TRANSPORTING MICROSTRUCTURE AND DISSIPATIVE EULER FLOWS

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1 TRANSPORTING MICROSTRUCTURE AND DISSIPATIVE EULER FLOWS TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Abstract. Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces arxiv: and arxiv: The motivation comes from Onsager s conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett arxiv: has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett s final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive ε there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class C 1/5 ε. 0. Introduction In what follows T 3 denotes the 3-dimensional torus, i.e. T 3 = S 1 S 1 S 1. In this note we give a proof of the following theorem. Theorem 0.1. Assume e : [0, 1] R is a positive smooth function and ε a positive number. Then there is a continuous vector field v C 1 /5 ε T 3 [0, 1], R 3 and a continuous scalar field p C 2 /5 2ε T 3 [0, 1] which solve the incompressible Euler equations t v + div v v + p = 0 1 div v = 0 in the sense of distributions and such that et = v 2 x, t dx t [0, 1]. 2 Results of this type are associated with the famous conjecture of Onsager. In a nutshell, the question is about whether or not weak solutions in a given regularity class satisfy the law of energy conservation or not. For classical 1

2 2 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. solutions say, v C 1 we can multiply 1 by v itself, integrate by parts and obtain the energy balance vx, t 2 dx = T 3 vx, 0 2 dx T 3 for all t > 0. 3 On the other hand, for weak solutions say, merely v L 2 3 might be violated, and this possibility has been considered for a rather long time in the context of 3 dimensional turbulence. In his famous note [17] about statistical hydrodynamics, Onsager considered weak solutions satisfying the Hölder condition vx, t vx, t C x x θ, 4 where the constant C is independent of x, x T 3 and t. He conjectured that a Any weak solution v satisfying 4 with θ > 1 3 conserves the energy; b For any θ < 1 3 there exist weak solutions v satisfying 4 which do not conserve the energy. This conjecture is also very closely related to Kolmogorov s famous K41 theory [16] for homogeneous isotropic turbulence in 3 dimensions. We refer the interested reader to [14, 18, 13]. Part a of the conjecture is by now fully resolved: it has first been considered by Eyink in [12] following Onsager s original calculations and proved by Constantin, E and Titi in [2]. Slightly weaker assumptions on v in Besov spaces were subsequently shown to be sufficient for energy conservation in [11, 1]. In this paper we are concerned with part b of the conjecture. Weak solutions violating the energy equality have been constructed for a long time, starting with the seminal work of Scheffer and Shnirelman [19, 20]. In [6, 7] a new point of view was introduced, relating the issue of energy conservation to Gromov s h-principle, see also [9]. In [10] and [8] the first constructions of continuous and Hölder-continuous weak solutions violating the energy equality appeared. In particular in [8] the authors proved Theorem 0.1 with Hölder exponent 1/10 ε replacing 1/5 ε. The threshold exponent 1 5 has been recently reached by P. Isett in [15] although strictly speaking he proves a variant of Theorem 0.1, since he shows the existence of nontrivial solutions which are compactly supported in time, rather than prescribing the total kinetic energy. Our aim in this note is to give a shorter proof of Isett s improvement in the Hölder exponent and isolate the main new ideas of [15] compared to [10, 8]. We observe in passing that the arguments given here can be easily modified to produce nontrivial solutions with compact support in time, but losing control on the exact shape of the energy. The question of producing a solution matching an energy profile e which might vanish is subtler. A similar issue has been recently treated in the paper [5] Euler-Reynolds system and the convex integration scheme. Let us recall the main ideas, on which the constructions in [10, 8] are based.

3 DISSIPATIVE EULER FLOWS 3 The proof is achieved through an iteration scheme. At each step q N we construct a triple v q, p q, R q solving the Euler-Reynolds system see [10, Definition 2.1]: t v q + div v q v q + p q = div R q 5 div v q = 0. The size of the perturbation w q := v q v q 1 will be measured by two parameters: q is the amplitude and λ q the frequency. More precisely, denoting the spatial Hölder norms by k see Section A for precise definitions, and similarly, w q 0 M q, 6 w q 1 M, 7 p q p q 1 0 M 2 δ q, 8 p q p q 1 1 M 2 δ, 9 where M is a constant depending only on the function e = et in the Theorem. In constructing the iteration, the new perturbation, w will be chosen so as to balance the previous Reynolds error R q, in the sense that cf. equation 5 we have w w 0 R q 0. This is formalized as R q 0 ηδ, 10 R q 1 Mδ λ q, 11 where η will be a small constant, again only depending on e = et in the Theorem. Estimates of type 6-11 appear already in the paper [8]: although the bound claimed for R 1 1 in the main proposition of [8] is the weaker one R q 1 M cf. [8, Proposition 2.2]: λ q here corresponds to Dδ/ δ 2 1+ε there, this was just done for the ease of notation and the actual bound achieved in the proof does in fact correspond to 11 cf. Step 4 in Section 9. In the language of [15] the estimates 6-11 correspond to the frequency energy levels of order 0 and 1 cf. Definition 9.1 therein. Along the iteration we will have δ q 0 and λ q at a rate that is at least exponential. On the one hand 6, 8 and 10 will imply the convergence of the sequence v q to a continuous weak solution of the Euler equations. On the other hand the precise dependence of λ q on δ q

4 4 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. will determine the critical Hölder regularity. Finally, the equation 2 will be ensured by et1 δ v q 2 x, t dx 1 4 δ et. 12 Note that, being an expression quadratic in v q, this estimate is consistent with 10. As for the perturbation, it will consist essentially of a finite sum of modulated Beltrami modes see Section 1 below, so that w q x, t = k a k x, t φ k x, t B k e iλqk x, where a k is the amplitude, φ k is a phase function i.e. φ k = 1 and B k e iλqk x is a complex Beltrami mode at frequency λ q. Having a perturbation of this form ensures that the oscillation part of the error div w q w q + R q 1 in the equation 5 vanishes, see [10] Isett in [15] calls this term high-high interaction. The main analytical part of the argument goes in to choosing a k and φ k correctly in order to deal with the so-called transport part of the error t w q + v q 1 w q. In [10, 8] a second large parameter = q was introduced to deal with this term. In some sense the role of is to interpolate between errors of order 1 in the transport term and errors of order λ 1 q in the oscillation term. The technique used in [8] for the transport term leads to the Hölder exponent In our opinion the key new idea introduced by Isett is to recognize that the transport error can be reduced by defining a k and φ k in such a way that adheres more closely to the transport structure of the equation. This requires two new ingredients. First, the phase functions φ k are defined using the flow map of the vector field v q, whereas in [8] they were functions of v q itself. With the latter choice, although some improvement of the exponent 1/10 is possible, the threshold 1/5 seems beyond reach. Secondly, Isett introduces a new set of estimates to complement 6-12 with the purpose of controlling the free transport evolution of the Reynolds error: t Rq + v q R q 0 δ. 13 These two ingredients play a key role also in the proof of Theorem 0.1 given here; however, compared to [15], we improve upon the simplicity of their implementation. In order to compare our proof to Isett s proof, it is worth to notice that the parameter corresponds to the inverse of the life-span parameter τ used in [15].

