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

Remarks on High Reynolds Number Hydrodynamics

Peter Constantin

Princeton University June 2017

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0,

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0.

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0 Euler: v| ∂Ω · n = 0

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0 Euler: v| ∂Ω · n = 0 Reynolds number Re = UL ν

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0 Euler: v| ∂Ω · n = 0 Reynolds number Re = UL ν U = LT −1, L, T length and time scales.

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0 Euler: v| ∂Ω · n = 0 Reynolds number Re = UL ν U = LT −1, L, T length and time scales. ν-kinematic viscosity.

The equations

Incompressible Navier-Stokes for u = uNS = SNS(t)u0: ∂tu + u · ∇u − ν∆u + ∇p = f, ∇ · u = 0, and incompressible Euler for v = uE = SE(t)u0: ∂tv + v · ∇v + ∇p = f, ∇ · v = 0. Boundary conditions: Navier-Stokes: u| ∂Ω = 0 Euler: v| ∂Ω · n = 0 Reynolds number Re = UL ν U = LT −1, L, T length and time scales. ν-kinematic viscosity. Flows with the same Reynolds number = the same behavior.

Energy dissipation

From NSE: ǫ = ν|∇u|2

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation.

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109,

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale.

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale. Large number of experiments, different situations, large number of numerical experiments:

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale. Large number of experiments, different situations, large number of numerical experiments: C bounded away from zero and from infinity, C ∈ [.3, 5].

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale. Large number of experiments, different situations, large number of numerical experiments: C bounded away from zero and from infinity, C ∈ [.3, 5]. Frisch:

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale. Large number of experiments, different situations, large number of numerical experiments: C bounded away from zero and from infinity, C ∈ [.3, 5]. Frisch: “If, in an experiment on turbulent flow, all the control parameters are kept the same, except for the viscosity, which is lowered as much as possible, the energy dissipation per unit mass dE

dt behaves in a way

consistent with a finite positive limit”

Energy dissipation

From NSE: ǫ = ν|∇u|2 u = turbulent fluctuation. Constancy of energy dissipation per unit volume ǫ = C U3 L High Reynolds numbers R ∼ 106 − 109, U = mean-square average velocity, L integral scale. Large number of experiments, different situations, large number of numerical experiments: C bounded away from zero and from infinity, C ∈ [.3, 5]. Frisch: “If, in an experiment on turbulent flow, all the control parameters are kept the same, except for the viscosity, which is lowered as much as possible, the energy dissipation per unit mass dE

dt behaves in a way

consistent with a finite positive limit”

Two-thirds Law, K’41

Frisch:

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.)

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd,

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd, Kolmogorov spectrum

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd, Kolmogorov spectrum E(k) = Cǫ

2 3 k− 5 3

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd, Kolmogorov spectrum E(k) = Cǫ

2 3 k− 5 3

k ≤ kd,

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd, Kolmogorov spectrum E(k) = Cǫ

2 3 k− 5 3

k ≤ kd, kd = ν3 ǫ − 1

4

= ℓ−1

d

Two-thirds Law, K’41

Frisch: “In a turbulent flow at very high Reynolds number, the mean square velocity increment (δv(ℓ))2 between two points separated by a distance ℓ behaves approximately as the two-thirds power of the distance” (in the inertial range.) |u(x + ℓ) − u(x)|2 ∼ (ǫ|ℓ|)

2 3

|ℓ| ≥ ℓd, Kolmogorov spectrum E(k) = Cǫ

2 3 k− 5 3

k ≤ kd, kd = ν3 ǫ − 1

4

= ℓ−1

d

Four-fifths law: homogeneous and isotropic turbulence, third order longitudinal moment: |u(x + ℓ) − u(x)|3 ∼ ǫ|ℓ|.

Turbulence Questions

Is the energy dissipation ǫ bounded away from zero?

Turbulence Questions

Is the energy dissipation ǫ bounded away from zero? Is the two-thirds law true?

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: |u(x + ℓ) − u(x)|p ∼ (ǫ|ℓ|)

p 3

|ℓ| L αp = CUp |ℓ| L ζp for |ℓ| ≥ ℓd

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: |u(x + ℓ) − u(x)|p ∼ (ǫ|ℓ|)

p 3

|ℓ| L αp = CUp |ℓ| L ζp for |ℓ| ≥ ℓd αp = 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: |u(x + ℓ) − u(x)|p ∼ (ǫ|ℓ|)

p 3

|ℓ| L αp = CUp |ℓ| L ζp for |ℓ| ≥ ℓd αp = intermittency “corrections”. Known rigorously (CF): If scaling, then ζ2 ≥ 2ζ1 ≥ 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. µ(Φ) = lim

Re→∞ lim T→∞

1 T ˆ T Φ(SNS(t))dt

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s.

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s. A gap (ν0, ν1). Artificial?

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s. A gap (ν0, ν1). Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space Hs, t < T.

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s. A gap (ν0, ν1). Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space Hs, t < T. Less smooth initial data: vortex patches (∇ × u0 ∈ L∞ ∩ L1).

