An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations
aa r X i v : . [ m a t h . A P ] O c t Communications in Mathematical Physics manuscript No. (will be inserted by the editor)
An Onsager Singularity Theorem for TurbulentSolutions of Compressible Euler Equations
Theodore D. Drivas and Gregory L. Eyink , Department of Applied Mathematics & StatisticsThe Johns Hopkins University, Baltimore, MD 21218, USAE-mail: [email protected] Department of Physics and AstronomyThe Johns Hopkins University, Baltimore, MD 21218, USAE-mail: [email protected] 26, 2017
Abstract:
We prove that bounded weak solutions of the compressible Eulerequations will conserve thermodynamic entropy unless the solution fields havesufficiently low space-time Besov regularity. A quantity measuring kinetic energycascade will also vanish for such Euler solutions, unless the same singularity con-ditions are satisfied. It is shown furthermore that strong limits of solutions ofcompressible Navier-Stokes equations that are bounded and exhibit anomalousdissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, withboth vanishing when solutions are above the critical degree of Besov regular-ity. Stationary, planar shocks in Euclidean space with an ideal-gas equation ofstate provide simple examples that satisfy the conditions of our theorems andwhich demonstrate sharpness of our L -based conditions. These conditions in-volve space-time Besov regularity, but we show that they are satisfied by Eulersolutions that possess similar space regularity uniformly in time.
1. Introduction
In a 1949 paper on turbulence in incompressible fluids [1], L. Onsager announceda result that spatial H¨older exponents ≤ / T. Drivas & G. Eyink solutions of the type conjectured by Onsager, beginning with pioneering work ofDeLellis & Sz´ekelyhidi, Jr. [6, 7] on the convex integration approach, that hassince culminated in constructions of solutions with the critical 1 / d ≥ . The basicstate variables are the mass density ̺ := ̺ ( x , t ), fluid velocity v := v ( x , t ) andinternal energy density u := u ( x , t ) (or specific internal energy u m = u/̺ ), withthe latter defined implicitly by the relation E := ̺ | v | + u in terms of thetotal energy density E . The Euler system then consists of the d + 2 dynamicalequations expressing conservation of mass, momentum and energy: ∂ t ̺ + ∇ x · ( ̺ v ) = 0 , (1) ∂ t ( ̺ v ) + ∇ x · ( ̺ vv + p I ) = 0 , (2) ∂ t E + ∇ x · (( p + E ) v ) = 0 . (3)We use the “dyadic product” notation vv of J. W. Gibbs for the tensor product v ⊗ v of space-vectors, which is convenient in this paper. The pressure is givenby a thermodynamic equation of state p := p ( u, ̺ ) as a function of u and ̺. Aprevious paper [10] has studied a similar problem, but under the assumptionof a barotropic equation of state, with pressure p = p ( ̺ ) a function only ofmass density and with no independent equation for the total energy density E. Our results are valid for a general equation of state p ( u, ̺ ) , assuming only thatthe fluid undergoes no phase transitions during its evolution (see Assumption2 for a more precise statement). We also consider strong limits of solutionsof the compressible Navier-Stokes equations for Reynolds and P´eclet numberstending to infinity. As we shall show, such strong limits are weak solutions ofthe compressible Euler system (1)–(3). This is a subclass of all Euler solutions,but arguably the one most relevant to compressible fluid turbulence.In order to state precisely our results, recall that the Navier-Stokes-Fouriersystem (or, simply, the compressible Navier-Stokes equations) for a viscous, heat-conducting fluid takes the form: ∂ t ̺ + ∇ x · ( ̺ v ) = 0 , (4) ∂ t ( ̺ v ) + ∇ x · ( ̺ vv + p I + T ) = 0 , (5) ∂ t E + ∇ x · (( p + E ) v + T · v + q ) = 0 . (6)The viscous stress tensor T is given by Newton’s rheological law : T := − η S − ζΘ I with S := 12 (cid:18) ∇ x v + ( ∇ x v ) ⊤ − d Θ I (cid:19) and Θ := div x v , (7)where η := η ( u, ̺ ) > ζ := ζ ( u, ̺ ) > heat flux q is given by Fourier’s law : q := − κ ∇ x T, (8)with thermal conductivity κ := κ ( u, ̺ ) > , where T := T ( u, ̺ ) is the tempera-ture of the fluid. For this system, see standard physics texts such as Landau & nsager Singularity Theorem 3 Lifshitz [11] ( §
49) or de Groot & Mazur [12], (Ch. XII, § § ∂ t (cid:18) ̺ | v | (cid:19) + ∇ x · (cid:18)(cid:18) p + 12 ̺ | v | (cid:19) v + T · v (cid:19) = p Θ − Q, (9) ∂ t u + ∇ x · ( u v + q ) = Q − p Θ, (10)where the rate of viscous heating of the fluid is explicitly: Q := − T : ∇ x v = 2 η | S | + ζΘ . (11)An essential role will be played in our analysis by the thermodynamic entropy .The entropy density s := s ( u, ̺ ) (or the specific entropy s m = s/̺ ) is related to u and ̺ through the first law of thermodynamics in the form: T d s = d u − µ d ̺, (12)with the chemical potential µ := µ ( u, ̺ ). The entropy s is a concave functionof ( u, ̺ ) , as a consequence of extensivity of the thermodynamic limit [17, 18]or macroscopically as an expression of thermodynamic stability [19, 20]. The fundamental equation s := s ( u, ̺ ) completely determines the thermodynamics ofany system, yielding by equilibrium thermodynamic relations all other functions,including temperature T ( u, ̺ ) , chemical potential µ ( u, ̺ ) , pressure p ( u, ̺ ) , etc.These functions satisfy the thermodynamic Gibbs relation : T s = u + p − µ̺, (13)by an application of the Euler theorem on homogeneous functions [19, 20]. Remark 1.
For concreteness, we mention here a couple of examples of thermo-dynamic fundamental equations of some standard fluids. First, an ideal gas has s ( u, ̺ ) = αk B ̺ (cid:20) log (cid:18) u̺ /α (cid:19) + s (cid:21) (14)for Boltzmann’s constant k B and parameter α = f / > , related to the numberof mechanical degrees of freedom f of individual gas molecules. For a simplemonatomic gas in d space dimensions, f = d. The constant s is determinedfrom microscopic statistical mechanics. This simple model with an appropriatechoice of α describes the thermodynamics of most gaseous systems at low density.Another standard example is the van der Waals fluid with entropy: s ( u, ̺ ) = conc . env . n αk B ̺ h log (cid:16) (1 /̺ − b ) /α ( u/̺ + a̺ ) (cid:17) + s io , (15)Here the notation “conc. env.” denotes the upper concave envelope of the func-tion inside the curly brackets, which is smooth but not a globally concave func-tion of ( u, ̺ ) . The van der Waals model incorporates some density correctionsthrough the new terms involving constants a, b > , but reduces to the ideal gaslaw in the low-density limit ρ → . This is the simplest example of a fluid model
T. Drivas & G. Eyink exhibiting a gas-liquid phase transition for low energies and high densities, atthe points in the ( u, ̺ )-plane of non-smoothness of the concave envelope in (15).For these models, see [19, 20]. Needless to say, our results apply not just tothese specific examples but very widely, because the relations (12) and (13) aregeneral results of equilibrium thermodynamics and statistical mechanics [17, 18].From the compressible Navier-Stokes system (4)–(6) and the thermodynamicrelation (12) follows the balance equation for the entropy density: ∂ t s + ∇ x · (cid:16) s v + q T (cid:17) = QT + Σ κ . (16)The entropy production rate Σ := Q/T + Σ κ involves a viscous heating contri-bution with Q again given by (11), and a term due to thermal conduction: Σ κ := − q · ∇ x TT = κ |∇ x T | T . (17)In accord with second law of thermodynamics, entropy is globally increased since: Σ := QT + Σ κ = 2 ηT | S | + ζT | Θ | + κ |∇ x T | T ≥ . (18)For these standard results see [11, 12].Smooth solutions of the compressible Euler system satisfy the same balanceequations as (9), (10), and (16), but with ζ, η, κ ≡ T , q = 0 and Q, Σ ≡
0. This need not be true, of course,for weak solutions. An important class of weak solutions that we consider arethose arising from limits of solutions ̺ ε , u ε , v ε of the Navier-Stokes system withtransport coefficients scaled as η ε = εη, ζ ε = εζ, κ ε = εκ, for ε → . Essentially,1 /ε represents the Reynolds and P´eclet numbers of the fluid. To avoid issuesinvolving boundary conditions, we consider only flows on space domains Ω either d -dimensional Euclidean space Ω = R d or the d -torus Ω = T d . We shall oftenuse the notation Γ = Ω × (0 , T ) for the space-time domain, T < ∞ or T = ∞ . We then make the following specific assumptions:
Assumption 1.
Given ε >
0, we assume that there exists a unique smoothsolution u ε , ̺ ε , v ε of the compressible Navier-Stokes system (4)–(6) on Ω × (0 , T )for a given equation of state. In fact, most of our analysis will apply to suitableweak Navier-Stokes solutions. We assume u ε , ̺ ε , v ε ∈ L ∞ ( Ω × (0 , T )) uniformlybounded for ε < ε and that for some 1 ≤ p < ∞ strong limits exist u ε → u, ̺ ε → ̺, v ε → v in L ploc ( Ω × (0 , T )) . (19)Here L ploc ( Γ ), as usual (see e.g. [21, 22]) , denotes the linear space of measurablefunctions which are locally p -integrable: L ploc ( Γ ) = { f : Γ → R meas . | f ∈ L p ( O ) , ∀ open O ⊂⊂ Γ } (20)where A ⊂⊂ B denotes that the closure ¯ A is compact and ¯ A ⊂ B . Strongconvergence f n → f in L ploc ( Γ ) is the requirement that for any open O ⊂⊂ Γ therestrictions converge f n (cid:12)(cid:12) O → f (cid:12)(cid:12) O strong in L p ( O ) . With this topology, L ploc ( Γ )is a complete metrizable space for all p ≥
1. Whenever ¯ Γ is itself compact nsager Singularity Theorem 5 (e.g. ¯ Γ = T d × [0 , T ] with T < ∞ ), L ploc ( Γ ) = L p ( Γ ). We remark also that,trivially, L ∞ ( Γ ) ⊂ L ploc ( Γ ) for all p ≥
1. Thus the convergence in (19) impliesconvergence pointwise almost everywhere for a subsequence ε k → u, ̺, v ∈ L ∞ ( Ω × (0 , T )) . The mode of convergence (19) permits limiting fields with jumpdiscontinuities. We also assume ̺ ε ≥ ̺ for some ̺ > ε < ε , so that thefluid nowhere approaches a vacuum state with zero density. Assumption 2.
We assume that the solutions involve thermodynamic states( u, ̺ ) strictly away from phase transitions, so that all thermodynamic functions h = p, T, µ, s , η, ζ, κ, etc. are smooth in u, ̺ . The set of states attainedby any solution is the essential range over space-time, R = ess . ran( u, ̺ ) and R ε = ess . ran( u ε , ̺ ε ) for ε > , which are compact sets in R [23]. The uniformboundedness in L ∞ ( Ω × (0 , T )) of u ε , ̺ ε for ε < ε implies that there exists acompact set K ⊂ R such that the closed convex hullconv[ R ε ∪ R ] ⊆ K, ∀ ε < ε . (21)We then assume for h that there is an open set U ⊂ R , with K ⊂ U and h ∈ C M ( U ) with smoothness exponent M ≥ . Assumption 3.
Assume that the dissipation terms defined in equations (11)and (18) converge as ε → Q εη := 2 η ε | S ε | , Q εζ := ζ ε ( Θ ε ) , Q ε := Q εη + Q εζ D ′ −→ Q, and Σ εη := Q εη T ε , Σ εζ := Q εζ T ε , Σ εκ := κ ε (cid:12)(cid:12)(cid:12)(cid:12) ∇ x T ε T ε (cid:12)(cid:12)(cid:12)(cid:12) , Σ ε := Σ εη + Σ εζ + Σ εκ D ′ −→ Σ. The limit distributions are obviously non-negative, and thus Radon measures.
Remark 2.
The set of compressible Navier-Stokes solutions on Euclidean space R d satisfying these three assumptions is non-empty and includes, in particular,shock solutions. See examples in [24] and [25]. Numerical simulations of com-pressible turbulence with the system (4)–(6) on the torus T d show that small-scale shocks (or “shocklets”) naturally develop. There is also some evidence,however, that at sufficiently high Mach numbers the limiting mass density ̺ as ε → Theorem 1.
