arXiv:1302.2815v3 [math.AP] 10 Dec 2013
TRANSPORTING MICROSTRUCTURE AND DISSIPATIVE EULER FLOWS TRISTAN BUCKMASTER, CAMILLO DE LELLIS, ´ ´ SZEKELYHIDI ´ AND LASZL O JR. Abstract. Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in H¨ older spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). 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:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better H¨ older 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 1 dissipate the total kinetic energy and belong to the H¨ older class C /5−ε .
0. Introduction In what follows T3 denotes the 3-dimensional torus, i.e. T3 = S1 ×S1 ×S1 . 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 (T3 × 2 [0, 1], R3 ) and a continuous scalar field p ∈ C /5−2ε (T3 × [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 ˆ e(t) = |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
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´ ´ SZEKELYHIDI ´ TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
solutions (say, v ∈ C 1 ) we can multiply (1) by v itself, integrate by parts and obtain the energy balance ˆ ˆ 2 |v(x, 0)|2 dx for all t > 0. (3) |v(x, t)| dx = T3
T3
On the other hand, for weak solutions (say, merely v ∈ L2 ) (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 H¨older condition |v(x, t) − v(x′ , t)| ≤ C|x − x′ |θ , (4) ′ 3 where the constant C is independent of x, x ∈ T and t. He conjectured that (a) Any weak solution v satisfying (4) with θ > 13 conserves the energy; (b) For any θ < 31 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 H¨older-continuous weak solutions violating the energy equality appeared. In particular in [8] the authors proved Theorem 0.1 with H¨older exponent 1/10 − ε replacing 1/5 − ε. The threshold exponent 15 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 H¨older 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]. 0.1. Euler-Reynolds system and the convex integration scheme. Let us recall the main ideas, on which the constructions in [10, 8] are based.
DISSIPATIVE EULER FLOWS
3
The proof is achieved through an iteration scheme. At each step q ∈ N ˚q ) solving the Euler-Reynolds system (see [10, we construct a triple (vq , pq , R Definition 2.1]): ˚q ∂t vq + div (vq ⊗ vq ) + ∇pq = div R (5) div vq = 0 .
The size of the perturbation
wq := vq − vq−1 1/2
will be measured by two parameters: δq is the amplitude and λq the frequency. More precisely, denoting the (spatial) H¨older norms by k · kk (see Section A for precise definitions), kwq k0 ≤ M δq/2 , 1
(6)
1/2
kwq k1 ≤ M δq λq ,
(7)
and similarly, kpq − pq−1 k0 ≤ M 2 δq , 2
kpq − pq−1 k1 ≤ M δq λq ,
(8) (9)
where M is a constant depending only on the function e = e(t) in the Theorem. In constructing the iteration, the new perturbation, wq+1 will be chosen so ˚q , in the sense that (cf. equation as to balance the previous Reynolds error R ˚q k0 . This is formalized as (5)) we have kwq+1 ⊗ wq+1 k0 ∼ kR ˚q k0 ≤ ηδq+1 , kR ˚q k1 ≤ M δq+1 λq , kR
(10) (11)
where η will be a small constant, again only depending on e = e(t) in the Theorem. Estimates of type (6)-(11) appear already in the paper [8]: ˚1 k1 in the main proposition of [8] is the although the bound claimed for kR 1/2 ˚ weaker one kRq k1 ≤ M δq λq (cf. [8, Proposition 2.2]): λq 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 δq → 0
and
λq → ∞
at a rate that is at least exponential. On the one hand (6), (8) and (10) will imply the convergence of the sequence vq to a continuous weak solution of the Euler equations. On the other hand the precise dependence of λq on δq
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´ ´ SZEKELYHIDI ´ TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
will determine the critical H¨older regularity. Finally, the equation (2) will be ensured by ˆ 2 e(t)(1 − δq+1 ) − |vq | (x, t) dx ≤ 1 δq+1 e(t) . (12) 4
Note that, being an expression quadratic in vq , 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 X wq (x, t) = ak (x, t) φk (x, t) Bk eiλq k·x , k
where ak is the amplitude, φk is a phase function (i.e. |φk | = 1) and Bk eiλq k· x is a complex Beltrami mode at frequency λq . Having a perturbation of this form ensures that the “oscillation part of the error” ˚q−1 ) div (wq ⊗ wq + R 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 ak and φk correctly in order to deal with the so-called transport part of the error ∂t wq + vq−1 · ∇wq . In [10, 8] a second large parameter µ(= µq ) was introduced to deal with this term. In some sense the role of µ 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 H¨older 1 . In our opinion the key new idea introduced by Isett is to exponent 10 recognize that the transport error can be reduced by defining ak and φk in such a way that adheres more closely to the transport structure of the equation. This requires two new ingredients. First, the phase functions φk are defined using the flow map of the vector field vq , whereas in [8] they were functions of vq 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: ˚q + vq · ∇R ˚q k0 ≤ δq+1 δ1/2 λq . k∂t R (13) q
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 µ corresponds to the inverse of the life-span parameter τ used in [15].
DISSIPATIVE EULER FLOWS
5
0.2. 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: ˚q in space only and then solve locally (a) We regularize the maps vq and R ˚q . in time the free-transport equation in order to approximate R (b) Our maps ak are then elementary algebraic functions of the approx˚q . imation of R (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. 0.3. The main iteration proposition and the proof of Theorem 0.1. Having outlined the general idea above, we proceed with the iteration, start˚0 ) = (0, 0, 0). We will construct new ing with the trivial solution (v0 , p0 , R ˚q ) inductively, assuming the estimates (6)-(13). triples (vq , pq , R Proposition 0.2. There are positive constants M and η depending only on e such that the following holds. For every c > 25 and b > 1, if a is suf˚q ) starting with ficiently large, then there is a sequence of triples (vq , pq , R ˚ (v0 , p0 , R0 ) = (0, 0, 0), solving (5) and satisfying the estimates (6)-(13), q+1 q+1 q where δq := a−b , λq ∈ [acb , 2acb ] for q = 0, 1, 2, . . . . In addition we claim the estimates k∂t (vq − vq−1 )k0 ≤ Cδq/2 λq 1
and
k∂t (pq − pq−1 )k0 ≤ Cδq λq
(14)
˚q ) be a Proof of Theorem 0.1. Choose any c > 25 and b > 1 and let (vq , pq , R sequence as in Proposition 0.2. It follows then easily that {(vq , pq )} converge uniformly to a pair of continuous functions (v, p) such that (1) and (2) hold. We introduce the notation k · kC ϑ for H¨older norms in space and time. From (6)-(9), (14) and interpolation we conclude 1/2
q+1 (2cbϑ−1)/2
kvq+1 − vq kC ϑ ≤ M δq+1 λϑq+1 ≤ Cab
kpq+1 − pq kC 2ϑ ≤ M 2 δq+1 λ2ϑ q+1 ≤ Ca Thus, for every ϑ
0 ¯ > 1 with the following property. There exist pairwise disjoint subsets and λ ¯ Λj ⊂ {k ∈ Z3 : |k| = λ} j ∈ {1, . . . , N }
and smooth positive functions (j)
γk ∈ C ∞ (Br0 (Id))
j ∈ {1, . . . , N }, k ∈ Λj
such that (j) (j) (a) k ∈ Λj implies −k ∈ Λj and γk = γ−k ; (b) For each R ∈ Br0 (Id) we have the identity 2 k k 1 X (j) γk (R) Id − ⊗ ∀R ∈ Br0 (Id) . R= 2 |k| |k|
(20)
k∈Λj
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 ∞ (T3 , R3 ) be a smooth vector field. We then define Rv to be the matrix-valued periodic function 3 1 1 ∇Pu + (∇Pu)T + ∇u + (∇u)T − (div u)Id, Rv := 4 4 2 where u ∈ C ∞ (T3 , R3 ) is the solution of ∆u = v −
v in T3 T3
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´ ´ SZEKELYHIDI ´ TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
ffl with T3 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 ∞ (T3 , R3 ) we have
(a) Rv(x) is a symmetric trace-free matrix for each x ∈ T3 ; ffl (b) div Rv = v − T3 v. 2. The inductive step
In this section we specify the inductive procedure which allows to con˚q+1 ) from (vq , pq , R ˚q ). Note that the choice of the sestruct (vq+1 , pq+1 , R quences {δq }q∈N and {λq }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: X X 1/2 X X 1/2 1 δj ≤ δj ≤ 2 . (21) δj λj ≤ 2δq λq , 1 ≤ δj λj ≤ 2δq/2 λq , j≤q
j
j≤q
j
Since we are concerned with a single step in the iteration, with a slight ˚ for (vq , pq , R ˚q ) and (v1 , p1 , R ˚1 ) for abuse of notation we will write (v, p, R) ˚q+1 ). Our inductive hypothesis implies then the following set (vq+1 , pq+1 , R of estimates: kvk0 ≤ 2M, ˚ 0 ≤ ηδq+1 , kRk 2
kpk0 ≤ 2M ,
kvk1 ≤ 2M δq/2 λq , ˚ 1 ≤ M δq+1 λq , kRk 1
2
kpk1 ≤ 2M δq λq ,
(22) (23) (24)
and ˚ 0 ≤ M δq+1 δ1/2 λq . k(∂t + v · ∇)Rk q
(25)
The new velocity v1 will be defined as a sum v1 := v + wo + wc , where wo is the principal perturbation and wc is a corrector. The “principal part” of the perturbation w will be a sum of Beltrami flows X ak φk Bk eiλq+1 k·x , wo (t, x) := |k|=λ0
where Bk eiλq+1 k·x is a single Beltrami mode at frequency λq+1 , with phase shift φk = φk (t, x) (i.e. |φk | = 1) and amplitude ak = ak (t, x). In the following subsections we will define ak and φk .
