Exit times of diffusions with incompressible drift Alexei Novikov - PowerPoint PPT Presentation

Exit times of diffusions with incompressible drift Alexei Novikov Department of Mathematics Penn State University, USA Sixth Workshop on Random Dynamical Systems November 1, 2013, Bielefeld, Germany.

Exit times of diffusions with incompressible drifts Transition from long-time Homogenization to strong-flow Freidlin-Wentzel Averaging, with Gautam Iyer, Tomasz Komorowski, and Lenya Ryzhik. Do incompressible drifts enhance transport? with Gautam Iyer, Lenya Ryzhik, Andrej Zlatos. √ dX x t = Au ( X x 2 dW t , X x t ) dt + 0 = x, ∇ · u ( x ) = 0 . Behavior of X x t , when P´ eclet number A ≫ 1 , and t ≫ 1 . Behavior of σ x ) , σ x = inf τ ( x ) = E ( X x t> 0 ( X x t ∈ ∂ Ω) , with | Ω | → ∞ .

Cellular vs. cat’s-eye incompressible flows � − ∂ H ( x ) , ∂ � u ( x ) = ∇ ⊥ H ( x ) = H ( x ) , x = ( x 1 , x 2 ) . ∂x 2 ∂x 1

Dissipation rate a.k.a. effective diffusivity. √ dX x t = Au ( X x t ) dt + 2 dW t , A is fixed, t → ∞ . E ( X x t × X x t ) lim = I. D A t →∞ t = D A d ˜ Then X x t ∼ Y x t , dY x W t . Let H = sin πx 1 sin πx 2 . PDE methods: √ D A ∼ C A, S.Childress ’79. D A √ lim = C > 0 , A.Fannjiang & G.Papanicolaou’94. A A → 0 Probabilistic methods: L.Koralov’01.

Averaging. Diffusion on graphs (after M. Freidlin & A.Wentzel ’78) √ dX x t = Au ( X x t ) dt + 2 dW t . Time t is fixed, A → ∞ Small Random Perturbation of Hamiltonian System ˙ X x t = Au ( X x t ) Figure taken from M.Freidlin, Reaction-Diffusion in Incompressible Fluid: Asymptotic Problems, 2002.

Boundary layer theory. Cellular flows Probability to exit through top, P´ eclet number is 30. ∆ φ − Au · ∇ φ = 0 , u = ∇ ⊥ H , H = sin πx 1 sin πx 2 , φ ( x 1 = − 1 , x 2 ) = 0 , φ ( x 1 = 1 , x 2 ) = 1 , ∂ ∂n φ ( x 1 , x 2 = ± 1) = 0 . · Level-set H ( x ) = 0 separates fluid motion into 4 eddies. · Large gradients near separatrices, boundary layers · Inside cells temperature is constant.

Boundary Layer Approximation Numerical simulation for cellular flows H = sin x 1 sin x 2 , A = 10 3 .

From Homogenization to Averaging in Cellular Flows • Let H ( x ) = 1 π sin( πx 1 ) sin( πx 2 ) . � − ∂ 2 H � • Let u ( x ) = ∇ ⊥ H = . ∂ 1 H • Let Ω = (0 , 1) 2 ⊂ R 2 . � −△ τ + A � x � ε v · ∇ τ = 1 in Ω , • ε τ = 0 on ∂ Ω • Here A and ε = 1 /L are two non-dimensional parameters. ⊲ A is the strength of stirring (the P´ eclet number). ⊲ ε is the cell size.

Contour plots of τ (a) A ‘small’ compared to L = 1 /ε . (b) A ‘large’ compared to L = 1 /ε .

√ Particle’s behaviour dXt = u ( Xt ) dt + 2 dWt (c) A ‘small’ compared to L . (d) A ‘large’ compared to L .

