A RATE OF CONVERGENCE FOR LAGRANGIAN AVERAGED NAVIER-STOKES - PowerPoint PPT Presentation

A RATE OF CONVERGENCE FOR LAGRANGIAN AVERAGED NAVIER-STOKES EQUATIONS ED WAYMIRE BASED ON JOINT WORK WITH LARRY CHEN RON GUENTHER SUN-CHUI KIM ENRIQUE THOMANN

Incompressible Navier-Stokes on a periodic domain D = [ − L, L ] 3 , L > 0 ∂v ∂t + v · ∇ v = ν ∆ v − ∇ p + g ∇ · v = 0 , v ( x, 0) = v 0 ( x ) , x ∈ D v = ( v 1 , v 2 , v 3 ) , x = ( x 1 , x 2 , x 3 )

Incompressible Navier-Stokes on a periodic domain ∇ · v = 0 v · ∇ v → ∇ ( v ⊗ v ) ⇒

Incompressible Navier-Stokes on a periodic domain ∇ · v = 0 v · ∇ v → ∇ ( v ⊗ v ) ⇒ ∂v ∂t + ∇ ( v ⊗ v ) = ν ∆ v − ∇ p + g ∇ · v = 0 , v ( x, 0) = v 0 ( x ) , x ∈ D

REGULARIZATION WHAT & WHY ? ∂v u ∂t + · ∇ v = ν ∆ v − ∇ p + g • SPATIAL FILTER u = G ∗ v • MATHEMATICAL TECHNIQUE: Leray existence theory for NS. • COMPUTATIONAL TECHNIQUE: Flow structure may exist below grid in high Reynolds NS.

Gallavotti Principle: Maintain Kelvin Circulation Theorem Time rate of change of momentum (per unit mass) around a closed material loop moving with the regularized fluid velocity should be an integral over viscous and external forces acting on the fluid. Leray’s regularization did not satisfy this principle. Foias, Holm, Titi (2002) DERIVED LANSalpha d � � v · dx = ( ν ∆ v + g ) dx dt γ ( u ) γ ( u )

Incompressible LANSalpha on a periodic domain D = [ − L, L ] 3 , L > 0 − LANSalpha as α ≥ 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 ) AVERAGING u = G ∗ v = ( I − α 2 ∆ ) − 1 v OPERATOR

CURRENT THEORY (BRIEF SURVEY) • Foias, Holm, Titi (2002), Marsden, Shkoller (2003): Existence and regularity theory based on energy estimates. • Kolmogorov Scaling and Attractor Dimension Estimates. • Foias, Holm,Titi (2002),Linshutz, Titi (2007): Convergence of subsequences as . α ↓ 0 • Computational numerical experiments.

Fourier Expansion: � v ( k, t ) e iβk · x v ( x, t ) = ˆ k ∈ Z 3 β = 2 π Aspect Ratio: 2 L

Fourier Expansion: � v ( k, t ) e iβk · x v ( x, t ) = ˆ k ∈ Z 3 β = 2 π Aspect Ratio: 2 L � � ∂ ˆ v ( k, t ) � � + iβ u ( l, t ) ⊗ ˆ ˆ v ( k − l, t ) + l ˆ u ( l, t ) · ˆ v ( k − l, t ) k ∂t l l = − ν | βk | 2 ˆ v − iβk ˆ p ( k, t ) + ˆ g. WHERE v ( k, t ) ˆ u ( k, t ) = ˆ 1 + α 2 | βk | 2

A SIMPLER PROBABILISTIC DRESS - FOR ILLUSTRATION - ∂ ˆ v dv ∂t = − | k | 2 ˆ v + ˆ g dt = ∆ v + g � t v 0 ( k ) e −| k | 2 t + e −| k | 2 s ˆ v ( k, t ) = ˆ ˆ g ( k, t − s ) ds 0 P ( S > t ) = e −| k | 2 t � t | k | 2 e −| k | 2 s ˆ g ( k, t − s ) v 0 ( k ) e −| k | 2 t + v ( k, t ) = ˆ ˆ ds | k | 2 0 � ˆ v 0 ( k ) if S > t X ( k, t ) = g ( k,t − S ) ˆ if S ≤ t | k | 2

