Dissipation in Onsager's critical classes and energy conservation in $BV\cap L^\infty$ with and without boundary

. This paper is concerned with the incompressible Euler equations. In Onsager’s critical classes we provide explicit formulas for the Duchon–Robert measure in terms of the regularization kernel and a family of vector-valued measures { µ z } z ⊂ M x,t , having some H¨older regularity with respect to the direction z ∈ B 1 . Then, we prove energy conservation for L ∞ x,t ∩ L 1 t BV x solutions, in both the absence or presence of a physical boundary. This result generalises the previously known case of Vortex Sheets, showing that energy conservation follows from the structure of L ∞ ∩ BV in-compressible vector ﬁelds rather than the ﬂow having “organized singularities”. The interior energy conservation features the use of Ambrosio’s anisotropic optimization of the convolution kernel and it diﬀers from the usual energy conservation arguments by heavily relying on the incompressibility of the vector ﬁeld. In particular the same argument fails to apply to solutions to the Burgers equation, coherently with compressible shocks having non-trivial entropy production. To run the boundary analysis we introduce a notion “normal Lebesgue trace” for general vector ﬁelds, very reminiscent of the one for BV functions. We show that having such a null normal trace is basically equivalent to have vanishing boundary energy ﬂux. This goes beyond the previous approaches, laying down a setup which apply to every Lipschitz bounded domain. Allowing any Lipschitz boundary introduces several technicalities to the proof, with a quite geometrical/measure-theoretical ﬂavour.


Introduction
Let Ω ⊂ R d be any domain with Lipschitz boundary, if any.We consider the incompressible Euler equations ∂ t u + (u • ∇) u + ∇p = 0 div u = 0, (1.1) in Ω × (0, T ).When ∂Ω = ∅, the system (1.1) has to be coupled with the boundary condition u • n| ∂Ω = 0, the latter to be interpreted in a suitable trace sense depending on the a priori regularity of u.
1.1.A brief background.Classical computations show that regular solutions of (1.1) conserve the kinetic energy, that is One of the possibilities to interpret the Euler system is as a model for turbulent flows in the infinite Reynolds number regime.Thus, since the celebrated work of Kolmogorov [25], later on discussed by Onsager [28], it has been clear that, at first approximation, a good understanding of incompressible turbulence is subject to the study of Euler solutions violating energy conservation.After the first works by Scheffer [30] and Shnirelman [31], the construction of non-conservative (in some cases truly dissipative) solutions has been put into a rigorous mathematical framework by the convex integration methods, introduced in the context of the incompressible Euler equations by De Lellis and Székelyhidi [5,14,23].The latter, together with the rigidity part proved by Constantin, E and Titi [12], soon after Eyink [19], provided an almost complete proof of the Onsager prediction about C 1 /3 being the sharp threshold determining energy conservation for (1.1) in the class of Hölder continuous weak solutions.See also [27] for an intermittency-based convex integration construction in Besov spaces.In view of observable turbulence, considering Besov regularity, thus not restricting to measure spatial increments in L ∞ , sounds indeed more appropriate.We refer to Frisch [21] for an extensive discussion on the mathematical theory of fully developed turbulence.
We remark that the proof of the Onsager's theorem is "almost complete" since the Hölder critical case C 1 /3 , as well as any critical Besov, is still open.Here, the term "critical" refers to any function space which makes the energy flux bounded and not vanishing.A precise definition of energy flux can be given by means of the Duchon-Robert measure from [17], which will be introduced below.Proving that a solution in a critical class conserves or dissipates energy is in general quite hard.The convex integration constructions available at the moment fail to provide Onsager's critical solutions because of infinitesimal losses along the iterations, while the energy conservation proof fails because of having non-vanishing energy flux.However, at least from a physical point of view, considering solutions having bounded energy flux is of main importance in the understanding of fully developed turbulence, since incompressible flows in the infinite Reynolds number limit inherit such property directly from Navier-Stokes.A rigorous mathematical counterpart of that has been given in [17], proving that anomalous Euler dissipation arise, for instance, as the limit of D ν := ν|∇u ν | 2 for a L 3 x,t -compact sequence of sufficiently regular vanishing viscosity solutions to the incompressible Navier-Stokes equations.The sequence D ν stays bounded in L 1 x,t in the natural class of solutions introduced by Leray [26].On the opposite side, it has been also proved by the first author and Tione [15] that energy dissipation for solutions in the supercritical class is highly unstable with respect to small perturbations.More precisely, [15] proves that solutions of Euler having supercritical regularity generically exhibit infinite energy flux, as opposed to what physics predicts.The emergence of this phenomenon was previously conjectured by Isett and Oh in [24].This gives further reasons to investigate Euler flows with critical regularity, thus not simply inheriting the finiteness of the energy flux from being vanishing viscosity limits.
