An Essay on A Statistical Theory of Turbulence

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1 53 特集 注目研究 in 年会 04 ɹ Պڀݚ Ӄཧ ژ দɹຊ ਓ Պ ཧ ɹ ɹ ɹ ɹوɹ೭ ɹ ࡔ Պڀݚ Ӄཧ ژ Kolmogorov Onsager Kármán-Howarth- Monin Euler An Essay on A Statistical Theory of Turbulence Takeshi MATSUMOTO, Faculty of Science, Kyoto University Hayato NAWA, Dept. of Math. School of Science and Technology, Meiji University Takashi SAKAJO, Faculty of Science, Kyoto University (Received DD Month, 04) Revisiting Kolmogorov s statistical laws (appearing in so-called Kolmogorov s Theory of 94) and Onsager s conjecture (949), we make an assessment of their mathematical relevance from the view point of stochastic processes. Then we need to examine the exact meaning of Kolmogorov s fundamental hypothesis, so that we introduce a new energy dissipation rate, which is inspired by the Kármán-Howarth-Monin relation. Our mathematical strategy viewing turbulence may not be a conventional one: we don t assume any fluid equations describing the turbulence at first, but we regard turbulence as an infinite dimensional probability measure on an ensemble of appropriate time-dependent vector fields on the flat torus T 3, which describes (a part of) Kolmogorov s statistical laws. We then consider necessary properties of the ensemble in which the desired probability should be constructed. Now we have a speculation that a family of incompressible Euler flows could be our candidate, according to a number of mathematical results on the Euler flows, e.g., Constantin-E-Titi (994), Duchon-Robert (000), Eyink (003), De Lellis-Székelyhidi (0), Isett (0), Buckmaster- De Lelis-Székelyhidi (04). This speculation could lead us to a (pseudo) Gibbs measure on the ensemble. (KEY WORDS): Turbulence, fluid mechanics, Kolmogorov, Onsager, Euler flow, Gibbs measure takeshi@kyoryu.scphys.kyoto-u.ac.jp nawa@meiji.ac.jp sakajo@math.kyoto-u.ac.jp 04 日本流体力学会 A reprint from symposium proceedings, not an original paper

2 54 T 3 Ω := C(T 3 0, T ]; ), B: Ω Borel P : B (Ω, B, P ) P Ω B P V x,t : Ω T x T 3 = E P f Ef(V x,t )] Ef(V x,t )] := f(v x,t (u))p (Du) Ω = f(v)p Vx,t (dv) 3 P Ω P Vx,t V x,t 4 f(u(x, t)) := Ef(V x,t )]. Kolmogorov Onsager Kolmogorov Onsager Kolmogorov ϵ := lim inf ν 0 ν u > 0 () u u(x, t) =: V x,t (u); (evaluation map) (projection) (x, t) V x,t x, y T 3 EV x,t ] = EV y,t ]. f C L ( ) F (0) = 0 F C(0, )) Ef(V x,t, V y,t )] = F ( x y ). h > 0 EV x,t ] = EV x,t+h ]. 5 Kolmogorov p S p ( p N ) ( ] p ) h S p u] : = E (V x+h,t (u) V x,t (u)) h ] p h = (u(x + h, t) u(x, t)). h K4 3 5) h p N 3 P Vx,t Lebesgue L 3 P V x,t (dv) L 3 (dv) () = p(v; x, t) p(v; x, t) v (x, t) f(v)p Vx,t (dv) = f(v)p(v; x, t)l 3 (dv) P Lebesgue dx 4 5 ν > 0 t T > 0 ν u = ν T 3 i,j 3 ( ) ui dx x j