5 DISSIPATIVE EULER FLOWS Improvements. Although the construction of Isett in [15] is essentially based on this same scheme outlined here, there are a number of further points of departure. For instance, Isett considers perturbations with a nonlinear phase rather than the simple stationary flows used here, and consequently, he uses a microlocal version of the Beltrami flows. This also leads to the necessity of appealing to nonlinear stationary phase lemmas. Our purpose here is to show that, although the other ideas exploited in [15] are of independent interest and might also, in principle, lead to better bounds in the future, with the additional control in 13, a scheme much more similar to the one introduced in [10] provides a substantially shorter proof of Theorem 0.1. To this end, however, we introduce some new devices which greatly simplifies the relevant estimates: a We regularize the maps v q and R q in space only and then solve locally in time the free-transport equation in order to approximate R q. b Our maps a k are then elementary algebraic functions of the approximation of R q. c The estimates for the Reynolds stress are still carried on based on simple stationary linear phase arguments. d The proof of 13 is simplified by one commutator estimate which, in spite of having a classical flavor, deals efficiently with one important error term The main iteration proposition and the proof of Theorem 0.1. Having outlined the general idea above, we proceed with the iteration, starting with the trivial solution v 0, p 0, R 0 = 0, 0, 0. We will construct new triples v q, p q, R q inductively, assuming the estimates Proposition 0.2. There are positive constants M and η depending only on e such that the following holds. For every c > 5 2 and b > 1, if a is sufficiently large, then there is a sequence of triples v q, p q, R q starting with v 0, p 0, R 0 = 0, 0, 0, solving 5 and satisfying the estimates 6-13, where δ q := a bq, λ q [a cb, 2a cb ] for q = 0, 1, 2,.... In addition we claim the estimates t v q v q 1 0 C and t p q p q 1 0 Cδ 14 Proof of Theorem 0.1. Choose any c > 5 2 and b > 1 and let v q, p q, R q be a sequence as in Proposition 0.2. It follows then easily that {v q, p q } converge uniformly to a pair of continuous functions v, p such that 1 and 2 hold. We introduce the notation C ϑ for Hölder norms in space and time. From 6-9, 14 and interpolation we conclude v v q C ϑ M λϑ Ca b 2cbϑ 1/2 15 p p q C 2ϑ M 2 δ λ 2ϑ Ca b 2cbϑ Thus, for every ϑ < 1 2bc, v q converges in C ϑ and p q in C 2ϑ.

6 6 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR Plan of the paper. In the rest of the paper we will use D and for differentiation in the space variables and t for differentiation in the time variable. After recalling in Section 1 some preliminary notation from the paper [10], in Section 2 we give the precise definition of the maps v, p, R assuming the triple v q, p q, R q to be known. The Sections 3, 4 and 5 will focus on estimating, respectively, w = v v q, v 2 x, t dx and R. These estimates are then collected in Section 6 where Proposition 0.2 will be finally proved. The appendix collects several technical and, for the most part, well-known estimates on the different classical PDEs involved in our construction, i.e. the transport equation, the Poisson equation and the bilaplace equation Acknowledgements. We wish to thank Phil Isett for several very interesting discussions and suggestions for improvements on this manuscript. T.B. and L.Sz. acknowledge the support of the ERC Grant Agreement No , C.dL. acknowledges the support of the SNF Grant Preliminaries 1.1. Geometric preliminaries. In this paper we denote by R n n, as usual, the space of n n matrices, whereas S n n and S0 n n denote, respectively, the corresponding subspaces of symmetric matrices and of trace-free symmetric matrices. The 3 3 identity matrix will be denoted with Id. For definitiveness we will use the matrix operator norm R := max v =1 Rv. Since we will deal with symmetric matrices, we have the identity R = max v =1 Rv v. Proposition 1.1 Beltrami flows. Let λ 1 and let A k R 3 be such that A k k = 0, A k = 1 2, A k = A k for k Z 3 with k = λ. Furthermore, let B k = A k + i k k A k C 3. For any choice of a k C with a k = a k the vector field W ξ = a k B k e ik ξ 17 k = λ is real-valued, divergence-free and satisfies Furthermore W W = W 2 div W W = T 3 W W dξ = 1 2 k = λ a k 2 Id k k k. 19 k

7 DISSIPATIVE EULER FLOWS 7 The proof of 18, which is quite elementary and can be found in [10], is based on the following algebraic identity, which we state separately for future reference: Lemma 1.2. Let k, k Z 3 with k = k = λ and let B k, B k C 3 be the associated vectors from Proposition 1.1. Then we have B k B k + B k B k k + k = B k B k k + k. Proof. The proof is a straight-forward calculation. Indeed, since B k k = B k k = 0, we have B k B k +B k B k k + k = B k kb k + B k k B k = B k k B k B k k B k + B k B k k + k = i λb k B k + B k B k + B k B k k + k, where the last equality follows from k B k = i λb k and k B k = i λb k. Another important ingredient is the following geometric lemma, also taken from [10]. Lemma 1.3 Geometric Lemma. For every N N we can choose r 0 > 0 and λ > 1 with the following property. There exist pairwise disjoint subsets and smooth positive functions Λ j {k Z 3 : k = λ} j {1,..., N} γ j k C B r0 Id j {1,..., N}, k Λ j such that a k Λ j implies k Λ j and γ j k = γ j k ; b For each R B r0 Id we have the identity R = 1 γ j 2 2 k R Id k k k R B r0 Id. 20 k k Λ j 1.2. The operator R. Following [10], we introduce the following operator in order to deal with the Reynolds stresses. Definition 1.4. Let v C T 3, R 3 be a smooth vector field. We then define Rv to be the matrix-valued periodic function Rv := 1 Pu + Pu T + 3 u + u T 1 div uid, where u C T 3, R 3 is the solution of u = v T 3 v in T 3

8 8 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. with ffl T 3 u = 0 and P is the Leray projection onto divergence-free fields with zero average. Lemma 1.5 R = div 1. For any v C T 3, R 3 we have a Rvx is a symmetric trace-free matrix for each x T 3 ; b div Rv = v ffl T 3 v. 2. The inductive step In this section we specify the inductive procedure which allows to construct v, p, R from v q, p q, R q. Note that the choice of the sequences {δ q } q N and {λ q } q N specified in Proposition 0.2 implies that, for a sufficiently large a > 1, depending only on b > 1 and c > 5/2, we have: δ j λ j 2δ, j q 1 j q j λ j 2, δ j j j j Since we are concerned with a single step in the iteration, with a slight abuse of notation we will write v, p, R for v q, p q, R q and v 1, p 1, R 1 for v, p, R. Our inductive hypothesis implies then the following set of estimates: and v 0 2M, v 1 2M, 22 R 0 ηδ, R 1 Mδ λ q, 23 p 0 2M 2, p 1 2M 2 δ, 24 t + v R 0 Mδ. 25 The new velocity v 1 will be defined as a sum v 1 := v + w o + w c, where w o is the principal perturbation and w c is a corrector. The principal part of the perturbation w will be a sum of Beltrami flows w o t, x := k =λ 0 a k φ k B k e iλk x, where B k e iλ k x is a single Beltrami mode at frequency λ, with phase shift φ k = φ k t, x i.e. φ k = 1 and amplitude a k = a k t, x. In the following subsections we will define a k and φ k.