To the best of our knowledge, the only known Onsager's critical class for which a complete understanding is currently available is that of Vortex-Sheets solutions to (1.1), that is solutions (in 2 or 3 dimensions) whose vorticity is concentrated on a regular surface (or a curve when d = 2).Shvydkoy [32] proved energy conservation for Vortex-Sheets, provided the regular evolution in time of the sheet.Indeed their energy flux is a priori non-vanishing, but the fact that they solve (1.1) forces the normal component of the velocity to be continuous across the sheet, together with having a continuous Bernoulli's pressure |u| 2  2 +p.This is sufficient to prove that there is no energy dissipation in any space-time region.An additional folklore point of view is the following: in Onsager's critical classes, if singularities happen to be "organized", then there is no possibility for energy gaps.
In view of the aforementioned issues, this paper proves two main results.The first is to provide a formula for the energy flux of Onsager's critical vector fields, not necessarily solving the incompressible Euler equations, in which the convolution kernel used to define the flux appears explicitly.In our second result, by optimizing the choice of the kernel, we prove local interior energy conservation for L 1 t BV x ∩ L ∞ x,t solutions to (1.1).Moreover, we analyse the role of physical boundaries, introducing a notion a normal boundary trace and studying how it is linked to the conservation of the total kinetic energy.The precise statements are given here below.
1.2.Main results.Recall from [17] that, whenever (u, p) ∈ L 3 loc × L 3 /2 loc , there exists a space-time distribution D[u] ∈ D ′ x,t such that where and where δ ξ u(x, t) It makes sense to define D ε [u] for every vector field u, thus not necessarily restricting to solutions of Euler.In this setting, the local energy flux of u can be defined as It is then natural to define Onsager's critical class any space X in which D ε [u] stays bounded in L 1 loc if u ∈ X, without having the limit in (1.5) equal to zero.Note that having D ε [u] bounded in L 1 loc guarantees, possibly passing to subsequences, that D[u] belongs to the space of (signed) Radon measures, the latter denoted by M loc .In view of the mathematical theory of fully developed turbulence, the most natural critical class is 3,∞ (see Section 2 for a precise definition), but more general spaces can be considered.Indeed, requiring more than 1 /3 of a derivative in L 3 forces the limit (1.5) to be zero [12,17].See also [11] for sharper, still subcritical, results.
Our first result provides a formula for the limit measure D[u] when u has critical regularity, in which the dependence on the kernel ρ is explicit.For p ∈ [1, ∞] we will denote by p ′ ∈ [1, ∞] its conjugate exponent, by B 1 the unit ball centered at the origin and by µ M the total variation of a measure µ (see Section 2 for the precise definition). ) ) for all I ⊂⊂ (0, T ) and O ⊂⊂ Ω open sets.Let D[u] be any weak limit, in the sense of measures, of the sequence (1.9) In particular, if D[u] does not depend on the convolution kernel, we have that Moreover, µ enjoys the following properties.
(i) Hölder continuity: for any compact set K ⊂ Ω×(0, T ) there exists a constant C = C(u, K) > 0 such that , (1.12) for all z 1 , z 2 ∈ B 1 , where (ii) Vanishing at the origin: µ 0 ≡ 0 as an element of M loc (Ω × (0, T ); R d ); (iii) Odd: for any ρ ∈ K it holds We prove Theorem 1.1 in Section 3, where we also present some results on the support of the dissipation measure as well as the case of vector fields with "bounded deformation".It will be clear from the proof that the constant C in the above statement can be written explicitly in terms of the corresponding local norms of u.We remark that p = 3 is the critical value that determines the expression of α p and β p , that is Hölder regularity of the measures µ z in the direction z.Indeed The proof of Theorem 1.1 is based on an Ascoli-Azelá argument on the cubic expression appearing in the sequence D ε [u].Indeed, we show that having any critical Besov regularity guarantees equicontinuity with respect to z of the operator in (1.3), being trilinear with respect to the displacements δ εz u.This allow us to define the family of vector-valued measures µ z , having Hölder continuity with respect to z.We emphasize that we are not able to prove that the measure µ z is unique, not even if u solves Euler, since it is obtained by compactness and up to subsequences.Further formulas for specific choices of ρ are discussed in Section 3.
Having an explicit dependence on ρ makes possible to optimize its choice whenever D[u] does not depend on the kernel, that is, for instance, the case in which u solves (1.1).Then, it is natural to wonder whether the infimum in (1.10) can be shown to be zero in some Onsager's critical class.We prove that this is the case for incompressible vector fields u ∈ L Theorem 1.2.Let u : Ω × (0, T ) → R d be a divergence-free vector field such that for all I ⊂⊂ (0, T ) and O ⊂⊂ Ω open sets.Let D[u] be any weak limit, in the sense of measures, of the sequence In the statement above we used the notation |µ| to denote the variation of a signed Radon measure µ (see Section 2).Since from [17] we know that the sequence D ε [u] has a unique limit which does not depend on the kernel ρ when u solves the incompressible Euler system, we deduce local energy conservation for solutions to (1.1) in the class (1.14), The proof of Theorem 1.2 is based on Ambrosio's anisotropic optimization of the kernel that has been introduced in [2] to prove the renormalization property for weak solutions to the transport equations with BV vector fields, and it heavily relies on the fact that u is divergence-free.In particular, the same approach would fail, and it has to do so, in the compressible context.For instance, Burgers shocks enjoy the same critical regularity L ∞ ∩ BV , and they enjoy an analogous version of the formulas given in Theorem 1.1, while having non-trivial dissipation measure.Thus, the approach we use here is quite different than the usual ones, since they all simultaneously apply to both compressible and incompressible models, with the exception of [32].Indeed, following Ambrosio's argument, we prove that for a positive measure ν and a matrix M x,t such that Tr M x,t = 0 for ν-a.e.(x, t) ∈ Ω × (0, T ).The measure ν is basically given by the variation of the gradient of u, while the matrix M x,t is the polar part of the Radon-Nikodym decomposition of ∇u with respect to |∇u|, which then has to be trace-free by the constraint div u = 0.Then, Alberti's lemma (see for instance [13,Lemma 2.13]) concludes the proof by showing that Strictly speaking, here we use a variation of Alberti's Lemma 2.7.Indeed, instead of M z, we consider any odd (or even), Lipschitz continuous, divergence-free vector field.This shows that having the linear structure M z is not necessary to achieve zero in the infimum, but any incompressible vector field would suffice.We believe this generalization to be useful to investigate further Onsager's critical classes in which the increments δ z u, when measured in an integral sense, may not behave linearly in z as it happens in the BV case, see (2.4).