χ ( k , t ) = ˆ v ( k , t ) ˆ g ( k , t ) ϕ ( k , t ) = h ( k ) , ν | β k | 2 h ( k ) q 3 exp[ − ν | β k | 2 t ] χ 0 ( k ) χ ( k , t ) = � t 2 0 ν | β k | 2 exp[ − ν | β k | 2 s ] m ( α ) � � + q l ( j , n ) Q l ( χ ( j , t − s ) , χ ( n , t − s ); j , n ) W ( j , n ; k ) ds l l =0 j , n � t 0 ν | β k | 2 exp[ − ν | β k | 2 s ] ϕ ( k , t − s ) ds + q 3 (2.12 MAIN INGREDIENTS Multipliers: m ( α ) ( j, n ) l (Branching) Quadratic Forms: Q l ( · , · ) Wave Number Transition Probabilities: W ( j, n : k ) Offspring Type Probabilities: q l

EXPECTED VALUE OF WHAT ? � ( α ) ( k � v � , t ) = �  χ 0 ( k � v � ) if S � v � ≥ t         ϕ ( k � v � , t − S � v � ) if S � v � < t, and κ � v � = 3   = (3   m ( α ) � � � ( α ) ( k � v 1 � , t − S � v � ) , � � ( α ) ( k � v 2 � , t − S � v � ); k � v 1 � , k � v 2 �  ( k � v 1 � , k � v 2 � ) Q l �   l    if S � v � < t, and κ � v � = l � = 3 .  

� � | ˆ v ( k, t ) | v ∈ L 2 : F h,T = sup < ∞ h ( k ) 0 ≤ t ≤ T, k � = 0 � ( α ) ( k , t ) | ) is finite for Theorem 3.1 Assume that ˆ v 0 ( k ) , ˆ g ( k , s ) and h ( k ) are such that E ( | � � ( α ) ( k , t )) is a mild solution of the LANS α all k ∈ Z 3 , 0 ≤ t ≤ T. Then ˆ v ( α ) ( k , t ) = h ( k ) E ( � equation. SLEDGE HAMMER APPROACH: MAKE ( α ) ( k, t ) ≤ |X | 1

q 0 (1 + α 2 | β j | 2 ) ≤ m ( k ) 1 m ( α ) 0 ( j , n ) = m ( k ) , q 0 α 2 | β j | l | β k 2 | 2 − l m ( α ) ( j , n ) = m ( k ) . l (1 + α 2 | β j | 2 )(1 + α 2 | β n | 2 q l m ( k ) = h ∗ h ( k ) h ( k ) ν | βk | LEMMA 3.1 The following inequality holds for any α , β > 0 and k ∈ Z 3 . α 2 | β k || β j | (1 + α 2 | β j | 2 )(1 + α 2 | β k − β j | 2 ) ≤ 1 .

m ( k ) = h ∗ h ( k ) W ( j, n ; k ) = h ( j ) j ( n ) h ∗ h ( k ) δ k ( j + n ) h ( k ) ν | βk | , Q 0 ( a , b ; j , n ) = − i ( ek · a ) π k ( b ) , Q 1 ( a , b ; j , n ) = − i π k ( en )( a · b ) , , ) , Q 2 ( a , b ; j , n ) = i π k ( en )( ej · en )( a · b ) . | Q l ( a , b ; j , n ) | ≤ | a || b | .

SMALL BALL APPROACH: CHOOSE A RADIUS R:

SMALL BALL APPROACH: CHOOSE A RADIUS R: NOTE ON ROLE OF MAJORIZING CONSTANTS: || · || ch = 1 c || · || h F h = F ch , c > 0 MAJORIZING KERNEL: h ∗ h ( k ) ≤ C | k | h ( k ) ch ∗ ch ( k ) ≤ cC | k | ch ( k )