9 DISSIPATIVE EULER FLOWS Space regularization of v and R. We fix a symmetric non-negative convolution kernel ψ Cc R 3 and a small parameter l whose choice will be specified later. Define v l := v ψ l and R l := R ψ l, where the convolution is in the x variable only. Standard estimates on regularizations by convolution lead to the following: v v l 0 C l, 26 R R l 0 C δ l, 27 and for any N 1 there exists a constant C = CN so that v l N C l 1 N, 28 R l N C δ λ q l 1 N Time discretization and transport for the Reynolds stress. Next, we fix a smooth cut-off function χ Cc 3 4, 3 4 such that χ 2 x l = 1, l Z and a large parameter N \ {0}, whose choice will be specified later. For any l [0, ] we define ρ l := 1 32π 3 el 1 1 δ q+2 v 2 x, l 1 dx. T 3 Note that 12 implies 1 32π 3 el δ δ q+2 ρ l 1 32π 3 el δ δ q+2. We will henceforth assume so that we obtain δ q δ, C 1 0 min eδ ρ l C 0 max eδ, 30 where C 0 is an absolute constant. Finally, define R l,l to be the unique solution to the transport equation { t Rl,l + v l R l,l = 0 R l,l x, l = R l x, l. 31 and set R l,l x, t := ρ l Id R l,l x, t. 32

10 10 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR The maps v 1, w, w o and w c. We next consider v l as a 2π-periodic function on R 3 [0, 1] and, for every l [0, ], we let Φ l : R 3 [0, 1] R 3 be the solution of t Φ l + v l Φ l = 0 33 Φ l x, l 1 = x Observe that Φ l, t is the inverse of the flow of the periodic vector-field v l, starting at time t = l 1 as the identity. Thus, if y 2πZ 3, then Φ l x, t Φ l x+y, t 2πZ 3 : Φ l, t can hence be thought as a diffeomorphism of T 3 onto itself and, for every k Z 3, the map T 3 [0, 1] x, t e iλ k Φ l x,t is well-defined. We next apply Lemma 1.3 with N = 2, denoting by Λ e and Λ o the corresponding families of frequencies in Z 3, and set Λ := Λ o + Λ e. For each k Λ and each l Z [0, ] we then set χ l t := χ t l a kl x, t := ρ l γ k Rl,l x, t ρ l, 34, 35 w kl x, t := a kl x, t B k e iλ k Φ l x,t. 36 The principal part of the perturbation w consists of the map w o x, t := l odd,k Λ o χ l tw kl x, t + l even,k Λ e χ l tw kl x, t. 37 From now on, in order to make our notation simpler, we agree that the pairs of indices k, l Λ [0, ] which enter in our summations satisfy always the following condition: k Λ e when l is even and k Λ o when l is odd. It will be useful to introduce the phase with which we obviously have φ kl x, t = e iλ k [Φ l x,t x], 38 φ kl e iλ k x = e iλ k Φ l. Since R l,l and Φ l are defined as solutions of the transport equations 31 and 33, we have t + v l a kl = 0 and t + v l e iλ k Φ l x,t = 0, 39 hence also t + v l w kl = 0. 40

11 DISSIPATIVE EULER FLOWS 11 The corrector w c is then defined in such a way that w := w o +w c is divergence free: w c := χ l k B k curl ia kl φ kl λ k 2 e iλ k x kl = i χ l a kl a kl DΦ l Idk k B k λ k 2 e iλ k Φ l 41 kl Remark 1. To see that w = w o +w c is divergence-free, just note that, since k B k = 0, we have k k B k = k 2 B k and hence w can be written as w = 1 k B k χ l curl ia kl φ kl λ k 2 e iλ k x 42 k,l For future reference it is useful to introduce the notation i L kl := a kl B k + a kl a kl DΦ l Idk k B k λ k 2, 43 so that the perturbation w can be written as w = χ l L kl e iλ k Φ l. 44 kl Moreover, we will frequently deal with the transport derivative with respect to the regularized flow v l of various expressions, and will henceforth use the notation D t := t + v l Determination of the constants η and M. In order to determine η, first of all recall from Lemma 1.3 that the functions a kl are well-defined provided R l,l Id r 0, ρ l where r 0 is the constant of Lemma 1.3. Recalling the definition of R l,l we easily deduce from the maximum principle for transport equations cf. 112 in Proposition B.1 that R l,l 0 R 0. Hence, from 10 and 30 we obtain R l,l Id ρ l C η 0 min e, and thus we will require that η C 0 min e r 0 4. The constant M in turn is determined by comparing the estimate 6 for q + 1 with the definition of the principal perturbation w o in 37. Indeed, using and 30 we have w o 0 C 0 Λ max e. We therefore set M = 2C 0 Λ max e,

12 12 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. so that w o 0 M 2 δ1 / The pressure p 1 and the Reynolds stress R 1. We set where R 1 = R 0 + R 1 + R 2 + R 3 + R 4 + R 5, R 0 = R t w + v l w + w v l 46 R 1 = Rdiv w o w o χ 2 l R l,l wo 2 2 Id 47 l R 2 = w o w c + w c w o + w c w c wc 2 +2 w o,w c 3 Id 48 R 3 = w v v l + v v l w 2 v v l,w 3 Id 49 R 4 = R R l 50 R 5 = l χ 2 l R l R l,l. 51 Observe that R 1 is indeed a traceless symmetric tensor. The corresponding form of the new pressure will then be p 1 = p w o w c w o, w c 2 3 v v l, w. 52 Recalling 32 we see that l χ2 l tr R l,l is a function of time only. Since also l χ2 l = 1, it is then straightforward to check that div R 1 p 1 = t w + div v w + w v + w w + div R p = t w + div v w + w v + w w + t v + div v v = t v 1 + div v 1 v 1. The following lemma will play a key role. Lemma 2.1. The following identity holds: w o w o = χ 2 l R l,l + χ l χ l w kl w k l. 53 l k,l,k,l,k k Proof. Recall that the pairs k, l, k, l are chosen so that k k if l is even and l is odd. Moreover χ l χ l = 0 if l and l are distinct and have the same parity. Hence the claim follows immediately from our choice of a kl in 35 and Proposition 1.1 and Lemma 1.3 cf. [10, Proposition 6.1ii].

13 DISSIPATIVE EULER FLOWS Conditions on the parameters - hierarchy of length-scales. In the next couple of sections we will need to estimate various expressions involving v l and w. To simplify the formulas that we arrive at, we will from now on assume the following conditions on, λ 1 and l 1: l 1, + 1 λ β and lλ 1/2 1 δ λ. 54 These conditions imply the following orderings of length scales, which will be used to simplify the estimates in Section 3: 1 λ 1 1 and 1 λ l 1 λ q. 55 One can think of these chains of inequalities as an ordering of various length scales involved in the definition of v 1. Remark 2. The most relevant and restrictive condition is δ1/2 q 1 λ q. Indeed, this condition can be thought of as a kind of CFL condition cf. [4], restricting the coarse-grained flow to times of the order of v 1 0, cf. Lemma 3.1 and in particular 57 below. Assuming only this condition on the parameters, essentially all the arguments for estimating the various terms would still follow through. The remaining inequalities are only used to simplify the many estimates needed in the rest of the paper, which otherwise would have a much more complicated dependence upon the various parameters. 3. Estimates on the perturbation Lemma 3.1. Assume 54 holds. For t in the range t l < 1 we have Moreover, DΦ l 0 C 56 DΦ l Id 0 C δ1 /2 57 DΦ l N C δ1 /2 l N, N 1 58 a kl 0 + L kl 0 C 59 a kl N C λ ql 1 N, N 1 60 L kl N C l N, N 1 61 φ kl N Cλ l N 1 + C N λ Cλ N1 β N 1. 62

