Critical regularity for conservation of energy: the 2D case Milton - - PowerPoint PPT Presentation

Critical regularity for conservation of energy: the 2D case

Milton C. Lopes Filho Instituto de Matemática, Universidade Federal do Rio de Janeiro MathFlows, Porquerolles, September 2015

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 1 / 25

Collaborators: Alexey Cheskidov (Univ. Illinois, Chicago) Helena J. Nussenzveig Lopes (IM-UFRJ) Roman Shvydkoy (Univ. Illinois, Chicago)

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 2 / 25

Plan of the Talk

1

Kolmogorov 41

2

Onsager conjecture

3

Convex integration counter-examples

4

Regularity threshold

5

Energy flux

6

Littlewood-Paley counterexample

7

Vanishing viscosity solutions

8

Conclusions

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 3 / 25

Kolmogorov 41

Statistical theory of stationary, homogeneous turbulence. Turbulence is described by a family of random variables {uν = uν(x)}ν>0, parametrized by the adimensional viscosity ν > 0, subject to the following hypothesis: H1 Displacements uν(x + v) − uν(x) are statistically independent of x and of the direction of v; H2 The flow description is self-similar, i.e. there exists an exponent δ > 0 such that, for any v in space and λ > 0, uν(x + λv) − uν(x) = λδ(uν(x + v) − uν(x)). H3 There is a positive dissipation rate ǫ0, independent of ν, i.e., in a control volume Q, lim

ν→0

ν |Q|

• Q

|∇uν|2dx = ǫ0.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 4 / 25

Kolmogorov 41

With these three hypothesis, one can derive all traditional features

• f homogeneous turbulence theory, including the fact that δ = 1/3,

and the energy cascade, with the well-known Kolmogorov’s −4/5 law.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25

Kolmogorov 41

With these three hypothesis, one can derive all traditional features

• f homogeneous turbulence theory, including the fact that δ = 1/3,

and the energy cascade, with the well-known Kolmogorov’s −4/5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, uν(x) be some sort

• f statistical solution of the Navier-Stokes system, for the sake of

consistency with first physical principles.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25

Kolmogorov 41

With these three hypothesis, one can derive all traditional features

• f homogeneous turbulence theory, including the fact that δ = 1/3,

and the energy cascade, with the well-known Kolmogorov’s −4/5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, uν(x) be some sort

• f statistical solution of the Navier-Stokes system, for the sake of

consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25

Kolmogorov 41

With these three hypothesis, one can derive all traditional features

• f homogeneous turbulence theory, including the fact that δ = 1/3,

and the energy cascade, with the well-known Kolmogorov’s −4/5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, uν(x) be some sort

• f statistical solution of the Navier-Stokes system, for the sake of

consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence. The existence of a nonvanishing dissipation rate is considered fundamental, in general, to turbulence theories, as well as being well-established experimentally.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25

Kolmogorov 41

With these three hypothesis, one can derive all traditional features

• f homogeneous turbulence theory, including the fact that δ = 1/3,

and the energy cascade, with the well-known Kolmogorov’s −4/5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, uν(x) be some sort

• f statistical solution of the Navier-Stokes system, for the sake of

consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence. The existence of a nonvanishing dissipation rate is considered fundamental, in general, to turbulence theories, as well as being well-established experimentally. See the 1995 book by U. Frisch for details.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25

Onsager Conjecture

Assuming that the realizations of the velocities uν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ0, by some mysterious means.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25

Onsager Conjecture

Assuming that the realizations of the velocities uν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ0, by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1/3 of a spatial derivative would be required to support inviscid dissipation.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25

Onsager Conjecture

Assuming that the realizations of the velocities uν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ0, by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1/3 of a spatial derivative would be required to support inviscid dissipation. His prediction became known as the "Onsager Conjecture".

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25

Onsager Conjecture

Assuming that the realizations of the velocities uν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ0, by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1/3 of a spatial derivative would be required to support inviscid dissipation. His prediction became known as the "Onsager Conjecture". Recent mathematical work on the Onsager conjecture has followed two fronts: proofs that Euler solutions with 1/3 of a derivative are conservative, and examples, mostly constructed by variations on the convex integration scheme, of Euler solutions with less than 1/3 of a derivative, exhibiting inviscid dissipation.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25

Convex integration

Examples of weak solutions of the Euler equations which exhibit inviscid dissipation.

1

Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not inviscid dissipation, but nonuniqueness and time-varying energy.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25

Convex integration

Examples of weak solutions of the Euler equations which exhibit inviscid dissipation.

1

Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not inviscid dissipation, but nonuniqueness and time-varying energy.