Having established interior local energy conservation, we also address the case in which a physical boundary ∂Ω is present.As previously noted in [4,16], if D[u] ≡ 0 in D ′ x,t , then by (1.2) kinetic energy conservation, for instance in the sense of Definition 5.1, follows as soon as the quantity vanishes when we let ϕ ∈ C ∞ c (Ω) converge to the indicator function of Ω.Assuming that P := As emphasised in [4,16], an assumption on the normal component of u when approaching ∂Ω is enough to have (1.15).For instance, the continuity of the normal velocity in a neighbourhood of the boundary has been shown to be sufficient, consistently with the possible formation of a Prandtl-type boundary layer in the vanishing viscosity limit.To this end, we give a notion of "normal Lebesgue trace" on ∂Ω (see Definition 5.2), which we will denote by u ∂Ω n .It turns out, see Proposition 5.3, that having such a notion of normal boundary trace vanishing on a portion of ∂Ω of H d−1 -full measure is basically equivalent to (1.15).Differently from the previous works [4,16] where a C 2 assumption on ∂Ω was required, our analysis applies to any domain with Lipschitz boundary.In particular, when ∂Ω is piece-wise C 2 , our approach shows that no assumption on the behaviour of the normal component of u around boundary corner points is required, since they would form a H d−1 -negligible subset of ∂Ω.(ii) for all I ⊂⊂ (0, T ) there exists ε 0 > 0 such that p ∈ L 1 (I; L ∞ ((∂Ω) ε 0 ∩ Ω)); (iii) for almost every t ∈ (0, T ), u(•, t) has zero Lebesgue normal boundary trace according to Definition 5.2, i.e. u(•, t) ∂Ω n ≡ 0. Then u conserves the kinetic energy in the sense of Definition 5.1.Some more effective conditions which imply u ∂Ω n ≡ 0 are given in Proposition 5.5.We emphasise that, differently from [4,16], our theorem provides kinetic energy conservation in an Onsager's critical class.Working on general Lipschitz domains introduces several technicalities in the proof, such as the study of the Minkowski content of closed rectifiable sets, the regularity of the distance function from a Lipschitz domain and covering arguments.We collect them in Section 2. To conclude, in Theorem 5.6 we also state a result in which the Bernoulli's pressure P = |u| 2 2 + p is not necessarily bounded in a neighbourhood of the boundary.Thus P is allowed to blow-up approaching ∂Ω, provided that the normal Lebesgue boundary trace is achieved in a quantitative (in terms on how P blows-up) fast enough way.
1.3.Organization of the paper.In Section 2 we collect the technical results needed in the paper, such as definitions and basic properties of Besov, BV and BD spaces, properties of the distance functions from Lipschitz sets, Minkowski content of closed rectifiable sets and Ambrosio-Alberti types lemmas.Section 3 is devoted to the proof, and the analysis of some consequences, of Theorem 1.1.In Section 4 we discuss the proof of Theorem 1.2.Finally, in Section 5 we introduce the notion of "normal Lebesgue trace" and we prove Theorem 1.3, together with its generalized version Theorem 5.6.In Proposition 5.5 we also discuss some effective conditions to have vanishing normal Lebesgue boundary trace, and thus energy conservation on bounded domains.

Notations and technical tools
In this section, we recall the main definitions and list the tools used throughout the manuscript.
Here Ω is any open set in R d .We will specify when we will restrict to consider bounded domains only.Moreover, we use the standard notation v • w to denote the Euclidean scalar product between the two vectors v and w.

Radon measures.
We denote by M loc (Ω; R m ) the space of Radon measures with values in R m , that is Borel measures, finite on compact subsets of Ω.When m = 1, we use the shorter notation M loc (Ω).Moreover, for a vector-valued measure µ we denote by |µ| ∈ M loc (Ω) its variation, namely the positive (scalar-valued) measure defined as We denote by M(Ω; R m ) the space of finite vector-valued Radon measures on Ω, i.e.Radon vector valued-measures on Ω such that |µ|(Ω) < ∞.M(Ω; R m ) is a Banach space, with the norm Then, the weak convergence in M loc (Ω; R m ) of µ k to µ is given by ) that is a separable space, we have sequential weak-star compactness for equi-bounded sequences.See for instance [18].