3 55 C p h (x, t) T 3 0, T ] S p u] = C p ( ϵ h ) p/3. (3) 6 Kolmogorov Onsager Hölder Hölder /3 (u(x + h, t) u(x, t)) h h h α, 0 < α /3. (4) (u(x + h, t) u(x, t)) h h h α, α>/3. (5) L u(x, t) dx = u(x, 0) dx. (6) T 3 T 3 Onsager Euler Besov 3, 4) Hölder Onsager Hölder /3,, 5, 0) Kolmogorov E V x,t V y,t ] Modulus Onsager 7) u u(x, t) u(y, t) Kolmogorov Onsager ˆω 7 ) Kolmogorov - Čentsov Modulus Hölder {X x,t ˆω,s } s 0,] p > Kolmogorov Hölder /3 Onsager (4) 0 < α < /3 Kolmogorov - Čentsov Kolmogorov Navier- Stokes Onsager Euler T 3 Ω P ν > 0 () Ω 3 Navier-Stokes Monin Kármán-Howarth-Monin 9) η η < ξ δ ξ u := u(x + ξ, t) u(x, t) ϵ = 4 div ξ δξ u δ ξ u. (9) X x,t ˆω,s (u) := V x+sˆω,t(u) ˆω = u(x + sˆω, t) ˆω, s > 0 t x s 0, ] (Ω, B, P ) {X x,t ˆω,s } s 0,] (3) p X x,t E p ] r p/3 (8) ˆω,s+r Xx,t ˆω,s (7) ν ξ = 0 Ω ϵu](x, t) := 4 div ξ = 4 div ξ ( ) δ ξ u δ ξ u ξ=0 ( ξ V x,t u] ξ V x,t u]) ξ=0 (0) 7 Kolmogorov - Čentsov 6 p = 3 C 3 = 4 Kolmogorov 4/5-5 (8) implicit constant (3)

4 56 4 Euler 8 ξ V x,t u] := V x+ξ,t u] V x,t u] = u(x + ξ, t) u(x, t) = δ ξ u Onsager Hölder /3 (0) C 0 φ ε (ε > 0) T 3 φ ε δ 0 (ε 0) ϵu](x, t) := 4 lim δ ξ u δ ξ u φ ε (ξ) dξ () ε 0 ϵu](x, t) := 4 lim r 0 L 3 δ ξ u δ ξ u (B(0; r)) ξ =r ξ ξ H (dξ) = 3 4 lim δ rˆω u δ rˆω u ˆω H (dˆω). r 0 4πr ˆω = 9 () 0 4 lim δ ξ u δ ξ u φ ε (ξ) dξ ε 0 = 3 4 lim δ rˆω u δ rˆω u ˆω H (dˆω). r 0 4πr ˆω = (3) 6) 8 ϵ ](x, t) Ω 9 H Hausdorff u B(0;r) ( div ξ δξ u δ ξ u ) dξ = δ ξ u ξ δ ξ u ξ =r ξ H (dξ). Duchon-Robert 6) u L 3 (T 3 (0, T )) Euler (3) D (T 3 (0, T )) Du](x, t) := 4 lim δ ξ u δ ξ u φ ε (ξ) dξ (4) ε 0 Du] = ( ) ( ( )) u u div u t + p. (5) Du] () ϵu] (5) Du] 0 (3) (3) 4 Du] = lim δ rˆω u δ rˆω u ˆω H (dˆω). 3 r 0 4πr ˆω = (6) Duchon-Robert 6) (6) Kármán- Howarth-Monin 4/3-4 3 ϵ ξ = δ ξ u δ ξ u ξ, η < ξ, ξ Eyink 7) Duchon-Robert 6) 4 lim δ ξ u δ ξ u φ ε (ξ) dξ ε 0 = 5 4 lim δ r ˆω u ˆω] 3 H (dˆω); r 0 4πr ˆω = (7) (4) Du] 4 Du] = lim 5 r 0 4πr δ rˆω u ˆω] 3 H (dˆω) (8) ˆω = Eyink 7) Duchon-Robert 6) (3) p = 3 Kolmogorov 4/5-4 5 ϵ ξ = δ ξ u ] 3 ξ, η < ξ, ξ Eyink-Sreenivasan 8) Onsager 0 Ω () () (4) - (5) Onsager