DISSIPATIVE EULER FLOWS
9
2.1. Space regularization of v and R. We fix a symmetric non-negative convolution kernel ψ ∈ Cc∞ (R3 ) and a small parameter ℓ (whose choice ˚ℓ := R ˚ ∗ ψℓ , where the will be specified later). Define vℓ := v ∗ ψℓ and R convolution is in the x variable only. Standard estimates on regularizations by convolution lead to the following: kv − vℓ k0 ≤ C δq/2 λq ℓ, ˚− R ˚ℓ k0 ≤ C δq+1 ℓ, kR 1
(26) (27)
and for any N ≥ 1 there exists a constant C = C(N ) so that kvℓ kN ≤ C δq/2 λq ℓ1−N , ˚ℓ kN ≤ C δq+1 λq ℓ1−N . kR 1
(28) (29)
2.2. Time discretization and transport for the Reynolds stress. Next, we fix a smooth cut-off function χ ∈ Cc∞ ((− 43 , 34 ) such that X χ2 (x − l) = 1, l∈Z
and a large parameter µ ∈ N \ {0}, whose choice will be specified later. For any l ∈ [0, µ] we define ˆ 1 2 −1 −1 |v| (x, lµ ) dx . ρl := e(lµ ) (1 − δq+2 ) − 3(2π)3 T3 Note that (12) implies 1 1 e(lµ−1 )( 34 δq+1 − δq+2 ) ≤ ρl ≤ e(lµ−1 )( 54 δq+1 − δq+2 ). 3(2π)3 3(2π)3 We will henceforth assume δq+2 ≤
1 δq+1 , 2
so that we obtain C0−1 (min e)δq+1 ≤ ρl ≤ C0 (max e)δq+1 ,
(30)
where C0 is an absolute constant. Finally, define Rℓ,l to be the unique solution to the transport equation ( ˚ℓ,l + vℓ · ∇R ˚ℓ,l = 0 ∂t R (31) l ˚ ˚ Rℓ,l (x, µ ) = Rℓ (x, µl ) . and set ˚ℓ,l (x, t). Rℓ,l (x, t) := ρl Id − R
(32)
´ ´ SZEKELYHIDI ´ 10 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
2.3. The maps v1 , w, wo and wc . We next consider vℓ as a 2π-periodic function on R3 × [0, 1] and, for every l ∈ [0, µ], we let Φl : R3 × [0, 1] → R3 be the solution of ∂t Φl + vℓ · ∇Φl = 0 (33) Φl (x, lµ−1 ) = x
Observe that Φl (·, t) is the inverse of the flow of the periodic vector-field vℓ , starting at time t = lµ−1 as the identity. Thus, if y ∈ (2πZ)3 , then Φl (x, t)− Φl (x+y, t) ∈ (2πZ)3 : Φl (·, t) can hence be thought as a diffeomorphism of T3 onto itself and, for every k ∈ Z3 , the map T3 × [0, 1] ∋ (x, t) → eiλq+1 k·Φl (x,t) is well-defined. We next apply Lemma 1.3 with N = 2, denoting by Λe and Λo the corresponding families of frequencies in Z3 , and set Λ := Λo + Λe . For each k ∈ Λ and each l ∈ Z ∩ [0, µ] we then set χl (t) := χ µ(t − l) , (34) Rℓ,l (x, t) √ akl (x, t) := ρl γk , (35) ρl wkl (x, t) := akl (x, t) Bk eiλq+1 k·Φl (x,t) .
The “principal part” of the perturbation w consists of the map X X wo (x, t) := χl (t)wkl (x, t) + χl (t)wkl (x, t) . l odd,k∈Λo
(36)
(37)
l even,k∈Λe
From now on, in order to make our notation simpler, we agree that the pairs of indices (k, l) ∈ Λ × [0, µ] which enter in our summations satisfy always the following condition: k ∈ Λe when l is even and k ∈ Λo when l is odd. It will be useful to introduce the “phase” φkl (x, t) = eiλq+1 k·[Φl (x,t)−x] ,
(38)
with which we obviously have φkl · eiλq+1 k·x = eiλq+1 k·Φl . Since Rℓ,l and Φl are defined as solutions of the transport equations (31) and (33), we have (∂t + vℓ · ∇)akl = 0
and
(∂t + vℓ · ∇)eiλq+1 k·Φl (x,t) = 0,
(39)
hence also (∂t + vℓ · ∇)wkl = 0.
(40)
DISSIPATIVE EULER FLOWS
11
The corrector wc is then defined in such a way that w := wo +wc is divergence free: X χl k × Bk curl iakl φkl eiλq+1 k·x wc := λq+1 |k|2 kl k×B X i k iλq+1 k·Φl (41) ∇akl − akl (DΦl − Id)k × e = χl λq+1 |k|2 kl
Remark 1. To see that w = wo + wc is divergence-free, just note that, since k · Bk = 0, we have k × (k × Bk ) = −|k|2 Bk and hence w can be written as 1 X k × Bk iλq+1 k·x w= χl curl iakl φkl e (42) λq+1 |k|2 (k,l)
For future reference it is useful to introduce the notation k×B i k ∇akl − akl (DΦl − Id)k × , Lkl := akl Bk + λq+1 |k|2
so that the perturbation w can be written as X w= χl Lkl eiλq+1 k·Φl .