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s. A gap (ν0, ν1). Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space Hs, t < T. Less smooth initial data: vortex patches (∇ × u0 ∈ L∞ ∩ L1). Convergence to Euler still holds but rate deteriorates (C-Wu ’95, Abidi-Danchin ’04, Masmoudi ’06.)

Finite time, no boundaries

Theorem

(C, ’86) If u0 and T are fixed, but arbitrary, if the solution SE(t; u0) is smooth on [0, T] (e.g. C(0, T; Hs(T3)), s > 5/2), then there exists ν0 = ν0(u0, T) such that S(ν)(t, u0) is smooth on the same time interval for all ν ≤ ν0 and SNS(t)u0 − SE(t)u0s′ = O(ν) s′ < s. A gap (ν0, ν1). Artificial? Swann’71, Kato ’72: short time. Masmoudi ’06: convergence in top Sobolev space Hs, t < T. Less smooth initial data: vortex patches (∇ × u0 ∈ L∞ ∩ L1). Convergence to Euler still holds but rate deteriorates (C-Wu ’95, Abidi-Danchin ’04, Masmoudi ’06.) Lions-DiPerna: Any L2 weak limit

• f 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. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

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. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

DeLellis- Sz´ ekelyhidi, Cs: convex integration, h-principle (Nash, Gromov), Beltrami flows, diminishing Reynolds fluxes. s <

1 10.

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

DeLellis- Sz´ ekelyhidi, Cs: convex integration, h-principle (Nash, Gromov), Beltrami flows, diminishing Reynolds fluxes. s <

1 10.

Isett, C

1 5+ : Material (Lagrangian) derivative better behaved than time

derivative, nonlinear phases.

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

DeLellis- Sz´ ekelyhidi, Cs: convex integration, h-principle (Nash, Gromov), Beltrami flows, diminishing Reynolds fluxes. s <

1 10.

Isett, C

1 5+ : Material (Lagrangian) derivative better behaved than time

derivative, nonlinear phases. Buckmaster, De Lellis, Sz´ ekelyhidi: L1(0, T; C

1 3 ). More careful

accounting.

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

DeLellis- Sz´ ekelyhidi, Cs: convex integration, h-principle (Nash, Gromov), Beltrami flows, diminishing Reynolds fluxes. s <

1 10.

Isett, C

1 5+ : Material (Lagrangian) derivative better behaved than time

derivative, nonlinear phases. Buckmaster, De Lellis, Sz´ ekelyhidi: L1(0, T; C

1 3 ). More careful

accounting. Isett C

1 3+ : gluing, Mikado flows.

Onsager Conjecture

Somewhat related to 2/3-law, but finite time IVP for 3D Euler. Onsager Conjecture: solutions conserve energy if smoother than C

1 3 .

For s < 1

3 there exist Cs solutions for which energy is dissipated.

Eyink, C-E-Titi, Duchon-Robert, C-Cheskidov-Friedlander-Shvydkoy: first part. u ∈ L3 dt; B

1 3

∞,c(N)

• ⇒ E(t) = const

Examples of wild Euler solutions: Scheffer: compactly supported in

• time. Shnirelman: dissipating energy, in L∞(dt; L2).

DeLellis- Sz´ ekelyhidi, Cs: convex integration, h-principle (Nash, Gromov), Beltrami flows, diminishing Reynolds fluxes. s <

1 10.

Isett, C

1 5+ : Material (Lagrangian) derivative better behaved than time

derivative, nonlinear phases. Buckmaster, De Lellis, Sz´ ekelyhidi: L1(0, T; C

1 3 ). More careful

accounting. Isett C

1 3+ : gluing, Mikado flows.

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: ∇u2

L2 bounded in time uniformly in ν.

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 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: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g Enstrophy balance: d 2dt ˆ

R2 |ω(x, t)|2dx + ν

ˆ

R2 |∇ω(x, t)|2dx =

ˆ

R2 gωdx

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g Enstrophy balance: d 2dt ˆ

R2 |ω(x, t)|2dx + ν

ˆ

R2 |∇ω(x, t)|2dx =

ˆ

R2 gωdx

Anomalous dissipation of enstrophy?

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g Enstrophy balance: d 2dt ˆ

R2 |ω(x, t)|2dx + ν

ˆ

R2 |∇ω(x, t)|2dx =

ˆ

R2 gωdx

Anomalous dissipation of enstrophy? lim

ν→0 ν∇ω2 L2 = χ > 0

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g Enstrophy balance: d 2dt ˆ

R2 |ω(x, t)|2dx + ν

ˆ

R2 |∇ω(x, t)|2dx =

ˆ

R2 gωdx

Anomalous dissipation of enstrophy? lim

ν→0 ν∇ω2 L2 = χ > 0

Kraichnan (’68): yes,

Long time, no boundaries

No results for 3d NS/Euler. 2DNS No anomalous dissipation of energy: ∇u2

L2 bounded in time uniformly in ν.

Vorticity: ω = ∇⊥ · u ∂tω + u · ∇ω − ν∆ω = ∇⊥ · f = g Enstrophy balance: d 2dt ˆ

R2 |ω(x, t)|2dx + ν

ˆ

R2 |∇ω(x, t)|2dx =

ˆ

R2 gωdx

Anomalous dissipation of enstrophy? lim

ν→0 ν∇ω2 L2 = χ > 0

Kraichnan (’68): yes, Bernard (’00): add damping, and then no.