14 14 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Consequently, for any N 0 w c N C λn, 63 w o 1 M 2 δ1 /2 λ + C λ1 β 64 w o N C λn, N 2 65 where the constants in depend only on M, the constant in 58 depends on M and N, the constants in 59 and 64 depend on M and e and the remaining constants depend on M, e and N. Proof. The estimates 56 and 57 are direct consequences of 115 in Proposition B.1, together with 55, whereas 116 in Proposition B.1 combined with the convolution estimate 28 implies 58. Next, 29 together with 112,113 and 114 in Proposition B.1 and 55 leads to R l,l 0 Cδ, 66 R l,l N Cδ λ q l 1 N, N The estimate 59 is now a consequence of 66, 57 and 30, whereas by 109 we obtain a kl N Cδ 1 /2 R l,l N C λ ql 1 N C l N. 68 Similarly we deduce 61 from L kl N C a kl N + Cλ 1 a kl N C a kl N DΦ l Id 0 + a kl 0 DΦ l N and once again using 55. In order to prove 62 we apply 110 with m = N to conclude φ kl N Cλ DΦ l N 1 + λ N DΦ l Id N 0, from which 62 follows using 57, 58 and 54. Using the formula 41 together with 57, 58, 59 and 61 we conclude w c 0 C a kl 1 + C a kl 0 DΦ l Id 0 C δ λ 1/2 q λ q

15 DISSIPATIVE EULER FLOWS 15 and, for N 1, w c N C 1 χ l a kl N+1 + a kl 0 DΦ l N + a kl N DΦ l Id 0 λ kl + C w c 0 χ l λ N DΦ l N 0 + λ DΦ l N 1 55 C λn l λ q + δ 1/2 λ C δ1 /2 λn. This proves 63. The estimates for w o follow analogously, using in addition the choice of M and 45. Lemma 3.2. Recall that D t = t +v l. Under the assumptions of Lemma 3.1 we have Proof. Estimate on D t v l. transport equation D t v l N Cδ l N, 69 D t L kl N C δ1 /2 l N, 70 D 2 t L kl N C δ qλ q l N 1, 71 D t w c N C δ1 /2 λ N, 72 D t w o N C λn. 73 Note that v l satisfies the inhomogeneous t v l + v l v l = p ψ l + div R l v v ψ l + v l v l. By hypothesis p ψ l N C p 1 l N Cδ l N and analogously div R ψ l Cδ λ q l N. On the other hand, by Proposition C.1: div v v ψ l v l v l N Cl 1 N v 2 1 Cl 1 N δ q λ 2 q. Thus 69 follows from 55. Estimates on L kl. Recall that L kl is defined as i L kl := a kl B k + a kl a kl DΦ l Idk λ Using that D t a kl = 0, D t Φ l = 0, D t a kl = Dv T l a kl, D t DΦ l = DΦ l Dv l, we obtain D t L kl = i Dvl T λ a kl + a kl DΦ l Dv l k k B k k 2. k B k k 2. 74

16 16 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Consequently, for times t l < 1 and N 0 we have D t L kl N C δ1 /2 l N λ q + λ q l + δ 1/2 q λ λ q + 1 C δ1 /2 l N, where we have used 107, Lemma 3.1 and 55. Taking one more derivative and using 74 again, we obtain Dt 2 L kl = i D t Dv l T a kl + i Dvl T λ λ DvT l a kl+ a kl DΦ l Dv l Dv l k + a kl DΦ l D t Dv l k k B k k 2. Note that D t Dv l = DD t v l Dv l Dv l, so that D t Dv l N D t v l N+1 + Dv l N Dv l 0 Cδ l N λ q l Cδ l N 1. It then follows from the product rule 107 and 55 that Dt 2 L kl N C δ qλ q l N 1 λ q + λ2 ql + λ q l + λ q l 2 + δ 1/2 q λ λ λ q + 1 C δ qλ q l N 1. Estimates on w c. Observe that w c = χ l L kl a kl B k e iλ k Φ l see 41 and 43. Differentiating this identity we then conclude D t w c = kl χ l D t L kl e iλ k Φ l + t χ l L kl a kl B k e iλ k Φ l = kl + kl χ l D t L kl φ kl e iλ k x + i akl t χ l a kl DΦ l Id k k B k λ k 2 φ kl e iλk x. Hence we obtain 72 as a consequence of Lemma 3.1 and 70. Estimates on w o. Using 74 we have D t w o = k,l χ l a klφ kl e iλ k x. Therefore 73 follows immediately from Lemma 3.1.

17 DISSIPATIVE EULER FLOWS Estimates on the energy Lemma 4.1 Estimate on the energy. et1 δ q+2 v 1 2 dx 1 T 3 + C δ δ 1/2 q λ q + C δ1 /2 δ1 /2 λ. 75 Proof. Define ēt := 32π 3 χ 2 l tρ l. l Using Lemma 2.1 we then have w o 2 = χ 2 l tr R l,l + χ l χ l w kl w k,l l k,l,k,l,k k = 2π 3 ē + χ l χ k a kl a k l φ klφ k l eiλk+k x. 76 k,l,k,l,k k Observe that ē is a function of t only and that, since k + k 0 in the sum above, we can apply Proposition E.1i with m = 1. From Lemma 3.1 we then deduce w o 2 dx ēt C δ δ 1/2 + C δ λ q. 77 T 3 λ Next we recall 42, integrate by parts and use 59 and 62 to reach v w dx C δ1/2 δ1 /2. 78 T 3 λ Note also that by 63 we have 1/2 w c 2 + w c w o dx C δ δ T Summarizing, so far we have achieved v 1 2 dx ēt + v 2 78 dx w 2 dx ēt + C δ1/2 δ1 /2 T 3 T 3 T 3 λ 79 w o 2 dx ēt + C δ1/2 δ1 /2 + C δ 1/2 δq λ q T 3 λ 77 C δ1 /2 δ1 /2 λ Next, recall that + C δ δ 1/2. 80 ēt = 32π 3 l = 1 δ q+2 l χ 2 l ρ l χ 2 l e l l χ 2 l T 3 vx, l 1 2 dx.

18 18 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Since t l < 1 on the support of χ l and since l χ2 l = 1, we have et l χ 2 l e l 1. Moreover, using the Euler-Reynolds equation, we can compute = T 3 t vx, t 2 v x, l 1 2 dx = l T 3 div v v 2 + 2p + 2 Thus, for t l 1 we conclude t l t l T 3 t v 2 T 3 v div R = 2 vx, t 2 vx, l 1 2 dx C δ δ 1/2 q T 3 Using again χ 2 l = 1, we then conclude et1 δ q+2 ēt + vx, t 2 dx 1 T 3 t l λ q. + C δ δ 1/2 q T 3 Dv : R. λ q. 81 The desired conclusion 75 follows from 80 and Estimates on the Reynolds stress In this section we bound the new Reynolds Stress R 1. The general pattern in estimating derivatives of the Reynolds stress is that: the space derivative gets an extra factor of λ when the derivative falls on the exponential factor, the transport derivative gets an extra factor when the derivative falls on the time cut-off. In fact the transport derivative is slightly more subtle, because in R 0 a second transport derivative of the perturbation w appears, which leads to an additional term see 92. Nevertheless, we organize the estimates in the following proposition according to the above pattern. Proposition 5.1. For any choice of small positive numbers ε and β, there is a constant C depending only upon these parameters and on e and M