(43)
(44)
kl
Moreover, we will frequently deal with the transport derivative with respect to the regularized flow vℓ of various expressions, and will henceforth use the notation Dt := ∂t + vℓ · ∇. 2.4. Determination of the constants η and M . In order to determine η, first of all recall from Lemma 1.3 that the functions akl are well-defined provided Rℓ,l ρ − Id ≤ r0 , l where r0 is the constant of Lemma 1.3. Recalling the definition of Rℓ,l we easily deduce from the maximum principle for transport equations (cf. (112) ˚ℓ,l k0 ≤ kRk ˚ 0 . Hence, from (10) and (30) we in Proposition B.1) that kR obtain Rℓ,l η ρl − Id ≤ C0 min e , and thus we will require that r0 η ≤ . C0 min e 4 The constant M in turn is determined by comparing the estimate (6) for q + 1 with the definition of the principal perturbation wo in (37). Indeed, 1/2 using (34)-(37) and (30) we have kwo k0 ≤ C0 |Λ|(max e)δq+1 . We therefore set M = 2C0 |Λ|(max e),
´ ´ SZEKELYHIDI ´ 12 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
so that kwo k0 ≤
M 1/2 δ . 2 q+1
(45)
˚1 . We set 2.5. The pressure p1 and the Reynolds stress R ˚1 = R0 + R1 + R2 + R3 + R4 + R5 , R where R0 = R (∂t w + vℓ · ∇w + w · ∇vℓ ) X R1 = Rdiv wo ⊗ wo − χ2l Rℓ,l −
(46) |wo 2
l
|2
Id
|wc |2 +2hwo ,wc i Id 3 2h(v−vℓ ),wi Id 3
R2 = wo ⊗ wc + wc ⊗ wo + wc ⊗ wc −
R3 = w ⊗ (v − vℓ ) + (v − vℓ ) ⊗ w − ˚− R ˚ℓ R4 = R X ˚ℓ − R ˚l,ℓ ) . R5 = χ2l (R
(47) (48) (49) (50) (51)
l
˚1 is indeed a traceless symmetric tensor. The corresponding Observe that R form of the new pressure will then be 2 2 |wo |2 1 − |wc |2 − hwo , wc i − hv − vℓ , wi . (52) 2 3 3 3 P 2 Recalling P 2 (32) we see that l χl tr Rℓ,l is a function of time only. Since also l χl = 1, it is then straightforward to check that p1 = p −
˚1 − ∇p1 = ∂t w + div (v ⊗ w + w ⊗ v + w ⊗ w) + div R ˚ − ∇p div R
= ∂t w + div (v ⊗ w + w ⊗ v + w ⊗ w) + ∂t v + div (v ⊗ v) = ∂t v1 + div v1 ⊗ v1 .
The following lemma will play a key role. Lemma 2.1. The following identity holds: X X wo ⊗ wo = χ2l Rℓ,l + l
(k,l),(k ′ ,l′ ),k6=−k ′
χl χl′ wkl ⊗ wk′ l′ .
(53)
Proof. Recall that the pairs (k, l), (k′ , l′ ) are chosen so that k 6= −k′ if l is even and l′ is odd. Moreover χl χl′ = 0 if l and l′ are distinct and have the same parity. Hence the claim follows immediately from our choice of akl in (35) and Proposition 1.1 and Lemma 1.3 (cf. [10, Proposition 6.1(ii)]).
DISSIPATIVE EULER FLOWS
13
2.6. Conditions on the parameters - hierarchy of length-scales. In the next couple of sections we will need to estimate various expressions involving vℓ and w. To simplify the formulas that we arrive at, we will from now on assume the following conditions on µ, λq+1 ≥ 1 and ℓ ≤ 1: 1/2
1/2
δq λq ℓ 1/2
δq+1
δq λq 1 + ≤ λ−β q+1 µ ℓλq+1
≤ 1,
and
1/2
1
≤
λq+1
δq+1 µ
.
(54)
These conditions imply the following orderings of length scales, which will be used to simplify the estimates in Section 3: 1 1 1 1 1 ≤ℓ≤ . (55) ≤ ≤ 1/2 and 1/2 µ λ λ q+1 q δ λq+1 δq λq q+1
One can think of these chains of inequalities as an ordering of various length scales involved in the definition of v1 . δ
1/2
Remark 2. The most relevant and restrictive condition is qµ ≤ λ1q . 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 k∇vk−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 |µt − l| < 1 we have kDΦl k0 ≤ C
(56) 1/2
kDΦl − Idk0 ≤ C
δq λq µ
(57)
1/2
kDΦl kN ≤ C
δq λq , µℓN
N ≥1
(58)
Moreover, 1/2
kakl k0 + kLkl k0 ≤ Cδq+1
(59)
1/2
kakl kN ≤ Cδq+1 λq ℓ1−N , 1/2
kLkl kN ≤ Cδq+1 ℓ−N , 1/2
kφkl kN ≤ Cλq+1
δq λq +C µℓN −1
N (1−β)
≤ Cλq+1
1/2
δq λq λq+1 µ
!N
N ≥1
(60)
N ≥1
(61)
N ≥ 1.
(62)
´ ´ SZEKELYHIDI ´ 14 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
Consequently, for any N ≥ 0 1/2
δq λq N kwc kN ≤ Cδq+1 λq+1 , µ M 1/2 1/2 δq+1 λq+1 + Cδq+1 λ1−β kwo k1 ≤ q+1 2 1/2 kwo kN ≤ Cδq+1 λN q+1 , 1/2
(63) (64) N ≥2
(65)
where the constants in (56)-(57) 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 kRℓ,l k0 ≤ Cδq+1 ,
(66)
kRℓ,l kN ≤ Cδq+1 λq ℓ1−N ,
N ≥ 1.
(67)
The estimate (59) is now a consequence of (66), (57) and (30), whereas by (109) we obtain −1/2
1/2
1/2
kakl kN ≤ Cδq+1 kRℓ,l kN ≤ Cδq+1 λq ℓ1−N ≤ Cδq+1 ℓ−N .
(68)
Similarly we deduce (61) from kLkl kN ≤Ckakl kN + Cλ−1 q+1 kakl kN +1 +
+ C (kakl kN kDΦl − Idk0 + kakl k0 kDΦl kN )
and once again using (55). In order to prove (62) we apply (110) with m = N to conclude N kφkl kN ≤ Cλq+1 kDΦl kN −1 + λN q+1 kDΦl − Idk0 ,
from which (62) follows using (57), (58) and (54). Using the formula (41) together with (57), (58), (59) and (61) we conclude kwc k0 ≤
C λq+1
1/2
kakl k1 + Ckakl k0 kDΦl − Idk0 ≤ C
δq λq µ
DISSIPATIVE EULER FLOWS
15
and, for N ≥ 1, X 1 kakl kN +1 + kakl k0 kDΦl kN + kakl kN kDΦl − Idk0 kwc kN ≤C χl λq+1 kl X N + Ckwc k0 χl λN q+1 kDΦl k0 + λq+1 kDΦl kN −1 l
(55)
≤
1/2
δq λq λq + λq+1 µ
1/2
Cδq+1 λN q+1
!
1/2
δq λq N ≤C λq+1 . µ
This proves (63). The estimates for wo follow analogously, using in addition the choice of M and (45). Lemma 3.2. Recall that Dt = ∂t + vℓ · ∇. Under the assumptions of Lemma 3.1 we have kDt vℓ kN ≤ Cδq λq ℓ−N , 1/2
(69)
kDt Lkl kN ≤ Cδq+1 δq/2 λq ℓ−N , 1
(70)
1/2
kDt2 Lkl kN ≤ Cδq+1 δq λq ℓ−N −1 , 1/2
(71)
kDt wc kN ≤ Cδq+1 δq/2 λq λN q+1 , 1
(72)
1/2
kDt wo kN ≤ Cδq+1 µλN q+1 . Proof. Estimate on Dt vℓ . transport equation
(73)
Note that vℓ satisfies the inhomogeneous
˚ℓ − (v ⊗ v) ∗ ψℓ + vℓ ⊗ vℓ ) . ∂t vℓ + vℓ · ∇vℓ = −∇p ∗ ψℓ + div (R By hypothesis k∇p ∗ ψℓ kN ≤ Ckpk1 ℓ−N ≤ Cδq λq ℓ−N and analogously ˚ ∗ ψℓ k ≤ Cδq+1 λq ℓ−N . On the other hand, by Proposition C.1: kdiv R kdiv ((v ⊗ v) ∗ ψℓ − vℓ ⊗ vℓ ) kN ≤ Cℓ1−N kvk21 ≤ Cℓ1−N δq λ2q . Thus (69) follows from (55). Estimates on Lkl . Recall that Lkl is defined as i k×B k Lkl := akl Bk + ∇akl − akl (DΦl − Id)k × . λq+1 |k|2
Using that
Dt akl = 0,
Dt Φl = 0,
Dt ∇akl = −DvℓT ∇akl ,
Dt DΦl = −DΦl Dvℓ ,
we obtain Dt Lkl =
−
i λq+1
DvℓT ∇akl
+ akl DΦl Dvℓ k ×
k × Bk . |k|2
(74)
´ ´ SZEKELYHIDI ´ 16 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
Consequently, for times |t − l| < µ−1 and N ≥ 0 we have 1/2
1/2
1/2
1
kDt Lkl kN ≤ Cδq+1 δq λq ℓ
−N
! 1/2 λq δq λq +1 + λq ℓ + λq+1 µ
≤ Cδq+1 δq/2 λq ℓ−N , where we have used (107), Lemma 3.1 and (55). Taking one more derivative and using (74) again, we obtain i i Dt2 Lkl = − (Dt Dvℓ )T ∇akl + Dv T DvℓT ∇akl + λq+1 λq+1 ℓ k×B k . − akl DΦl Dvℓ Dvℓ k + akl DΦl Dt Dvℓ k × |k|2 Note that Dt Dvℓ = DDt vℓ − Dvℓ Dvℓ , so that kDt Dvℓ kN ≤ kDt vℓ kN +1 + kDvℓ kN kDvℓ k0
≤ Cδq λq ℓ−N −1 (1 + λq ℓ) ≤ Cδq λq ℓ−N −1 .