Let u, ̺, v ∈ L ∞ ( Ω × (0 , T )) be any weak solution of the compress-ible Euler system (1)–(3) satisfying ̺ ≥ ̺ > and Assumption 2. Let Q flux ℓ bethe “energy flux” defined by (70) below and Σ inert ∗ ℓ the “inertial entropy produc-tion” defined by (95). Assuming that the distributional limit of Q flux ℓ exists, Q flux = D ′ - lim ℓ → Q flux ℓ (22) T. Drivas & G. Eyink then local energy and entropy balance equations hold in the sense of distributionson Ω × (0 , T ) : ∂ t (cid:18) ̺ | v | (cid:19) + ∇ x · (cid:18)(cid:18) p + 12 ̺ | v | (cid:19) v (cid:19) = p ◦ Θ − Q flux , (23) ∂ t u + ∇ x · ( u v ) = Q flux − p ◦ Θ, (24) ∂ t s + ∇ x · ( s v ) = Σ inert . (25) where Σ inert and p ◦ Θ necessarily exist and are defined by the distributionallimits Σ inert = D ′ - lim ℓ → Σ inert ∗ ℓ , p ◦ Θ = D ′ - lim ℓ → ( p ∗ G ℓ )( Θ ∗ G ℓ ) , (26) with G ℓ , ℓ > a space-time mollifying sequence.Remark 3. This result is analogous to Proposition 2 of [4] for weak solutions ofincompressible Euler with v ∈ L ( T d × (0 , T )). In their theorem, the assumptionon the existence of Q flux was unnecessary. We need to add this as an additionalhypothesis, because of the new term p ◦ Θ that appears in the energy balanceequations. Of course, p ◦ Θ = 0 assuming incompressibility. Remark 4.
Note that the second equation in (26) for p ◦ Θ is a standard definitionof a generalized distributional product of p and Θ [27]. This standard definitionrequires that the limit be independent of the chosen mollifier G . We note thatfor the purposes of Theorem 1, one could alternatively assume existence of p ◦ Θ and then deduce it for Q flux . The combination p ◦ Θ − Q flux always exists.Our next results concern the strong limits of Navier-Stokes solutions satisfyingAssumptions 1 – 3. First, we prove that these limits are necessarily weak solutionsof the Euler equations, even if the limit dissipation measures in Assumption 3remain positive: Q >
Σ > . Moreover, we show that such solutions satisfyweak energy and entropy balance laws which include possible anomalies:
Theorem 2.
The strong limits u, ̺, v of compressible Navier-Stokes solutionsunder Assumptions 1 – 3 are weak solutions of the compressible Euler system(1)–(3) on Ω × (0 , T ) . Furthermore, the following local energy and entropy equa-tions hold in the sense of distributions on Ω × (0 , T ) : ∂ t (cid:18) ̺ | v | (cid:19) + ∇ x · (cid:18)(cid:18) p + 12 ̺ | v | (cid:19) v (cid:19) = p ∗ Θ − Q, (27) ∂ t u + ∇ x · ( u v ) = Q − p ∗ Θ, (28) ∂ t s + ∇ x · ( s v ) = Σ, (29) with Q ≥ and Σ ≥ given by Assumption 3 and with p ∗ Θ := D ′ - lim ε → p ε Θ ε , (30) where this distributional limit necessarily exists. nsager Singularity Theorem 7 Remark 5.
Theorem 2 is analogous to Proposition 4 of [4] for the strong limits ofsolutions of the incompressible Navier-Stokes equation with viscosity tending tozero. Again, in their theorem, the analogue of our Assumption 3 was unnecessary,whereas we needed to add this as an additional hypothesis because of the newterm p ∗ Θ defined by (30) that appears in the energy balance equations. Remark 6.
Euler solutions obtained from Theorem 2 for vanishing viscosity nec-essarily satisfy Theorem 1 for general weak Euler solutions. It follows that: Σ inert = Σ ≥ Q inert := Q flux + τ ( p, Θ ) = Q ≥ , (31)where τ ( p, Θ ) is the “pressure-dilatation defect” defined by τ ( p, Θ ) = p ∗ Θ − p ◦ Θ. (32)The lefthand sides in (31) are “inertial-range” expressions for Q and Σ , analo-gous to those established in Proposition 1 and Section 5 of [4] for incompressiblefluids. In particular, Σ inert and Q flux describe “cascade” and can be expressedin terms of increments of the variables u, ̺, v by analogues of the Kolmogorov“4/5th-law” for compressible turbulence. Whereas Σ inert , Q flux can have anysigns for general weak Euler solutions, they are constrained by (31) for zero-viscosity solutions. The pressure-dilation defect in (32) is an additional sourceof anomalous energy dissipation, with no analogue for incompressible fluids. Remark 7.
Shock solutions on Euclidean space R d , as discussed in [24] and [25],provide examples for which Q >
Σ > Q = τ ( p, Θ ) > , sothat the entire contribution to Q is from the pressure-dilatation defect. See [25]for this result. Although shock solutions with discontinuous state variables u,̺, v provide the simplest examples of weak Euler solutions with Q, Σ positive,presumably positive anomalies can occur even with continuous solutions.We now state an analogue of the Onsager singularity theorem. We provenecessary conditions for anomalous dissipation involving Besov space exponents,as in the improvement by [3] of Onsager’s H¨older-space statement. Here we notethat the Besov space B σ, ∞ p ( O ) for a general open set O ⊂⊂ Γ is made up ofmeasurable functions f : Γ → R which are finite in the norm: k f k B σ, ∞ p ( O ) := k f k L p ( O ) + sup h ∈ R D , | h | 1) and where h O = dist( O, ∂Γ ). See [10] and, for a generaldiscussion, [28], § B σ, ∞ p,loc ( Γ ) := { f : Γ → R meas . | f ∈ B σ, ∞ p ( O ) , ∀ open O ⊂⊂ Γ } . (34)Again, whenever ¯ Γ is itself compact (e.g. ¯ Γ = T d × [0 , T ]), B σ, ∞ p,loc ( Γ ) = B σ, ∞ p ( Γ ). Theorem 3. Let u, ̺, v ∈ L ∞ ( Ω × (0 , T )) be any weak solution of the compress-ible Euler system (1)–(3) satisfying ̺ ≥ ̺ > , Assumption 2, and additionally u ∈ B σ up , ∞ p,loc ( Ω × (0 , T )) , ̺ ∈ B σ ̺p , ∞ p,loc ( Ω × (0 , T )) , v ∈ B σ vp , ∞ p,loc ( Ω × (0 , T )) , T. Drivas & G. Eyink with all three of the following conditions satisfied { σ up , σ ̺p } + σ vp > , (35)min { σ up , σ ̺p } + 2 σ vp > , (36)3 σ vp > , (37) for some p ≥ . Then Q flux , Σ flux necessarily exist and equal zero. Further,inviscid limit solutions from Theorem 2 satisfying exponent conditions (35)-(37)have Q = Σ = 0 and p ∗ Θ = p ◦ Θ. Thus, it is only possible that Q > or Σ > if at least one of (35)–(37) fails tohold for each p ≥ . Remark 8. Our proof of Theorem 3 generalizes the argument of [3], which em-ployed a simple mollification of the weak Euler solution. In fact, this idea canbe exploited to give a new notion of “coarse-grained Euler solution” which weintroduce in section 2 and show there to be equivalent to the standard notion of“weak solution,” not only for compressible Euler equations but for very generalbalance relations. As discussed in [25], the concept of “coarse-grained solution”makes connection with renormalization-group methods in physics. We employthis notion to prove both our Theorems 2 and 3. Our analysis of compressibleNavier-Stokes and Euler solutions was directly motivated by the earlier work ofAluie [29], and our theorems generalize previous results for barotropic compress-ible flow [10]. It is worth noting that all of our results generalize to relativisticEuler equations in Minkowski spacetime, following the discussion in [30]. Remark 9. Our Theorem 3 is formulated in terms of space-time regularity, whereasthe original statement of Onsager and most following works have given necessaryconditions for anomalous dissipation in terms of space-regularity only. Note thatour proof of Theorem 3 requires mollification/coarse-graining in time as well asspace, and thus space-time regularity is natural for the proof (and also in the rel-ativistic setting). However, we obtain conditions involving space-regularity onlyfrom the next theorem. Adapting standard definitions, we set: L ∞ ((0 , T ); B s, ∞ p,loc ( Ω )) := { f : Γ → R meas . | (38)sup t ∈ (0 ,T ) k f ( · , t ) k B s, ∞ p ( O ) < ∞ , ∀ open O ⊂⊂ Ω } . With this convention, we have the following result: Theorem 4. Let u, ̺, v be any weak Euler solution satisfying ̺ ≥ ̺ > and ̺, u, v ∈ L ∞ ( Ω × (0 , T )) together with: u ∈ L ∞ ((0 , T ); B σ up , ∞ p,loc ( Ω )) , ̺ ∈ L ∞ ((0 , T ); B σ ̺p , ∞ p,loc ( Ω )) , v ∈ L ∞ ((0 , T ); B σ vp , ∞ p,loc ( Ω )) , for Besov exponents ≤ σ up , σ ̺q , σ vq ≤ . Then the solutions are also Besov regularlocally in space-time: u ∈ B min { σ ̺p ,σ vp ,σ up } , ∞ p,loc ( Ω × (0 , T )) , (39) ̺ ∈ B min { σ ̺p ,σ vp } , ∞ p,loc ( Ω × (0 , T )) , (40) v ∈ B min { σ ̺p ,σ vp ,σ up } , ∞ p,loc ( Ω × (0 , T )) . (41) nsager Singularity Theorem 9 Remark 10. This result is very similar to that obtained in recent work of P.Isett for H¨older-continuous weak solutions of incompressible Euler [31], and theproof is almost the same. In fact, we shall derive Theorem 4 as a consequenceof a more general result which derives time-regularity from space-regularity fora wide class of weak balance equations. Remark 11. It is interesting to know how sharp are the necessary conditionsfor anomalous dissipation following from Theorems 3 and 4. While answeringthis question for the incompressible case has required more sophisticated tools[6, 8, 32, 33], we have a very cheap argument showing that our conditions aresharp for p = 3 and Ω = R d . In fact, the stationary planar shock solutions foran ideal gas in [24, 25] are obtained as strong limits of compressible Navier-Stokes solutions for vanishing viscosity and satisfy u, ̺, v ∈ ( BV loc ∩ L ∞ )( R d ) . These provide a simple example of dissipative Euler solutions saturating ourbounds, since ( BV loc ∩ L ∞ )( Ω ) ⊂ B /p, ∞ p,loc ( Ω ) , p ≥ Ω = T d , but the proofrests on a standard approximation theorem for BV functions that