Absence of anomalous dissipation

C-Ramos ’07: If you add damping ut + γu + νAu + B(u, u) = f then there is no anomalous dissipation of enstrophy: lim

ν→0 lim sup T→∞

ν T ˆ T ˆ

• ∇ω(ν)(x, t)
• 2

dxdt = 0

Absence of anomalous dissipation

C-Ramos ’07: If you add damping ut + γu + νAu + B(u, u) = f then there is no anomalous dissipation of enstrophy: lim

ν→0 lim sup T→∞

ν T ˆ T ˆ

• ∇ω(ν)(x, t)
• 2

dxdt = 0 C-Tarfulea-Vicol ’13. No anomalous dissipation of energy in critical SQG.

Absence of anomalous dissipation

C-Ramos ’07: If you add damping ut + γu + νAu + B(u, u) = f then there is no anomalous dissipation of enstrophy: lim

ν→0 lim sup T→∞

ν T ˆ T ˆ

• ∇ω(ν)(x, t)
• 2

dxdt = 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 = λf.

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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0.

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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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(uE, uE) = 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 = λf. Exact solutions u(t) = y(t)f. Navier-Stokes: y(t) = y0e−νλt + 1 νλ

• 1 − e−νλt

Euler: y(t) = y0 + t Any finite t: SNS(t)u0 → SE(t)u0. Solution bounded in ν, locally in

• time. But: t → ∞: u(t) → uf =

1 νλf

νAuf = f, B(uf, uf) = 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(uE, uE) = 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. uNS → uE inside the domain.

Finite time, with boundaries: known results

Inviscid limit: same initial data, fixed time, smooth domain. uNS → uE inside the domain. Weakly.

Finite time, with boundaries: known results

Inviscid limit: same initial data, fixed time, smooth domain. uNS → uE inside the domain. Weakly. Or better.

Finite time, with boundaries: known results

Inviscid limit: same initial data, fixed time, smooth domain. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2.

Finite time, with boundaries: known results

Inviscid limit: same initial data, fixed time, smooth domain. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2. Energy dissipation rate ν ´ T ´

Ω |∇u|2dxdt vanishes

Finite time, with boundaries: known results

Inviscid limit: same initial data, fixed time, smooth domain. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2. Energy dissipation rate ν ´ T ´

Ω |∇u|2dxdt vanishes

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. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2. Energy dissipation rate ν ´ T ´

Ω |∇u|2dxdt vanishes

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. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2. Energy dissipation rate ν ´ T ´

Ω |∇u|2dxdt vanishes

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. uNS → uE inside the domain. Weakly. Or better. Kato 84: If (and oly if) dissipation ν ˆ

BL(ν)

|∇u|2dx → 0 then the finite time inviscid limit holds in strong L2. Energy dissipation rate ν ´ T ´

Ω |∇u|2dxdt vanishes

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} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP solves Prandtl equations. Note that Kato’s BL is much smaller.

Prandtl

Prandtl: u(x) = v(x)1{dist(x,∂Ω)>√ν} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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).

Prandtl

Prandtl: u(x) = v(x)1{dist(x,∂Ω)>√ν} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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). Prandtl are well posed if

◮ If there is no turning point of the Euler flow on the boundary, and

if the initial vorticity is bounded below by a positive constant. Oleinik ’66, Masmoudi-Wong in Sobolev space, ’12.

Prandtl

Prandtl: u(x) = v(x)1{dist(x,∂Ω)>√ν} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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). Prandtl are well posed if

◮ If there is no turning point of the Euler flow on the boundary, and

if the initial vorticity is bounded below by a positive constant. Oleinik ’66, Masmoudi-Wong in Sobolev space, ’12.

◮ If the initial velocity is real analytic in all variables.

Sammartino-Caflisch ’98.

Prandtl

Prandtl: u(x) = v(x)1{dist(x,∂Ω)>√ν} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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). Prandtl are well posed if

◮ If there is no turning point of the Euler flow on the boundary, and

if the initial vorticity is bounded below by a positive constant. Oleinik ’66, Masmoudi-Wong in Sobolev space, ’12.

◮ If the initial velocity is real analytic in all variables.

Sammartino-Caflisch ’98.

◮ If the initial velocity is real analytic with respect to tangential

variables, smooth transversally. Cannone, Lombardo, Sammartino ’01, Kukavica, Vicol ’13.

Prandtl

Prandtl: u(x) = v(x)1{dist(x,∂Ω)>√ν} + uP1{dist(x,∂Ω)<√ν} + O(√ν). and uP 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). Prandtl are well posed if

◮ If there is no turning point of the Euler flow on the boundary, and

if the initial vorticity is bounded below by a positive constant. Oleinik ’66, Masmoudi-Wong in Sobolev space, ’12.

◮ If the initial velocity is real analytic in all variables.

Sammartino-Caflisch ’98.

◮ If the initial velocity is real analytic with respect to tangential

variables, smooth transversally. Cannone, Lombardo, Sammartino ’01, Kukavica, Vicol ’13.