It then follows from the product rule (107) and (55) that 1/2
kDt2 Lkl kN ≤ Cδq+1 δq λq ℓ−N −1
! 1/2 λ2q ℓ δ λ λq q q +1 + + λq ℓ + (λq ℓ)2 + λq+1 λq+1 µ
1/2
≤ Cδq+1 δq λq ℓ−N −1 . P Estimates on wc . Observe that wc = χl (Lkl − akl Bk )eiλq+1 k·Φl (see (41) and (43)). Differentiating this identity we then conclude X Dt wc = χl (Dt Lkl ) eiλq+1 k·Φl + (∂t χl ) (Lkl − akl Bk ) eiλq+1 k·Φl kl
=
X
χl (Dt Lkl )φkl eiλq+1 k·x +
kl
X i∇akl k × Bk + (∂t χl ) − akl (DΦl − Id) k × φkl eiλq+1 k·x . λq+1 |k|2 kl
Hence we obtain (72) as a consequence of Lemma 3.1 and (70). Estimates on wo . Using (74) we have X Dt wo = χ′l akl φkl eiλq+1 k·x . k,l
Therefore (73) follows immediately from Lemma 3.1.
DISSIPATIVE EULER FLOWS
17
4. Estimates on the energy Lemma 4.1 (Estimate on the energy). 1/2 1/2 ˆ 1/2 δq+1 δq λq δ δ λ 1 q q+1 q 2 e(t)(1 − δq+2 ) − +C . |v1 | dx ≤ + C µ µ λq+1 T3 Proof. Define
e¯(t) := 3(2π)3
X
(75)
χ2l (t)ρl .
l
Using Lemma 2.1 we then have X X |wo |2 = χ2l tr Rℓ,l + l
(k,l),(k ′ ,l′ ),k6=−k ′
X
= (2π)−3 e¯ +
χl χl′ wkl · wk,l′ ′
χl χk′ akl ak′ l′ φkl φk′ l′ eiλq+1 (k+k )·x .
(76)
(k,l),(k ′ ,l′ ),k6=−k ′
Observe that e¯ is a function of t only and that, since (k + k′ ) 6= 0 in the sum above, we can apply Proposition E.1(i) with m = 1. From Lemma 3.1 we then deduce ˆ 1/2 δq+1 λq δq+1 δq λq 2 +C . (77) |w | dx − e ¯ (t) ≤ C 3 o µ λq+1 T
Next we recall (42), integrate by parts and use (59) and (62) to reach 1/2 1/2 ˆ δq+1 δq λq (78) 3 v · w dx ≤ C λq+1 . T
Note also that by (63) we have ˆ 1/2 δq+1 δq λq . (79) |wc |2 + |wc wo | dx ≤ C µ T3 Summarizing, so far we have achieved 1/2 1/2 ˆ (78) ˆ ˆ δq+1 δq λq 2 2 2 3 |v1 | dx − e¯(t) + 3 |v| dx ≤ 3 |w| dx − e¯(t) + C λq+1 T T T 1/2 1/2 ˆ 1/2 (79) δq+1 δq λq δq+1 δq λq 2 ≤ |wo | dx − e¯(t) + C +C λq+1 µ T3 1/2
1/2
/2 δq+1 δq λq δq+1 δq λq ≤ C +C . λq+1 µ
(77)
1
(80)
Next, recall that e¯(t) = 3(2π)3
X
χ2l ρl
l
= (1 − δq+2 )
X l
χ2l e
µ l
−
X l
χ2l
ˆ
T3
|v(x, lµ−1 )|2 dx .
´ ´ SZEKELYHIDI ´ 18 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
P Since t − µl < µ−1 on the support of χl and since l χ2l = 1, we have X l χ2l e e(t) − ≤ µ−1 . µ l
Moreover, using the Euler-Reynolds equation, we can compute ˆ
T3
=−
ˆ tˆ 2 |v(x, t)|2 − v x, lµ−1 dx = l µ
ˆ tˆ l µ
2
T3
div v |v| + 2p
+2
ˆ tˆ l µ
T3
T3
∂t |v|2
˚ = −2 v · div R
ˆ tˆ l µ
˚. Dv : R
T3
l Thus, for t − µ ≤ µ−1 we conclude ˆ
1/2 δq+1 δq λq −1 2 . |v(x, t)| − v(x, lµ ) dx ≤ C µ 3 2
T
Using again
P
χ2l = 1, we then conclude
ˆ e(t)(1 − δq+2 ) − e¯(t) +
1/2 1 δq+1 δq λq |v(x, t)| dx ≤ + C . µ µ T3 2
The desired conclusion (75) follows from (80) and (81).
(81)
5. Estimates on the Reynolds stress ˚1 . The general pattern In this section we bound the new Reynolds Stress R in estimating derivatives of the Reynolds stress is that: • the space derivative gets an extra factor of λq+1 (when the derivative falls on the exponential factor), • the transport derivative gets an extra factor µ (when the derivative falls on the time cut-off). In fact the transport derivative is slightly more subtle, because in R0 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 )
DISSIPATIVE EULER FLOWS
19
such that, if µ, λq+1 and ℓ satisfy the conditions (54), then we have 1/2
1/2
δq+1 µ δq+1 δq λq 1 kR k1 + kDt R0 k0 ≤ C 1−ε + 1−ε kR k0 + λq+1 µ λq+1 λq+1 µℓ 1
0
0
(82)
1/2
δq+1 δq λq λεq+1 1 kR k1 + kDt R1 k0 ≤ C kR k0 + λq+1 µ µ 1
1
1
(83)
1/2
1
1 δq+1 δq λq kDt R2 k0 ≤ C λq+1 µ µ 1 1 1/2 1 kR3 k1 + kDt R3 k0 ≤ Cδq+1 δq/2 λq ℓ kR3 k0 + λq+1 µ kR2 k0 +
kR2 k1 +
(85)
1/2
1
kR4 k0 +
(84)
λq+1
kR4 k1 +
δq+1 δq λq 1 kDt R4 k0 ≤ C + Cδq+1 λq ℓ µ µ
(86)
1/2
δq+1 δq λq 1 kR k1 + kDt R5 k0 ≤ C . kR k0 + λq+1 µ µ 1
5
5
(87)
Thus 1
˚1 k0 + kR
λq+1 1/2
≤C
δq+1 µ λ1−ε q+1
˚1 k1 + kR
1 ˚1 k0 ≤ kDt R µ 1/2
1/2
δq+1 δq λq δq+1 δq λq λεq+1 1/2 1 + + δq+1 δq/2 λq ℓ + 1−ε µ λq+1 µℓ
!
(88) ,
and, moreover, ˚1 + v1 · ∇R ˚1 k0 ≤ k∂t R 1/2
1/2
≤ Cδq+1 λq+1
δq+1 µ λ1−ε q+1
1/2
1/2
δq+1 δq λq δq+1 δq λq λεq+1 1/2 1 + + δq+1 δq/2 λq ℓ + 1−ε µ λq+1 µℓ
!