2

Shnirelman 2000, L∞

t L2 x, De Lellis, Szekelyhidi 2013, C1/10−ǫ x,t

, Isset 2014 C1/5−ǫ

x,t

, Buckmaster, De Lellis, Szekelyhidi 2014, L1

t C1/3−ǫ x

. These do not adapt to 2D.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25

Convex integration

Examples of weak solutions of the Euler equations which exhibit inviscid dissipation.

1

Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not inviscid dissipation, but nonuniqueness and time-varying energy.

2

Shnirelman 2000, L∞

t L2 x, De Lellis, Szekelyhidi 2013, C1/10−ǫ x,t

, Isset 2014 C1/5−ǫ

x,t

, Buckmaster, De Lellis, Szekelyhidi 2014, L1

t C1/3−ǫ x

. These do not adapt to 2D.

3

Szekelyhidi, Wiedemann 2012, L2 dense subset of initial data with nonincreasing energy. Choffrut, De Lellis, Szekelyhidi 2012, solution with decreasing energy in C0

x,t. Constructions work in 2D.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25

Convex integration

Examples of weak solutions of the Euler equations which exhibit inviscid dissipation.

1

Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not inviscid dissipation, but nonuniqueness and time-varying energy.

2

Shnirelman 2000, L∞

t L2 x, De Lellis, Szekelyhidi 2013, C1/10−ǫ x,t

, Isset 2014 C1/5−ǫ

x,t

, Buckmaster, De Lellis, Szekelyhidi 2014, L1

t C1/3−ǫ x

. These do not adapt to 2D.

3

Szekelyhidi, Wiedemann 2012, L2 dense subset of initial data with nonincreasing energy. Choffrut, De Lellis, Szekelyhidi 2012, solution with decreasing energy in C0

x,t. Constructions work in 2D.

In addition, Szekelyhidi 2011 observed that there are infinitely many dissipative weak solutions with a flat vortex sheet as initial data, but that the only solution obtained as vanishing viscosity limit with that initial data is the flat sheet itself. Vanishing viscosity limit as a selection

• criterion. See also Bardos, Titi and Wiedemann 2012.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25

Regularity threshold

The other side of Onsager conjecture. Basic result: if a weak solution has the proposed regularity, then it is conservative.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 8 / 25

Regularity threshold

The other side of Onsager conjecture. Basic result: if a weak solution has the proposed regularity, then it is conservative. Any space dimensions, Frisch-Sulem 1975, H5/6+ǫ, Eyink 94, C1/3+ǫ, Constantin, E, Titi 1994, B1/3+ǫ

3.∞

. State of the art, Cheskidov, Constantin, Friedlander, Shvydkoy 2008.B1/3

3,c0, any spatial dimension.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 8 / 25

Regularity threshold

The other side of Onsager conjecture. Basic result: if a weak solution has the proposed regularity, then it is conservative. Any space dimensions, Frisch-Sulem 1975, H5/6+ǫ, Eyink 94, C1/3+ǫ, Constantin, E, Titi 1994, B1/3+ǫ

3.∞

. State of the art, Cheskidov, Constantin, Friedlander, Shvydkoy 2008.B1/3

3,c0, any spatial dimension.

Specifically for 2D, initial vorticity in W 1,p, for p > 3/2 implies conservation of energy, Duchon and Robert 2000. Extension to p = 3/2 follows from the work of Cheskidov, Constantin, Friedlander and Shvydkoy.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 8 / 25

Regularity threshold

The other side of Onsager conjecture. Basic result: if a weak solution has the proposed regularity, then it is conservative. Any space dimensions, Frisch-Sulem 1975, H5/6+ǫ, Eyink 94, C1/3+ǫ, Constantin, E, Titi 1994, B1/3+ǫ

3.∞

. State of the art, Cheskidov, Constantin, Friedlander, Shvydkoy 2008.B1/3

3,c0, any spatial dimension.

Specifically for 2D, initial vorticity in W 1,p, for p > 3/2 implies conservation of energy, Duchon and Robert 2000. Extension to p = 3/2 follows from the work of Cheskidov, Constantin, Friedlander and Shvydkoy. Involves studying optimal conditions for the energy flux to vanish. This is already the basis for Onsager’s original heuristics.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 8 / 25

Regularity threshold

The other side of Onsager conjecture. Basic result: if a weak solution has the proposed regularity, then it is conservative. Any space dimensions, Frisch-Sulem 1975, H5/6+ǫ, Eyink 94, C1/3+ǫ, Constantin, E, Titi 1994, B1/3+ǫ

3.∞

. State of the art, Cheskidov, Constantin, Friedlander, Shvydkoy 2008.B1/3

3,c0, any spatial dimension.