holds for anyopen O ⊂ R d (see e.g. [22], Thm. 2 of § p = 3 thismeans that we may take σ u = σ ̺ = σ v = 1 / p > 3, the sharpness of our results for solutions on R d remains anopen issue. Note that a standard Besov embedding gives B σ, ∞ p,loc ( Ω ) ⊂ C σ − d/ploc ( Ω )and B σ, ∞ p,loc ( Ω × (0 , T )) ⊂ C σ − ( d +1) /ploc ( Ω × (0 , T )) (see [28], § p must be H¨older-continuous.No stationary Euler solution can illustrate the sharpness of our results, if afinite entropy S = R d d x s and bounded velocities are required. If (1 ∧| x | − ) s v ∈ L ( R d ) , then ∇ x · ( s v ) = Σ ≥ Σ ≡ . This follows by smearing thestationary entropy balance with φ ( | x | /R ) for φ ∈ C ∞ c ( R + , R + ) with φ ( r ) = 1 for r < , φ ( r ) = 0 for r > , so R d d x Σ = lim R →∞ − R R< | x | < R d d x R φ ′ (cid:16) | x | R (cid:17) sv r , with v r the radial component of v . Thus, R d d x Σ = 0 with the integrabilityassumption on s v , e.g. for v ∈ L ∞ ( R d ) and s ∈ L ( R d ) . The sharpness of ourconditions thus remains open for all p ≥ R d . Likewise,the question remains open for Euler solutions on T d . No stationary shock exam-ples of the type discussed in [24, 25] can exist on the torus, since the anomalousentropy production in a stationary solution must arise from positivity of thespace-divergence of the entropy current, which necessarily vanishes for periodicsolutions. (We owe both of the above observations to an anonymous referee). Onthe other hand, turbulent solutions of the compressible Navier-Stokes equationobserved in numerical simulations on the torus appear to exhibit non-stationaryshocks (e.g. [26]). We therefore expect that such shock solutions again illustratesharpness of our results for p = 3 and Ω = R d or T d , but the rigorous mathe-matical construction of such non-stationary solutions will be more involved.The detailed contents of the present paper are as follows: In section 2 weintroduce the space-time coarse-graining operation and prove the equivalenceof distributional and coarse-grained solutions. In section 3 we derive balanceequations for the coarse-grained compressible Navier-Stokes system. In section4 we establish auxiliary commutator estimates necessary for our main theorems.In sections 5–8 we prove Theorems 1–4. 2. Coarse-Grained Solutions and Weak Solutions We are concerned in this section with general balance equations of the form ∂ t u + ∇ x · F = (42)on a space-time domain Ω × R where again either Ω = T d or R d , for sim-plicity, and u ∈ R m and F ∈ R d × m . As usual, one defines ( u , F ) to be a weak/distributional solution of (42) iff h ∂ t ϕ, u i + h∇ x ϕ ; F i = , ∀ ϕ ∈ D ( Ω × R ) , (43)where the space D ( Ω × R ) = C ∞ c ( Ω × R ) of test functions consists of C ∞ functions ϕ compactly supported in space-time, provided the topology definedby uniform convergence of functions and all their derivatives on compact setscontaining all the supports. Components u a , F ia belong to the space D ′ ( Ω × R )of continuous linear functionals on D ( Ω × R ), with h ∂ t ϕ, u i a = h ∂ t ϕ, u a i and h∇ x ϕ ; F i a = P di =1 h∇ x i ϕ, F ia i for a = 1 , . . . , m. For these standard notions, e.g.see [35, 36]. We offer here a slightly different point of view on these topics.Let G be a standard space-time mollifier , with G ∈ D ( Ω × R ) , G ≥ , andalso R Ω d d r R R d τ G ( r , τ ) = 1 . To simplify certain estimates we also assume,without loss of generality, that supp( G ) is contained in the Euclidean unit ballin ( d + 1) dimensions. Define the dilatation G ℓ ( r , τ ) = ℓ − ( d +1) G ( r /ℓ, τ /ℓ ) andspace-time reflection ˇ G ( r , τ ) = G ( − r , − τ ). For any u ∈ D ′ ( Ω × R ) we define its coarse-graining at scale ℓ by¯ u ℓ = ˇ G ℓ ∗ u ∈ C ∞ ( Ω × R ) . (44)Here ∗ denotes the convolution defined by( ˇ G ℓ ∗ u )( x , t ) = h S x ,t G ℓ , u i (45)for shift operator ( S x ,t G ℓ )( r , τ ) = G ℓ ( r − x , τ − t ) or, equivalently, by h ϕ, ˇ G ℓ ∗ u i = h ϕ ∗ G ℓ , u i (46)for all test functions ϕ ∈ D ( Ω × R ) . See [36]. We say that ( u , F ) are a (space-time)coarse-grained solution of (42) iff ∂ t ¯ u ℓ + ∇ x · ¯ F ℓ = (47)holds pointwise in space-time for all ℓ > . We then have: Proposition 1. ( u , F ) are a distributional solution of (42) on Ω × R iff ( u , F ) are a coarse-grained solution of (42) on Ω × R Proof. If ( u , F ) satisfy (42) weakly, then taking ϕ = S x ,t G ℓ in (43) for anyspace-time point ( x , t ) implies (47) by the definition (45) of the convolution.On the other hand, suppose that ( u , F ) are a coarse-grained solution of (42).Smearing (47) with an arbitrary test function ϕ ∈ D ( Ω × R ) , then gives by thesecond definition (46) of convolution that h ( ∂ t ϕ ) ∗ G ℓ , u i + h ( ∇ x ϕ ) ∗ G ℓ ; F i = 0 . (48)However, in the limit ℓ → , then ( ∂ t ϕ ) ∗ G ℓ → ∂ t ϕ and ( ∇ x ϕ ) ∗ G ℓ → ∇ x ϕ in the standard Fr´echet topology on test functions. Since u , F ∈ D ′ ( Ω × R )are, by definition, continuous functionals on D ( Ω × R ) , the equation (43) of thestandard weak formulation immediately follows. (cid:3) nsager Singularity Theorem 11 This equivalence extends to solutions with prescribed initial-data. A standardapproach to define weak solutions ( u , F ) of (42) on space-time domain Ω × [0 , ∞ )with initial data u ∈ D ′ ( Ω ) is to require that h ∂ t ϕ, u i + h∇ x ϕ ; F i + h ϕ ( · , , u i = , ∀ ϕ ∈ D ( Ω × [0 , ∞ )) . (49)Here the space D ( Ω × [0 , ∞ )) is taken to consist of piecewise-smooth functionsof the form ϕ ( x , t ) = θ ( t ) φ ( x , t ) , products of the Heaviside step function θ ( t )and some φ ∈ D ( Ω × R ) . Such test functions ϕ ∈ D ( Ω × [0 , + ∞ )) are causal ,with ϕ ( x , t ) = 0 for t < . In order to make the lefthand side of (49) meaningful,a stronger assumption is required than only ( u , F ) ∈ D ′ ( Ω × R ). A very generalassumption is that distributional products θ ⊙ u , θ ⊙ F exist defined by θ ⊙ f := D ′ - lim ℓ → θf ℓ for f ∈ D ′ ( Ω × R ) [27]. In that case, we can take h ∂ t ϕ, u i := h ∂ t φ, θ ⊙ u i , h∇ x ϕ ; F i := h∇ x φ ; θ ⊙ F i . (50)Because limit distributions θ ⊙ f clearly have support in Ω × [0 , ∞ ) , the definition(50) does not depend upon the choice of φ such that ϕ = θφ. In the special casewhen f = u , F ∈ L loc ( Ω × [0 , ∞ )), then strong convergence of f ℓ → f in L loc (e.g.see Lemma 7.2 of [21]) implies that the definitions (50) reduce to their standardinterpretation. In addition,to make the definition (49) meaningful, one mustrequire weak- ∗ continuity of the distribution u in time, so that t 7→ h ψ, u ( · , t ) i is continuous for all ψ ∈ D ( Ω ) . Initial data is then achieved in the sense thatlim t → h ψ, u ( · , t ) i = h ψ, u i , ∀ ψ ∈ D ( Ω ) . (51)The coarse-graining approach can be also carried over with only minor changes.The mollifier G must now be chosen to be strictly causal , with G ∈ D ( Ω × (0 , ∞ ))and thus G ( r , τ ) ≡ τ ≤ . The definition (44) of coarse-graining still applies,noting that the convolution in time is ( χ ∗ χ )( t ) = R t d s χ ( s ) χ ( t − s ) for causalfunctions χ , χ . We can again define ( u , F ) to be a coarse-grained solution of(42) if (47) holds pointwise in space-time for all ℓ > . Since u ℓ ∈ C ∞ ( Ω × [0 , ∞ )),the functions u ℓ ( · , ∈ C ∞ ( Ω ) are well-defined and the coarse-grained solutionis naturally said to take on initial data u ∈ D ′ ( Ω ) when D ′ - lim ℓ → u ℓ ( · , 0) = u . (52)It is straightforward to see for all ψ ∈ D ( Ω ) that h ψ, u ℓ i = Z d d r Z ∞ dτ G ℓ ( r , τ ) Ψ ( r , t ) , Ψ ( r , τ ) := h S r ψ, u ( · , τ ) i . (53)Suppose that one requires not only weak- ∗ continuity of u in time, but also thestronger statement that Ψ ( r , τ ) defined in (53) is jointly continuous in ( r , τ ) forall ψ ∈ D ( Ω ) . The initial data prescribed by (50) and (52) are then the same.This leads to: Proposition 2. If ( u , F ) is a coarse-grained solution of (42) on Ω × [0 , ∞ ) withinitial data u , then it is a distributional solution with the same initial data.If also h S r ψ, u ( · , τ ) i is jointly continuous in ( r , τ ) for all ψ ∈ D ( Ω ) , then adistributional solution ( u , F ) of (42) on Ω × [0 , ∞ ) with initial data u is acoarse-grained solution with the same initial data. Proof. To prove the first statement, multiply the coarse-grained equation (47)with the Heaviside function θ and then smear with an arbitrary φ ∈ D ( Ω × R ) . An integration-by-parts in time gives that h ( ∂ t φ ) , θ u ℓ i + h ( ∇ x φ ); θ F ℓ i + h φ ( · , , u ℓ i = 0 . Taking the limit ℓ → ϕ = S x ,t G ℓ ∈ D ( Ω × (0 , ∞ )) for any x ∈ Ω and t ≥ . We see that ϕ is strictly causal, i.e. ϕ ( · , 0) = 0 . The equation (49)of the weak formulation thus yields the coarse-grained equation (47) for thatchoice of ( x , t ) and ℓ. Furthermore, because of (53) and the joint continuity of h S r ψ, u ( · , τ ) i in ( r , τ ) , u ℓ ( · , D ′ −→ u holds for the same u given by (51). (cid:3) Remark 12. If u ∈ C ([0 , ∞ ); L p ( Ω )) with continuity in the strong L p -norm topol-ogy for some p ≥ 1, then the joint continuity follows from the obvious continuityof Ψ ( r , τ ) in r for each τ and the H¨older inequality | Ψ ( r , τ ) − Ψ ( r , τ ′ ) | ≤ k ψ k q k u ( · , τ ) − u ( · , τ ′ ) k p , q = p/ ( p − , which implies continuity of Ψ ( r , τ ) in τ uniform in r ∈ Ω. Remark 13. In Lemma 8 of [6] it was