. (89)
Proof. Estimates on R0 . We start by calculating X ∂t w + vℓ · ∇w + w · ∇vℓ = χ′l Lkl + χl Dt Lkl + χl Lkl · ∇vℓ eik·Φl .
kl ′ Define Ωkl := (χl Lkl + χl Dt Lkl + identity φkl eiλq+1 k·x = eiλq+1 k·Φl ),
χl Lkl · ∇vℓ ) φkl and write (recalling the
∂t w + vℓ · ∇w + w · ∇vℓ = Using Lemmas 3.1 and 3.2 and (55)
X
1/2
kl
δq λq kΩkl k0 ≤ Cδq+1 µ 1 + µ 1/2
and similarly, for N ≥ 1
Ωkl eiλq+1 k·x .
!
1/2
≤ Cδq+1 µ
1/2 1/2 N (1−β) kΩkl kN ≤ Cδq+1 µ ℓ−N + kφkl kN ≤ Cδq+1 µ λq+1 .
(90)
´ ´ SZEKELYHIDI ´ 20 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O 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.1(ii) we obtain X ε−1 −N +ε kR0 k0 ≤ λq+1 kΩkl k0 + λq+1 [Ωkl ]N + λ−N [Ω ] N +ε kl q+1 kl
1/2
β+ε −1+ε ≤Cδq+1 µ λq+1 + λ−N q+1
(91)
It suffices to choose N so that N β ≥ 1 in order to achieve 1/2
ε−1 kR0 k0 ≤ Cδq+1 µλq+1 .
As for kR0 k1 , we differentiate (90). We therefore conclude ! X iλq+1 k·x 0 . (iλq+1 kj Ωkl + ∂j Ωkl )e ∂j R = R kl
1/2
Applying Proposition E.1(ii) as before we conclude kR0 k1 ≤ Cδq+1 µλεq+1 . Estimates on Dt R0 . We start by calculating X ∂t2 χl Lkl + 2∂t χl Dt Lkl + χl Dt2 Lkl + Dt (∂t w + vℓ · ∇w + w · ∇vℓ ) = kl
+ ∂t χl Lkl · ∇vℓ + χl Dt Lkl · ∇vℓ + χl Lkl · ∇Dt vℓ − χl Lkl · ∇vℓ · ∇vℓ eik·Φl X =: Ω′kl eiλq+1 k·x . kl
As before, we have kΩ′kl k0
δq λ2q δq λq 1 + δq/2 λq + ≤ Cδq+1 µ µ + µℓ µ 1/2
!
δq λq 1/2 ≤ Cδq+1 µ + (92) µℓ
and, for any N ≥ 1 kΩ′kl kN
δq λ2q δq λq 1 ≤ Cδq+1 µℓ µ+ + δq/2 λq + µℓ µ δq λq 1/2 ≤ Cδq+1 µ µ + ℓ−N + kφkl kN µℓ δq λq 1/2 N (1−β) λq+1 . ≤ Cδq+1 µ µ + µℓ 1/2
−N
!
+ kΩ′kl k0 kφkl kN
DISSIPATIVE EULER FLOWS
21
Next, observe that we can write Dt R0 = [Dt , R] + RDt (∂t w + vℓ · ∇w + w · ∇vℓ ) = [vℓ , R]∇ + RDt (∂t w + vℓ · ∇w + w · ∇vℓ ) X = [vℓ , R](∇Ωkl eiλq+1 k·x ) + iλq+1 [vℓ · k, R](Ωkl eiλq+1 k·x ) + R(Ω′kl eiλq+1 k·x ) kl
(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.1(ii), we obtain [Ω′kl ]N +ε kΩ′kl k0 [Ω′kl ]N + + N −ε λN λ1−ε λq+1 q+1 q+1 1/2 δq+1 µ δq λq β ≤ C 1−ε µ + 1 + λ1−N q+1 µℓ λq+1
kR(Ω′kl eiλq+1 k·x )k0 ≤
Furthermore, applying Proposition F.1 we obtain
kΩkl kN +1+ε kvℓ k2+ε + kΩkl k3+ε kvℓ kN +ε
[vℓ , R](∇Ωkl eiλq+1 k·x ) ≤ C 0 λN q+1 +C
kΩkl kN +1 kvℓ k2 + kΩkl k3 kvℓ kN kΩkl k1 kvℓ k1 +C N −ε λ2−ε λq+1 q+1
1/2
≤C
δq+1 µ λ1−ε q+1
1/2
δq
1/2 δq+1 µ2 3−β(N +1) 3−β−βN λq 1 + λq+1 ≤ C 1−ε 1 + λq+1 λq+1
and similarly 1/2
δq+1 µ2
3−β(N +1) iλq+1 k·x ) ≤ C 1−ε 1 + λq+1 λq+1 [vℓ · k, R](Ωkl e . 0 λq+1
By choosing N ∈ N sufficiently large so that β(N + 1) ≥ 3
and
βN ≥ 1,
we deduce 1/2
0
kDt R k0 ≤ C This concludes the proof of (82).
δq+1 µ λ1−ε q+1
δq λq µ+ µℓ
´ ´ SZEKELYHIDI ´ 22 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
Estimates on R1 . Using Lemma 2.1 we have ! X |wo |2 2˚ Id = div wo ⊗ wo − χl Rℓ,l − 2 l X wkl · wk′ l′ = χl χl′ div wkl ⊗ wk′ l′ − Id = I + II 2 ′ ′ (k,l),(k ,l ) k+k ′ 6=0
where, setting fklk′ l′ := χl χl′ akl ak′ l′ φkl φk′ l′ , X ′ Bk ⊗ Bk′ − 12 (Bk · Bk′ )Id ∇fklk′ l′ eiλq+1 (k+k )·x I= (k,l),(k ′ ,l′ ) k+k ′ 6=0
II =iλq+1
X
(k,l),(k ′ ,l′ ) k+k ′ 6=0
′ fklk′ l′ Bk ⊗ Bk′ − 12 (Bk · Bk′ )Id (k + k′ )eiλq+1 (k+k )·x .
Concerning II, recall that the summation is over all l ∈ Z ∩ [0, µ] and all k ∈ Λe if l is even and all k ∈ Λo if l is odd. Furthermore, both ¯ 2 ∩ Z3 satisfy the conditions of Lemma 1.3. Therefore we may Λe , Λo ⊂ λS symmetrize the summand in II in k and k′ . On the other hand, recall from Lemma 1.2 that (Bk ⊗ Bk′ + Bk′ ⊗ Bk )(k + k′ ) = (Bk · Bk′ )(k + k′ ). From this we deduce that II = 0. Concerning I, we first note, using the product rule, (59) and (60), that for N ≥ 1 . [fklk′ l′ ]N ≤ Cδq+1 λq ℓ1−N + kφkl φk′ l′ kN
By Lemma (62) and (54) (cf. (55)) we then conclude ! 1/2 1/2 δq λq δq λq ≤ Cδq+1 λq+1 , [fklk′ l′ ]1 ≤ Cδq+1 λq + λq+1 µ µ N (1−β)
[fklk′ l′ ]N ≤ Cδq+1 λq+1
,
N ≥ 2.
Applying Proposition E.1(ii) to I we obtain X −N +ε ε−1 ′ ′ ] [f [fklk′ l′ ]N +1 + λ−N λq+1 [fklk′ l′ ]1 + λq+1 kR1 k0 ≤ q+1 klk l N +1+ε (k,l),(k ′ ,l′ ) k+k ′ 6=0
≤ Cδq+1
1/2
δq λq β+ε λεq+1 + λ1−N q+1 µ
!
By choosing N sufficiently large we deduce 1/2
δq+1 δq λq λεq+1 kR k0 ≤ C µ 1
(93)
DISSIPATIVE EULER FLOWS
23
as required. Moreover, differentiating we conclude ∂j R1 = R(∂j I) where X Bk ⊗ Bk′ − 12 (Bk · Bk′ )Id · ∂j I = (k,l),(k ′ ,l′ ) k+k ′ 6=0
′
(iλq+1 (k + k′ )j ∇fklk′ l′ + ∂j ∇fklk′ l′ )eiλq+1 (k+k )·x .
(94)
Therefore we apply again Proposition E.1(ii) to conclude the desired estimate for kR1 k1 .