Homogenization vs. Averaging • Large amplitude. (Freidlin-Wentzel Averaging) ⊲ For fixed cell size ε , and amplitude A → ∞ . ⊲ τ is nearly constant on stream lines of u . ⊲ Forces τ to be almost identical in each cell. Separatrices are highways. ⊲ M.Friedlin, A.Wentzell; Yu.Kifer; H.Berestycki, F.Hamel, N.Nadirashvili. • Small amplitude. (Homogenization) ⊲ For fixed amplitude A , and cell size ε → 0 . ⊲ τ converges to the solution of an ‘effective’ enhanced diffusion equation. ⊲ No difference whether you start near or away from separatrices. ⊲ S.Childress; A.Fannjiang; G.Papanicolaou; L. Koralov.

Large amplitude, and a large number of cells. With T.Komorowski, G.Iyer, L.Ryzhik’13 Send both A → ∞ , ε → 0 . Let τ = τ A,ε as before. Theorem. (Homogenization; A ≪ 1 /ε 4 ) • Suppose α > 0 , and A ≈ 1 /ε 4 − α . • If Ω = B (0 , 1) , then τ ( x ) ≈ τ eff ( x ) = 1 − | x | 2 2 D eff ( A ) ≈ 1 − | x | 2 √ . c A 1 c • If Ω = (0 , 1) 2 , only have √ � τ ( x ) � √ on the interior of Ω . c A A Theorem. (Averaging; A ≫ 1 /ε 4 ) ε 2 √ A • Suppose lim log A log(1 /ε ) = ∞ . A →∞ • Oscillation of τ along streamlines tends to 0 . • On cell boundaries τ ( x ) � log A log(1 /ε ) √ → 0 . A

Let ϕ = ϕ ε,A be the (positive) principal eigenfunction: −△ ϕ + A � x � � in Ω = (0 , 1) 2 , ε v · ∇ ϕ = λϕ ε ϕ = 0 on ∂ Ω . Theorem. There exists c 1 , c 2 independent of L and A such that. A ≫ (log A ) 2 (log(1 /ε )) 2 (A) If then λ ≈ λ avg . , ε 4 1 λ ≈ λ eff . (H) If then A ≈ ε 4 − α , • λ avg = c 0 ε 2 , for some explicitly computable c 0 . √ • λ eff = λ 0 ( ∇ · D eff ( A ) ∇ ) ≈ c 1 A , for explicitly computable c 1 .

Asymptotics of the transition. • The transition should occur when τ avg ≈ τ eff . • Freidlin-Wentzel Averaging τ avg = τ one cell ∼ ε 2 . √ √ A ˜ • Homogenization X x t ∼ Y x t = W t τ eff ∼ 1 / A . √ A ≈ 1 ε 2 , or A ≈ 1 • Transition should occur for ε 4 .

Three scales of diffusion in cellular flows √ dX x t = Au ( X x 2 dW t , X x t ) dt + 0 = x. • t avg ≪ t rw ≪ t eff • Freidlin-Wentzell averaging time t avg . • Random walk on separatrices time t rw . • Effective diffusion t eff

Three scales of diffusion in cellular flows (e) A ‘small’ compared to L . (f) A ‘large’ compared to L .

Universality of diffusion in periodic fluid flows dX t = u ( X t ) dt + dV t , X 0 = 0 . • t avg ≪ t rw ≪ t eff • Time of V t . • Random walk time t rw . • Effective diffusion t eff Theorem (T.Komorowski, A.N., L.Ryzhik, ’13) If V t is a fractional Brownian motion with H < 1 / 2 , u ( x ) is shear u = ( u 1 ( x 2 ) , 0) then εX t/ε 2 → W t , as ε → 0 .

Do incompressible flows improve mixing? Suppose λ u is the principal eigenvalue of L u = − ∆ + u · ∇ . If ∂ t φ + L u φ = 0 , φ | ∂ Ω = 0 then || φ || L 2 ∼ e − λ u t as t → ∞ . If ∇ · u = 0 , principal eigenvalue of L u = − ∆ + u · ∇ (with φ | ∂ Ω = 0 ) is larger than that of L 0 = − ∆ .

