Remarks on High Reynolds Number Hydrodynamics Peter Constantin - PowerPoint PPT Presentation

Turbulence Questions Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true? four-fifths law?

Turbulence Questions Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true? four-fifths law? Are there high Reynolds number universal asymptotics?

Turbulence Questions Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true? four-fifths law? Are there high Reynolds number universal asymptotics? Structure functions: � α p � ζ p � | ℓ | � | ℓ | p �| u ( x + ℓ ) − u ( x ) | p � ∼ ( ǫ | ℓ | ) = CU p 3 L L for | ℓ | ≥ ℓ d

Turbulence Questions Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true? four-fifths law? Are there high Reynolds number universal asymptotics? Structure functions: � α p � ζ p � | ℓ | � | ℓ | p �| u ( x + ℓ ) − u ( x ) | p � ∼ ( ǫ | ℓ | ) = CU p 3 L L for | ℓ | ≥ ℓ d α p = intermittency “corrections”.

Turbulence Questions Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true? four-fifths law? Are there high Reynolds number universal asymptotics? Structure functions: � α p � ζ p � | ℓ | � | ℓ | p �| u ( x + ℓ ) − u ( x ) | p � ∼ ( ǫ | ℓ | ) = CU p 3 L L for | ℓ | ≥ ℓ d α p = intermittency “corrections”. Known rigorously (CF): If scaling, then ζ 2 ≥ 2 ζ 1 ≥ 2 3

Inviscid limit Infinite time and zero viscosity limits do not commute.

Inviscid limit Infinite time and zero viscosity limits do not commute. Time interval fixed. In the absence of boundaries the finite time inviscid limit leads to the initial value problem for Euler equations.

Inviscid limit Infinite time and zero viscosity limits do not commute. Time interval fixed. In the absence of boundaries the finite time inviscid limit leads to the initial value problem for Euler equations. Time → ∞ first, only then Reynolds number UL ν → ∞

Inviscid limit Infinite time and zero viscosity limits do not commute. Time interval fixed. In the absence of boundaries the finite time inviscid limit leads to the initial value problem for Euler equations. Time → ∞ first, only then Reynolds number UL ν → ∞ Limits: selected stationary statistical solutions.

Inviscid limit Infinite time and zero viscosity limits do not commute. Time interval fixed. In the absence of boundaries the finite time inviscid limit leads to the initial value problem for Euler equations. Time → ∞ first, only then Reynolds number UL ν → ∞ Limits: selected stationary statistical solutions. ˆ T 1 Φ( S NS ( t )) dt µ (Φ) = Re →∞ lim lim T T →∞ 0

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s.

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s. A gap ( ν 0 , ν 1 ) . Artificial?

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s. A gap ( ν 0 , ν 1 ) . Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space H s , t < T .

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s. A gap ( ν 0 , ν 1 ) . Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space H s , t < T . Less smooth initial data: vortex patches ( ∇ × u 0 ∈ L ∞ ∩ L 1 ).

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s. A gap ( ν 0 , ν 1 ) . Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space H s , t < T . Less smooth initial data: vortex patches ( ∇ × u 0 ∈ L ∞ ∩ L 1 ). Convergence to Euler still holds but rate deteriorates (C-Wu ’95, Abidi-Danchin ’04, Masmoudi ’06.)

Finite time, no boundaries Theorem (C, ’86) If u 0 and T are fixed, but arbitrary, if the solution S E ( t ; u 0 ) is smooth on [ 0 , T ] (e.g. C ( 0 , T ; H s ( T 3 )) , s > 5 / 2 ), then there exists ν 0 = ν 0 ( u 0 , T ) such that S ( ν ) ( t , u 0 ) is smooth on the same time interval for all ν ≤ ν 0 and � S NS ( t ) u 0 − S E ( t ) u 0 � s ′ = O ( ν ) s ′ < s. A gap ( ν 0 , ν 1 ) . Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space H s , t < T . Less smooth initial data: vortex patches ( ∇ × u 0 ∈ L ∞ ∩ L 1 ). Convergence to Euler still holds but rate deteriorates (C-Wu ’95, Abidi-Danchin ’04, Masmoudi ’06.) Lions-DiPerna: Any L 2 weak limit of NSE is a dissipative solution of Euler.