We denote by µ C the restriction of the measure µ to the set C, that is µ C(A) := µ(A ∩ C) for every Borel set A.
2.2.Besov, bounded variation and bounded deformation spaces.For θ ∈ (0, 1) and p ∈ The full norm is then given by With a slight abuse of notation, we define Besov functions on a bounded domain Ω as follows: for any O ⊂⊂ Ω, define B θ p,∞ (O) by replacing R d in (2.1) with O, and compute the supremum all over |h| ≤ dist(O, ∂Ω), so that f (x + h) is well defined.For p = ∞ we identify B θ ∞,∞ (Ω) with C θ (Ω), the space of Hölder continuous functions.
We denote by BV (Ω) the space of functions with bounded variations, that is functions f whose distributional gradient is a finite Radon measure on Ω, i.e.
The latter definition generalises to vector fields f : Ω → R d .The space of functions with bounded deformation is given by vector fields whose symmetric gradient is a finite measure, i.e.

BD(Ω
Throughout the paper, we will denote the symmetric part of the gradient by Recall from [3, Theorem 3.87] that functions in BV (Ω) admit a notion of trace, on any domain Ω with Lipschitz boundary.
Moreover, the extension f of f to zero outside Ω belongs to BV (R d ; R m ) and In the lemma below we make use of the notation for the ε tubular neighbourhood of a Borel set A ⊂⊂ Ω, with the implicit restriction 2ε < dist(A, ∂Ω) whenever Ω has a boundary.

Lemma 2.2 (BV and BD
For any A ⊂⊂ Ω Borel set, and any z ∈ B 1 , we have where z • ∇f is the m-dimensional measure, whose m-th component is given by Proof.We start by proving the validity of (2.3) and (2.4) for smooth functions.The general case follows by density.
Let f : Ω → R d be smooth, A ⊂⊂ Ω and 2ε < dist(A, ∂Ω).For any x ∈ A and z ∈ B 1 we have Thus If f ∈ BD loc (Ω), we consider its mollification f δ := f * ρ δ , for any 2δ < dist(A, ∂Ω).Since f δ is smooth, by writing (2.3) for f δ and the properties of the convolution, we have Since f δ → f strongly in L 1 loc (Ω) and δ>0 ((A) ε ) δ = (A) ε , the latter inequality implies (2.3) for f , The proof of (2.4) follows by very similar considerations, by first showing that whenever f is smooth, and then by considering the mollification f δ in the general case.
and since we assumed x to be a differentiability point for d ∂Ω , by letting t → 0 + , we deduce We state and prove a result which will be used in the proof of Proposition 5.5 to show that any divergence-free vector field u ∈ BV (Ω; R d ), tangent to the boundary, has zero normal Lebesgue trace according to Definition 5.2.To the best of the authors knowledge the theorem below is not standard in the literature.We thank Camillo De Lellis for the argument of the proof.where n : ∂Ω → S d−1 denotes the inward unit normal to ∂Ω.
Let us notice that when Ω is piece-wise C 2 there is a shorter and quite classical argument to prove (2.8).Indeed in this case, up to a H d−1 -negligible set, ∂Ω is locally C 2 , and thus d ∂Ω ∈ C 2 (locally) by the classical theory (see for instance [22,Lemma 14.16]).Then, a straightforward computation shows that every point x ∈ ∂Ω in the corresponding local neighbourhood is a Lebesgue point for ∇d ∂Ω .As we will see, the proof for general Lipschitz domains is more delicate.Differently from Lemma 2.3, here the Lipschitz regularity of ∂Ω plays a crucial role.
Proof.For sake of clarity, let us divide the proof into steps.Let us show how the claim gives (2.9).We postpone the proof of the claim soon after.
3. Proof of (2.9) assuming the claim.Fix η = ε > 0. From Lemma 2.3 together with the above claim we have where y(x) is any point on ∂Ω such that d ∂Ω (x) = |x − y(x)|.This proves that lim sup from which (2.9) directly follows by the arbitrariness of ε > 0.
4. Proof of the claim.We run the proof by contradiction.If the conclusion would be false, we could find ε, η > 0 and x k , y k , r k such that , and from ∇ψ(0) = 0, ψ(0) = 0 and Thus, by (ii) we obtain if k is chosen large enough.Moreover, by (ii) and (2.10), we also have (2.12) In particular, (2.10) and (2.11) imply that lim Then, together with (2.12), we infer the expansion Then (iv) implies that necessarily lim inf for some ε = ε(η, ε) > 0. But (2.13) contradicts the minimality of y k .Indeed, letting z k := (x k , ψ(x k )), since ∇ψ(0) = 0 and ψ(0) = 0, we have On the other hand, (2.13) implies where in the first equality we have also used (2.12).It is clear that (2.14), together with (2.15), yields to a contradiction, since Moreover, there exists a dimensional constant c > 0 such that (2.17) Pick any δ > 0, which will be fixed at the end of the proof.By Egorov's theorem we can find A δ ⊂ C1 /2 , closed in the induced topology of ∂Ω, such that Consider the covering and extract a disjoint Vitali subcovering {B ε (x i )} Nε i=1 such that Thus, since A δ ⊂ R d is closed and (d − 1)-rectifiable, by (2.16) we obtain for some dimensional constant c > 0. By (2.18) and since the family {B ε (x i )} Nε i=1 is disjoint, we find k 0 ∈ N sufficiently large (recall that ε k → 0 + ) such that, for all k ≥ k 0 , we bound from below In particular, by (2.19) we obtain Hence, since the sequence ε k → 0 + was arbitrary, we obtain (2.17) by choosing 2δ = H d−1 (C).

