | β k | 2 s ] = (1 + α 2 | β j | 2 ) ν | β k | 2 h ∗ h (k) 0 j � �� � � �� � h (k) q 0 m ( α ) W (j , n;k) (j , k − j) 0 � � e k · ˆ π k (ˆ v (j , t − s ) v (k − j , t − s ) ( − i ) ) ds h (j) h (k − j) � �� � Q 0 ( a , b ;j , n) Special Semester on Stochastics with Emphasis on Finance

Fourier transform rescaled by h(k). χ (k , t ) = ˆ v (k , t ) g (k , t ) ˆ ϕ (k , t ) = ν | β k | 2 h (k) q 3 h (k) , Equivalent LANS-alpha formulation exp[ − ν | β k | 2 t ] χ 0 (k) χ (k , t ) = � t 2 � ν | β k | 2 exp[ − ν | β k | 2 s ] + q l 0 l =0 � m ( α ) (j , n) Q l ( χ (j , t − s ) , χ (n , t − s ); j , n) W (j , n; k) ds l � �� � � �� � j , n branching multipliers � t ν | β k | 2 exp[ − ν | β k | 2 s ] ϕ (k , t − s ) ds + q 3 0 Special Semester on Stochastics with Emphasis on Finance

3 Offspring Type � 0 < q i < 1 , q i = 1 , | Q l ( a , b ; j , n) | ≤ | a || b | Probabilities i =0 W (j , n; k) = h (j) h (n) Branching distribution h ∗ h (k) δ k (j , n) , | , Random Variables and distribution S v , k v , κ v Indexed by v ∈ V = ∪ ∞ n =0 { 1 , 2 } n . L = Exp ( ν | k v | 2 ) , P ( κ v = i ) = q i , i = 0 , 1 , 2 , 3 . S v k v 1 + k v 2 = k v W (j , n : k ) = h (j) h (n) ( h ∗ h )(k) δ k (j , n) Special Semester on Stochastics with Emphasis on Finance