◮ If the initial vorticity has a single curve of nondegenerate critical

points and it is Gevrey 7

4 in the tangential directions, smooth

• transversally. G´

erard-Varet, Masmoudi ’13.

Set up, half space

The domain Ω = {(x1, x2) | x1 ∈ R, x2 > 0}. The Euler solution v = v(x1, x2, t) is assumed to be smooth, bounded and Hs, s > 2.

Set up, half space

The domain Ω = {(x1, x2) | x1 ∈ R, x2 > 0}. The Euler solution v = v(x1, x2, t) is assumed to be smooth, bounded and Hs, s > 2. The restriction of the tangential component

• f the Euler solution to the boundary is denoted U:

v1(x1, x2, t) = U(x1, t).

Set up, half space

The domain Ω = {(x1, x2) | x1 ∈ R, x2 > 0}. The Euler solution v = v(x1, x2, t) is assumed to be smooth, bounded and Hs, s > 2. The restriction of the tangential component

• f the Euler solution to the boundary is denoted U:

v1(x1, x2, t) = U(x1, t). The Navier-Stokes solution is u ∈ Hs. Vorticity of the NS solution is ω: ω(x1, x2, t) = ∂1u2(x1, x2, t) − ∂2u1(x1, x2, t)

Set up, half space

The domain Ω = {(x1, x2) | x1 ∈ R, x2 > 0}. The Euler solution v = v(x1, x2, t) is assumed to be smooth, bounded and Hs, s > 2. The restriction of the tangential component

• f the Euler solution to the boundary is denoted U:

v1(x1, x2, t) = U(x1, t). The Navier-Stokes solution is u ∈ Hs. Vorticity of the NS solution is ω: ω(x1, x2, t) = ∂1u2(x1, x2, t) − ∂2u1(x1, x2, t) The initial data for both u and v are the same: u(x1, x2, 0) = v(x1, x2, 0) = u0(x1, x2).

Set up, half space

The domain Ω = {(x1, x2) | x1 ∈ R, x2 > 0}. The Euler solution v = v(x1, x2, t) is assumed to be smooth, bounded and Hs, s > 2. The restriction of the tangential component

• f the Euler solution to the boundary is denoted U:

v1(x1, x2, t) = U(x1, t). The Navier-Stokes solution is u ∈ Hs. Vorticity of the NS solution is ω: ω(x1, x2, t) = ∂1u2(x1, x2, t) − ∂2u1(x1, x2, t) The initial data for both u and v are the same: u(x1, x2, 0) = v(x1, x2, 0) = u0(x1, x2). The boundary conditions are thus u| x2=0 = 0, v2| x2=0 = 0, v1| x2=0 = U(x1, t)

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0,

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0, and lim

ν→0

ˆ T Dτ,βdt = 0,

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0, and lim

ν→0

ˆ T Dτ,βdt = 0, with Dτ,β = ν ˆ

x2<β

|ωτ|2dx,

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0, and lim

ν→0

ˆ T Dτ,βdt = 0, with Dτ,β = ν ˆ

x2<β

|ωτ|2dx, ωτ = min{ω + 1 τ ; 0}

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0, and lim

ν→0

ˆ T Dτ,βdt = 0, with Dτ,β = ν ˆ

x2<β

|ωτ|2dx, ωτ = min{ω + 1 τ ; 0} and lim

ν→0 ν

ˆ T 1 τ dt = 0, lim

ν→0

β ν = ∞.

Result CKV (Kukavica, Vicol, C)

Theorem

(CKV ’14) Asume: U(x1, t) ≥ 0, and lim

ν→0

ˆ T Dτ,βdt = 0, with Dτ,β = ν ˆ

x2<β

|ωτ|2dx, ωτ = min{ω + 1 τ ; 0} and lim

ν→0 ν

ˆ T 1 τ dt = 0, lim

ν→0

β ν = ∞. Then lim

ν→0 sup t≤T

u(t) − v(t)L2 = 0.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries. ◮ Uses better corrector, with sign condition.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries. ◮ Uses better corrector, with sign condition. ◮ Condition on viscous flow is satisfied by viscous shear flow,

(eνt∂yy v(y), 0), v(0) = 0, v′(y) ≤ 0.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries. ◮ Uses better corrector, with sign condition. ◮ Condition on viscous flow is satisfied by viscous shear flow,

(eνt∂yy v(y), 0), v(0) = 0, v′(y) ≤ 0.

◮ Condition U ≥ 0 persists for short time from ID.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries. ◮ Uses better corrector, with sign condition. ◮ Condition on viscous flow is satisfied by viscous shear flow,

(eνt∂yy v(y), 0), v(0) = 0, v′(y) ≤ 0.

◮ Condition U ≥ 0 persists for short time from ID. ◮ In particular, U ≥ 0, ω ≥ − 1 ν1−0 in x2 < ν1−0 implies

u − vL2 → 0.

Remarks

◮ Motivation: Oleinik’s sign condition, to give a one-sided Kato

condition for inviscid limit when Prandtl is well-posed.

◮ Result works in bounded domains with smooth boundaries. ◮ Uses better corrector, with sign condition. ◮ Condition on viscous flow is satisfied by viscous shear flow,

(eνt∂yy v(y), 0), v(0) = 0, v′(y) ≤ 0.