Estimates on Dt R1 . As in the estimate for Dt R0 , we again make use of the identity Dt R = [vℓ , R]∇ + RDt in order to write X ′ Dt R1 = [vℓ , R] ∇Uklk′ l′ eiλq+1 (k+k )·x (k,l),(k ′ ,l′ ) k+k ′ 6=0
′ iλq+1 (k+k ′ )·x ′ , + iλq+1 [vℓ · (k + k′ ), R] Uklk′ l′ eiλq+1 (k+k )·x + R Uklk ′ l′ e where we have set Uklk′ l′ = Bk ⊗ Bk′ − 12 (Bk · Bk′ )Id ∇fklk′l′ and 2 X X ′ iλq+1 (k+k ′ )·x ˚ℓ,l − |wo | Id = Uklk . Dt div wo ⊗ wo − χ2l R ′ l′ e 2 ′ ′ l
′ In order to further compute Uklk ′ l′ , we write
(k,l),(k ,l ) k+k ′ 6=0
′
′
∇fklk′ l′ eiλq+1 (k+k )·x = χl χl′ (akl ∇ak′ l′ + ak′ l′ ∇akl ) eiλq+1 (k·Φl +k ·Φl′ ) + ′ + iλq+1 χl χl′ akl ak′ l′ (DΦl − Id)k + (DΦl′ − Id)k′ eiλq+1 (k·Φl +k ·Φl′ )
and hence, using (74), ′ ′ Dt ∇fklk′ l′ eiλq+1 (k+k )·x = (χl χl′ )′ (akl ∇ak′ l′ + ak′ l′ ∇akl ) eiλq+1 (k·Φl +k ·Φl′ ) ′ + iλq+1 (χl χl′ )′ akl ak′ l′ (DΦl − Id)k + (DΦl′ − Id)k′ eiλq+1 (k·Φl +k ·Φl′ ) ′ − χl χl′ akl DvℓT ∇ak′ l′ + ak′ l′ DvℓT ∇akl eiλq+1 (k·Φl +k ·Φl′ ) ′ − χl χl′ akl ak′ l′ DΦl DvℓT k + DΦl′ DvℓT k′ eiλq+1 (k·Φl +k ·Φl′ ) ′ =: Σ1klk′ l′ + Σ2klk′ l′ + Σ3klk′ l′ + Σ4klk′ l′ eiλq+1 (k·Φl +k ·Φl′ ) ′
=: Σklk′ l′ eiλq+1 (k·Φl +k ·Φl′ ) .
Ignoring the subscripts we can use (107), Lemma 3.1 and Lemma 3.2 to estimate (55)
kΣkN ≤ Cδq+1 λq ℓ−N (µ + λq+1 δq/2 + δq/2 λq + δq/2 ) ≤ Cδq+1 λq+1 δq/2 λq ℓ−N . 1
1
1
1
We thus conclude
′ kUklk ′ l′ kN ≤ CkΣklk ′ l′ kN + CkΣklk ′ l′ k0 kφkl φk ′ l′ kN 1 1 N (1−β) 1+N (1−β) ≤ Cδq+1 λq+1 δq/2 λq ℓ−N + λq+1 ≤ Cδq+1 δq/2 λq λq+1
´ ´ SZEKELYHIDI ´ 24 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
The estimate on kDt R1 k0 now follows exactly as above for Dt R0 applying Proposition F.1 to the commutator terms. This concludes the verification of (83). Estimates on R2 and Dt R2 . Using Lemma 3.1 we have 1/2
2
kR k0 ≤
C(kwc k20
δq λq , + kwo k0 kwc k0 ) ≤ Cδq+1 µ 1/2
kR2 k1 ≤ C(kwc k1 kwc k0 + kwo k1 kwc k0 + kwo k0 kwc k1 ) ≤ Cδq+1
δq λq λq+1 . µ
Similarly, with the Lemmas 3.1 and 3.2 we achieve
Dt R2 ≤ C kDt wc k (kwo k + kwc k ) + CkDt wo k0 kwc k0 ≤ Cδq+1 δq1/2 λq . 0 0 0 0
Estimates on R3 and Dt R3 . The estimates on kR3 k0 and kR3 k1 are a direct consequence of the mollification estimates (26) and (28) as well as Lemma 3.1. Moreover, kDt R3 k0 ≤ kv − vℓ k0 kDt wk0 + (kDt vk0 + kDt vℓ k)kwk0
= kv − vℓ k0 (kDt wc k0 + kDt wo k) + (kDt vk0 + kDt vℓ k)kwk0 (95)
Concerning Dt v, note that, by our inductive hypothesis
kDt vk0 ≤ k∂t v + v · ∇vk0 + kv − vℓ k0 kvk1 ˚q k1 + Cδq λ2q ℓ ≤ Cδq λq . ≤ kpq k1 + kR
Thus the required estimate on Dt R3 follows from Lemma 3.2. Estimates on R4 and Dt R4 . From the mollification estimates (27) and (29) we deduce ˚ 1 ℓ ≤ Cδq+1 λq ℓ kR4 k0 ≤ CkRk ˚ 1 ≤ Cδq+1 λq . kR4 k1 ≤ 2kRk
As for Dt R4 , observe first that, using our inductive hypothesis, ˚ 0 ≤ k∂t R ˚ + v · ∇Rk ˚ 0 + kvℓ − vk0 kRk ˚ 1 ≤ Cδq+1 δq1/2 λq + Cδq+1 δq1/2 λ2q ℓ kDt Rk
Moreover, ˚ℓ = (Dt R) ˚ ∗ ψℓ + vℓ · ∇R ˚ℓ − (vℓ · ∇R) ˚ ∗ ψℓ Dt R ˚ ∗ ψℓ + div vℓ ⊗ R ˚ℓ − (v ⊗ R) ˚ ∗ ψℓ +[(v − vℓ ) · ∇R] ˚ ∗ ψℓ , =(Dt R)
(96)
where we have used that div v = 0. Using Proposition C.1 we deduce ˚ℓ − (v ⊗ R) ˚ ∗ ψℓ k1 ≤ Cδq+1 δ1/2 λq λq ℓ . (97) kvℓ ⊗ R q Gathering all the estimates we then achieve ˚ 0 + kDt R ˚ℓ k0 ≤ Cδq+1 λq δq1/2 . kDt R4 k0 ≤ kDt Rk
The estimate (86) follows now using (55).
DISSIPATIVE EULER FLOWS
25
˚ℓ,l = 0. Therefore, using the arguEstimates on R5 . Recall that Dt R ments from (96) ˚ℓ − R ˚ℓ,l )k0 = kDt R ˚ℓ k0 ≤ Cδq+1 δq1/2 λq . kDt (R On the other hand, using again the identity (96) and Proposition C.1 ˚ℓ − R ˚ℓ,l )k1 = kDt R ˚ℓ k1 ≤ Cℓ−1 kDt Rk ˚ 0 + kvℓ ⊗ R ˚ℓ − (v ⊗ R) ˚ ∗ ψℓ k2 kDt (R ˚ 1 ≤ Cδq+1 δ1/2 λq ℓ−1 . + Cℓ−1 kv − vℓ k0 kRk q
˚ℓ,l (x, tµ−1 ) = R ˚ℓ (x, tµ−1 ), the difference R ˚ℓ − R ˚ℓ,l vanishes at t = Since R −1 lµ . From Proposition B.1 we deduce that, for times t in the support of χl (i.e. |t − lµ−1 | < µ−1 ), ˚ℓ − R ˚ℓ,l k0 ≤ Cµ−1 kDt (R ˚ℓ − R ˚ℓ,l )k0 ≤ Cµ−1 δq+1 δq1/2 λq kR ˚ℓ − R ˚ℓ,l k1 ≤ Cµ−1 kDt (R ˚ℓ − R ˚ℓ,l )k1 ≤ Cµ−1 δq+1 δ1/2 λq ℓ−1 . kR q
The desired estimates on kR5 k0 and kR5 k1 follow then easily using (55). Estimate on Dt R5 . In this case we compute X X ˚ℓ − R ˚ℓ,l ) + ˚ℓ . Dt R5 = 2χl ∂t χl (R χ2l Dt R l
l
The second summand has been estimate above and, since k∂t χl k0 ≤ Cµ, 1/2 the first summand can be estimated by Cµδq+1 δq λq µ−1 (again appealing to the arguments above). Proof of (89). To achieve this last inequality, observe that ˚1 + v1 · ∇R ˚1 k0 ≤ kDt R ˚1 k0 + (kv − vℓ k0 + kwk0 ) kR ˚1 k1 . k∂t R 1/2
On the other hand, by (26), kv − vℓ k0 ≤ Cδq λq ℓ. Moreover, by (45), (63) 1/2 and (55) kwk ≤ kwo k0 + kwc k0 ≤ Cδq+1 . Thus, by (88) we conclude ˚1 + v1 · ∇R ˚1 k0 ≤ C µ + δ1/2 λq+1 k∂t R q+1 ! 1/2 1/2 1/2 δq+1 δq λq δq+1 µ δq+1 δq λq λεq+1 1/2 1/2 + + δq+1 δq λq ℓ + 1−ε µ λ1−ε λq+1 µℓ q+1 1/2
Since by (55) µ ≤ δq+1 λq+1 , (89) follows easily.