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler.

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N )

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time.

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) .

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) . ekelyhidi, C s : convex integration, h-principle (Nash, DeLellis- Sz´ 1 Gromov), Beltrami flows, diminishing Reynolds fluxes. s < 10 .

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) . ekelyhidi, C s : convex integration, h-principle (Nash, DeLellis- Sz´ 1 Gromov), Beltrami flows, diminishing Reynolds fluxes. s < 10 . 1 5 + : Material (Lagrangian) derivative better behaved than time Isett, C derivative, nonlinear phases.

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) . ekelyhidi, C s : convex integration, h-principle (Nash, DeLellis- Sz´ 1 Gromov), Beltrami flows, diminishing Reynolds fluxes. s < 10 . 1 5 + : Material (Lagrangian) derivative better behaved than time Isett, C derivative, nonlinear phases. 1 ekelyhidi: L 1 ( 0 , T ; C 3 ) . More careful Buckmaster, De Lellis, Sz´ accounting.

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) . ekelyhidi, C s : convex integration, h-principle (Nash, DeLellis- Sz´ 1 Gromov), Beltrami flows, diminishing Reynolds fluxes. s < 10 . 1 5 + : Material (Lagrangian) derivative better behaved than time Isett, C derivative, nonlinear phases. 1 ekelyhidi: L 1 ( 0 , T ; C 3 ) . More careful Buckmaster, De Lellis, Sz´ accounting. 1 3 + : gluing, Mikado flows. Isett C

Onsager Conjecture Somewhat related to 2/3-law, but finite time IVP for 3D Euler. 1 3 . Onsager Conjecture: solutions conserve energy if smoother than C 3 there exist C s solutions for which energy is dissipated. For s < 1 Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L 3 � 1 � dt ; B ⇒ E ( t ) = const 3 ∞ , c ( N ) Examples of wild Euler solutions: Scheffer: compactly supported in time. Shnirelman: dissipating energy, in L ∞ ( dt ; L 2 ) . ekelyhidi, C s : convex integration, h-principle (Nash, DeLellis- Sz´ 1 Gromov), Beltrami flows, diminishing Reynolds fluxes. s < 10 . 1 5 + : Material (Lagrangian) derivative better behaved than time Isett, C derivative, nonlinear phases. 1 ekelyhidi: L 1 ( 0 , T ; C 3 ) . More careful Buckmaster, De Lellis, Sz´ accounting. 1 3 + : gluing, Mikado flows. Isett C Buckmaster, De Lellis, Sz´ ekelyhidi, Vicol: dissipative.

Long time, no boundaries No results for 3d NS/Euler.

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν .

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g Enstrophy balance: d ˆ ˆ ˆ R 2 | ω ( x , t ) | 2 dx + ν R 2 |∇ ω ( x , t ) | 2 dx = R 2 g ω dx 2 dt

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g Enstrophy balance: d ˆ ˆ ˆ R 2 | ω ( x , t ) | 2 dx + ν R 2 |∇ ω ( x , t ) | 2 dx = R 2 g ω dx 2 dt Anomalous dissipation of enstrophy?

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g Enstrophy balance: d ˆ ˆ ˆ R 2 | ω ( x , t ) | 2 dx + ν R 2 |∇ ω ( x , t ) | 2 dx = R 2 g ω dx 2 dt Anomalous dissipation of enstrophy? ν → 0 ν ��∇ ω � 2 lim L 2 � = χ > 0

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g Enstrophy balance: d ˆ ˆ ˆ R 2 | ω ( x , t ) | 2 dx + ν R 2 |∇ ω ( x , t ) | 2 dx = R 2 g ω dx 2 dt Anomalous dissipation of enstrophy? ν → 0 ν ��∇ ω � 2 lim L 2 � = χ > 0 Kraichnan (’68): yes,

Long time, no boundaries No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: �∇ u � 2 L 2 bounded in time uniformly in ν . Vorticity: ω = ∇ ⊥ · u ∂ t ω + u · ∇ ω − ν ∆ ω = ∇ ⊥ · f = g Enstrophy balance: d ˆ ˆ ˆ R 2 | ω ( x , t ) | 2 dx + ν R 2 |∇ ω ( x , t ) | 2 dx = R 2 g ω dx 2 dt Anomalous dissipation of enstrophy? ν → 0 ν ��∇ ω � 2 lim L 2 � = χ > 0 Kraichnan (’68): yes, Bernard (’00): add damping, and then no.