Incompressible flows improve mixing in L 2 -sense Ω φ 2 = 1 then � Suppose φ ∈ H 1 0 (Ω) and L u φ = λ u φ , with || φ || 2 L 2 = � � � φ 2 = λ u φL u φ = |∇ φ | 2 . Ω Ω Ω On the other hand the Raleigh quotient characterizes the principal eigen- value of L 0 : � � λ 0 = |∇ φ | 2 = λ u . |∇ ψ | 2 � inf || ψ || L 2 =1 Ω Ω

Exit time problem (with G.Iyer, L.Ryzhik & A.Zlatos’10) For any incompressible u − ∆ τ u + u · ∇ τ u = 1 , τ u | ∂ Ω = 0 . τ u ( x ) = E ( X x σ x ) is the expected exit time from Ω of the diffusion: √ 2 dW t , σ x = inf dX x t = u ( X x t> 0 ( X x t ) dt + t ∈ ∂ Ω) . Theorem 1 Let Ω ⊂ R 2 be a bounded, simply connected and Lipschitz domain. Then u ≡ 0 maximizes || τ u || L ∞ (Ω) if and only if Ω is a disk. Theorem 2 Let D ⊂ R n be a ball. Then || τ u || L p ( D ) � || τ 0 || L p ( D ) for all incompressible u , and all 1 � p � ∞ .

The 2-dimensional case. General domain The stream function H for the “worst” flow u = ∇ ⊥ H solves � − 1 � � �� dσ � − 2∆ H = 1 + |∇ H | 2 |∇ H | dσ , |∇ H | ∂ Ω h ∂ Ω h where Ω h = { x ∈ Ω , H ( x ) � h } .

Streamfunction H for the “worst” flow, and T 0

Streamfunction H for the “worst” flow, and T 0

Remarks • If ∇ · u = 0 is dropped, the problem is trivial (a flow with a sink). • Recall that presence of incompressible flow always improves mixing in the sense of increasing the first eigenvalue. • Using fast flows, we can always make T u arbitrarily small. − ∆ τ u + u · ∇ τ u = f in Ω ⊂ R n , � • A more general question: Then τ u = 0 on ∂ Ω there is an L p → L ∞ bound: || τ u || L ∞ � C || f || L p , p > n/ 2 , where C = C ( n, p, Ω) , but C is independent of u (see Berestycki, Kiselev, Novikov, Ryzhik ’09). Find an optimal C . Theorem (A.N.’13) If Ω is a disk, then optimal C arises when f = g ( | x | , p, n ) , and g is a certain optimal non-increasing function.

Exit times in a ball. Proposition. Let Ω ⊂ R n be bounded, simply connected and Lipschitz domain, and u be any divergence free vector field which is tangential on ∂ Ω . Then || τ u || L p (Ω) � || τ 0 ,D || L p ( D ) where D ⊂ R n is a ball with | D | = | Ω | , and τ 0 ,D is the expected exit time from D with 0 drift.

Proof of Proposition • Given any τ = τ u , consider its symmetric rearrangement τ ∗ : – D is a ball with | D | = | Ω | , and τ ∗ : D → R + is radial. – For all h , |{ τ > h }| = |{ τ ∗ > h }| . – || τ || L p (Ω) = || τ ∗ || L p ( D ) for all p . h = { τ ∗ > h } . • Denote Ω h = { τ > h } , Ω ∗

Proof of Proposition � � 1 h | 2 � | ∂ Ω h | 2 � |∇ τ ∗ | dσ |∇ τ ∗ | dσ = | ∂ Ω ∗ ∂ Ω ∗ ∂ Ω ∗ h h � � 1 |∇ τ | dσ |∇ τ | dσ. ∂ Ω h ∂ Ω h � • Integrating the equation on Ω h we obtain ∂ Ω h |∇ τ | dσ = | Ω h | . |∇ τ | dσ = − d 1 dh | Ω h | = − d 1 � dh | Ω ∗ � • Co-area implies h | = |∇ τ ∗ | dσ ∂ Ω h ∂ Ω ∗ h � h |∇ τ ∗ | dσ � � ∂ Ω h |∇ τ | dσ = | Ω h | = | Ω ∗ • So h | . ∂ Ω ∗ • Using τ ∗ is radial and h | we conclude that τ ∗ � τ 0 ,D � h |∇ τ ∗ | dσ � | Ω ∗ ∂ Ω ∗ point-wise, where τ 0 ,D is a solution of the exit time problem in the ball D with no flow.