19 DISSIPATIVE EULER FLOWS 19 such that, if, λ and l satisfy the conditions 54, then we have Thus R R λ D tr 0 0 C δ 1/2 λ 1 ε R R λ D tr 1 0 C δ 1/2 δ λ ε R R λ D tr 2 0 C δ δ 1/2 q + δ1 /2 δ qλ q λ 1 ε l 82 λ q R λ R D tr 3 0 C δ1 /2 l 85 R R λ D tr 4 0 C δ δ R R λ D tr 5 0 C δ δ 1/2 q 1/2 q λ q λ q + Cδ λ q l R R λ D t R 1 0 1/2 δ C λ 1 ε + δ λ ε + δ1 /2 l + δ1/2 δ qλ q λ 1 ε l, and, moreover, 88 t R1 + v 1 R 1 0 1/2 C δ λ λ 1 ε + δ λ ε + δ1 /2 l + δ1/2 δ qλ q λ 1 ε l. 89 Proof. Estimates on R 0. We start by calculating t w + v l w + w v l = χ l L kl + χ l D t L kl + χ l L kl v l e ik Φ l. kl Define Ω kl := χ l L kl + χ l D t L kl + χ l L kl v l φ kl and write recalling the identity φ kl e iλ k x = e iλ k Φ l, t w + v l w + w v l = kl Using Lemmas 3.1 and 3.2 and 55 Ω kl 0 C 1 + δ1 /2 q and similarly, for N 1 λ q Ω kl e iλ k x. 90 C Ω kl N C l N + φ kl N Cδ 1/2 λn1 β.

20 20 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Moreover, observe that although this estimate has been derived for N integer, by the interpolation inequality 108 it can be easily extended to any real N 1 besides, this fact will be used frequently in the rest of the proof. Applying Proposition E.1ii we obtain R 0 0 λ ε 1 Ω kl 0 + λ N+ε [Ω kl ] N + λ N [Ω kl] N+ε kl C λ 1+ε + λ Nβ+ε It suffices to choose N so that Nβ 1 in order to achieve R 0 0 C λε 1. As for R 0 1, we differentiate 90. We therefore conclude j R 0 = R iλ k j Ω kl + j Ω kl e iλ k x. kl Applying Proposition E.1ii as before we conclude R 0 1 C λε. Estimates on D t R 0. We start by calculating D t t w + v l w + w v l = t 2 χ l L kl + 2 t χ l D t L kl + χ l Dt 2 L kl + kl 91 + t χ l L kl v l + χ l D t L kl v l + χ l L kl D t v l χ l L kl v l v l e ik Φ l =: kl Ω kl eiλ k x. As before, we have Ω kl 0 C + δ qλ q l + δ1 /2 + δ qλ 2 q C + δ qλ q l 92 and, for any N 1 Ω kl N C l N + δ qλ q l + δ1 /2 + δ qλ 2 q C + δ qλ q l N + φ kl N l C + δ qλ q l λ N1 β. + Ω kl 0 φ kl N

21 DISSIPATIVE EULER FLOWS 21 Next, observe that we can write D t R 0 = [D t, R] + RD t t w + v l w + w v l = [v l, R] + RD t t w + v l w + w v l = kl [v l, R] Ω kl e iλ k x + iλ [v l k, R]Ω kl e iλ k x + RΩ kl eiλ k x where, as it is customary, [A, B] denotes the commutator AB BA of two operators A and B. Using the estimates for Ω kl derived above, and applying Proposition E.1ii, we obtain RΩ kl eiλ k x 0 Ω kl 0 λ 1 ε C δ1 /2 λ 1 ε + δ qλ q l + [Ω kl ] N λ N ε 1 + λ 1 Nβ Furthermore, applying Proposition F.1 we obtain + [Ω kl ] N+ε λ N [v l, R] Ω kl e iλk x C Ω kl N+1+ε v l 2+ε + Ω kl 3+ε v l N+ε 0 C δ1 /2 + C Ω kl N+1 v l 2 + Ω kl 3 v l N λ N ε λ 1 ε and similarly 1 + λ 3 β βn C δ1 /2 2 λ N + C Ω kl 1 v l 1 λ 2 ε λ 1 ε [vl λ k, R]Ω kl e iλk x C δ1 /2 0 By choosing N N sufficiently large so that 2 λ 1 ε βn and βn 1, 1 + λ 3 βn λ 3 βn+1. we deduce D t R 0 0 C δ1 /2 λ 1 ε + δ qλ q l This concludes the proof of 82.

22 22 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Estimates on R 1. Using Lemma 2.1 we have div w o w o χ 2 R l l,l w o 2 2 Id = l = χ l χ l div w kl w k l w kl w k l 2 k,l,k,l k+k 0 Id = I + II where, setting f klk l := χ lχ l a kl a k l φ klφ k l, I = Bk B k 1 2 B k B k Id f klk l eiλ k+k x k,l,k,l k+k 0 II =iλ k,l,k,l k+k 0 f klk l Bk B k 1 2 B k B k Id k + k e iλ k+k x. Concerning II, recall that the summation is over all l Z [0, ] and all k Λ e if l is even and all k Λ o if l is odd. Furthermore, both Λ e, Λ o λs 2 Z 3 satisfy the conditions of Lemma 1.3. Therefore we may symmetrize the summand in II in k and k. On the other hand, recall from Lemma 1.2 that B k B k + B k B k k + k = B k B k k + k. From this we deduce that II = 0. Concerning I, we first note, using the product rule, 59 and 60, that [f klk l ] N Cδ λq l 1 N + φ kl φ k l N for N 1. By Lemma 62 and 54 cf. 55 we then conclude [f klk l ] 1 Cδ λ q + λ Cδ λ, [f klk l ] N Cδ λ N1 β, N 2. Applying Proposition E.1ii to I we obtain R 1 0 λ ε 1 [f klk l ] 1 + λ N+ε [f klk l ] N+1 + λ N [f klk l ] N+1+ε k,l,k,l k+k 0 Cδ λ ε + λ 1 Nβ+ε By choosing N sufficiently large we deduce R 1 0 C δ λ ε 93

23 DISSIPATIVE EULER FLOWS 23 as required. Moreover, differentiating we conclude j R 1 = R j I where j I = Bk B k 1 2 B k B k Id k,l,k,l k+k 0 iλ k + k j f klk l + j f klk l eiλ k+k x. 94 Therefore we apply again Proposition E.1ii to conclude the desired estimate for R 1 1. Estimates on D t R 1. As in the estimate for D t R 0, we again make use of the identity D t R = [v l, R] + RD t in order to write D t R 1 = [v l, R] U klk l eiλ k+k x k,l,k,l k+k 0 + iλ [v l k + k, R] U klk l eiλ k+k x + R U klk l eiλ k+k x, where we have set U klk l = B k B k 1 2 B k B k Id f klk l D t div w o w o l χ 2 l R l,l w o 2 2 Id = In order to further compute U klk l, we write k,l,k,l k+k 0 and U klk l eiλ k+k x. f klk l eiλ k+k x = χ l χ l a kl a k l + a k l a kl e iλ k Φ l +k Φ l + + iλ χ l χ l a kl a k l DΦl Idk + DΦ l Idk e iλ k Φ l +k Φ l and hence, using 74, D t f klk l eiλ k+k x = χ l χ l a kl a k l + a k l a kl e iλ k Φ l +k Φ l + iλ χ l χ l a kl a k l DΦl Idk + DΦ l Idk e iλ k Φ l +k Φ l χ l χ l akl Dvl T a k l + a k l DvT l a kl e iλ k Φ l +k Φ l χ l χ l a kl a k l DΦl Dv T l k + DΦ l DvT l k e iλ k Φ l +k Φ l =: Σ 1 klk l + Σ2 klk l + Σ3 klk l + Σ4 klk l e iλ k Φ l +k Φ l =: Σ klk l eiλ k Φ l +k Φ l. Ignoring the subscripts we can use 107, Lemma 3.1 and Lemma 3.2 to estimate Σ N Cδ λ q l N + λ q + + q 55 Cδ λ l N. We thus conclude U klk l N C Σ klk l N + C Σ klk l 0 φ kl φ k l N Cδ λ l N + λ N1 β Cδ λ 1+N1 β