Specifically for 2D, initial vorticity in W 1,p, for p > 3/2 implies conservation of energy, Duchon and Robert 2000. Extension to p = 3/2 follows from the work of Cheskidov, Constantin, Friedlander and Shvydkoy. Involves studying optimal conditions for the energy flux to vanish. This is already the basis for Onsager’s original heuristics. Note that Onsager conjecture in 3D is basically settled, with weak solutions with 1/3 of a derivative conserving energy and examples of dissipative weak solutions with derivatives arbitrarily close to 1/3. Provided that convex integration solutions are seen as physically reasonable.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 8 / 25

Energy flux

2D Euler equations on the torus T2 ≡ [0, 2π]2, with initial data u0 ∈ L2(T2),    ∂tu + (u · ∇)u = −∇p div u = 0 u(t = 0) = u0. (1) We are interested in weak solutions for which the vorticity, ω ≡ curl u, is p-th power integrable, for some p > 1.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 9 / 25

Definition

Fix T > 0 and u0 ∈ L2(T2) with initial vorticity in Lp(T2), for some p > 1. Let u ∈ Cweak(0, T; L2(T2)) with ω ∈ L∞(0, T; Lp(T2)). We say u is a weak solution of the incompressible Euler equations with initial velocity u0 if

1

for every test vector field Φ ∈ C∞([0, T) × T2) such that divΦ(t, ·) = 0 the following identity holds true: T

• T2 ∂tΦ · u + u · DΦu dxdt +
• T2 Φ(0, ·) · u0 dx = 0.

2

For almost every t ∈ (0, T), div u(t, ·) = 0, in the sense of distributions. Existence of such weak solutions is known (DiPerna, Majda 87), but uniqueness is open, except for the case p = ∞. We call a weak solution conservative if the L2-norm of velocity is constant in time.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 10 / 25

Theorem

Fix T > 0 and let u ∈ Cweak(0, T; L2(T2)) be a weak solution with ω ≡ curl u ∈ L∞(0, T; L3/2(T2)). Then u is conservative. Moreover, the following local energy balance law holds in the sense of distributions: ∂t |u|2 2

• + div
• u

|u|2 2 + p

• = 0.

(2) This result is contained in the main result in Cheskidov et alli 2008, but we outline an elementary proof.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 11 / 25

Idea of the proof of the Theorem

Let ζε(x) = ξε(x), for x ∈ [−1/2, 1/2) × [−1/2, 1/2), extended periodically, be a family of mollifiers in C∞(T2). Take the convolution of the Euler equations with ζε and write uε = ζε ∗ u, pε = ζε ∗ p. We find: ∂tuε + (uε · ∇)uε = −∇pε + Rε, (3) with Rε ≡ (uε · ∇)uε − ζε ∗ [(u · ∇)u]. We may multiply the equation by uε to obtain: ∂t |uε|2 2

• + div
• uε

|uε|2 2 + pε

• = uε · Rε.

(4)

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 12 / 25

We claim that, as ε → 0, we have: (A) ∂t

• |uε|2

2

• → ∂t
• |u|2

2

• in the sense of distributions;

(B) div

• uε

|uε|2 2

+ pε → div

• u
• |u|2

2 + p

• in the sense of

distributions; (C) uε · Rε → 0 strongly in L∞(0, T; L1(T2)). The convergences of (A) and (B) are subcritical for ω ∈ L3/2. In fact, they require ω ∈ L6/5. It is the convergence of the energy flux term, which is (C) that requires ω ∈ L3/2. Good behavior of the energy flux term is the key point in all results along these lines. So, conclusion, conservation of energy for weak solutions follows from enough regularity to guarantee good behavior, and vanishing of the energy flux.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 13 / 25

Let us argue the convergence of the flux term. We show that Rε → 0 strongly in L∞(0, T; L6/5(T2)). This is enough to prove what we want, since uε is bounded in L∞(0, T; L6(T2)). We have: RεL∞(L6/5) = (uε · ∇)uε − ζε ∗ [(u · ∇)u]L∞(L6/5) ≤ (uε · ∇)(uε − u)L∞(L6/5) + (uε − u) · ∇uL∞(L6/5)+ +(u · ∇)u − ζε ∗ [(u · ∇)u]L∞(L6/5) ≤ uεL∞(L6)∇uε − ∇uL∞(L3/2) + uε − uL∞(L6)∇uL∞(L3/2) +(u · ∇)u − ζε ∗ [(u · ∇)u]L∞(L6/5) → 0, because uε → u in L∞(L6(T2)), ∇uε = ζε ∗ ∇u → ∇u in L∞(L3/2(T2)) and u · ∇u ∈ L∞(L6/5(T2)).