6. Conclusion of the proof In Sections 2-5 we showed the construction for a single step, referring to ˚q ) as (v, p, R) ˚ and to (vq+1 , pq+1 , R ˚q+1 ) as (v1 , p1 , R ˚1 ). From now (vq , pq , R 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 {δq }q∈N and {λq }q∈N are chosen to satisfy q
δq = a−b ,
q+1
acb
q+1
≤ λq ≤ 2acb
´ ´ SZEKELYHIDI ´ 26 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O 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 1/2
2
1/5
δq/2 λq/5 ≤ δq+1 λq+1 , 1
1
δq+1 ≤ δq ,
b+1 . λq ≤ λq+1
and
(98)
6.1. Choice of the parameters µ and ℓ. We start by specifying the parameters µ = µq and ℓ = ℓq : we determine them optimizing the right hand side of (88). More precisely, we set 1/4
1/2
µ := δq+1 δq/4 λq/2 λq+1 1
1
(99)
so that the first two expressions in (88) are equal, and then, having determined µ, set −1/8
−3/4
ℓ := δq+1 δq/8 λq− /4 λq+1 1
1
(100)
so that the last two expressions in (88) are equal (up to a factor λεq+1 ). In turn, these choices lead to ˚q+1 k0 + kR
1
3/4
ε−1/2
3/8
ε−3/4
/8 /4 ˚q+1 k1 ≤ Cδ δ /4 λ /2 λ kR q q+1 + Cδq+1 δq λq λq+1 q+1 q λq+1 !3/4 1/2 1/3 δq λq 3/4 1 1 ε−1/2 = Cδq+1 δq/4 λq/2 λq+1 1 + 1/2 1/3 δq+1 λq+1 (98)
3/4
1
1
3
5
ε−1/2
≤ Cδq+1 δq/4 λq/2 λq+1 . 1
1
(101)
Observe also that by (89), we have ˚q+1 + vq+1 · ∇R ˚q+1 k0 ≤ Cδ1/2 λq+1 δ3/4 δq1/4 λq1/2 λε−1/2 . k∂t R q+1 q+1 q+1
(102)
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 1/2
1/2
δq λq ℓ 1/2
δq+1
=
1 = ℓλq+1
3/5
δq λq 1/2
3/5
!5/4
δq+1 λq+1 !1/4 1/2 δq+1 λq 1/2
δq λq+1
,
1/2
,
1/2
δq λq = µ µ 1/2
δq+1 λq+1
δq λq 1/2
δq+1 λq+1
!1/2
1/2
=
δq λq 1/2
δq+1 λq+1
Hence the conditions (54) follow from (98) choosing β =
,
!1/2
b−1 5b+5 .
.
DISSIPATIVE EULER FLOWS
27
6.2. Proof of Proposition 0.2. Fix the constants c > 52 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 ˚0 ) = (0, 0, 0). Given now (vq , pq , R ˚q ) satisfying the estimates (6)(v0 , p0 , R ˚q+1 ) constructed above satisfies (13), we claim that the triple (vq+1 , pq+1 , R again all the corresponding estimates. ˚q+1 . Note first of all that, using the form of the estimates Estimates on R 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 3/4
ε−1/2
Cδq+1 δq/4 λq/2 λq+1 ≤ ηδq+2 . 1
1
Using our choice of δq and λq from Proposition 0.2, we see that this inequality is equivalent to 1 q 2 C ≤ a 4 b (1+3b−2cb+(2c−4−4εc)b ) , which, since b > 1, is satisfied for all q ≥ 1 for a sufficiently large fixed constant a > 1, provided 1 + 3b − 2cb + (2c − 4 − 4εc)b2 > 0.
Factorizing, we obtain the inequality (b − 1)((2c − 4)b − 1) − 4εcb2 > 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 ˚q+1 . sufficiently large so that (10), and hence also (11) and (13) hold for R Estimates on vq+1 − vq . By (45), Lemma 3.1 and (54) we conclude M 1/2 −β + λq+1 (103) kvq+1 − vq k0 ≤ kwo k0 + kwc k0 ≤ δq+1 2 M 1/2 −β + λq+1 . (104) kvq+1 − vq k1 ≤ kwo k1 + kwc k1 ≤ δq+1 2 2
Since λq+1 ≥ λ1 ≥ acb , for a sufficiently large we conclude (6) and (7). Estimate on the energy. Recall Lemma 4.1 and observe that, by (54), 1/2
1/2
1/2
δq+1 δq λq λq+1
≤
δq+1 µ λq+1 .
Moreover, q+1 −bq /2+cbq+1
δq+1 δq/2 λq = a−b 1
q ((c−1)b−1/2)
= ab
δ
δ
≥ a ≥ 1. 1/2
λ
δ
1/2
µ
So the right hand side of (75) is smaller than C q+1 µq q + C λq+1 , i.e. q+1 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 pq+1 − p1 . From the definition of pq+1 in (52) we deduce 1 kpq+1 − pq k0 ≤ (kwo k0 + kwc k0 )2 + Cℓkvq k1 kwk0 2
´ ´ SZEKELYHIDI ´ 28 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR. 1/2
As already argued in the estimate for (6), kwo k + kwc k ≤ M δq . Moreover 1/2 1/2 Cℓkv1 k1 kwk0 ≤ CM δq+1 δq λq ℓ, which is smaller than the right hand side of (88). Having already argued that such quantity is smaller than ηδq+2 we 2 can obviously bound Cℓkvq k1 kwk0 with M2 δq+1 . This shows (8). Moreover, differentiating (52) we achieve the bound 1/2
kpq+1 − pq k1 ≤ (kwo k1 + kwc k1 )(kwo k0 + kwc k0 ) + Cδq+1 δq/2 λq λq+1 ℓ 1
and arguing as above we conclude (9). Estimates (14). Here we can use the obvious identity ∂t wq = Dt wq − (vq )ℓ · ∇wq together with Lemmas 3.1 and 3.2 to obtain k∂t vq+1 − ∂t vq k0 ≤ 1/2 1/2 Cδq+1 λq+1 Then, using (21), we conclude k∂t vq k0 ≤ Cδq λq . To handle ∂t pq+1 − ∂t pq observe first that, by our construction, k∂t (pq+1 − pq )k0 ≤ (kwc k0 + kwo k0 )(k∂t wc k0 + k∂t wo k0 ) + 2kwk0 k∂t vq k0 + ℓkvq k1 k∂t wk0 .