24 24 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. The estimate on D t R 1 0 now follows exactly as above for D t R 0 applying Proposition F.1 to the commutator terms. This concludes the verification of 83. Estimates on R 2 and D t R 2. Using Lemma 3.1 we have R 2 0 C w c w o 0 w c 0 Cδ, R 2 1 C w c 1 w c 0 + w o 1 w c 0 + w o 0 w c 1 Cδ λ. Similarly, with the Lemmas 3.1 and 3.2 we achieve D t R 2 0 C D t w c 0 w o 0 + w c 0 + C D t w o 0 w c 0 Cδ. Estimates on R 3 and D t R 3. The estimates on R 3 0 and R 3 1 are a direct consequence of the mollification estimates 26 and 28 as well as Lemma 3.1. Moreover, D t R 3 0 v v l 0 D t w 0 + D t v 0 + D t v l w 0 = v v l 0 D t w c 0 + D t w o + D t v 0 + D t v l w 0 95 Concerning D t v, note that, by our inductive hypothesis D t v 0 t v + v v 0 + v v l 0 v 1 p q 1 + R q 1 + Cδ q λ 2 ql Cδ. Thus the required estimate on D t R 3 follows from Lemma 3.2. Estimates on R 4 and D t R 4. From the mollification estimates 27 and 29 we deduce R 4 0 C R 1 l Cδ λ q l R R 1 Cδ λ q. As for D t R 4, observe first that, using our inductive hypothesis, D t R 0 t R + v R 0 + v l v 0 R 1 Cδ + Cδ q λ 2 ql Moreover, D t Rl = D t R ψl + v l R l v l R ψ l =D t R ψl + div v l R l v R ψ l +[v v l R] ψ l, 96 where we have used that div v = 0. Using Proposition C.1 we deduce v l R l v R ψ l 1 Cδ λ q l. 97 Gathering all the estimates we then achieve D t R 4 0 D t R 0 + D t Rl 0 Cδ λ q q. The estimate 86 follows now using 55.

25 DISSIPATIVE EULER FLOWS 25 Estimates on R 5. Recall that D t Rl,l = 0. Therefore, using the arguments from 96 D t R l R l,l 0 = D t Rl 0 Cδ. On the other hand, using again the identity 96 and Proposition C.1 D t R l R l,l 1 = D t Rl 1 Cl 1 D t R 0 + v l R l v R ψ l 2 + Cl 1 v v l 0 R 1 Cδ l 1. Since R l,l x, t 1 = R l x, t 1, the difference R l R l,l vanishes at t = l 1. From Proposition B.1 we deduce that, for times t in the support of χ l i.e. t l 1 < 1, R l R l,l 0 C 1 D t R l R l,l 0 C 1 δ R l R l,l 1 C 1 D t R l R l,l 1 C 1 δ l 1. The desired estimates on R 5 0 and R 5 1 follow then easily using 55. Estimate on D t R 5. In this case we compute D t R 5 = 2χ l t χ l R l R l,l + χ 2 l D t R l. l l The second summand has been estimate above and, since t χ l 0 C, the first summand can be estimated by Cδ 1 again appealing to the arguments above. Proof of 89. To achieve this last inequality, observe that t R1 + v 1 R 1 0 D t R1 0 + v v l 0 + w 0 R 1 1. On the other hand, by 26, v v l 0 C l. Moreover, by 45, 63 and 55 w w o 0 + w c 0 C. Thus, by 88 we conclude t R1 + v 1 R 1 0 C + λ 1/2 δ λ 1 ε + δ λ ε + δ1 /2 l + δ1/2 δ qλ q λ 1 ε l Since by 55 λ, 89 follows easily. 6. Conclusion of the proof In Sections 2-5 we showed the construction for a single step, referring to v q, p q, R q as v, p, R and to v, p, R as v 1, p 1, R 1. From now on we will consider the full iteration again, hence using again the indices q and q + 1. In order to proceed, recall that the sequences {δ q } q N and {λ q } q N are chosen to satisfy δ q = a bq, a cb λ q 2a cb

26 26 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. for some given constants c > 5/2 and b > 1 and for a > 1. Note that this has the consequence that if a is chosen sufficiently large depending only on b > 1 then q λ 1 /5 q λ1 /5, δ δ q, and λ q λ 2 b Choice of the parameters and l. We start by specifying the parameters = q and l = l q : we determine them optimizing the right hand side of 88. More precisely, we set := δ 1 /4 δ1 /4 q λ 1 /2 q λ 1 /2 99 so that the first two expressions in 88 are equal, and then, having determined, set l := δ 1/8 δ1 /8 q λ 1 /4 q λ 3/4 100 so that the last two expressions in 88 are equal up to a factor λ ε. In turn, these choices lead to R R 1 Cδ 3 /4 λ δ1 /4 q λ 1 /2 q λ ε 1 /2 + Cδ 3 /8 δ5 /8 q λ 3 /4 q λ ε 3 /4 3/4 Observe also that by 89, we have = Cδ 3 /4 δ1 /4 q λ 1 /2 t R + v R 0 C λ q λ ε 1 /2 1 + q λ 1 /3 q λ1 /3 98 Cδ 3 /4 δ1 /4 q λ 1 /2 q λ ε 1/ δ 3 /4 δ1 /4 q λ 1 /2 q λ ε 1/ Let us check that the conditions 54 are satisfied for some β > 0 remember that β should be independent of q. To this end we calculate l = 1 lλ = q λ 3 /5 q λ3 /5 δ 1/2 λ q q λ 5/4, 1/4, = λ = λ λ Hence the conditions 54 follow from 98 choosing β = b 1 5b+5. 1/2, 1/2.