proved that, if ( u , F ) is a weak solution with u ∈ L ∞ ([0 , ∞ ) , L ( Ω )) and F ∈ L ( Ω × [0 , ∞ )) , then u can always be alteredon a zero measure set of times so that u ∈ C w ([0 , ∞ ) , L ( Ω )) , with continuityin the weak topology of L ( Ω ) . In that case, Ψ ( r , τ ) defined for any ψ ∈ D ( Ω )by (53) is continuous in τ for each r ∈ Ω. By Cauchy-Schwartz, |∇ r Ψ ( r , τ ) | ≤ k∇ ψ k k u k L ∞ ([0 , ∞ ); L ( Ω )) , so that Ψ ( r , τ ) is also (Lipschitz) continuous in r uniformly in τ, and thus isjointly continuous in ( r , τ ) under the same assumptions as in [6]. Remark 14. The above results hold with only minor modifications for solutionson Ω × [0 , T ) with 0 < T < ∞ . Coarse-grained solutions are required now tosatisfy equations (47) only for x , t and ℓ such that S x ,t G ℓ ∈ D ( Ω × (0 , T )) . On theother hand, for any ϕ ∈ D ( Ω × [0 , T )) , then T ϕ = max { t : ( x , t ) ∈ supp( ϕ ) } < T .Since supp( G ) is contained in the unit ball, then S x ,t G ℓ ∈ D ( Ω × (0 , T )) for any ℓ < T − T ϕ and ( x , t ) ∈ supp( ϕ ) and our previous arguments on equivalence ofthe two notions of solution can be repeated without change. Remark 15. In the paper [3], only space mollification was employed. One canalso define a space coarse-graining with a standard mollifier G ℓ ( r ) = ℓ − d G ( r /ℓ ) , that is, ˆ u ℓ = ˇ G ℓ ∗ u . This is a smooth function of space but only a distributionin time. In that case, we say that ( u , F ) are a (space) coarse-grained solution ofthe balance relation (42) iff ∂ t ˆ u ℓ + ∇ x · ˆ F ℓ = (54)holds pointwise in space and distributionally in time for all ℓ > 0. This is alsoequivalent to the standard notion of weak solution, as can be seen by argumentsvery similar to those given above. If furthermore u , F ∈ L loc ( Ω × (0 , T )) , thenstandard approximation arguments show that the time-derivative in (54) can betaken to be a classical derivative at Lebesgue almost all times. nsager Singularity Theorem 13 In many applications, including those considered in this paper, u is not merelya distribution but a measurable function of space-time, and F := F ( u ) is apointwise nonlinear function of u . A key aspect of the coarse-graining operation isthat coarse-graining nonlinear functions of fields generally gives a result differentfrom evaluating the function at the coarse-grained fields, i.e. the operations ofcoarse-graining and function-evaluation do not commute. For simple products ofthe form f f · · · f n , this non-commutation can be measured by coarse-grainingcumulants , which are defined iteratively in n by τ ℓ ( f ) = ¯ f ℓ and( f · · · f n ) ℓ = X Π | Π | Y p =1 τ ℓ ( f i ( p )1 , . . . , f i ( p ) np ) , (55)where the sum is over all partitions Π of the set { , , . . . , n } into | Π | disjointsubsets { i ( p )1 , . . . , i ( p ) n p } , p = 1 , . . . , | Π | . See e.g. [37, 38]. For example, for n = 2( f g ) ℓ = f ℓ g ℓ + τ ℓ ( f, g ) or τ ℓ ( f, g ) = ( f g ) ℓ − f ℓ g ℓ . (56)For general composed functions h = h ( f , · · · , f n ) with h a smooth nonlinearfunction on R n , the non-commutation is measured by the quantity ∆ ℓ h := h ( f , · · · , f n ) ℓ − h (( f ) ℓ , · · · , ( f n ) ℓ ) . (57)To simplify the writing of various expressions, we shall often use an “under-bar”notation to indicate the function evaluated at coarse-grained fields: h ℓ := h (( f ) ℓ , · · · , ( f n ) ℓ ) , (58)whereas h ℓ = h ( f , · · · , f n ) ℓ . Remark 16. If, as in Remark 14 above, we consider space-time domains with afinite time interval Γ = Ω × (0 , T ), T < ∞ (or a semi-infinite interval Ω × (0 , ∞ )for mollifiers which are not causal), coarse-graining cumulants τ ℓ ( f , · · · , f n ) andsmooth functions h ℓ of coarse-grained fields are not defined everywhere on Γ for ℓ > 0. Instead, they are defined only for ( x , t ) ∈ Γ such that S x ,t G ℓ ∈D ( Ω × (0 , T )) , e.g. when the distance of ( x , t ) to ∂Γ is less than ℓ. They are thuswell-defined for every ( x , t ) ∈ Ω × (0 , T ) at sufficiently small ℓ. 3. Coarse-Grained Navier-Stokes and Balance Equations We now discuss the results of coarse-graining the solutions of the compressibleNavier-Stokes system. None of the results in this section depend upon the par-ticular type of coarse-graining and are valid whether coarse-graining is in space,time, space-time or using some other averaging procedure (such as as weightedcoarse-graining). We drop the superscript ε in this section to simplify notations.The coarse-grained Navier-Stokes equations for mass density ̺, momentumdensity j = ̺ v , and energy density E are ∂ t ̺ ℓ + ∇ x · ℓ = 0 , (59) ∂ t ℓ + ∇ x · (cid:16) ( jv ) ℓ + p ℓ I + T ℓ (cid:17) = , (60) ∂ t E ℓ + ∇ x · (cid:16) (( E + p ) v ) ℓ + ( T · v ) ℓ + q ℓ (cid:17) = 0 . (61) It is useful to rewrite the equations (59) and (60) employing the Favre (density-weighted) averaging : ˜ f ℓ = ( ̺f ) ℓ /̺ ℓ . (62)One may likewise define cumulants ˜ τ ℓ ( f i , . . . , f n ) with respect to this Favre fil-tering. See [29, 39]. With this new averaging, (59)–(60) may be rewritten: ∂ t ̺ ℓ + ∇ x · ( ̺ ℓ ˜ v ℓ ) = 0 , (63) ̺ ℓ ( ∂ t + ˜ v ℓ · ∇ x )˜ v ℓ + ∇ x · (cid:0) ̺ ℓ ˜ τ ℓ ( v , v ) + p ℓ I + T ℓ (cid:1) = 0 . (64)We emphasize that our use of Favre coarse-graining is mathematically only amatter of convenience, in order to reduce the number of terms in our coarse-grained equations (and to provide them with simple physical interpretations[25, 29]). Favre cumulants of f , . . . , f n may always be rewritten in terms ofunweighted cumulants of f , . . . , f n and ̺. For example [29, 40]:˜ f ℓ = f ℓ + 1 ̺ ℓ τ ℓ ( ̺, f ) , (65)˜ τ ℓ ( f, g ) = τ ℓ ( f, g ) + 1 ̺ ℓ τ ℓ ( ̺, f, g ) − ̺ ℓ τ ℓ ( ̺, f ) τ ℓ ( ̺, g ) , (66)˜ τ ℓ ( f, g, h ) = τ ℓ ( f, g, h ) + 1 ̺ ℓ τ ℓ ( ̺, f, g, h ) (67) − ̺ ℓ [ τ ℓ ( ̺, f ) τ ℓ ( ̺, g, h ) + cyc. perm. f, g, h ] + 2 ̺ ℓ τ ℓ ( ̺, f ) τ ℓ ( ̺, g ) τ ℓ ( ̺, h ) . We next derive various balance equations for the coarse-grained fields. Resolved Kinetic Energy: Following Aluie [29], we consider a resolved kinetic en-ergy ̺ ℓ | ˜ v | = | | ℓ / ̺ ℓ . Using (63) and (64) one can derive its balance equation: ∂ t (cid:18) ̺ ℓ | ˜ v ℓ | (cid:19) + ∇ x · J vℓ = p ℓ Θ ℓ − Q flux ℓ − D vℓ , (68)where the various terms are defined by: J vℓ := (cid:18) ̺ ℓ | ˜ v ℓ | + p ℓ (cid:19) ˜ v ℓ + ̺ ℓ ˜ v ℓ · ˜ τ ℓ ( v , v ) − p ℓ ̺ ℓ τ ℓ ( ̺, v ) + ˜ v ℓ · T ℓ , (69) Q flux ℓ := ∇ x p ℓ ̺ ℓ · τ ℓ ( ̺, v ) − ̺ ℓ ∇ x ˜ v ℓ : ˜ τ ℓ ( v , v ) , (70) D vℓ := −∇ x ˜ v ℓ : T ℓ . (71)Equation (68) may be rewritten as ∂ t (cid:18) ̺ ℓ | ˜ v ℓ | (cid:19) + ∇ x · J vℓ = ( pΘ ) ℓ − Q inert ℓ − D vℓ , (72)where the “inertial dissipation” is defined by Q inert ℓ := Q flux ℓ + τ ℓ ( p, Θ ) . (73) nsager Singularity Theorem 15 Unresolved Kinetic Energy . We define this quantity (with summation over re-peated i indices) as k ℓ := 12 ̺ ℓ ˜ τ ℓ ( v i , v i ) . (74)Note that ̺ ℓ | ˜ v ℓ | + k ℓ = ( ̺ | v | ) ℓ , whose integral over Ω is a time-mollificationof the total kinetic energy. Taking the difference of the coarse-grained kinetic-energy Eq. (9) governing ( ̺ | v | ) ℓ and Eq. (68) for ̺ ℓ | ˜ v ℓ | , one obtains: ∂ t k ℓ + ∇ · J kℓ = ( τ ℓ ( p, Θ ) − Q ℓ ) + Q flux ℓ + D kℓ , (75)where J kℓ : = 12 ̺ ℓ ˜ τ ℓ ( v i , v i )˜ v ℓ + τ ℓ ( p, v ) + 12 ̺ ℓ ˜ τ ℓ ( v i , v i , v ) (76)+( T · v ) ℓ − T ℓ · ˜ v ℓ ,D kℓ : = − T ℓ : ∇ x ˜ v ℓ . (77) Resolved Internal Energy: Directly coarse-graining equation (10), one finds thefollowing balance equation for the resolved internal energy: ∂ t u ℓ + ∇ x · J uℓ = Q ℓ − ( pΘ ) ℓ , (78)where J uℓ = ( u v ) ℓ + q ℓ = u ℓ v ℓ + τ ℓ ( u, v ) + q ℓ . (79)A more important quantity for our analysis is u ∗ ℓ := u ℓ + k ℓ , which we termthe “intrinsic resolved internal energy”. It is defined more fundamentally by theimplicit relation E ℓ = 12 ̺ ℓ | ˜ v ℓ | + u ∗ ℓ , (80)in terms of the resolved quantities ̺ ℓ , ˜ v ℓ , and E ℓ . One thus derives a balanceequation for this intrinsic internal energy by subtracting the resolved kineticenergy balance (68) from the coarse-grained total energy equation (61): ∂ t u ∗ ℓ + ∇ x · J u ∗ ℓ = Q flux ℓ − p ℓ Θ ℓ + D kℓ , (81)where D kℓ is defined by equation (77) and J u ∗ ℓ = u ℓ v ℓ + τ ℓ ( h, v ) + 12 ̺ ℓ ˜ τ ℓ ( v i , v i )˜ v ℓ + 12 ̺ ℓ ˜ τ ℓ ( v i , v i , v )+ q ℓ + ( T · v ) ℓ − T ℓ · ˜ v ℓ , (82)with h := u + p defining the standard thermodynamic enthalpy. Resolved Entropy : We derive an equation for s ℓ := s ( u ℓ , ̺ ℓ ) using (78), also (59)rewritten as ∂ t ̺ ℓ + ∇ x · ( ̺ ℓ v ℓ + τ ℓ ( ̺, v )) = 0 , (83)the homogeneous Gibbs relation T ℓ s ℓ = ( u ℓ + p ℓ ) − µ ℓ ̺ ℓ , and the first law ofthermodynamics: T ℓ D t s ℓ = D t u ℓ − µ ℓ D t ̺ ℓ , (84) with D t = ∂ t + v ℓ · ∇ being the material derivative along the smoothed flow.One then finds that the resolved entropy satisfies: ∂ t s ℓ + ∇ x · J sℓ = Q ℓ − τ ℓ ( p, Θ ) T ℓ − I flux ℓ + Σ flux ℓ + D sℓ , (85)where J sℓ := s ℓ v ℓ + β ℓ ( τ ℓ ( u, v ) + q ℓ ) − λ ℓ τ ℓ ( ̺, v ) , (86) I flux ℓ := β ℓ ( p ℓ − p ℓ ) Θ ℓ , (87) Σ flux ℓ := ∇ x β ℓ · τ ℓ ( u, v ) − ∇ x λ ℓ · τ ℓ ( ̺, v ) , (88) D sℓ := − q ℓ · ∇ x T ℓ T ℓ , (89)with β := 1 /T and λ := µ/T . Considering the source terms on the righthandside of (85), we shall see that all of the terms marked “flux” satisfy simplebounds, and the direct dissipation term D sℓ will be seen to