2.5.
A one-sided Egorov lemma.We have the following "one side" Egorov-type convergence result.
Lemma 2.6 (One-sided Egorov).Let (X, µ) be a finite measure space, L ∈ R and f n a sequence of measurable functions such that lim inf n→∞ f n (x) ≥ L for µ-a.e.x ∈ X.For every δ > 0 there exists B δ ⊂ X such that µ(B c δ ) < δ and lim inf Proof.Since lim inf n→∞ f n ≥ L µ-a.e. in X, for every k ∈ N, the set has full measure in X.For every k fixed, the sets form an increasing family with respect to N ≥ 1, and since µ(X) < ∞, we deduce that Fix δ > 0. For every k ∈ N we can find Thus define B δ := ∞ k=1 A N k ,k .Clearly B δ ⊂ A Nm,m for every m ∈ N. Thus let x ∈ B δ and m ∈ N be arbitrary.In particular x ∈ A Nm,m , that is Since N m does not depend on x, we deduce The proof is concluded since m ∈ N was arbitrary.
2.6.Alberti-Ambrosio anisotropic kernel optimization.Here we prove a variation of the so called Alberti's lemma, which was introduced to prove the renormalization property of weak solutions to the transport equations with BV vector fields, first established by Ambrosio in [2].Actually, in the original proof, Ambrosio performed the optimization of the kernel by exploiting the rank-one structure of the singular part of the gradient of BV function, a deep result in Geometric Measure Theory also due to Alberti [1].Alberti's lemma provides a shortcut to optimize "by hands" the choice of the kernel.
Note that the kernels used in the next lemma are compactly supported, but not necessarily in B 1 , and not smooth.
Lemma 2.7 (Alberti).Let η : R d → R d be a Lipschitz vector field.Assume that η is either odd or even and divergence-free.Let Then, we have that Usually Alberti's lemma (see for instance [13, Lemma 2.13]) assumes that η(z) = M z, for a given matrix M , and proves that the infimum equals | Tr M |.Here we preferred to state the version with a general divergence-free vector field η since it might be of better use in context in which the linearity in z is missing.For instance, this might be the case in proving energy conservation for incompressible Euler in an Onsager's critical class different from BV .
Proof.Letting X : [0, +∞) × R d → R d be the flow generated by the vector field η, by the classical Cauchy-Lipschitz theory we have that X t is a bi-Lipschitz transformation with Lipschitz constants proportional to e Lt , where L is the Lipschitz constant of η.Since η is incompressible, then X t , X −1 t are measure preserving.Since η is odd (or even), we infer that X t , X −1 t are odd (or even) for any time slice t ∈ [0, +∞).Fix a convolution kernel θ ∈ K (see (1.4)) and, for any T > 0, define It is immediate to see that ρ T is even, non-negative and Lipschitz, with Lipschitz constant proportional to e LT and ρ T has compact support.Moreover, we have that Therefore, we have that ρ T ∈ K c .Since X is the flow generated by η, we have that
Following the idea used in [2], a consequence of Lemma 2.7 is the following result, which will be used in Section 4 to prove the energy conservation in L 1 t BV x ∩ L ∞ x,t of Theorem 1.2.The set K denotes kernels supported in the unit ball as introduced in (1.4).(iii) Divergence-free: div z η x = 0 for ν-a.e.x ∈ U ; (iv) Uniformly ν-integrable: η In particular |λ| ≪ ν and, denoting by D ν |λ| the Radon-Nikodym derivative of |λ| with respect to ν, we have We claim that Indeed, for any ρ ∈ K W there exists a sequence {ρ n } n ⊂ K such that ρ n → ρ in W 1,1 0 (B 1 ).Thus, (2.25) remains valid for every ρ ∈ K W .Then, pick KW ⊂ K W a countable dense subset (with respect to the W 1,1 topology).For any ρ ∈ KW , we find a ν-negligible set A ρ ⊂ K such that (2.25) is satisfied for any x ∈ A c ρ .Since KW is countable, we have that A = ρ∈ KW A ρ is ν-negligible and for any x ∈ A c we have that Moreover, since the functional ρ → ´B1 |∇ρ(z)η x (z)| dz is continuous with respect to the W 1,1 topology, by density we have that Thus, (2.26) is proved.