27 DISSIPATIVE EULER FLOWS Proof of Proposition 0.2. Fix the constants c > 5 2 and b > 1 and also an ε > 0 whose choice, like that of a > 1, will be specified later. The proposition is proved inductively. The initial triple is defined to be v 0, p 0, R 0 = 0, 0, 0. Given now v q, p q, R q satisfying the estimates 6-13, we claim that the triple v, p, R constructed above satisfies again all the corresponding estimates. Estimates on R. Note first of all that, using the form of the estimates in 88 and 89, the estimates 11 and 13 follow from 10. On the other hand, in light of 101, 10 follows from the recursion relation Cδ 3 /4 δ1 /4 q λ 1 /2 q λ ε 1/2 ηδ q+2. Using our choice of δ q and λ q from Proposition 0.2, we see that this inequality is equivalent to C a 1 4 bq 1+3b 2cb+2c 4 4εcb 2, which, since b > 1, is satisfied for all q 1 for a sufficiently large fixed constant a > 1, provided 1 + 3b 2cb + 2c 4 4εcb 2 > 0. Factorizing, we obtain the inequality b 12c 4b 1 4εcb 2 > 0. It is then easy to see that for any b > 1 and c > 5/2 there exists ε > 0 so that this inequality is satisfied. In this way we can choose ε > 0 and β above depending solely on b and c. Consequently, this choice will determine all the constants in the estimates in Sections 2-5. We can then pick a > 1 sufficiently large so that 10, and hence also 11 and 13 hold for R. Estimates on v v q. By 45, Lemma 3.1 and 54 we conclude v v q 0 w o 0 + w c 0 M 2 + λ β v v q 1 w o 1 + w c 1 M 2 + λ β Since λ λ 1 a cb2, for a sufficiently large we conclude 6 and 7. Estimate on the energy. Recall Lemma 4.1 and observe that, by 54, δ 1/2 δ1/2 λ δ1/2 λ. Moreover, δ = a b b q /2+cb = a bq c 1b 1/2 a 1. So the right hand side of 75 is smaller than C δ δ 1/2 + C δ1/2 λ, i.e. smaller up to a constant factor than the right hand side of 88. Thus, the argument used above to prove 10 gives also 12. Estimates on p p 1. From the definition of p in 52 we deduce p p q w o 0 + w c Cl v q 1 w 0

28 28 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. As already argued in the estimate for 6, w o + w c M q. Moreover Cl v 1 1 w 0 CM δ1 /2 l, which is smaller than the right hand side of 88. Having already argued that such quantity is smaller than ηδ q+2 we can obviously bound Cl v q 1 w 0 with M 2 2 δ. This shows 8. Moreover, differentiating 52 we achieve the bound p p q 1 w o 1 + w c 1 w o 0 + w c 0 + C and arguing as above we conclude 9. δ1 /2 q λ q λ l Estimates 14. Here we can use the obvious identity t w q = D t w q v q l w q together with Lemmas 3.1 and 3.2 to obtain t v t v q 0 C λ Then, using 21, we conclude t v q 0 C. To handle t p t p q observe first that, by our construction, t p p q 0 w c 0 + w o 0 t w c 0 + t w o w 0 t v q 0 + l v q 1 t w 0. As above, we can derive the estimates t w o 0 + t w c 0 C λ from Lemmas 3.1 and 3.2. Hence t p p q 0 Cδ λ + C δ1 /2 + C l λ. 105 Since l λ 1 q and λ, the desired inequality follows. This concludes the proof. Appendix A. Hölder spaces In the following m = 0, 1, 2,..., α 0, 1, and β is a multi-index. We introduce the usual spatial Hölder norms as follows. First of all, the supremum norm is denoted by f 0 := sup T 3 [0,1] f. We define the Hölder seminorms as [f] m = max β =m Dβ f 0, [f] m+α = max sup D β fx, t D β fy, t β =m x y,t x y α, where D β are space derivatives only. The Hölder norms are then given by m f m = [f] j j=0 f m+α = f m + [f] m+α. Moreover, we will write [ft] α and ft α when the time t is fixed and the norms are computed for the restriction of f to the t-time slice. Recall the following elementary inequalities: [f] s C ε r s [f] r + ε s f 0 106

29 DISSIPATIVE EULER FLOWS 29 for r s 0, ε > 0, and [fg] r C [f] r g 0 + f 0 [g] r 107 for any 1 r 0. From 106 with ε = f 1 r 0 [f] 1 r r interpolation inequalities we obtain the standard [f] s C f 1 s r 0 [f] s r r. 108 Next we collect two classical estimates on the Hölder norms of compositions. These are also standard, for instance in applications of the Nash- Moser iteration technique. Proposition A.1. Let Ψ : Ω R and u : R n Ω be two smooth functions, with Ω R N. Then, for every m N \ {0} there is a constant C depending only on m, N and n such that [Ψ u] m C[Ψ] 1 [u] m + [Ψ] m u m 1 0 [u] m 109 [Ψ u] m C[Ψ] 1 [u] m + [Ψ] m [u] m Appendix B. Estimates for transport equations In this section we recall some well known results regarding smooth solutions of the transport equation: { t f + v f = g, 111 f t0 = f 0, where v = vt, x is a given smooth vector field. We denote the material derivative t + v by D t. We will consider solutions on the entire space R 3 and treat solutions on the torus simply as periodic solution in R 3. Proposition B.1. Any solution f of 111 satisfies ft 0 f t t 0 gτ 0 dτ, 112 [ft] 1 [f 0 ] 1 e t t 0[v] 1 + e t τ[v] 1 [gτ] 1 dτ, 113 t 0 and, more generally, for any N 2 there exists a constant C = C N so that [ft] N [f 0 ] N + Ct t 0 [v] N [f 0 ] 1 e Ct t 0[v] 1 + t + e Ct τ[v] 1 [gτ] N + [v] N [gτ] 1 dτ. 114 t 0 Define Φt, to be the inverse of the flux X of v starting at time t 0 as the d identity i.e. dt X = vx, t and Xx, t 0 = x. Under the same assumptions as above: t DΦt Id 0 e t t 0[v] 1 1, 115 [Φt] N Ct t 0 [v] N e Ct t 0[v] 1 N

30 30 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Proof. We start with the following elementary observation for transport equations: if f solves 111, then d dtfxt, x, t = gxt, x, t and consequently t ft, x = f 0 Φx, t + gxφt, x, τ, τ dτ. t 0 The maximum principle 112 follows immediately. Next, differentiate 111 in x to obtain the identity Applying 112 to Df yields D t Df = t + v Df = Dg DfDv. [ft] 1 [f 0 ] 1 + t t 0 [gτ] 1 + [v] 1 [fτ] 1 dτ. An application of Gronwall s inequality then yields 113. More generally, differentiating 111 N times yields N 1 t D N f + v D N f = D N g + c j,n D j+1 f : D N j v 117 where : is a shorthand notation for sums of products of entries of the corresponding tensors. Also, using 112 and the interpolation inequality 108 we can estimate [ft] N [f 0 ] N + t j=0 t 0 [gτ]n + C [v] N [fτ] 1 + [v] 1 [fτ] N dτ. Plugging now the estimate 113, Gronwall s inequality leads after some elementary calculations to 114. The estimate 116 follows easily from 114 observing that Φ solves 111 with g = 0 and D 2 Φ, t 0 = 0. Consider next Ψx, t = Φx, t x and observe first that t Ψ + v Ψ = v. Since DΨ, t 0 = 0, we apply 113 to conclude [Ψt] 1 t t 0 e t τ[v] 1 [v] 1 dτ = e t t 0[v] 1 1. Since DΨx, t = DΦx, t Id, 115 follows. Appendix C. Constantin-E-Titi commutator estimate Finally, we recall the quadratic commutator estimate from [2] cf. also with [3, Lemma 1]: Proposition C.1. Let f, g C T 3 T and ψ the mollifier of Section 2. For any r 0 we have the estimate r f χ l g χ l fg χ l Cl 2 r f 1 g 1, where the constant C depends only on r.