vanish as ε → , butthe quantity Q ℓ − τ ℓ ( p, Θ ), which originates from the D t u ℓ term in (84), is moredifficult to estimate. Fortunately, the same term appears in the balance equationfor “unresolved kinetic energy.” Intrinsic Resolved Entropy : In order to cancel the difficult term Q ℓ − τ ℓ ( p, Θ ),we introduce an “intrinsic resolved entropy density” by s ∗ ℓ := s ( u ℓ , ̺ ℓ ) + β ℓ k ℓ . This quantity is defined more fundamentally by s ∗ ℓ = β ℓ ( u ∗ ℓ + p ℓ ) − λ ℓ ̺ ℓ , (90)where u ∗ ℓ is the intrinsic resolved internal energy defined in (80). The two def-initions are seen to be the same using the homogenous Gibbs relation (13), or s ℓ = β ℓ ( u ℓ + p ℓ ) − λ ℓ ̺ ℓ . By means of (90) and (81), together with the standardthermodynamic relation D t ( β ℓ p ℓ ) = ̺ ℓ D t λ ℓ − u ℓ D t β ℓ , one obtains D t s ∗ ℓ = ( D t β ℓ ) k ℓ + β ℓ D t u ∗ ℓ − λ ℓ D t ̺ ℓ . (91)rather than (84). Note that D t u ∗ ℓ appears here rather than D t u ℓ . It is straight-forward using (91) to derive the balance equation for s ∗ ℓ : ∂ t s ∗ ℓ + ∇ x · J s ∗ ℓ = − I flux ℓ + Σ flux ∗ ℓ + D sℓ + β ℓ D kℓ (92)with J s ∗ ℓ := J sℓ + β ℓ J kℓ , (93) Σ flux ∗ ℓ := Σ flux ℓ + β ℓ Q flux ℓ + ∂ t β ℓ k ℓ + ∇ x β ℓ · J kℓ . (94)We also then write Σ inert ∗ ℓ = − I flux ℓ + Σ flux ∗ ℓ (95)for the net “inertial” production of the intrinsic entropy. The balance equation(92) of the intrinsic entropy turns out to be the key identity for the proof ofTheorem 3. On the righthand side, the direct dissipation terms will be shownto vanish as ε → u, ̺, v . This latter result follows fromcommutator estimates of Section 4. nsager Singularity Theorem 17 Remark 17. Note that the balance equations (68) for resolved kinetic energy,(81) for intrinsic resolved internal energy and (92) for intrinsic resolved entropyare valid for general weak Euler solutions after setting T = q = , without theneed for considering the viscous regularization with ε > ε → . Onthe other hand, the balance equations (75) for unresolved kinetic energy, (78) forresolved internal energy, and (85) for resolved entropy are valid with T = q = only for weak Euler solutions obtained from the inviscid limit. In fact, the latterequations contain the quantities Q ℓ and τ ℓ ( p, Θ ) which are a priori undefinedfor general weak Euler solutions. 4. Commutator Estimates The estimates that we derive in this section are valid for coarse-graining in space,time, or space-time. We state them here for the space-time coarse-graining thatwe use in our proofs of Theorems 1–3. The need for coarse-graining in time aswell as in space is due to the time-derivative term in expression (94) for Σ flux ∗ ℓ . In order to present the estimates, it is useful to employ a “space-time vector”notation, with X = ( x , ct ) , R = ( r , cτ ) where c is a constant with dimensionsof velocity which is fixed independent of ǫ and ℓ. For example, we may take c to be the speed of sound (or, in the relativistic case, the speed of light). Wecorrespondingly take the ( d +1)-dimensional domain Γ = Ω × (0 , T ) and considercoarse-graining of functions f i ∈ L ∞ ( Γ ) , i = 1 , , , . . . with a non-negative,standard mollifier G ∈ C ∞ ( Γ ) which can, but need not, be causal. We assume,for convenience, that supp( G ) is contained in the Euclidean unit ball. Recallthat since L ∞ ( Γ ) ⊂ L ploc ( Γ ) for p ≥ 1, the functions f i are locally p –integrable, f i ∈ L ploc ( Γ ). For any open O ⊂⊂ Γ, let k · k p,O represent the standard L p ( O )-norm on the restriction f i (cid:12)(cid:12) O . All estimates assume ℓ sufficiently small for fixed O ⊂⊂ Γ , in particular ℓ < ℓ O = dist( O, ∂Γ ).A basic result is the following: Lemma 1. For n > , the coarse-graining cumulants are related to cumulantsof the difference fields δf ( R ; X ) := f ( X + R ) − f ( X ) as follows: τ ℓ ( f , . . . , f n ) = h δf , . . . , δf n i cℓ , (96) where h·i ℓ denotes average over the displacement vector R with density G ℓ ( R ) and the superscript c indicates the cumulant with respect to this average. This result is proved in [3] for n = 2 and, in the more general form quoted here,in [41] or [40], Appendix B. The proof is an easy application of the invariance ofcumulants of “random variables” to shifts of those variables by “non-random”constants. A direct consequence of Lemma 1 is: Proposition 3. (cumulant estimates) For open O ⊂⊂ Γ, p ∈ [1 , ∞ ] and n > k τ ℓ ( f , . . . , f n ) k p,O = O n Y i =1 k δf i ( ℓ ) k p i ,O ! with 1 p = n X i =1 p i , (97) where k δf ( ℓ ) k p,O := sup | R | <ℓ k δf ( R ) k p,O . Assuming f i ∈ B σ i , ∞ p i ,loc ( Γ ) with <σ i ≤ for i = 1 , . . . , n : k τ ℓ ( f , . . . , f n ) k p,O = O (cid:16) ℓ P ni =1 σ i (cid:17) , (98) If only f i ∈ L ∞ ( Γ ) , then at least lim ℓ → k τ ℓ ( f , . . . , f n ) k p,O = 0 , ≤ p < ∞ , (99) but without an estimate of the rate. Here “big- O ” notation, as usual, means inequality up to a constant independentof ℓ , which in this case depends on the details of the mollifier G . The finalstatement is a consequence of the bound (97) and the strong continuity of theshift operators ( S − r f )( x ) = f ( x + r ) in the L p ( O )-norm, a standard fact whichfollows from a simple density argument.We also need bounds on space-time derivatives of the cumulants. This can beaccomplished using the fact that all derivatives of cumulants with respect to X can be transferred to space-derivatives of the filter kernels G ℓ ( R ) with respect to R . This is another consequence of the invariance of cumulants to constant shifts;see [41] or [40]. For example, with ∂∂X k τ ℓ ( f i ) = ∂ ( f i ) ℓ ∂X k = − ℓ Z d d +1 R (cid:18) ∂ G ∂R k (cid:19) ℓ ( R ) δf i ( R ) , (100) ∂∂X k τ ℓ ( f i , f j ) = − ℓ (cid:26)Z d d +1 R (cid:18) ∂ G ∂R k (cid:19) ℓ ( R ) δf i ( R ) δf j ( R ) − Z d d +1 R (cid:18) ∂ G ∂R k (cid:19) ℓ ( R ) δf i ( R ) Z d R ′ G ℓ ( r ′ ) δf j ( R ′ ) − Z d d +1 R G ℓ ( R ) δf i ( R ) Z d R ′ (cid:18) ∂ G ∂R ′ k (cid:19) ℓ ( R ′ ) δf j ( R ′ ) (cid:27) , (101)and so forth. Using expressions of this type, one obtains bounds of the form: Proposition 4. (cumulant-derivative estimates) For open O ⊂⊂ Γ, n ≥ and ∂ k = ∂/∂X k k ∂ k · · · ∂ k m τ ℓ ( f , . . . , f n ) k p,O = O ℓ − m n Y i =1 k δf i ( ℓ ) k p i ,O ! with 1 p = n X i =1 p i . (102) Assuming f i ∈ B σ i , ∞ p i ,loc ( Γ ) with < σ i ≤ for i = 1 , . . . , n : k ∂ k · · · ∂ k m τ ℓ ( f , . . . , f n ) k p,O = O (cid:16) ℓ − m + P ni =1 σ i (cid:17) . (103)For the “unresolved” or “fluctuation” part of a field f ′ ℓ := f − f ℓ , we have thesimple formula f ′ ℓ ( X ) = − Z d d +1 R G ℓ ( R ) δf ( R ; X ) , (104)which gives Proposition 5. (fluctuation estimates) For open O ⊂⊂ Γ and p ∈ [1 , ∞ ] , k f ′ ℓ k p,O = O ( k δf ( ℓ ) k p,O ) and k f ′ ℓ k p,O = O ( ℓ σ ) when also f ∈ B σ, ∞ p,loc ( Γ ) for < σ ≤ . nsager Singularity Theorem 19 Finally, we will also require estimates on ∆ ℓ h = h ℓ − h ℓ for composite functionsof the form h ( f, g ), where f, g ∈ L ∞ ( Γ ) and h is a smooth function of twovariables. We have the following Lemma: Lemma 2. For p ≥ , let f ∈ ( B σ fp , ∞ p,loc ∩ L ∞ )( Γ ) and g ∈ ( B σ gp , ∞ p,loc ∩ L ∞ )( Γ ) .Let U ⊂ R be open and containing the closed convex hull of R = ess . ran( f, g ) ,the essential range of the measurable function ( f, g ) ∈ L ∞ ( Γ, R ) . Consider H := h ( f, g ) with h ∈ C ( U, R ) . Then H ∈ ( B min { σ fp ,σ gp } , ∞ p,loc ∩ L ∞ )( Γ ) . Proof. Clearly, H ∈ L ∞ ( Γ ) . Since h ∈ C ( U, R ), the mean value theorem gives: δH ( R ; X ) := h ( f ( X + R ) , g ( X + R )) − h ( f ( X ) , g ( X ))= ( δf ( R ; X ) , δg ( R ; X )) · ∂ h ( f ∗ , g ∗ ) (105)for ( f ∗ , g ∗ ) on the line segment joining ( f ( X ) , g ( X )), ( f ( X + R ) , g ( X + R )). Wehave used the notation ∂ = ( ∂/∂f, ∂/∂g ). Since R ⊂ U is compact, then so alsois its closed convex hull conv( R ) ⊂ U and ∂ h is bounded on conv( R ). It followsfor any open O ⊂⊂ Γ, | R | < ℓ O , p ≥ k δH ( R ) k p,O = O (cid:16) | R | min { σ fp ,σ gp } (cid:17) . (cid:3) Corollary 1. Let f, g be as in Lemma 2. Then f g ∈ ( B min { σ fp ,σ gp } , ∞ p,loc ∩ L ∞ )( Γ ) . The estimate on ∆ ℓ h = h ℓ − h ℓ is as follows: Proposition 6. Let h ∈ C ( U ) with f, g, U as in Lemma 2. For open O ⊂⊂ Γ k ∆ ℓ h k p/ ,O = O (cid:16) ℓ { σ fp ,σ gp } (cid:17) , p ≥ Assuming only f, g ∈ L ∞ ( Γ ) , then at least lim ℓ → k ∆ ℓ h k p/ ,O = 0 , ≤ p < ∞ , (107) but without an estimate of the rate.Proof. Using the notation ∂ = ( ∂/∂f, ∂/∂g ), we have: ∆ ℓ h := h ( f, g ) ℓ − h ( f ℓ , g ℓ )= (cid:16) h ( f, g ) ℓ − h ( f, g ) + ( f ′ ℓ , g ′ ℓ ) · ∂ h ( f, g ) (cid:17) + (cid:16) h ( f, g ) − h ( f ℓ , g ℓ ) − ( f ′ ℓ , g ′ ℓ ) · ∂ h ( f, g ) (cid:17) . The first term can be rewritten as h ( f, g ) ℓ − h ( f, g ) + ( f ′ ℓ , g ′ ℓ ) · ∂ h ( f, g )= Z d d +1 R G ℓ ( R ) (cid:16) h ( f ( X + R ) , g ( X + R )) − h ( f ( X ) , g ( X )) − ( δf ( R ; X ) , δg ( R ; X )) · ∂ h ( f ( X ) , g ( X )) (cid:17) = Z d d +1 R G ℓ ( R ) ( ∂∂ ) h | ( f ∗ ,g ∗ ) : ( δf ( R ; X ) , δg ( R ; X ))( δf ( R ; X ) , δg ( R ; X )) , where in the second equality the Taylor theorem with remainder was employedand ( f ∗ , g ∗ ) is defined similarly as in Lemma 2. Likewise, using f = f ℓ + f ′ ℓ , thesecond term can be rewritten as h ( f, g ) − h ( f ℓ , g ℓ ) − ( f ′ ℓ , g ′ ℓ ) · ∂ h ( f, g )= ( ∂∂ ) h | ( f ⋆ ,g ⋆ ) : ( f ′ ℓ , g ′ ℓ )( f ′ ℓ , g ′ ℓ ) , and ( f ⋆ , g ⋆ ) is a point on the line segment connecting ( f ℓ ( X ) , g ℓ ( X )),( f ( X ) , g ( X )) . Note that ( f ℓ ( X ) , g ℓ ( X )) ∈ conv( R ) because the coarse-grained field with anon-negative mollifier G ℓ is a limit of averages of values in ess . ran . ( f, g ) . Thus,( ∂∂ ) h | ( f ⋆ ,g ⋆ ) is uniformly bounded, since ( ∂∂ ) h is bounded on conv( R ) . It fol-lows from the above formulas, the H¨older inequality, and Proposition 5 that k ∆ ℓ h k p/ ,O = O (cid:0) max {k δf ( ℓ ) k p,O , k δg ( ℓ ) k p,O } (cid:1) . (108)The above estimate immediately yields k ∆ ℓ h k p/ ,O = O (cid:16) ℓ { σ fp ,σ gp } (cid:17) assum-ing the appropriate Besov regularity.The final statement of the proposition is obtained from the estimate (108)and the strong continuity of the shift operators in the L p ( O )-norm. (cid:3) One last estimate will be needed: Proposition 7. Let h ∈ C ( U ) with f, g, U as in Lemma 2. For open O ⊂⊂ Γ k∇ x h ℓ k p,O = O (cid:16) ℓ min { σ fp ,σ gp }− (cid:17) , p ≥ . (109) Proof. By the chain rule, ∇ x h = ∂ h ( f ℓ , g ℓ ) · ( ∇ x f ℓ , ∇ x g ℓ ) . Since ( f ℓ , g ℓ ) is inthe closed convex hull of R , one immediately obtains from Proposition 4 that k∇ x h ℓ k p,O = O (cid:18) ℓ max {k δf ( ℓ ) k p,O , k δg ( ℓ ) k p,O } (cid:19) , (110)which gives the claimed estimate for the assumed Besov regularity. (cid:3) 5. Proof of Theorem 1 By assumption u, ̺, v ∈ L ∞ ( Ω × (0 , T )) ⊂ L ploc ( Ω × (0 , T )). We shall obtainestimates in L p ( O ) for any open set O ⊂⊂ Γ . To simplify expressions in theproof, we let O be implicit in this section and everywhere use k · k p to denotethe L p ( O )-norm k · k p,O . Also, all estimates assume ℓ < ℓ O = dist( O, ∂Γ ). Weconsider in order the three balance equations (23)–(25) in Theorem 1. Kinetic Energy: Setting ε = 0 , the coarse-grained kinetic energy balance (68)for compressible Navier-Stokes simplifies, because terms involving T ε vanish: ∂ t (cid:18) ̺ ℓ | ˜ v ℓ | (cid:19) + ∇ x · J vℓ = p ℓ Θ ℓ − Q flux ℓ , (111) nsager Singularity Theorem 21 where the various terms are defined by: J vℓ := (cid:18) ̺ ℓ | ˜ v ℓ | + p ℓ (cid:19) ˜ v ℓ + ̺ ℓ ˜ v ℓ · ˜ τ ℓ ( v , v ) − p ℓ ̺ ℓ τ ℓ ( ̺, v ) , (112) Q flux ℓ := ∇ x p ℓ ̺ ℓ · τ ℓ ( ̺, v ) − ̺ ℓ ∇ x ˜ v ℓ : ˜ τ ℓ ( v , v ) . (113)We now consider the limit as ℓ → u ℓ , ̺ ℓ , v ℓ , p ℓ → u, ̺, v , p strong in L ploc for any 1 ≤ p < ∞ (see e.g.[21], Lemma 7.2 or [22], § v ℓ = v ℓ + τ ℓ ( ̺, v ) /̺ ℓ , (114)which implies for any p ≥ k ˜ v ℓ − v k p ≤ k v ℓ − v k p + k /̺ k ∞ k τ ℓ ( ̺, v ) k p , so that ˜ v ℓ → v strongly as well. Here (99) of Proposition 3 was used. We inferthat ̺ ℓ | ˜ v ℓ | converges to ̺ | v | strong in L ploc for any p ≥ , and thus ∂ t (cid:18) ̺ ℓ | ˜ v ℓ | (cid:19) D ′ −→ ∂ t (cid:18) ̺ | v | (cid:19) (115)as ℓ → . Using the special case of (66)˜ τ ℓ ( v , v ) = τ ℓ ( v , v ) + 1 ̺ ℓ τ ℓ ( ̺, v , v ) − ̺ ℓ τ ℓ ( ̺, v ) τ ℓ ( ̺, v ) , (116)one obtains by exactly similar arguments with Proposition 3 that ∇ x · J vℓ D ′ −→ ∇ x (cid:18) ( 12 ̺ | v | + p ) v (cid:19) . (117)Also, under our assumptions, Q flux ℓ has a distributional limit: Q flux ℓ D ′ −→ Q flux . (118)Thus, all of the terms in (111) except p ℓ Θ ℓ have been proved to have distribu-tional limits as ℓ → . It follows that the limit of p ℓ Θ ℓ also exists and equals − Q flux − ∂ t (cid:0) ̺ | v | (cid:1) − ∇ x (cid:0) ( ̺ | v | + p ) v (cid:1) , independent of choice of G . Thus, p ℓ Θ ℓ D ′ −→ p ◦ Θ (119)which completes the derivation of the kinetic energy balance (23). Internal Energy : From (23), the internal energy constructed as u = E − ̺ | v | ,satisfies (24) distributionally. This could be alternatively deduced by consideringthe ℓ → ε = 0. Entropy : Setting ε = 0 in the intrinsic resolve entropy equation (92), we obtain ∂ t s ∗ ℓ + ∇ x · J s ∗ ℓ = Σ inert ∗ ℓ , (120) for J s ∗ ℓ := J sℓ + β ℓ J kℓ , (121) J sℓ := s ℓ v ℓ + β ℓ τ ℓ ( u, v ) − λ ℓ τ ℓ ( ̺, v ) , (122) J kℓ := 12 ̺ ℓ ˜ τ ℓ ( v i , v i )˜ v ℓ + τ ℓ ( p, v ) + 12 ̺ ℓ ˜ τ ℓ ( v i , v i , v ) , (123)and, with Σ inert ∗ ℓ = − I flux ℓ + Σ flux ∗ ℓ , for I flux ℓ := β ℓ ( p ℓ − p ℓ ) Θ ℓ , (124) Σ flux ∗ ℓ := Σ flux ℓ + β ℓ Q flux ℓ + ∂ t β ℓ k ℓ + ∇ x β ℓ · J kℓ , (125) Σ flux ℓ := ∇ x β ℓ · τ ℓ ( u, v ) − ∇ x λ ℓ · τ ℓ ( ̺, v ) . (126)We next show that ∂ t s ∗ ℓ + ∇ x · J s ∗ ℓ D ′ −→ ∂ t s + ∇ x · ( s v ) as ℓ → . Note that k s ( u ℓ , ̺ ℓ ) − s ( u, ̺ ) k p ≤ k s ( u, ̺ ) ℓ − s ( u, ̺ ) k p + k s ( u, ̺ ) ℓ − s ( u ℓ , ̺ ℓ ) k p . Obviously s ℓ → s strong in L ploc for p ≥ , but also k ∆ ℓ s k p → s ℓ → s strong in L ploc . Also, k β ℓ k ℓ k p → s ∗ ℓ → s strong in L ploc for p ≥ ∂ t s ∗ ℓ D ′ −→ ∂ t s ( u, ̺ ) . Using the formula (116) for ˜ τ ℓ ( u , u ) and the similar formula for ˜ τ ℓ ( u , u , u ) thatfollows from (67), then similar arguments with Propositions 3 and 6 show that J s ∗ ℓ D ′ −→ s v strong in L ploc for p ≥ ∇ x · J s ∗ ℓ D ′ −→ ∇ x · ( s ( u, ̺ ) v ) . We infer from (120) that the distributional limit of Σ inert ∗ ℓ as ℓ → Σ flux := ∂ t s + ∇ x · ( s v ) . Thus, entropy balance (25) holds, with Σ inert ∗ ℓ D ′ −→ Σ flux . (127)This completes the proof of Theorem 1. (cid:3) 6. Proof of Theorem 2 To prove that the strong limits of u ε , ̺ ε , v ε in L ploc ( Γ ) for some 1 ≤ p < ∞ as ε → ε appears the same as (59)–(61) except thatthere is now a factor ε implicitly contained in the terms T ε and q ε whereverthey appear. Our strategy shall be to show that, pointwise in space-time, theseterms indeed vanish as ε → , while all of the other terms in the coarse-grainedNavier-Stokes equation converge pointwise as ε → u, ̺, v . nsager Singularity Theorem 23 Here again, we let the open set O ⊂⊂ Γ be implicit in the estimates below anduse k·k p to represent the L p ( O )-norm. We also assume that ℓ < ℓ O = dist( O, ∂Γ ).We first note that the properties that (i) k f ε k ∞ is bounded uniformly in ε and(ii) f ε → f in L ploc ( Γ ) for 1 ≤ p < ∞ as ε → f ε = u ε ,̺ ε , v ε immediately implies that the same is true for simple product functionssuch as j ε = ̺ ε v ε , ̺ ε | v ε | , ̺ ε | v ε | v ε , etc. For compositions h ε := h ( u ε , ̺ ε )with thermodynamic functions such as h = T, p, µ , η, ζ, κ we need the preciseAssumption 2 on smoothness of h with M = 1. Of course, R ε , R ⊂ K for ε < ε , so that k h ε k ∞ is bounded uniformly for ε < ε and k h k ∞ satisfies thesame bound. Furthermore, we can write h ( u ε ( X ) , ̺ ε ( X )) − h ( u ( X ) , ̺ ( X ))= ∂ h ( u ∗ , ̺ ∗ ) · ( u ε ( X ) − u ( X ) , ̺ ε ( X ) − ̺ ( X )) , (128)where ( u ∗ , ̺ ∗ ) is on the line segment between ( u ε ( X ) , ̺ ε ( X )) and ( u ( X ) , ̺ ( X )).Since ( u ∗ , ̺ ∗ ) ∈ K, then, by Assumption 2, the 2-vector ℓ q -norm | ∂ h ( u ∗ , ̺ ∗ ) | q with q = p/ ( p − 1) is bounded by the maximum value C h,q of | ∂ h | q on K. Itthus follows easily that k h ( u ε , ̺ ε ) − h ( u, ̺ ) k p ≤ C h,q [ k u ε − u k pp + k ̺ ε − ̺ k pp ] /p , (129)so that h ε = h ( u ε , ̺ ε ) also satisfies k h ε − h k p → p as ε → . Thus h ε → h in L ploc ( Γ ). Next note from the identity (100) that ∂∂X k ( f ε − f ) ℓ ( X ) = − ℓ Z d d +1 R (cid:18) ∂ G ∂R k (cid:19) ℓ ( R − X )( f ε ( R ) − f ( R )) , (130)Hence, for each X, | ∂ k ( f ε − f ) ℓ ( X ) | ≤ ( c ℓ,p /ℓ ) k f ε − f k p (131)with c ℓ,p = k ( ∂ G ) ℓ k q for q = p/ ( p − 1) and thus ∂ k ( f ε ) ℓ ( X ) → ∂ k f ℓ as ε → f ε → f in L ploc ( Γ ). Applying this result with f = ̺, j , jv , p, E, ( E + p ) v , we get that pointwise in space-time ∂ t ̺ εℓ + ∇ x · εℓ −→ ∂ t ̺ ℓ + ∇ x · ℓ , (132) ∂ t εℓ + ∇ x · (cid:16) ( j ε v ε ) ℓ + p εℓ I (cid:17) −→ ∂ t ℓ + ∇ x · (cid:16) ( jv ) ℓ + p ℓ I (cid:17) , (133) ∂ t E εℓ + ∇ x · (cid:16) (( E ε + p ε ) v ε ) ℓ (cid:17) −→ ∂ t E ℓ + ∇ x · (cid:16) (( E + p ) v ) ℓ (cid:17) , (134)as ε → . The coarse-grained Euler equations ∂ t ̺ ℓ + ∇ x · ℓ = 0 , (135) ∂ t ℓ + ∇ x · (cid:16) ( jv ) ℓ + p ℓ I (cid:17) = , (136) ∂ t E ℓ + ∇ x · (cid:16) (( E + p ) v ) ℓ (cid:17) = 0 , (137)follow for u, ̺, v if ∇ x · ( T ε ) ℓ , ∇ x · ( T ε · v ε ) ℓ , and ∇ x · ( q ε ) ℓ all vanish as ε → . We first consider the shear-viscosity contribution to ∇· ( T ε ) ℓ . With the short-hand notation η ε ( X ) := εη ( u ε ( X ) , ̺ ε ( X )) , we can bound this using Cauchy-Schwartz inequality as (cid:12)(cid:12)(cid:12) ∇ x · (2 η ε S ε ) ℓ ( X ) (cid:12)(cid:12)(cid:12) = 2 ℓ (cid:12)(cid:12)(cid:12)(cid:12)Z d d +1 R ( ∇ x G ) ℓ ( R ) · η ε ( X + R ) S ε ( X + R ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ sZ supp( G ℓ ) d d +1 R η ε ( X + R ) × Z | ( ∂ G ) ℓ ( R − X ) | Q εη ( dR ) , (138)with Q εη (d R ) = 2 η ε ( R ) | S ( R ) | d d +1 R denoting the kinetic-energy dissipationmeasure for ε > . Finally, because Q εζ ≥ , (cid:12)(cid:12)(cid:12) ∇ x · (2 η ε S ε ) ℓ ( X ) (cid:12)(cid:12)(cid:12) ≤ ℓ sZ supp( G ℓ ) d d +1 R η ε ( X + R ) × Z | ( ∂ G ) ℓ ( R − X ) | Q ε ( dR )(139)with Q ε = Q εη + Q εζ . Since G ℓ ∈ D ( Γ ) implies that S X | ∂ G ℓ | ∈ D ( Γ ) alsowhenever dist( X, ∂Γ ) < ℓ , thenlim ε → Z | ( ∂ G ) ℓ ( R − X ) | Q ε ( dR ) = Z | ( ∂ G ) ℓ ( R − X ) | Q ( dR ) (140)by Assumption 3. On the other hand, because η ( u ε , ̺ ε ) ∈ L ∞ ( Γ ) when η satisfiesthe smoothness Assumption 2 with M = 0 , then the upper bound in (138) isproportional to ε / . Thus, ∇ x · (2 η