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 14 / 25

Littlewood-Paley counterexample

Conservation of energy for weak solutions hinges upon a scaling argument that has very little to do with the dynamics of the Euler

• equations. Therefore, to show that the exponent 3/2 is optimal in the

argument above, we exhibit a vector field which just fails to be W 1,3/2 for which the energy flux does not vanish. We introduce the Littlewood-Paley truncation Sq by Sq[f] = f(0,0) +

• p≤q−1

∆pf =

• α∈Z2

χ(λ−1

q α)

f(α)e2πiα·x. The operator Sq is actually a convolution with a mollifier, and the argument for the vanishing of the energy flux with Sq in place of the convolution with a mollifier is an immediate adaptation of the argument presented for f ∈ W 1,3/2.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 15 / 25

By testing the Euler system with Sq[Sq[u]], it is easy to see that the proof of the global energy conservation reduces to showing that the energy flux Πq[u] =

• T2 Sq[u] · Sq[(u · ∇)u] dx

vanishes on average in time as q → ∞. This holds, in fact, pointwise in time for any divergence-free field with curl in L3/2.

Theorem

There exists a divergence free vector field u ∈ B1/3

3,∞ ∩ W 1,p(T2), for

any 1 ≤ p < 3/2, such that lim supq→∞ Πq[u] = 0.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 16 / 25

Idea of proof of Theorem

1

The construction begins by building a skeleton, made up with at most four Fourier modes in each dyadic shell, chosen in a way that Πqj[Sqju] = 4π, for a well-chosen sequence qj → ∞.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 17 / 25

Idea of proof of Theorem

1

The construction begins by building a skeleton, made up with at most four Fourier modes in each dyadic shell, chosen in a way that Πqj[Sqju] = 4π, for a well-chosen sequence qj → ∞.

2

Next we choose qj widely separated, so that the interaction between dyadic shells in computing the energy flux is small. We also scale the dyadic shells components consistently with a vector field in B1/3

p,∞, for p > 1 arbitrary. The end result is a vector field in

B1/3

p,∞, with at most four modes in each of a sequence of widely

separated dyadic shells, with lim sup Πqj[u] > 0.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 17 / 25

Idea of proof of Theorem

1

The construction begins by building a skeleton, made up with at most four Fourier modes in each dyadic shell, chosen in a way that Πqj[Sqju] = 4π, for a well-chosen sequence qj → ∞.

2

Next we choose qj widely separated, so that the interaction between dyadic shells in computing the energy flux is small. We also scale the dyadic shells components consistently with a vector field in B1/3

p,∞, for p > 1 arbitrary. The end result is a vector field in

B1/3

p,∞, with at most four modes in each of a sequence of widely

separated dyadic shells, with lim sup Πqj[u] > 0.

3

Finally, we substitute each isolated mode in the skeleton by a small lattice block around each Fourier mode, in such a way as to gain just enough regularity, without destroying the flux estimate. This gives a modified vector field in W 1,3/2

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 17 / 25

Idea of proof of Theorem

1

The construction begins by building a skeleton, made up with at most four Fourier modes in each dyadic shell, chosen in a way that Πqj[Sqju] = 4π, for a well-chosen sequence qj → ∞.

2

Next we choose qj widely separated, so that the interaction between dyadic shells in computing the energy flux is small. We also scale the dyadic shells components consistently with a vector field in B1/3

p,∞, for p > 1 arbitrary. The end result is a vector field in

B1/3

p,∞, with at most four modes in each of a sequence of widely

separated dyadic shells, with lim sup Πqj[u] > 0.

3

Finally, we substitute each isolated mode in the skeleton by a small lattice block around each Fourier mode, in such a way as to gain just enough regularity, without destroying the flux estimate. This gives a modified vector field in W 1,3/2 This construction is both complicated and rather technical, but it is essentially based on a similar construction in 3D, done previously by Cheskidov and Shvydkoy.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 17 / 25

Vanishing viscosity solutions

The main point of this work is that the issue of conservation of energy for weak solutions does not end with Onsager’s conjecture.

Definition

Let u ∈ C(0, T; L2(T2)). We say that u is a physically realizable weak solution of the incompressible 2D Euler equations with initial velocity u0 ∈ L2(T2) if the following conditions hold.

1

u is a weak solution of the Euler equations in the sense of Definition 1;

2

there exists a family of solutions of the incompressible 2D Navier-Stokes equations with viscosity ν > 0, {uν}, such that, as ν → 0,

1

uν ⇀ u weakly∗ in L∞(0, T; L2(T2));

2

uν(0, ·) ≡ uν

0 → u0 strongly in L2(T2).