◮ Condition U ≥ 0 persists for short time from ID. ◮ In particular, U ≥ 0, ω ≥ − 1 ν1−0 in x2 < ν1−0 implies

u − vL2 → 0.

◮ Bardos-Titi (’13) : if νωNS = ν∂2u1 → 0 on boundary.

Result CEIV (Elgindi, Ignatova, Vicol, C)

Theorem

(CEIV ’15) Assume u1u2 equicontinuous at x2 = 0: ∃γ(x1, t) ∈ L1(dx1dt), such that

Result CEIV (Elgindi, Ignatova, Vicol, C)

Theorem

(CEIV ’15) Assume u1u2 equicontinuous at x2 = 0: ∃γ(x1, t) ∈ L1(dx1dt), such that ∀ǫ > 0 ∃ρ(ǫ) > 0, |u1(x1, y, t)u2(x1, y, t)| ≤ ǫγ(x1, t), ∀|y| ≤ ρ(ǫ), x1, t.

Result CEIV (Elgindi, Ignatova, Vicol, C)

Theorem

(CEIV ’15) Assume u1u2 equicontinuous at x2 = 0: ∃γ(x1, t) ∈ L1(dx1dt), such that ∀ǫ > 0 ∃ρ(ǫ) > 0, |u1(x1, y, t)u2(x1, y, t)| ≤ ǫγ(x1, t), ∀|y| ≤ ρ(ǫ), x1, t. Then lim

ν→0 sup 0≤t≤T

u(t) − v(t)L2 = 0.

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

and ∂1u1 uniformly integrable near x2 = 0:

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

and ∂1u1 uniformly integrable near x2 = 0: ∀ǫ > 0, ∀L > 0,

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

and ∂1u1 uniformly integrable near x2 = 0: ∀ǫ > 0, ∀L > 0, ∃ρ(ǫ, L) > 0 such that

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

and ∂1u1 uniformly integrable near x2 = 0: ∀ǫ > 0, ∀L > 0, ∃ρ(ǫ, L) > 0 such that

• ˆ

x2≤ρ

ˆ

|x1|≤L

|∂1u1(x1, x2, t)| dx1dx2

• L2(dt)

≤ ǫ

Corollary CEIV

Corollary

Assume sup

0<ν≤ν0

ˆ T u2

L∞dt ≤ C

and ∂1u1 uniformly integrable near x2 = 0: ∀ǫ > 0, ∀L > 0, ∃ρ(ǫ, L) > 0 such that

• ˆ

x2≤ρ

ˆ

|x1|≤L

|∂1u1(x1, x2, t)| dx1dx2

• L2(dt)

≤ ǫ Then lim

ν→0 sup 0≤t≤T

u(t) − v(t)L2 = 0.

New 3D result

Theorem

Let un be a sequence of weak solutions of the Navier-Stokes equations ∂tun + un · ∇un − νn∆un + ∇pn = fn in Ω ⊂ R3 bounded, with ∇ · un = 0, fn bounded in L2(0, T; L2(Ω)), converging weakly to f, un(0) divergence-free and bounded in L2(Ω) and νn → 0. We assume that for any K ⊂⊂ Ω there exists a constant EK, a constant ǫ > 0 and a constant ζ2 > 0 such that sup

n

ˆ T ˆ

K

|un(x + y, t) − un(x, t)|2dxdt ≤ EK|y|ζ2 holds for |y| < dist(K, ∂Ω) in the inertial range |y| ≥ ǫ− 1

4 ν 3 4

n = η(n)

New 3D result

Theorem

Let un be a sequence of weak solutions of the Navier-Stokes equations ∂tun + un · ∇un − νn∆un + ∇pn = fn in Ω ⊂ R3 bounded, with ∇ · un = 0, fn bounded in L2(0, T; L2(Ω)), converging weakly to f, un(0) divergence-free and bounded in L2(Ω) and νn → 0. We assume that for any K ⊂⊂ Ω there exists a constant EK, a constant ǫ > 0 and a constant ζ2 > 0 such that sup

n

ˆ T ˆ

K

|un(x + y, t) − un(x, t)|2dxdt ≤ EK|y|ζ2 holds for |y| < dist(K, ∂Ω) in the inertial range |y| ≥ ǫ− 1

4 ν 3 4

n = η(n)

Assume that un(t) converge weakly in L2(Ω) to u∞(t) for almost all t ∈ (0, T). Then u∞ is a weak solution of the Euler equations.

Remarks

• 1. The result can be proved for suitable weak solutions in exterior

domains as well.

Remarks

• 1. The result can be proved for suitable weak solutions in exterior

domains as well.

• 2. Obviously, the scaling assumption does not imply regularity,

because it is L2 and also limited to y bounded away from zero. Also, the exact Kolmogorov form of η(n) is not needed. All that is used is that η(n) converges to zero as n → ∞.

Remarks

• 1. The result can be proved for suitable weak solutions in exterior

domains as well.