Absence of anomalous dissipation C-Ramos ’07: If you add damping u t + γ u + ν Au + B ( u , u ) = f then there is no anomalous dissipation of enstrophy: ˆ T ν ˆ � 2 � � ∇ ω ( ν ) ( x , t ) ν → 0 lim sup lim dxdt = 0 � � T � T →∞ 0

Absence of anomalous dissipation C-Ramos ’07: If you add damping u t + γ u + ν Au + B ( u , u ) = f then there is no anomalous dissipation of enstrophy: ˆ T ν ˆ � 2 � � ∇ ω ( ν ) ( x , t ) ν → 0 lim sup lim dxdt = 0 � � T � T →∞ 0 C-Tarfulea-Vicol ’13. No anomalous dissipation of energy in critical SQG.

Absence of anomalous dissipation C-Ramos ’07: If you add damping u t + γ u + ν Au + B ( u , u ) = f then there is no anomalous dissipation of enstrophy: ˆ T ν ˆ � 2 � � ∇ ω ( ν ) ( x , t ) ν → 0 lim sup lim dxdt = 0 � � T � T →∞ 0 C-Tarfulea-Vicol ’13. No anomalous dissipation of energy in critical SQG. Method of proof: statistical solutions.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f .

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t :

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 .

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in time.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded. Forgets initial data.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded. Forgets initial data. Same true with Dirichlet BC ( f = sin nx sin my ).

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded. Forgets initial data. Same true with Dirichlet BC ( f = sin nx sin my ). Open problem: construct example of f so that a stationary condensate exists.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded. Forgets initial data. Same true with Dirichlet BC ( f = sin nx sin my ). Open problem: construct example of f so that a stationary condensate exists. B ( u E , u E ) = f is easily solvable.

Long time, undamped Gallet-Young (’13): “Condensate” in 2D periodic, undamped forced by Kolmogorov forcing. Inverse cascade. Numerical, “semi-analytical”. And wrong for some solutions. Kolmogorov forcing: Af = λ f . Exact solutions u ( t ) = y ( t ) f . Navier-Stokes: y ( t ) = y 0 e − νλ t + 1 1 − e − νλ t � � νλ Euler: y ( t ) = y 0 + t Any finite t : S NS ( t ) u 0 → S E ( t ) u 0 . Solution bounded in ν , locally in 1 time. But: t → ∞ : u ( t ) → u f = νλ f ν Au f = f , B ( u f , u f ) = 0 Unbounded. Forgets initial data. Same true with Dirichlet BC ( f = sin nx sin my ). Open problem: construct example of f so that a stationary condensate exists. B ( u E , u E ) = f is easily solvable. Can any steadily forced undamped Navier-Stokes solutions remain bounded for all time as viscosity is removed?

Basic Questions

Basic Questions ◮ How does the 2D inverse cascade work? Does deterministic 3D eddy diffusity exist? (i.e. can B ( u , u ) provide average friction at low wave numbers in 2D, at high wave numbers in 3D?).

Basic Questions ◮ How does the 2D inverse cascade work? Does deterministic 3D eddy diffusity exist? (i.e. can B ( u , u ) provide average friction at low wave numbers in 2D, at high wave numbers in 3D?). ◮ Is there a well defined (weak) inviscid limit, when energy dissipation rate does not necessarily vanish?

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain.

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly.

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better.

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 .

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 . Energy ´ T Ω |∇ u | 2 dxdt vanishes ´ dissipation rate ν 0

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 . Energy ´ T Ω |∇ u | 2 dxdt vanishes ´ dissipation rate ν 0 Here BL : { x ∈ Ω | dist ( x , ∂ Ω) ≤ O ( ν ) } Assumed: smooth regime, smooth boundary. Method: corrector.