31 DISSIPATIVE EULER FLOWS 31 Appendix D. Schauder Estimates We recall here the following consequences of the classical Schauder estimates cf. [10, Proposition 5.1]. Proposition D.1. For any α 0, 1 and any m N there exists a constant Cα, m with the following properties. If φ, ψ : T 3 R are the unique solutions of then φ = f ffl φ = 0 ψ = div F ffl ψ = 0, φ m+2+α Cm, α f m,α and ψ m+1+α Cm, α F m,α. 118 Moreover we have the estimates Rv m+1+α Cm, α v m+α 119 Rdiv A m+α Cm, α A m+α 120 Appendix E. Stationary phase lemma We recall here the following simple facts for a proof we refer to [10] Proposition E.1. i Let k Z 3 \ {0} and λ 1 be fixed. For any a C T 3 and m N we have axe iλk x dx [a] m T 3 λ m. 121 ii Let k Z 3 \ {0} be fixed. For a smooth vector field a C T 3 ; R 3 let F x := axe iλk x. Then we have RF α where C = Cα, m. C λ 1 α a 0 + C λ m α [a] m + C λ m [a] m+α, Appendix F. One further commutator estimate Proposition F.1. Let k Z 3 \ {0} be fixed. For any smooth vector field a C T 3 ; R 3 and any smooth function b, if we set F x := axe iλk x, we then have [b, R]F α C λ 2 α a 0 b 1 + C λ m α a m b 2 + a 2 b m + C λ m a m+α b 2+α + a 2+α b m+α, 122 where C = Cα, m.

32 32 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. Proof. Step 1 First of all, given a vector field v define the operator First observe that Sv := v + v t 2 div vid. 3 div Sv = 0 v const. 123 One implication is obvious. Next, assume div Sv = 0. This is equivalent to the equations Differentiating and summing in j we then conclude v j jdiv v = div v = 0. Thus div v must be constant and, since any divergence has average zero, we conclude that div v = 0. Thus 124 implies that v i = 0 for every i, which in turn gives the desired conclusion. From this observation we conclude the identity Rv = Sv + Rv div Sv for all v C T 3, R Indeed, observe first that Rv = Sw, where w = 1 4 Pv v. Thus, applying the argument above, since both sides of 125 have zero averages, it suffices to show that they have the same divergence. But since div Rv = v ffl v, applying the divergence we obtain which is obviously true. v v = div Sv + v v div Sv, Step 2 Next, for a C T 3, R 3, k Z 3 \ {0} and λ N \ {0}, consider 3 S ae iλk x a := S eiλk x 1 a kk 4 λ 2 k 2 + 4λ 2 k 2 a k 2 e iλk x. Observe that S bae iλk x bs ae iλk x = aab λ 2 e iλk x, 126 where A is an homogeneous differential operator of order one with constant coefficients all depending only on k. Moreover, ae iλk x div S ae iλk x = B 1a eiλk x + B 2a λ λ 2 e iλk x, where B 1 and B 2 are homogeneous differential operators of order 1 and 2 respectively with constant coefficients again all depending only on k.

33 We use then the identity 125 to write DISSIPATIVE EULER FLOWS 33 [b, R]F = RbF brf = S bae iλk x bs ae iλk x + R bf div S bae iλk x br F div S ae iλk x = aab λ 2 e iλk x B1 ab + R e iλk x + B 2ab λ λ 2 e iλk x B1 a br eiλk x + B 2a λ λ 2 e iλk x. 127 Using the Leibniz rule we can write B 1 ab = B 1 ab+ab 1 b and B 2 ab = B 2 ab+ab 2 b+c 1 ac 1 b, where C 1 is an homogeneous operator of order 1. We can then reorder all terms to write [b, R]F = aab λ 2 e iλk x ab1 b + R λ e iλk x ab2 b + C 1 ac 1 b + R λ 2 e iλk x 1 λ [b, R] B 1 ae iλk x 1 λ 2 [b, R] bb 2 ae iλk x. 128 In the first two summands appear only derivatives of b, but there are no zero order terms in b. We can then estimate the two terms in the second line applying Proposition E.1, with m = N 1 to the first summand and with m = N 2 to the second summand. Applying in addition interpolation identities, we conclude [b, R]F α C a 0 b 1 λ 2 α + C a 2 b N + a N b 2 λ N α + C a 2+α b N+α + a N+α b 2+α λ N + 1 [b, R] bb 1 ae iλk x α + 1 [b, R] bb λ }{{} λ 2 2 ae iλk x α. 129 II Indeed the above estimate is sub-optimal as we get, for instance, terms of type a 0 b N + a 1 b N 1 + a 2 b N 2 instead of a 2 b N ; however, this crude estimate is still sufficient for our purposes. Step 3 We can now apply the same idea to the term II in 129 to reach the estimate [b, R]F α C a 0 b 1 λ 2 α + C a 2 b N + a N b 2 λ N α + C a 2+α b N+α + a N+α b 2+α λ N + 1 [b, R] λ 2 bb 2ae iλk x α. where B 2 = B 2 + B 1 B 1 is second order and this time we have applied Proposition E.1 with m = N 2 and m = N 3 to handle the corresponding

34 34 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LÁSZLÓ SZÉKELYHIDI JR. two terms of order λ 2 and λ 3 arising from II. Proceeding now inductively, we end up with RbF brf 0 a 0 b 1 λ 2 + C a 2 b N + a N b 2 λ N α + C a 2+α b N+α + a N+α b 2+α λ N + 1 R λ N bb Nae iλk x br B Nae iλk x α. 130 Finally, applying Proposition D.1 to the final term we reach the desired estimate. References [1] Cheskidov, A., Constantin, P., Friedlander, S., and Shvydkoy, R. Energy conservation and Onsager s conjecture for the Euler equations. Nonlinearity 21, , [2] Constantin, P., E, W., and Titi, E. S. Onsager s conjecture on the energy conservation for solutions of Euler s equation. Comm. Math. Phys. 165, , [3] Conti, S., De Lellis, C., and Székelyhidi, Jr., L. h-principle and rigidity for C 1,α isometric embeddings. In Nonlinear Partial Differential Equations, vol. 7 of Abel Symposia. Springer, 2012, pp [4] Courant, R., Friedrichs, K., and Lewy, H. On the partial difference equations of mathematical physics. IBM J. Res. Develop , [5] Daneri, S. Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations. Preprint. 2013, [6] De Lellis, C., and Székelyhidi, Jr., L. The Euler equations as a differential inclusion. Ann. of Math , , [7] De Lellis, C., and Székelyhidi, Jr., L. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, , [8] De Lellis, C., and Székelyhidi, Jr., L. Dissipative Euler flows and Onsager s conjecture. Preprint. 2012, [9] De Lellis, C., and Székelyhidi, Jr., L. The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. N.S. 49, , [10] De Lellis, C., and Székelyhidi, Jr., L. Dissipative continuous Euler flows. To appear in Inventiones 2013, [11] Duchon, J., and Robert, R. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13, , [12] Eyink, G. L. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78, , [13] Eyink, G. L., and Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence. Rev. Modern Phys. 78, , [14] Frisch, U. Turbulence. Cambridge University Press, Cambridge, The legacy of A. N. Kolmogorov. [15] Isett, P. Hölder continuous Euler flows in three dimensions with compact support in time. Preprint 2012, [16] Kolmogorov, A. N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. Roy. Soc. London Ser. A 434, , Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov s ideas 50 years on. [17] Onsager, L. Statistical hydrodynamics. Nuovo Cimento 9 6, Supplemento, 2 Convegno Internazionale di Meccanica Statistica 1949,