From now on, fix x ∈ K such that (2.26) is satisfied.For any ε > 0, Lemma 2.7 provides a Since Spt ρ ε is not necessarily contained in B 1 , that is ρ ε ∈ K c , we have to consider a rescaling, which, together with the 1-homogeneity of η x , will conclude the proof.Indeed, if B Rε is a ball containing the support of ρ ε , we define ρε (z) := R d ε ρ ε (R ε z).Clearly ρε ∈ K c , and moreover by (ii) Together with (2.26), and being ε > 0 arbitrary, this proves that and the arbitrariness of K ⊂⊂ U concludes the proof.

Duchon-Robert measure in critical Onsager's classes
This section is devoted to the proof of Theorem 1.1 and discuss some of its consequences.
Proof of Theorem 1.1.By looking at the expression of D ε [u] in (1.3), it is natural to define the trilinear operator T (u, v, w) := u(v • w).Then, we define Pick any K := A × I ⊂⊂ Ω × (0, T ).By Hölder inequality it is clear that under any of the assumptions (1.6)-(1.8)we have sup Moreover, by the trilinear structure of T ε,z we deduce equicontinuity of the sequence with respect to z. Indeed we have The three terms in the above formula have to be estimated in a different, although quite similar, way depending on which is the assumption on u among (1.6)- (1.8).Thus let us divide the cases.
. By the definition of Besov seminorm (2.1), we estimate each term in the right hand side of (3.2) by Hölder inequality, always taking δ εz 1 u − δ εz 2 u in L p and each where in the last line we used Young's inequality 2ab ≤ a 2 + b 2 .
Assuming (1.6) for p > 3: In this case we always take δ εz 1 u − δ εz 2 u in L 2p ′ , one of the δ εz i u in L 2p ′ and the other in L p .Thus, By the definition of Besov seminorm (2.1), we have The last inequality follows by Young's inequality ab ≤ a r r + b s s , with r = p+1 2 and s = p+1 p−1 .
Assuming (1.7):In this case we always put δ εz 1 u − δ εz 2 u in L 2 , one of the δ εz i u in L 2 and the other in L ∞ .Hence, using again the definition of Besov seminorm (2.1), we have Assuming (1.8):In this case we estimate δ εz 1 u − δ εz 2 u in L 1 by Lemma 2.2, and both of the To summarize, under any of the assumption of the theorem, we have that for some γ ∈ (0, 1] depending on the regularity of u, where the implicit constant depends on K, u and it is uniform with respect to ε. We are ready to apply an Ascoli-Arzelà argument.Pick {z k } k ⊂ B 1 dense.By (3.1) we have T ε,z 1 bounded in L 1 loc (Ω × (0, T ); R d ), thus we can find a subsequence T j,z 1 ⇀ µ z 1 for some µ z 1 ∈ M loc (Ω × (0, T ); R d ).Evaluating the above subsequence on z 2 , i.e.T j,z 2 , we can extract a further (non relabelled) subsequence such that T j,z 2 ⇀ µ z 2 for some µ z 2 ∈ M loc (Ω × (0, T ); R d ).Hence, by a diagonal argument, we find a subsequence {T n,z } n and measures µ z k ∈ M loc (Ω × (0, T ); R d ) for any k, such that We aim to extend By the compactness of B 1 , together with the equicontinuity estimate (3.7), we check that , by a standard argument we check that T n,z , ϕ is a Cauchy sequence in C 0 (B 1 ; R d ).Indeed, fix ε > 0 and by the equicontinuity estimate (3.7) find δ > 0 such that Clearly the family {B δ (z k )} k covers B 1 .By compactness we find N ∈ N such that B 1 ⊂ N i=1 B δ (z i ).Then, by the weak convergence on {z k } k , we find M ∈ N such that Hence, for any z ∈ B 1 , pick j ∈ {1, . . ., N } such that |z − z j | < δ.Thus, for n 1 , n 2 ≥ M we have that Thus, there exists a vector The map ϕ → V ϕ (z) is linear and satisfies By Riesz theorem we find proving (1.9).Again by (3.8) we have T n,z ⇀ µ z , for every z ∈ B 1 , and since T n,z enjoys (3.3)-(3.6)if u is assumed to satisfy (1.6)-(1.8)respectively, the very same estimate transfers to µ z by lower semicontinuity of the total variation under weak convergence of measures.This proves (i).Since T n,0 ≡ 0 for all n, we also get (ii).Finally, (iii) directly follows by the change of variable z → −z, together with the fact that the kernel ρ is chosen to be even.
Theorem 1.1 shows that, given any test function ϕ ∈ C 0 c (Ω × (0, T )), the dissipation measure in Onsager's critical spaces is characterized as for some vector V ϕ ∈ C 0 (B 1 ; R d ), which is moreover γ-Hölder continuous in z, γ depending on the Besov regularity of u.This leads to the following definition.
is the local energy flux of u, averaged on the Spt ϕ, in the direction z.
Note that here we are slightly abusing terminology, since z, being any element in the unit ball, is not normalized to |z| = 1.
Assume that D[u] does not depend on ρ.In [17] Duchon and Robert investigated the case in which the kernel is chosen to be radial.With this choice, our formula (1.10) gives and additional equivalent expression.Indeed, letting ρ(z) = ρ rad (|z|), by direct computations we deduce n being the inward unit normal.Alternatively, one could let As a byproduct of the formula (1.10) we get the following result.