• 2. Obviously, the scaling assumption does not imply regularity,

because it is L2 and also limited to y bounded away from zero. Also, the exact Kolmogorov form of η(n) is not needed. All that is used is that η(n) converges to zero as n → ∞.

• 3. It is possible to remove the assumption of almost all time L2(Ω)

convergence, and replace it with the weak convergence in L2(0, T; L2(Ω)), at the price of demanding space-time second order structure function scaling. ˆ T ˆ

K

|un(x + y, t + s) − un(x, t)|2dxdt ≤ EK(|y|ζ2 + |s|β) for η(n) ≤ |y| < dist(K; ∂Ω), t + s ∈ [0, T], |s| ≥ τ(n), τ(n) → 0, and β > 0. If τ(n) = 0 the requirement is strong, it implies the sequence bounded in Cβ(0, T; L2(Ω)), and in particular the L2(Ω) convergence

• n each time slice.

New 2D Result

Theorem

Let Ω ⊂ R2 be a bounded open set with smooth boundary. Let un be a sequence of solutions of Navier-Stokes equations with viscosities νn → 0. We assume that the solutions are driven by forces fn ∈ H1(Ω) that are uniformly bounded in H1(Ω) and converge weakly in H1(Ω) to

• f. We assume that the initial data un(0) are divergence-free and are

in H1

0(Ω) and uniformly bounded in L2(Ω). Moreover, we assume that

for any K ⊂⊂ Ω, sup

0≤t≤T

ˆ

K

|ωn|2dx ≤ EK uniformly in n.

New 2D Result

Theorem

Let Ω ⊂ R2 be a bounded open set with smooth boundary. Let un be a sequence of solutions of Navier-Stokes equations with viscosities νn → 0. We assume that the solutions are driven by forces fn ∈ H1(Ω) that are uniformly bounded in H1(Ω) and converge weakly in H1(Ω) to

• f. We assume that the initial data un(0) are divergence-free and are

in H1

0(Ω) and uniformly bounded in L2(Ω). Moreover, we assume that

for any K ⊂⊂ Ω, sup

0≤t≤T

ˆ

K

|ωn|2dx ≤ EK uniformly in n. Then any weak limit in L2(0, T; L2(Ω)) of the sequence un, u∞, is a weak solution of the Euler equations ∂tω∞ + u∞ · ∇ω∞ = g = ∇⊥ · f with ω∞ = ∇⊥ · u∞.

2D result, continued

The solution has bounded energy, u∞ ∈ L∞(0, T; L2(Ω)). and for any compact K ⊂⊂ Ω, sup

t∈[0,T]

ˆ

K

|ω∞(x, t)|2dx ≤ EK holds.

2D result, continued

The solution has bounded energy, u∞ ∈ L∞(0, T; L2(Ω)). and for any compact K ⊂⊂ Ω, sup

t∈[0,T]

ˆ

K

|ω∞(x, t)|2dx ≤ EK holds.

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz.

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz. From CET (uv)r(x) − ur(x)vr(x) = ρr(u, v)(x) is good.

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz. From CET (uv)r(x) − ur(x)vr(x) = ρr(u, v)(x) is good. NΦ(n) = ˆ T ˆ

Ω

(un ⊗ un) : ∇Φ dxdt

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz. From CET (uv)r(x) − ur(x)vr(x) = ρr(u, v)(x) is good. NΦ(n) = ˆ T ˆ

Ω

(un ⊗ un) : ∇Φ dxdt

• NΦ(n) −

ˆ T ˆ

Ω

(un)r ⊗ (un)r : ∇Φ dxdt

• ≤ CΦ(EKr ζ2 + Er)

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz. From CET (uv)r(x) − ur(x)vr(x) = ρr(u, v)(x) is good. NΦ(n) = ˆ T ˆ

Ω

(un ⊗ un) : ∇Φ dxdt

• NΦ(n) −

ˆ T ˆ

Ω

(un)r ⊗ (un)r : ∇Φ dxdt

• ≤ CΦ(EKr ζ2 + Er)

(un)r converges pointwise.

Idea of Proof for 3D result

Smooth function j(z) supported in the annulus 1 < |z| < 2, with ´

R3 j(z)dz = 1, j(−z) = j(z). We fix a compact K ⊂⊂ Ω and denote,

for a function u, for x ∈ K and 2r < dist(K, ∂Ω), ur(x) = ˆ

1≤|z|≤2

u(x − rz)j(z)dz. From CET (uv)r(x) − ur(x)vr(x) = ρr(u, v)(x) is good. NΦ(n) = ˆ T ˆ

Ω

(un ⊗ un) : ∇Φ dxdt

• NΦ(n) −

ˆ T ˆ

Ω

(un)r ⊗ (un)r : ∇Φ dxdt

• ≤ CΦ(EKr ζ2 + Er)

(un)r converges pointwise. Lebegue dominated, and continuity of translation: lim

n→∞ NΦ(n) =

ˆ T ˆ

Ω

(u∞) ⊗ (u∞) : ∇Φ dxdt

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n.

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n. Localize vorticity equation:

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n. Localize vorticity equation: ∂t(χωn) ∈ L∞(0, T; H−2(Ω)).