5 57 Duchon-Robert 6) Eyink 7) Euler Du] > 0 (6) (8) Du] > 0 ϵ ] Ω Duchon-Robert 6) Eyink 7) Ω P Euler 5 Euler Euler T 0 T 3 Du](x, t) dxdt > 0 (9) 3 Euler 8, 9) Onsager,, 5, 0) Hölder /3 ε ( ε > 0), ) Hölder /3 4 Hölder /3 Onsager (5) (6) 3 (4) Du] 4 L e = e(t) > 0 u(t) = e(t) Euler Euler Duchon-Robert 6) Eyink 7) T 3 0, T ] t x Hölder Hölder /3 (6) (8) Du] = 0 (5) (9) Euler P Kolmogorov (3) p = 3 6 Kato ) Navier-Stokes (ν 0 ) u L T 3 0, T ] 5 Ω () () () Duchon-Robert 6) (4) (9) ξ Euler t t, t + ] t Du] ϵ x,t u] := t t+ t/ t t/ ds Du](y, s) dy. x+ t t, t + ] t Minimal Flow Unit (MFU ) T 3 0, T ] N 6 5 6

6 58 (x + ) t t, t + t ] MFU (x, t) T 3 (0, T ) N (x, t) MFU 7 (x,t) MF U { (x + ) t t, t + t ]} (x, t) (y, s) { (x + ) t t { (y + ), t + t ]} s t, s + t = T 3 (0, T ), ]} =. MFU MFU 8 N MFU T 3 0, T ] A 9 N MFU T 3 0, T ] A N 0 0) Shannon- McMillan N 7 8 Shannon-McMillan A N Shannon-McMillan A N Shannon-McMillan Shannon-McMillan A MFU ϵ x,t, ϵ > 0 ϵ > 0 Ω Dv MFU Z x,t Ω β := β exp t t+ t/ t t/ ] ds Dv](y, s)dy Dv. x+ β (0) (β > 0 ) ϵ x,t v] Hölder /3 Hölder Ω (0) Ω Z x,t β MFU N MFU Ω 9 0 A N N MFU A Ω MFU Shannon-McMillan MFU

7 59 Shannon-McMillan β t+ t/ ] Theorem (Shannon - McMillan). exp ds Dv](y, s)dy t A = {a (x,t) MFU t t/ x+, a,..., a k } Ω = A N N X i = X i (ω) := ω i ( ω = (ω, ω,..., ω N ) Ω ) () (i =,,..., N) Nβ T ] = exp ds Dv](y, s)dy. () T 0, T ] 0 T A 3 = a a... a k p p p... p k β MFU N ϵ 3 N H H := k p j ln p j. j= JSPS (B) This work is partially supported by Grant-in-Aid for Scientific Research (B) # of JSPS. Kolmogorov-Čentsov Theorem (Kolmogorov - Čentsov ). (Ω, B, P ) X = { X t 0 t T } α β C E X t X s α ] C t s +β, 0 s, t T, X Hölder (modification) X { } = Xt 0 t T Hölder γ γ (0, β/α) P ω Ω sup 0<t s<h(ω) s, t 0,T ] δ t s γ X t (ω) X s (ω) =. h(ω) δ > 0 (7) {X x,t ˆω,s } s 0,] p > Kolmogorov (8) α = p β = p 3 Hölder /3 Brown n α = n β = n Brown Hölder Hölder {X x,t ˆω,s } s 0,] Hölder 3 Onsager 3 α β N(α, β) N N(α, β) Ω Ω N Shannon-McMillan () P (Ω N ) > α, () ω Ω N e N(H+β) P ({ω}) e N(H β), (3) Ω N (events) #Ω N e N(H β) #Ω N e N(H+β). Ω N δ k ln p j β δ > 0 { Ω N := ω Ω Ni(ω) N pi j= } < δ, i k. N i(ω) ω A N a i P N () () (3) {A N } {B N } 3 A N B N def lim N ln A N ln B N = A a a... a k =. p p(a ) p(a )... p(a k ) 0 p(a i ) (i =,,..., k) k p(a i ) = i= Ω = A N P

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