Corollary 3.2 (Support restriction).Denote by U x,t ⊂ Ω × (0, T ) any open set containing the point (x, t), and assume that D[u] does not depend on ρ.It holds It is clear that under any of the assumptions ( . Indeed, under the BV assumption (1.8), we have that ˆ|δ εz u(x, t)| 3 dx dt ε.
If we restrict the kernels ρ to be radial, i.e.
an assumption on the symmetric part of ∇u is enough to deduce (3.9), as opposite to the one on the full gradient in (1.8).Thus, instead of BV , it is enough to consider BD, the space of "bounded deformation" vector fields.However, the assumption on the symmetric gradient does not seem enough to get equicontinuity with respect to z of the trilinear expression which is used to generate the measure µ z .For this reason, in this case, we can only conclude that D[u] ∈ M loc (Ω × (0, T )).Indeed, building on (2.3) from Lemma 2.2, we have the following result.
for all 2ε < dist(O, ∂Ω) and some dimensional constant C > 0. The right hand side of the previous estimate stays bounded as ε → 0 + , and the proof is concluded.
where in the last inequality we have used Fatou's lemma.For any z ∈ B 1 , define the scalar function v z := u • ∇ z ρ(z), v z : Ω × (0, T ) → R. Note that the distributional derivative of v z is given by for a.e.t ∈ (0, T ), where in the last equality we have used the Radon-Nikodym theorem to decompose the matrix-valued measure ∇u(•, t) with respect to its variation |∇u( We remark that if u ∈ L 1 t BD x ∩ L ∞ x,t , by using (2.3), one can prove an estimate similar to (2.23) restricting to radial kernels.However, considering ρ ∈ K rad prevents the use of the Alberti's optimization of Lemma 2.7, being the latter anisotropic.Details are left to the reader.

Energy conservation on Lipschitz bounded domains
This section aims to prove global energy conservation on bounded domains Ω ⊂ R d , with Lipschitz boundary, that is Theorem 1.3 together with its more general version Theorem 5.6.To begin, we clarify on the precise meaning of kinetic energy conservation in this setting.Moreover, we say that the kinetic energy of u is conserved if It is clear that if u ∈ C 0 ([0, T ]; L 2 (Ω)), then kinetic energy conservation is equivalent to e u (t) = e u (0) for all t ∈ [0, T ].
But clearly, if r j → 0 + is any sequence, we have which implies Whenever it exists, the normal Lebesgue trace is unique.Indeed, since Ω has Lipschitz boundary, see for instance [3,Theorem 3.61], we have ) are two Lebesgue normal traces of u in the sense of Definition 5.2, then, for H d−1 -a.e.x ∈ ∂Ω, we have In the computations above we have used that We refer the interested reader to the recent works [6][7][8][9][10]29] which build a notion of normal trace for "measure divergence" vector fields, together with several applications to systems of conservation laws, in setting with very low regularity.
In the next proposition we prove that having zero normal Lebesgue trace implies that the term appearing in the proof of the energy conservation on bounded domains vanishes.In fact, modulo considering subsequences, the two properties are substantially equivalent.
where ).Indeed, u ∂Ω n is the point-wise limit, as k → ∞, of the sequence Proof.We prove the first implication.We fix a sequence ε k → 0 + .We have to show that lim k→∞ Define the sequence of functions f ε k : ∂Ω → R as By assumption, f 5ε k (x) → 0 for H d−1 -a.e.x ∈ ∂Ω.Fix any δ > 0. By Egorov's theorem we find A δ ⊂ ∂Ω, closed in the induced topology on ∂Ω, such that H d−1 (∂Ω \ A δ ) < δ and f 5ε k → 0 uniformly on A δ .By the uniform convergence on A δ , we can find k 0 = k 0 (δ) > 0 such that, for all k ≥ k 0 it holds sup For any fixed k ≥ k 0 consider the family of balls {B ε k (x)} x∈A δ .By Vitali's covering theorem we find a finite and disjoint sub-family of balls F := {B ε k (x j )} j∈Jε k , such that Since F is a disjoint family and ∂Ω has Minkowski dimension d − 1, by possibly choosing k 0 even larger, it must hold #(F) ≤ cε 1−d k , (5.7) for some constant c > 0 which depends only on Ω.Thus we split By (5.6) and (5.7) we have Moreover, since u ∈ L ∞ (Ω; R d ), we can bound Since both ∂Ω, A δ ⊂ R d are closed (d − 1)-rectifiable sets, by (2.16) from Proposition 2.5 we have lim for some purely dimensional constant c > 0. Thus, since (∂Ω , by possibly choosing k 0 larger, we deduce that I 2 ε k ≤ Cδ, for all k ≥ k 0 .Summing up, we have proved that for any δ > 0, we can find a k 0 ∈ N large enough such that I ε k ≤ Cδ for all k ≥ k 0 , for some constant C > 0 depending only on the domain Ω.This gives (5.5) and concludes the proof of the first part.