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n. Localize vorticity equation: ∂t(χωn) ∈ L∞(0, T; H−2(Ω)). Aubin-Lions: χωn converge strongly in L2(0, T; H−1(Ω)).

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n. Localize vorticity equation: ∂t(χωn) ∈ L∞(0, T; H−2(Ω)). Aubin-Lions: χωn converge strongly in L2(0, T; H−1(Ω)). NΦ(n) = ˆ T ˆ

Ω

(un · ∇Φ)ωndxdt

Idea of proof or 2D result

Localize with cutoff χ: ∇ · (χun) ∈ L2(Ω), ∇⊥ · (χun) ∈ L2(Ω) ⇒ χun ∈ H1

0(Ω)

bounded uniformly in time and n. Localize vorticity equation: ∂t(χωn) ∈ L∞(0, T; H−2(Ω)). Aubin-Lions: χωn converge strongly in L2(0, T; H−1(Ω)). NΦ(n) = ˆ T ˆ

Ω

(un · ∇Φ)ωndxdt NΦ(n) = ˆ T ˆ

Ω

ΛD(χun · ∇Φ)Λ−1

D (χωn)dxdt.

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt,

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt, and η ∈ C∞

0 (R+), supported in [1, 2] and with integral equal to 1 √ 2π.

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt, and η ∈ C∞

0 (R+), supported in [1, 2] and with integral equal to 1 √ 2π.

φ2 = − ˆ x2 ∂1φ1(x1, z)dz

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt, and η ∈ C∞

0 (R+), supported in [1, 2] and with integral equal to 1 √ 2π.

φ2 = − ˆ x2 ∂1φ1(x1, z)dz Properties:

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt, and η ∈ C∞

0 (R+), supported in [1, 2] and with integral equal to 1 √ 2π.

φ2 = − ˆ x2 ∂1φ1(x1, z)dz Properties: ∇ · φ = 0, (φ + v)| x2=0 = 0, (∂t − ν∆)φ = small and

Corrector CEIV

φ1 = −U(x1, t)

• 2

ˆ ∞

x2

gσ(t)(z)dz − 2δ(t)η(x2)

• ,

with gσ(z) = 1 √ 2πσ e− z2

2σ ,

σ(t) = δ(t)2 = 2νt, and η ∈ C∞

0 (R+), supported in [1, 2] and with integral equal to 1 √ 2π.

φ2 = − ˆ x2 ∂1φ1(x1, z)dz Properties: ∇ · φ = 0, (φ + v)| x2=0 = 0, (∂t − ν∆)φ = small and φ1Lp + ∂1φ1Lp ≤ Cδ

1 p

φ2Lp + ∂1φ2Lp ≤ Cδ ∂2φ1Lp ≤ Cδ

1 p −1

Ideea of proof of CEIV result

w = u − v − φ

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0.

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with T1 =

• ˆ

v(∂t − ν∆)φdx

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with T1 =

• ˆ

v(∂t − ν∆)φdx

• T2 =
• ˆ

(u · ∇φ)udx

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with T1 =

• ˆ

v(∂t − ν∆)φdx

• T2 =
• ˆ

(u · ∇φ)udx

• and

T3 =

• ˆ

[(u · ∇v)φ + (φ · ∇v)w]dx

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with T1 =

• ˆ

v(∂t − ν∆)φdx

• T2 =
• ˆ

(u · ∇φ)udx

• and

T3 =

• ˆ

[(u · ∇v)φ + (φ · ∇v)w]dx

• T3 is ok.

Ideea of proof of CEIV result

w = u − v − φ Note w| x2=0 = 0. d dt w2

L2 + 2ν∇w2 L2 ≤ C∇vL∞w2 L2 + T1 + T2 + T3

with T1 =

• ˆ

v(∂t − ν∆)φdx

• T2 =
• ˆ

(u · ∇φ)udx

• and

T3 =

• ˆ

[(u · ∇v)φ + (φ · ∇v)w]dx

• T3 is ok.

Recall: v smooth, φ small, u bounded in L2.

Bound of T1

The corrector was designed so that (∂t − ν∆)φL2 ≤ C[δ

1 2 + ˙

δ]

Bound of T1

The corrector was designed so that (∂t − ν∆)φL2 ≤ C[δ

1 2 + ˙

δ] which is due to the fact that (∂t − ν(∂x2)2)gσ = 0, and bounds on U, Ut.

Bound of T1

The corrector was designed so that (∂t − ν∆)φL2 ≤ C[δ

1 2 + ˙

δ] which is due to the fact that (∂t − ν(∂x2)2)gσ = 0, and bounds on U, Ut. Recall: δ = √ 2νt.

Bound of T1

The corrector was designed so that (∂t − ν∆)φL2 ≤ C[δ

1 2 + ˙

δ] which is due to the fact that (∂t − ν(∂x2)2)gσ = 0, and bounds on U, Ut. Recall: δ = √ 2νt. and thus ˆ t T1dt = O(νt)

1 4

for 0 ≤ t ≤ T.

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ.

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

The integral for x2 ≥ ρ(ǫ) is bounded using the Gaussian gσ(ρ) and L2 bounds on u. Its time integarl rapidly vanishes.