ε S ε ) ℓ ( X ) → ε → ℓ > dist( X, ∂Γ ).An identical argument using Q εη ≥ ∇ x ( ζ ε Θ ε ) ℓ ( X ) → ε → , and both results together imply that ∇ · ( T ε ) ℓ → ∇ x · ( T ε · v ε ) ℓ can bebounded as (cid:12)(cid:12)(cid:12) ∇ x · (2 η ε S ε · v ε ) ℓ ( X ) (cid:12)(cid:12)(cid:12) = 2 ℓ (cid:12)(cid:12)(cid:12)(cid:12)Z d d +1 R ( ∇ x G ) ℓ ( R ) · η ε ( X + R ) S ε ( X + R ) · v ε ( X + R ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ sZ supp( G ℓ ) d d +1 R η ε ( X + R ) | v ε ( X + R ) | × sZ | ( ∂G ) ℓ ( R − X ) | Q ε ( dR ) , (141)and an analogous bound holds for ∇ x · (2 ζ ε Θ ε v ε ) ℓ . Thus, by Assumption 3 ∇ x · ( T ε · v ε ) ℓ → ε → . Finally, ∇ x · ( q ε ) ℓ = −∇ · ( κ ε ∇ x T ε ) ℓ and the entropy-production measuredue to thermal conductivity is defined by Σ εκ (d R ) = κ ε ( R ) (cid:12)(cid:12)(cid:12) ∇ x T ε ( R ) T ε ( R ) (cid:12)(cid:12)(cid:12) d d +1 R for ε > . Because Q ε /T ε ≥ , thus Σ εκ ≤ Σ ε . Writing κ ε ∇ x T ε = √ κ ε T ε ·√ κ ε ∇ x T ε T ε and using a Cauchy-Schwartz estimate similar to (141), it follows nsager Singularity Theorem 25 from the convergence Σ ε D ′ −→ Σ in Assumption 3 that ∇ x · ( q ε ) ℓ → ε → ℓ > dist( X, ∂Γ ).In conclusion, the coarse-grained Euler equations (135)–(137) hold for all X with dist( X, ∂Γ ) < ℓ and for all ℓ > . By Proposition 1 in section 2, we havethus proved that ( u, ̺, v ) form a weak Euler solution. As an aside, we note thatit would clearly suffice for this statement to have in Assumption 3 only thecondition on entropy-production Σ ε D ′ −→ Σ and not the additional assumption Q ε D ′ −→ Q . If in Theorem 2 only the statement (29) on entropy balance weremade, then this would be more economical in terms of hypotheses. However, toderive the balance equations (27) and (28) we need the additional convergencestatement in Assumption 3 for Q ε as we now show.To derive the balance equations of kinetic energy, internal energy and entropyfor the weak Euler solutions, we start with the corresponding eqs.(9),(10),(16)for compressible Navier-Stokes. Then, because the basic fields u ε , ̺ ε , v ε andtheir compositions with functions h ε := h ( u ε , ̺ ε ) satisfying the smoothness as-sumptions converge strongly in L ploc for some 1 ≤ p < ∞ to the correspondingfields u, ̺, v and h ( u, ̺ ) , it follows directly that ∂ t (cid:18) ̺ ε | v ε | (cid:19) + ∇ x · (cid:18)(cid:18) p ε + 12 ̺ ε | v ε | (cid:19) v ε (cid:19) D ′ −→ ∂ t (cid:18) ̺ | v | (cid:19) + ∇ x · (cid:18)(cid:18) p + 12 ̺ | v | (cid:19) v (cid:19) ,∂ t u ε + ∇ x · ( u ε v ε ) D ′ −→ ∂ t u + ∇ x · ( u v ) ,∂ t s ε + ∇ x · ( s ε v ε ) D ′ −→ ∂ t s + ∇ x · ( s v ) . (142)To see that ∇ x · ( T ε · v ε ) , ∇ x · q ε , ∇ x · (cid:18) q ε T ε (cid:19) D ′ −→ , note that this is equivalent to ∇ x ( T ε · v ε ) ℓ , ∇ x q εℓ , ( q ε /T ε ) ℓ → q ε /T ε = −√ κ ε · √ κ ε ∇ x T ε /T ε . Because of the condition Σ ε D ′ −→ Σ in Assumption 3, all of the terms in theNavier-Stokes entropy balance (16) converge distributionally and thus one ob-tains in the limit ε → Q ε D ′ −→ Q in Assumption 3, all of the termsin the Navier-Stokes kinetic energy and internal energy balances (9),(10) areproved to converge distributionally, except p ε Θ ε . Thus, this term also converges D ′ - lim ε → p ε Θ ε = ∂ t (cid:18) ̺ | v | (cid:19) + ∇ x · (cid:18)(cid:18) p + 12 ̺ | v | (cid:19) v (cid:19) + Q = Q − [ ∂ t u + ∇ x · ( u v )] . With the notation p ∗ Θ := D ′ - lim ε → p ε Θ ε we thus obtain the balances (27),(28)of kinetic and internal energy for the limiting weak Euler solution. (cid:3) 7. Proof of Theorem 3 The strategy to prove Theorem 2 is to use the commutator estimates developedin Section 4 to show that Q flux and Σ flux vanish when the Euler solutions possesssuitable Besov regularity. Then, we use the “inertial-range” expressions (31) toshow the dissipation measures Q and Σ also vanish, and that p ∗ Θ = p ◦ Θ . Weagain make implicit the open set O ⊂⊂ Γ , let k · k p represent the L p ( O )-norm,and assume that ℓ < ℓ O = dist( O, ∂Γ ). Energy Flux: We first show that Q flux defined by (22),(70) necessarily exists andvanishes for weak Euler solutions satisfying the exponent inequalities (35)–(37).To show this, simple bounds can be derived for Q flux ℓ using the expressions (114),(116) and Propositions 3 and 4. One obtains k (1 /̺ ℓ ) ∇ x p ℓ · τ ℓ ( ̺, v ) k p/ = O (cid:18) k /̺ k ∞ ℓ k δp ( ℓ ) k p k δ̺ ( ℓ ) k p k δ v ( ℓ ) k p (cid:19) , p ≥ , k∇ x ˜ v ℓ k p = 1 ℓ k δ v ( ℓ ) k p (cid:2) O (1) + O ( k /̺ k ∞ k ̺ k ∞ ) + O ( k /̺ k ∞ k ̺ k ∞ ) (cid:3) , p ≥ , k ˜ τ ℓ ( v , v ) k p/ = k δ v ( ℓ ) k p (cid:2) O (1) + O ( k /̺ k ∞ k ̺ k ∞ ) + O ( k /̺ k ∞ k ̺ k ∞ ) (cid:3) , p ≥ , and thus k Q flux ℓ k p/ = O (cid:18) ℓ k δp ( ℓ ) k p k δ̺ ( ℓ ) k p k δ v ( ℓ ) k p (cid:19) + O k δ v ( ℓ ) k p ℓ ! , p ≥ . (143)In this latter estimate we absorb the dependence upon the maximum-to-minimummass ratio k /̺ k ∞ k ̺ k ∞ into the constant factor, since this ratio is ℓ -independent.Assuming the Besov regularity of u, ̺, v in Theorem 3 and using Lemma 2 toget the Besov regularity of p , one thus obtains k Q flux ℓ k p/ = O (cid:16) ℓ min { σ up ,σ ̺p } + σ ̺p + σ vp − (cid:17) + O (cid:16) ℓ σ vp − (cid:17) , p ≥ . It follows that2 min { σ up , σ ̺p } + σ vp > , σ vp > , for some p ≥ ⇒ D ′ - lim ℓ → Q flux ℓ = 0 . This is enough to infer the first statement of Theorem 3 that Q flux exists andvanishes for weak Euler solutions, but not enough to conclude that the viscousanomaly vanishes, Q = 0 . Recall by (31) that Q = Q flux + τ ( p, Θ ) . (144)Therefore, with the exponent inequalities assumed above, we can only conclude Q = τ ( p, Θ ) := p ∗ Θ − p ◦ Θ. (145)In order to show that Q = 0 , we must make use of the entropy balance, whichwe consider next. Entropy Anomaly : We show that Σ flux defined by (26) necessarily exists andvanishes for weak Euler solutions satisfying the exponent inequalities (35)–(37). nsager Singularity Theorem 27 To accomplish this, we next derive bounds on Σ inert ∗ ℓ using (124)–(126) andPropositions 3, 4, 6, and 7. Expression (124) and Propositions 4, 6 give: k I flux ℓ k p/ = O (cid:18) ℓ max {k δu ( ℓ ) k p , k δ̺ ( ℓ ) k p } k δ v ( ℓ ) k p (cid:19) . Expression (126) and Propositions 3, 7 give: k Σ flux ℓ k p/ = O (cid:0) k∇ x β ℓ k p k δu ( ℓ ) k p k δ v ( ℓ ) k p (cid:1) + O ( k∇ x λ ℓ k p k δ̺ ( ℓ ) k p k δ v ( ℓ ) k p )= O (cid:18) ℓ max {k δu ( ℓ ) k p , k δ̺ ( ℓ ) k p } k δ v ( ℓ ) k p (cid:19) , (146)while Propositions 3, 7 give for the added terms to Σ flux ∗ ℓ in (125) the estimates k ∂ t β ℓ k ℓ k p/ = O (cid:0) k ∂ t β ℓ k p k δ v ( ℓ ) k p (cid:1) = O (cid:18) ℓ max {k δu ( ℓ ) k p , k δ̺ ( ℓ ) k p }k δ v ( ℓ ) k p (cid:19) , k∇ x β ℓ · J kℓ k p/ = O (cid:0) k∇ x β ℓ k p k δ v ( ℓ ) k p (cid:1) = O (cid:18) ℓ max {k δu ( ℓ ) k p , k δ̺ ( ℓ ) k p }k δ v ( ℓ ) k p (cid:19) . To estimate k ℓ and J kℓ we here used the expressions (114) for ˜ v ℓ , (116) for ˜ τ ℓ ( v , v )and the similar expression for ˜ τ ℓ ( v , v , v ) that follows from (67). Assuming theBesov regularity of u, ̺, v in Theorem 3, one thus obtains from these estimatesand the estimate of β ℓ Q fluxℓ using (143) that for any p ≥ k Σ inert ∗ ℓ k p/ = O (cid:16) ℓ { σ up ,σ ̺p } + σ vp − (cid:17) + O (cid:16) ℓ min { σ up ,σ ̺p } +2 σ vp − (cid:17) + O (cid:16) ℓ σ vp − (cid:17) . The inequalities (35)–(37) thus imply that Σ inert ∗ ℓ → L p/ loc as ℓ → p ≥ . Because of (31), it follows that the non-ideal entropyproduction also vanishes Σ ≡ . Viscous Energy Dissipation Anomaly: We now show that Σ = 0 implies that Q = 0 . First note Σ ε ≥ β ε Q ε ≥ Q ε / k T ε k ∞ . Because k T ε k ∞ by Assumption 1 is bounded by some constant T uniformly in ε < ε , we thus find that Σ ε ≥ Q ε /T ≥ , ε < ε , and one obtains in the limit ε → Σ ≥ Q/T ≥ . Thus, the inequalities (35)–(37) in Theorem 3 for some p ≥ Q ≡ Pressure-Dilatation Defect: Lastly, the result Q = τ ( p, Θ ) in (145) together with Q ≡ p ∗ Θ = p ◦ Θ, as was claimed. (cid:3) 8. Proof of Theorem 4 We derive Theorem 4 from a result for more general balance equations (42).We consider cases where u ∈ L ∞ ( Ω × (0 , T ); R m ) , so that R = ess . ran . ( u ) isa compact subset of R m with K = conv( R ) also compact, and F = F ( u ) isa C function on an open set U, K ⊂ U ⊂ R m . Furthermore, the individualcomponents of F ia of F for i = 1 , . . . , d and a = 1 , . . . , m may not depend uponall of the components u a , a = 1 , . . . , m of u but only upon a subset. We assumethat for each a = 1 , . . . , m the d -vector F a = ( F a , . . . , F da ) is a function of theform F a ( u ) = ˜ F a ( u b ( a )1 , . . . , u b ( a ) ma ) , a = 1 , . . . , m (147)where the subset M a = { b ( a )1 , . . . , b ( a ) m a } ⊂ { , . . . , m } has cardinality m a ≤ m, and thus F a is constant in the variables u b for b / ∈ M a We then have the following general result: Theorem 4* Suppose that u ∈ L ∞ ( Ω × (0 , T ); R m ) is a weak solution of (42)where F ∈ C ( U ) with U open and conv(ess . ran . ( u )) ⊂ U ⊂ R m , and that also F a satisfies the condition (147) for each a = 1 , . . . , m. If for some p ≥ u a ∈ L ∞ ((0 , T ); B σ ap , ∞ p,loc ( Ω )) , < σ ap ≤ a = 1 , . . . , m, (148) where the above spaces are defined by (38), then u a ∈ B ¯ σ ap , ∞ p,loc ( Ω × (0 , T )) , ¯ σ ap = min { σ ap , min