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 . Energy ´ T Ω |∇ u | 2 dxdt vanishes ´ dissipation rate ν 0 Here BL : { x ∈ Ω | dist ( x , ∂ Ω) ≤ O ( ν ) } Assumed: smooth regime, smooth boundary. Method: corrector. Improvements: only tangential gradient in slightly thicker boundary layer (Temam-Wang ’97, Wang ’01).

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 . Energy ´ T Ω |∇ u | 2 dxdt vanishes ´ dissipation rate ν 0 Here BL : { x ∈ Ω | dist ( x , ∂ Ω) ≤ O ( ν ) } Assumed: smooth regime, smooth boundary. Method: corrector. Improvements: only tangential gradient in slightly thicker boundary layer (Temam-Wang ’97, Wang ’01). Inviscid unconditionally: only for very short time and real analytic data (Caflisch-Sammartino ’98), vorticity supported away from the boundary ( Maekawa ’14) and restrictive symmetries (Lopes Filho-Mazzucato-Nussenzveig-Lopes-M. Taylor,’08, Kelliher ’09).

Finite time, with boundaries: known results Inviscid limit: same initial data, fixed time, smooth domain. u NS → u E inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ˆ |∇ u | 2 dx → 0 ν BL ( ν ) then the finite time inviscid limit holds in strong L 2 . Energy ´ T Ω |∇ u | 2 dxdt vanishes ´ dissipation rate ν 0 Here BL : { x ∈ Ω | dist ( x , ∂ Ω) ≤ O ( ν ) } Assumed: smooth regime, smooth boundary. Method: corrector. Improvements: only tangential gradient in slightly thicker boundary layer (Temam-Wang ’97, Wang ’01). Inviscid unconditionally: only for very short time and real analytic data (Caflisch-Sammartino ’98), vorticity supported away from the boundary ( Maekawa ’14) and restrictive symmetries (Lopes Filho-Mazzucato-Nussenzveig-Lopes-M. Taylor,’08, Kelliher ’09). All results: rate of dissipation vanishes.

Prandtl Prandtl: u ( x ) = v ( x ) 1 { dist ( x ,∂ Ω) > √ ν } + u P 1 { dist ( x ,∂ Ω) < √ ν } + O ( √ ν ) . and u P solves Prandtl equations. Note that Kato’s BL is much smaller.

Prandtl Prandtl: u ( x ) = v ( x ) 1 { dist ( x ,∂ Ω) > √ ν } + u P 1 { dist ( x ,∂ Ω) < √ ν } + O ( √ ν ) . and u P solves Prandtl equations. Note that Kato’s BL is much smaller. Prandtl are ill posed (Gerard-Varet-Dormy ’10, Guo-Nguyen ’11, Gerard-Varet-Nguyen ’12),

Prandtl Prandtl: u ( x ) = v ( x ) 1 { dist ( x ,∂ Ω) > √ ν } + u P 1 { dist ( x ,∂ Ω) < √ ν } + O ( √ ν ) . and u P solves Prandtl equations. Note that Kato’s BL is much smaller. Prandtl are ill posed (Gerard-Varet-Dormy ’10, Guo-Nguyen ’11, Gerard-Varet-Nguyen ’12), the expansion is not valid (Grenier ’00, Grenier-Guo-Nguyen ’14)

Prandtl Prandtl: u ( x ) = v ( x ) 1 { dist ( x ,∂ Ω) > √ ν } + u P 1 { dist ( x ,∂ Ω) < √ ν } + O ( √ ν ) . and u P solves Prandtl equations. Note that Kato’s BL is much smaller. Prandtl are ill posed (Gerard-Varet-Dormy ’10, Guo-Nguyen ’11, Gerard-Varet-Nguyen ’12), the expansion is not valid (Grenier ’00, Grenier-Guo-Nguyen ’14) and the equations blow up (E-Engquist ’97, Kukavica-Vicol-Wang ’16).