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

The integral for x2 ≥ ρ(ǫ) is bounded using the Gaussian gσ(ρ) and L2 bounds on u. Its time integarl rapidly vanishes. The integral for x2 ≤ ρ(ǫ) is essentially (ignoring the small term due to η) and rescaling in x2

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

The integral for x2 ≥ ρ(ǫ) is bounded using the Gaussian gσ(ρ) and L2 bounds on u. Its time integarl rapidly vanishes. The integral for x2 ≤ ρ(ǫ) is essentially (ignoring the small term due to η) and rescaling in x2

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

The integral for x2 ≥ ρ(ǫ) is bounded using the Gaussian gσ(ρ) and L2 bounds on u. Its time integarl rapidly vanishes. The integral for x2 ≤ ρ(ǫ) is essentially (ignoring the small term due to η) and rescaling in x2

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• which is bounded uniformly by

Behavior of T2

• (j,i)=(2,1)

ˆ uj∂jφiuidx

• ≤ Cu2

L∞δ(t)

because of properties of φ. Main contribution is from

• ˆ

u2∂2φ1u1dx

• .

The integral for x2 ≥ ρ(ǫ) is bounded using the Gaussian gσ(ρ) and L2 bounds on u. Its time integarl rapidly vanishes. The integral for x2 ≤ ρ(ǫ) is essentially (ignoring the small term due to η) and rescaling in x2

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• which is bounded uniformly by

2ǫ ˆ

ρ(ǫ) δ

ˆ

R

g1(x2)|U(x1, t)|γ(x1, t)dx1dx2

Ideea of proof of Corollary CEIV

We have the same problem term:

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand 2 ˆ

|x1|≤L,x2≤L

dx1dx2U(x1)g1(x2)u1(x1, δx2, t) ˆ δx2 (−∂1u1(x1, z, t))dz

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand 2 ˆ

|x1|≤L,x2≤L

dx1dx2U(x1)g1(x2)u1(x1, δx2, t) ˆ δx2 (−∂1u1(x1, z, t))dz is bounded by

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand 2 ˆ

|x1|≤L,x2≤L

dx1dx2U(x1)g1(x2)u1(x1, δx2, t) ˆ δx2 (−∂1u1(x1, z, t))dz is bounded by 2UL∞u1(t)L∞ ˆ

|x1|≤L

ˆ δL |∂1u1(x1, z, t)|dx1dz,

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand 2 ˆ

|x1|≤L,x2≤L

dx1dx2U(x1)g1(x2)u1(x1, δx2, t) ˆ δx2 (−∂1u1(x1, z, t))dz is bounded by 2UL∞u1(t)L∞ ˆ

|x1|≤L

ˆ δL |∂1u1(x1, z, t)|dx1dz, and so, if δL ≤ ρ(ǫ, L)

Ideea of proof of Corollary CEIV

We have the same problem term: B(t) =

• 2

ˆ U(x1, t)u1(x1, δx2, t)u2(x1, δx2, t)g1(x2)dx1dx2

• For given ǫ there exists L > 0 such that

u(t)2

L∞

ˆ

|x1|≥Lor x2≥L

|U(x1, t)|g1(x2)dx1dx2 ≤ ǫu(t)2

L∞.

On the other hand 2 ˆ

|x1|≤L,x2≤L

dx1dx2U(x1)g1(x2)u1(x1, δx2, t) ˆ δx2 (−∂1u1(x1, z, t))dz is bounded by 2UL∞u1(t)L∞ ˆ

|x1|≤L

ˆ δL |∂1u1(x1, z, t)|dx1dz, and so, if δL ≤ ρ(ǫ, L) ˆ T B(t)dt ≤ 2ǫ  UL∞ ˆ T u1(t)2

L∞dt +

ˆ T u2

L∞dt

 

Discussion

◮ All previous results require some uniform assumptions of

gradients of Navier-Stokes solutions at the boundary.

Discussion

◮ All previous results require some uniform assumptions of

gradients of Navier-Stokes solutions at the boundary.

◮ All but the new results require regular Euler solutions and derive

strong L2 convergence

Discussion

◮ All previous results require some uniform assumptions of

gradients of Navier-Stokes solutions at the boundary.

◮ All but the new results require regular Euler solutions and derive

strong L2 convergence

◮ The strong stability and the regularity of the Euler solution imply

the vanishing of the energy dissipation rate in the zero viscosity limit.

Discussion

◮ All previous results require some uniform assumptions of

gradients of Navier-Stokes solutions at the boundary.

◮ All but the new results require regular Euler solutions and derive

strong L2 convergence

◮ The strong stability and the regularity of the Euler solution imply

the vanishing of the energy dissipation rate in the zero viscosity limit.

◮ The vanishing of the dissipation rate follows from weak

convergence in L2(Ω) for all times only if the Euler equation is

• conservative. We proved results of emergence of weak, possibly

dissipative solutions of Euler equations in 3D if the ensemble of Navier-Stokes solutions obeys a local-in-space but uniform in the ensemble second order structure function scaling from above. In two dimensions, we proved the emergence of weak solutions form arbitrary families of strong solutions of Navier-Stokes equations with uniform interior (local) enstrophy bounds. There might be dissipative solutions among them, although an example is not available at this time.