b ∈ M a σ bp } ; a = 1 , . . . , m. (149) Proof. We use the notation Γ = Ω × (0 , T ) and R = ( r , τ ) ∈ Γ. Since L ∞ ( Γ ) ⊂ L ploc ( Γ ) and p ≥ , we must only bound the requisite L p ( O )-norm in the def-inition (33) of the local space-time Besov norm for any open O ⊂⊂ Γ . For R = ( r , τ ) with | R | < R O = dist( O, ∂Γ ) , Minkowski’s inequality gives: k u a ( · + R ) − u a k L p ( O ) ≤ k u a ( · , · + τ ) − u a k L p ( O ′ ) + k u a ( · + r , · ) − u a k L p ( O ) (150)where O ′ = S r O := { ( x + r , t ) : ( x , t ) ∈ O } ⊂⊂ Γ . The assumed uniformregularity (148) guarantees that k u a ( · + r , · ) − u a k L p ( O ) = O ( | r | σ ap ). To estimatethe time-increment term, fix an 0 < ℓ ≤ | τ | and decompose u = ˆ u ℓ + u ′ ℓ withˆ u ℓ = u ∗ ˇ G ℓ for a spatial mollifier G ℓ . Applying Minkowski’s inequality again, k u a ( · , · + τ ) − u a k L p ( O ′ ) ≤ k ˆ u a,ℓ ( · , · + τ ) − ˆ u a,ℓ k L p ( O ′ ) + k u ′ a,ℓ ( · , · + τ ) − u ′ a,ℓ k L p ( O ′ ) . (151)In order to estimate these terms, it is convenient to assume that O = O r × O t , aspace-time product of open sets, and thus O ′ = O ′ r × O ′ t as well. It clearly sufficesto consider product sets, because any other pre-compact open set can be strictlyincluded in such a product set. Since ∂ t u a + ∇ x · F a = 0 is satisfied in thesense of distributions or, equivalently, pointwise after space-time mollification(see Proposition 1), standard approximation arguments show: k ˆ u a,ℓ ( · , · + τ ) − ˆ u a,ℓ k L p ( O ′ r × O ′ t ) ≤ | τ |k∇ x · ˆ F a,ℓ k L ∞ ( O ′ t ; L p ( O ′ r )) = O ( ℓ µ ap − | τ | ) , µ ap = min b ∈ M a σ bp . nsager Singularity Theorem 29 Here we have used the inherited spatial Besov regularity of F a with exponent µ ap ,which follows from a straightforward generalization of Lemma 2, and the spatialversion of Proposition 4. On the other hand, the term involving the fluctuationfields can be bounded using the spatial analogue of Proposition 5 as: k u ′ a,ℓ ( · , · + τ ) − u ′ a,ℓ k L p ( O ′ r × O ′ t )) ≤ k u ′ a,ℓ k L ∞ ( O ′ t ) ,L p ( O ′ r )) = O ( ℓ σ ap ) . (152)From equations (151)–(152) we obtain k u a ( · , · + τ ) − u a k L p ( O ′ ) = O ( ℓ µ ap − | τ | ) + O ( ℓ σ ap ) . (153)Since ℓ ≤ | τ | < µ ap and σ ap with their minimum, ¯ σ ap , in (149). The resulting boundis then optimized by choosing the arbitrary scale ℓ ≤ | τ | to be ℓ ∝ | τ | . Altogether, k u a ( · , · + τ ) − u a k L p ( O ′ ) = O ( | τ | ¯ σ ap ) , (154) k u a ( · + r , · ) − u a k L p ( O ) = O ( | r | ¯ σ ap ) . (155)It follows from (150) and (154),(155) that u a ∈ B ¯ σ ap , ∞ p,loc ( Ω × (0 , T )) . (cid:3) Proof (Theorem 4). The result is proved as a corollary of Theorem 4*, specializedto the compressible Euler system with ( u , u , . . . , u d , u d +1 ) := ( ̺, j , . . . , j d , E )and F i, := u i ,F i,j := u − u i u j + p ( u, u ) δ ij ,F i,d +1 := ( u d +1 + p ( u, u )) u − u i . for i, j = 1 , . . . , d and u := u d +1 − u + ··· + u d u . The assumed strict positivity of ̺ ≥ ̺ > 0, space-time boundedness of u , and smoothness of p implies that F possesses the requisite regularity. It follows that: ̺ ∈ B min { σ ̺p ,σ jp } , ∞ p,loc ( Ω × (0 , T )) , j , E ∈ B min { σ ̺p ,σ jp ,σ Ep } , ∞ p,loc ( Ω × (0 , T )) , Recalling that the fields j and E are algebraically related to u, ̺ , v by j := ̺ v and E := ̺ | v | + u , an application of Corollary 1 shows that we may take σ jp = min { σ ̺p , σ vp } and σ Ep = min { σ up , σ ̺p , σ vp } . The inverse relations v = ̺ − j and u = E − ̺ − | j | and another application of Corollary 1 yields the space-timeregularity (40)–(41) claimed in Theorem 3. (cid:3) Remark 18. Theorem 4* applies also to solutions of the incompressible Eulerequations with velocity v and (kinematic) pressure P satisfying v , P ∈ L ∞ ( Γ ) , for Γ = T d × (0 , T ) . Assuming for q ≥ v ∈ L ∞ ((0 , T ) , B σ q , ∞ q ( T d )), ellipticregularization of the solutions of the Poisson equation −△ P = ∂ ( v i v j ) /∂x i ∂x j implies that P ∈ L ∞ ((0 , T ) , B σ q , ∞ q ( T d )). Alternatively, this regularity of P fol-lows from boundedness of Calder´on-Zygmund operators in Besov-space norms.Theorem 4* yields v ∈ B σ q , ∞ q ( T d × (0 , T )) , so that v is as regular in time as itis in space. Acknowledgements. We would like to thank Hussein Aluie for sharing his unpublished work.T.D. would like to thank Daniel Ginsberg for useful discussions. Research of T.D. is partiallysupported by NSF-DMS grant 1703997 and a Fink Award from the Department of AppliedMathematics & Statistics at the Johns Hopkins University. References 1. Onsager, L. Statistical hydrodynamics. Nuovo Cim. Suppl. , VI:279–287,1949.2. Eyink, G. L. Energy dissipation without viscosity in ideal hydrodynamicsi. Fourier analysis and local energy transfer. Physica D , 78(3-4):222–240,1994.3. Constantin, P., Weinan, E., and Titi, E. S. Onsager’s conjecture on theenergy conservation for solutions of Euler’s equation. Commun. Math. Phys. ,165(1):207–209, 1994.4. Duchon, J. and Robert, R. Inertial energy dissipation for weak solutions ofincompressible Euler and Navier-Stokes equations. Nonlinearity , 13(1):249,2000.5. Eyink, G. L. and Sreenivasan, K. R. Onsager and the theory of hydrody-namic turbulence. Rev. Mod. Phys. , 78:87–135, 2006.6. De Lellis, C. and Sz´ekelyhidi, Jr., L. On admissibility criteria for weaksolutions of the Euler equations. Arch. Ration. Mech. and Anal. , 195:225–260, 2010.7. De Lellis, C. and Sz´ekelyhidi Jr, L. The h-principle and the equations offluid dynamics. Bull. Am. Math. Soc , 49(3):347–375, 2012.8. Buckmaster, T. Onsager’s conjecture almost everywhere in time. arXive-prints , April 2013.9. Isett, P. A proof of Onsager’s conjecture. arXiv preprint arXiv:1608.08301 ,2016.10. Feireisl, E., Gwiazda, P., ´Swierczewska-Gwiazda, A., and Wiedemann, E.Regularity and energy conservation for the compressible Euler equations. arXiv preprint arXiv:1603.05051 , 2016.11. Landau, L. and Lifshitz, E. Fluid Mechanics . Pergamon Press, 2nd edition,1987.12. de Groot, S. and Mazur, P. Non-equilibrium Thermodynamics . Dover Pub-lications, 1984.13. Gallavotti, G. Foundations of Fluid Dynamics . Springer Science & BusinessMedia, 2013.14. Feireisl, E. Dynamics of Viscous Compressible Fluids , volume 26. OxfordUniversity Press, 2004.15. Feireisl, E. and Novotn`y, A. Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Commun. Math. Phys. , 321(3):605–628, 2013.16. Lions, P.-L. Mathematical Topics in Fluid Mechanics: Volume 2: Compress-ible Models . Oxford University Press, 1998.17. Martin-L¨of, A. Statistical Mechanics and the Foundations of Thermodynam-ics . Lecture Notes in Physics. Springer-Verlag, 1979.18. Ruelle, D. Statistical Mechanics: Rigorous Results . World Scientific, 1999.19. Callen, H. Thermodynamics and an Introduction to Thermostatistics . Wiley,1985. nsager Singularity Theorem 31 20. Evans, L. C. Entropy and partial differential equations, 2004. URL http://math.berkeley.edu/evans/entropy.and.PDE.pdf .21. Gilbarg, D. and Trudinger, N. S. Elliptic Partial Differential Equations ofSecond Order . Springer, 2015.22. Evans, L. C. and Gariepy, R. F. Measure Theory and Fine Properties ofFunctions . CRC press, 2015.23. Rudin, W. Real and Complex Analysis . McGraw-Hill, 1987.24. Johnson, B. M. Closed-form shock solutions. J. Fluid Mech. , 745:R1, 2014.25. Eyink, G. L. and Drivas, T. D. Cascades and dissipative anomalies in com-pressible fluid turbulence. arXiv preprint arXiv:1704.03532 , 2017.26. Kim, J. and Ryu, D. Density power spectrum of compressible hydrodynamicturbulent flows. Astrophys. J. Lett , 630(1):L45, 2005.27. Oberguggenberger, M. Multiplication of Distributions and Applications toPartial Differential Equations , volume 259 of Pitman research notes in math-ematics series . Longman Scientific & Technical, 1992.28. Triebel, H. Theory of Function Spaces III . Birkh¨auser Basel, 2006.29. Aluie, H. Scale decomposition in compressible turbulence. Physica D , 247(1):54–65, 2013.30. Eyink, G. L. and Drivas, T. D. Cascades and dissipative anomalies in rela-tivistic fluid turbulence. arXiv preprint arXiv:1704.03541 , 2017.31. Isett, P. Regularity in time along the coarse scale flow for the incompressibleEuler equations. arXiv preprint arXiv:1307.0565 , 2013.32. Isett, P. H¨older continuous Euler flows in three dimensions with compactsupport in time. arXiv preprint arXiv:1211.4065 , 2012.33. Isett, P. and Oh, S.-J. On nonperiodic Euler flows with H¨older regularity. Arch. Ration. Mech. Anal. , 221(2):725–804, 2016.34. Ziemer, W. Weakly Differentiable Functions . Graduate Text in Mathematics120, Springer-Verlag, 1989.35. Showalter, R. Hilbert Space Methods in Partial Differential Equations . DoverPublications, 2011.36. Rudin, W. Functional Analysis . McGraw-Hill, 2006.37. Huang, K. Introduction to Statistical Physics . CRC press, 2009.38. Stuart, A. and Ord, K. Kendall’s Advanced Theory of Statistics: Volume 1:Distribution Theory . Wiley, 2009.39. Favre, A. Statistical equations of turbulent gases. In Lavrentiev, M. A.,editor, Problems of Hydrodynamics and Continuum Mechanics , pages 37–44. SIAM, Philadelphia, 1969.40. Eyink, G. L. Turbulent general magnetic reconnection. Astrophys. J , 807(2):137, 2015.41. Eyink, G. L. Turbulence Theory. course notes , 2015. URL ..