SMALL BALL APPROACH: CHOOSE A RADIUS R: NOTE ON ROLE OF MAJORIZING CONSTANTS: || · || ch = 1 c || · || h F h = F ch , c > 0 MAJORIZING KERNEL: h ∗ h ( k ) ≤ C | k | h ( k ) Rh ∗ Rh ( k ) ≤ RC | k | Rh ( k ) g ( k , t ) | ≤ ν | β k | 2 Rh ( k ) q 3 , | ˆ v 0 ( k ) | ≤ Rh ( k ) , | ˆ m ( k ) = h ∗ h ( k ) h ( k ) ν | βk | m ( α ) ( k , j ) ≤ 1 , l = 0 , 1 , 2 . l

RECALL SLEDGE HAMMER CONDITION (FOLLOWING IS A COROLLARY) Theorem 3.2 Let h be a standardized majorizing kernel. Take q 3 = 1 2 , and q 0 = q 1 = q 2 = 1 6 . Let B R ⊆ F h denote the ball of radius R centered at 0 , where R = (2 L ) 3 νβ . If the v 0 ∈ B R 6 and ∆ − 1 g ∈ B ν R 2 then the solution of each LANS α , ˆ v α ( k , t ) exists and is unique for all t > 0 . Moreover, for each k ∈ Z 3 one has α → 0 v ( α ) ( k , t ) = v (0) ( k , t ) . lim

RATE OF CONVERGENCE Theorem 5.1 Let h ∈ l 1 ( Z 3 ) be a standardized majorizing kernel satisfying the following further moment conditions: k ∈ Z 3 , l = 2 , 3 . � � | j | l h ( j ) h ( k − j ) < ∞ , | j | h ( j ) < ∞ , j j νβ 2 1 1 νβ Take q 0 = q 1 = q 2 = 6 and q 3 = 2 . Let γ = 2 . Let R = 6 and suppose v 0 ∈ B R , ∆ − 1 g ∈ B ν R 2 . Then LANS α has a unique global solution for all α ≥ 0 . Moreover, there is a positive constant A ( T ) , not depending on α , such that � T 0 || v ( α ) ( · , t ) − v (0) ( · , t ) || L 2 ( T 3 ) dt ≤ A ( T ) α .

MOMENTS OF ALL ORDERS: Corollary 4.1 The function h ( k ) = e − | k | k ∈ Z 3 , k � = 0 , | k | , h ( 0 ) = 0 , defines a majorizing kernel. In fact, h ∈ l 1 is normalizable to a probability. LEJAN-SZNITMAN -- BESSEL--HELMHOLTZ TYPE !

Proposition 4.1 For measurable h : R 3 → [0 , ∞ ) , define � ξ ∈ R 3 , h ∗ c h ( ξ ) := d h ( ξ − η ) h ( η ) d η , R and k ∈ Z 3 . � h ∗ d h ( k ) := h ( k − j ) h ( j ) , 3 k ∈ Z Suppose ξ ∈ R 3 . h ∗ c h ( ξ ) ≤ c | ξ | h ( ξ ) , Let Q k (1) denote the unit cube centered at k ∈ Z 3 . If there are constants c 1 , c 2 such that c 2 h ( k ) ≤ h ( η ) ≤ c 1 h ( k ) , ∀ η ∈ Q k (1) , then c 2 2 h ∗ d h ( k ) ≤ h ∗ c h ( k ) ≤ c 2 k ∈ Z 3 . 1 h ∗ d h ( k ) , In particular, h ∗ d h ( k ) ≤ c | k | h ( k ) , k � = 0 . c 2 2

APPROACH TO RATES: Gronwall inequality to an integral equation for the difference ^ ^ δ ( k , t ) = v ( α ) ( k , t ) − v (0) ( k , t ) , k ∈ Z 3 , ∆ ( t ) := sup | δ ( k , t ) | , t ≥ 0 . k ˜ γ = νβ 2 ∆ ( t ) = e γ t ∆ ( t ) , t ≥ 0 . 2   ˜ � t ∆ ( s ) ds  α 2 e γ t + ˜  , ∆ ( t ) ≤ M ∗ t ≥ 0 . � ν ( t − s ) 0 re-root” case of the Abel transform term appearing in this INVERT ABEL TRANSFORM: � t � � + M ∗ 2 π e γ t + 1 ˜ ˜ ∆ ( t ) ≤ α 2 M ∗ √ ν c ( t ) ∆ ( s ) ds. ν 0 