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 18 / 25

Theorem

Let u ∈ C(0, T; L2(T2)) be a physically realizable weak solution of the incompressible 2D Euler equations. Suppose that u0 ∈ L2 is such that curl u0 ≡ ω0 ∈ Lp(T2), for some p > 1. Then u conserves energy.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 19 / 25

proof:

Assume that ω0 ∈ Lp(T2) for some p < 2, and that ω0 / ∈ L2(T2) as,

• therwise, the result is trivial. As u is physically realizable, ∃ family

{uν} of solutions of Navier-Stokes satisfying the corresponding

• conditions. Let ων = curl uν. The vorticity equation reads:

∂tων + uν · ∇ων = ν∆ων. Multiplying the vorticity equation by ων and integrating on the torus yields d dt ων2

L2 = −2ν∇ων2 L2.

Use the Gagliardo-Nirenberg inequality to obtain, for any 1 < p < 2, ωνL2 ≤ ∇ων

1− p

2

L2

ων

p 2

Lp.

Then, −2ν∇ων2

L2 ≤ −2νων

4 2−p

L2 ων − 2p

2−p

Lp

.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 20 / 25

If we multiply the vorticity equation by |ων|p−2ων and integrate on the torus, we obtain a maximum principle for the Lp norm of vorticity, namely: ων(t, ·)Lp ≤ ων

0Lp,

for any t ≥ 0. Therefore we obtain: d dt ων2

L2 ≤ −2νων

4 2−p

L2 ων − 2p

2−p

Lp

. Write y = y(t) = ων2

L2 and C0 = ων − 2p

2−p

Lp

. Then, integrating in time, starting from δ > 0, we obtain [y(t)]

−p 2−p − [y(δ)] −p 2−p ≥ 2νC0p

2 − p (t − δ). Taking the limit as δ → 0 and using that limδ→0 ων(δ, ·)2

L2 = +∞ we

find that ων(t, ·)2

L2 ≤

2νpC0t 2 − p − 2−p

p

.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 21 / 25

Next recall that solutions of the Navier-Stokes equations satisfy the energy identity in two space dimensions: d dt uν2

L2 = −2ν∇uν2 L2.

(5) Rewriting the right-hand-side above in terms of vorticity yields d dt uν2

L2 = −2νων2 L2.

(6) Hence, integrating the energy estimate in time and using the estimate for vorticity we deduce that 0 ≥ uν(t, ·)2

L2 − uν 02 L2

≥ −2ν t 2νpC0s 2 − p − 2−p

p

ds = −2ν 2νpC0 2 − p − 2−p

p

p 2(p − 1)t

2(p−1) p

,

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 22 / 25

Hence, 0 ≥ uν(t, ·)2

L2 − uν 02 L2 ≥ −(2ν)

2(p−1) p

pC0 2 − p − 2−p

p

p 2(p − 1)t

2(p−1) p

. Now, since p > 1 the right-hand-side of this inequality vanishes as ν → 0. Therefore, lim

ν→0 uν(t, ·)2 L2 − uν 02 L2 = 0.

Using strong convergence of initial data, together with the known fact that there are no energy concentrations for the vanishing viscosity limit with vorticity in Lp, p > 1, we complete the proof.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 23 / 25

Conclusions

Vorticity transport is a relevant physical restriction on incompressible flow behavior that is ignored by wild solutions.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 24 / 25

Conclusions

Vorticity transport is a relevant physical restriction on incompressible flow behavior that is ignored by wild solutions. The Onsager scaling is not the last word on inviscid dissipation

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 24 / 25

Conclusions

Vorticity transport is a relevant physical restriction on incompressible flow behavior that is ignored by wild solutions. The Onsager scaling is not the last word on inviscid dissipation No comment on 2D turbulence.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 24 / 25

Conclusions

Vorticity transport is a relevant physical restriction on incompressible flow behavior that is ignored by wild solutions. The Onsager scaling is not the last word on inviscid dissipation No comment on 2D turbulence. Analysis easily extended to bounded domains with ω = 0 at the boundary.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 24 / 25

Conclusions

Vorticity transport is a relevant physical restriction on incompressible flow behavior that is ignored by wild solutions. The Onsager scaling is not the last word on inviscid dissipation No comment on 2D turbulence. Analysis easily extended to bounded domains with ω = 0 at the boundary. Extend to p = 1? It would be routine to extend to spaces like L1,p, p > 2.

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 24 / 25

Thank you!

MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 25 / 25