Let us now prove the second implication by contradiction.If (5.4) does not hold, then we find a sequence of radii Since C m ր C, we find m large enough so that (5.9) Moreover, by interior approximation of Radon measures, we can also find a set M ⊂ Cm , closed in the induced topology on ∂Ω, such that Then, cover (M ) ε with {B ε (x)} x∈M and extract a finite and disjoint Vitali subcovering such that (5.10) From (5.10) and (2.16), if ε is sufficiently small, we obtain from which lim inf Recall that solutions of Euler, say u ∈ L 2 (Ω×(0, T )), on a Lipschitz bounded domain Ω satisfy ˆΩ u(x, t) • ∇ϕ(x) dx = 0 ∀ϕ ∈ H 1 (Ω), for a.e.t ∈ (0, T ).The latter condition is indeed a way to define, in a weak sense, L 2 divergence-free vector fields tangent to ∂Ω.However, this does not automatically imply that the Lebesgue normal boundary trace of u, in the sense of Definition 5.2, vanishes.In fact, it is not even clear in general whether such a notion of trace exists.We give some conditions which make this true.(5.12) Then u ∂Ω n ≡ 0 if one of the following conditions holds: (i) For H d−1 -a.e.x ∈ ∂Ω, ∃ε 0 > 0 such that u • ∇d ∂Ω ∈ C 0 B ε 0 (x) ∩ Ω ; (ii) u ∈ BV (Ω; R d ).
In particular, we emphasise that the assumption (i) relaxes the usual one in which u • ∇d ∂Ω is continuous in a full open neighbourhood of ∂Ω.For instance, when the domain is piece-wise C 2 , this allows us to deduce kinetic energy conservation on Lipschitz domains, without imposing anything in neighbourhoods of corner points of ∂Ω.
Assume that u ∈ BV (Ω; R d ).Letting ũ be the zero extension of u outside Ω, by Theorem 2.1, we get that ũ ∈ BV (R d ; R d ) and the distributional gradient is given by ∇ũ = (∇u) Ω + (u where n is the inward unit normal to ∂Ω.Note that (5.12) is equivalent to the fact that ũ is divergence-free in R d .Hence, we have that  where P := |u| 2 2 + p denotes the Bernoulli pressure.In particular, P L ∞ ((Ω ε \Ω 2ε )×I) is also allowed to blow-up in the limit as ε → 0 + , provided that the last term ´T 0 ´Ω |u • ∇ϕ ε | compensate.This happens, for instance, if u achieves its Lebesgue normal boundary trace in a fast enough way.

Theorem 2 . 1 (
Boundary trace).Let Ω ⊂ R d be an open set with bounded Lipschitz boundary and

2. 3 .
Lipschitz sets and properties of the distance function.For a bounded open Lipschitz set Ω ⊂ R d , we denote by d ∂Ω : Ω → [0, ∞) the interior distance to the boundary, i.e. d ∂Ω (x) := dist(x, ∂Ω).The distance function d ∂Ω satisfies the properties listed in the following lemma.

2. 4 .Proposition 2 . 5 (
Minkowski content and rectifiable sets.For all m ≥ 1, we let ω m := H m (B 1 ) to be the m-dimensional volume of the unit ball B 1 ⊂ R m .We also keep the notation (A) ε introduced in (2.2) for the ε-tubular neighbourhood.Minkowski content of rectifiable sets).Let Ω ⊂ R d be a bounded domain with Lipschitz boundary ∂Ω.If C ⊂ ∂Ω is closed in the induced topology of ∂Ω, then
5.1.Normal Lebesgue boundary trace.In the spirit of traces for BV function given in Theorem 2.1, we now introduce a notion of trace for the normal component of a vector field u : Ω → R d , that we call "normal Lebesgue trace".Such quantity will play a major role in the boundary analysis.Definition 5.2 (Normal Lebesgue boundary trace).Let Ω ⊂ R d be a Lipschitz domain and u ∈ L 1 (Ω; R d ) a vector field.We say that u has a Lebesgue normal trace on ∂Ω if there exists a function u ∂Ω n ∈ L 1 (∂Ω; H d−1 ) such that, for every sequence r k → 0 + , it holdsThe choice of taking sequence r k → 0 + instead of continuous radii r → 0 + as in Theorem 2.1 is made for convenience.It is easy to see that Definition 5.2 is a slightly weaker notion with respect to that using continuous radii, i.e.
.3) Moreover, if (5.2) is true, then, for every sequence r k → 0 + , it must hold We remark that as soon as u ∈ L ∞ (Ω; R d ) and it admits a Lebesgue normal trace in the sense of Definition 5.2, then necessarily u ∂Ω n ∈ L ∞ (∂Ω; H d−1 ).Thus, Theorem 2.4 concludes the proof.It is clear that Theorem 5.6 assumes a more general condition with respect to Theorem 1.3, where both u and p are assumed to be bounded in a neighbourhood of ∂Ω.Indeed, one can estimateˆT 0 ˆΩ |u| 2 2 + p u • ∇ϕ ε dx dt ≤ P L ∞ ((Ω ε \Ω 2ε )×I) ˆT 0 ˆΩ |u • ∇ϕ ε | dx dt,