1/2
As above, we can derive the estimates k∂t wo k0 + k∂t wc k0 ≤ Cδq+1 λq+1 from Lemmas 3.1 and 3.2. Hence 1/2
1/2
k∂t (pq+1 − pq )k0 ≤ Cδq+1 λq+1 + Cδq+1 δq/2 λq + Cδq/2 λq ℓδq+1 λq+1 . (105) 1/2
1
1
1/2
Since ℓ ≤ λ−1 q and δq λq ≤ δq+1 λq+1 , the desired inequality follows. This concludes the proof. ¨ lder spaces Appendix A. Ho In the following m = 0, 1, 2, . . . , α ∈ (0, 1), and β is a multi-index. We introduce the usual (spatial) H¨older norms as follows. First of all, the supremum norm is denoted by kf k0 := supT3 ×[0,1] |f |. We define the H¨older seminorms as [f ]m = max kD β f k0 , |β|=m
|D β f (x, t) − D β f (y, t)| , |x − y|α |β|=m x6=y,t
[f ]m+α = max sup
where D β are space derivatives only. The H¨older norms are then given by kf km
m X [f ]j = j=0
kf km+α = kf km + [f ]m+α . Moreover, we will write [f (t)]α and kf (t)kα 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: (106) [f ]s ≤ C εr−s [f ]r + ε−s kf k0
DISSIPATIVE EULER FLOWS
for r ≥ s ≥ 0, ε > 0, and
[f g]r ≤ C [f ]r kgk0 + kf k0 [g]r − r1
1
for any 1 ≥ r ≥ 0. From (106) with ε = kf k0r [f ]r interpolation inequalities 1− rs
[f ]s ≤ Ckf k0
29
(107)
we obtain the standard
s
[f ]rr .
(108)
Next we collect two classical estimates on the H¨older norms of compositions. These are also standard, for instance in applications of the NashMoser iteration technique. Proposition A.1. Let Ψ : Ω → R and u : Rn → Ω be two smooth functions, with Ω ⊂ RN . 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 kukm−1 [u]m ) 0 [Ψ]m [u]m 1 ).
[Ψ ◦ u]m ≤ C([Ψ]1 [u]m +
(109) (110)
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 = f0 ,
where v = v(t, x) is a given smooth vector field. We denote the material derivative ∂t + v · ∇ by Dt . We will consider solutions on the entire space R3 and treat solutions on the torus simply as periodic solution in R3 . Proposition B.1. Any solution f of (111) satisfies ˆ t kf (t)k0 ≤ kf0 k0 + kg(τ )k0 dτ ,
(112)
t0
[f (t)]1 ≤ [f0 ]1 e(t−t0 )[v]1 +
ˆ
t
e(t−τ )[v]1 [g(τ )]1 dτ ,
(113)
t0
and, more generally, for any N ≥ 2 there exists a constant C = CN so that [f (t)]N ≤ [f0 ]N + C(t − t0 )[v]N [f0 ]1 eC(t−t0 )[v]1 + ˆ t (114) + eC(t−τ )[v]1 [g(τ )]N + [v]N [g(τ )]1 dτ. t0
Define Φ(t, ·) to be the inverse of the flux X of v starting at time t0 as the d X = v(X, t) and X(x, t0 ) = x). Under the same assumptions identity (i.e. dt as above: kDΦ(t) − Idk0 ≤ e(t−t0 )[v]1 − 1 ,
[Φ(t)]N ≤ C(t − t0 )[v]N eC(t−t0 )[v]1
(115) ∀N ≥ 2.
(116)
´ ´ SZEKELYHIDI ´ 30 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
Proof. We start with the following elementary observation for transport d f (X(t, x), t) = g(X(t, x), t) and conequations: if f solves (111), then dt sequently ˆ t f (t, x) = f0 (Φ(x, t)) + g(X(Φ(t, x), τ ), τ ) dτ. t0
The maximum principle (112) follows immediately. Next, differentiate (111) in x to obtain the identity Dt Df = (∂t + v · ∇)Df = Dg − Df Dv.
Applying (112) to Df yields
[f (t)]1 ≤ [f0 ]1 +
ˆ
t
([g(τ )]1 + [v]1 [f (τ )]1 ) dτ.
t0
An application of Gronwall’s inequality then yields (113). More generally, differentiating (111) N times yields ∂t D N f + (v · ∇)D N f = D N g +
N −1 X
cj,N D j+1 f : D N −j v
(117)
j=0
(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 ˆ t [g(τ )]N + C [v]N [f (τ )]1 + [v]1 [f (τ )]N dτ. [f (t)]N ≤ [f0 ]N + t0
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(·, t0 ) = 0. Consider next Ψ(x, t) = Φ(x, t) − x and observe first that ∂t Ψ + v · ∇Ψ = −v. Since DΨ(·, t0 ) = 0, we apply (113) to conclude ˆ t [Ψ(t)]1 ≤ e(t−τ )[v]1 [v]1 dτ = e(t−t0 )[v]1 − 1 . t0
Since DΨ(x, t) = DΦ(x, 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 ∞ (T3 × T) and ψ the mollifier of Section 2. For any r ≥ 0 we have the estimate
(f ∗ χℓ )(g ∗ χℓ ) − (f g) ∗ χℓ ≤ Cℓ2−r kf k1 kgk1 , r
where the constant C depends only on r.
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 C(α, m) with the following properties. If φ, ψ : T3 → R are the unique solutions of ∆φ = f ∆ψ = div F , ffl ffl φ=0 ψ=0 then
kφkm+2+α ≤ C(m, α)kf km,α
and
kψkm+1+α ≤ C(m, α)kF km,α . (118)
Moreover we have the estimates kRvkm+1+α ≤ C(m, α)kvkm+α kR(div A)km+α ≤ C(m, α)kAkm+α
(119) (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 ∈ Z3 \ {0} and λ ≥ 1 be fixed. For any a ∈ C ∞ (T3 ) and m ∈ N we have ˆ [a]m iλk·x (121) dx ≤ m . 3 a(x)e λ T
(ii) Let k ∈ Z3 \ {0} be fixed. For a smooth vector field a ∈ C ∞ (T3 ; R3 ) let F (x) := a(x)eiλk·x . Then we have kR(F )kα ≤
C λ1−α
kak0 +
C λm−α
[a]m +
C [a]m+α , λm
where C = C(α, m). Appendix F. One further commutator estimate Proposition F.1. Let k ∈ Z3 \ {0} be fixed. For any smooth vector field a ∈ C ∞ (T3 ; R3 ) and any smooth function b, if we set F (x) := a(x)eiλk·x , we then have C C k[b, R](F )kα ≤ 2−α kak0 kbk1 + m−α (kakm kbk2 + kak2 kbkm ) λ λ C (122) + m (kakm+α kbk2+α + kak2+α kbkm+α ), λ where C = C(α, m).
´ ´ SZEKELYHIDI ´ 32 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
Proof. Step 1 First of all, given a vector field v define the operator 2 S(v) := ∇v + (∇v)t − (div v)Id . 3 First observe that div S(v) = 0
⇐⇒
v ≡ const.
(123)
One implication is obvious. Next, assume div S(v) = 0. This is equivalent to the equations 1 (124) ∆vj + ∂j div v = 0 3 Differentiating and summing in j we then conclude 4 ∆div v = 0 . 3 Thus div v must be constant and, since any divergence has average zero, we conclude that div v = 0. Thus (124) implies that ∆vi = 0 for every i, which in turn gives the desired conclusion. From this observation we conclude the identity R(v) = S(v) + R(v − div S(v))
for all v ∈ C ∞ (T3 , R3 ).
(125)
Indeed, observe first that R(v) = S(w), where w = 41 P(v) + 34 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 R(v) = ffl v − v, applying the divergence we obtain v−
v = div S(v) + v −
v − div S(v) ,
which is obviously true. Step 2 Next, for a ∈ C ∞ (T3 , R3 ), k ∈ Z3 \ {0} and λ ∈ N \ {0}, consider 3 a 1 (a · k)k iλk·x iλk·x iλk·x S (ae ) := −S e + 2 2 a− e . 4 λ2 |k|2 4λ |k| |k|2 Observe that S (baeiλk·x ) − bS (aeiλk·x ) =
aA(b) iλk·x e , λ2
(126)
where A is an homogeneous differential operator of order one with constant coefficients (all depending only on k). Moreover, aeiλk·x − div S (aeiλk·x ) =
B1 (a) iλk·x B2 (a) iλk·x e + e , λ λ2
where B1 and B2 are homogeneous differential operators of order 1 and 2 (respectively) with constant coefficients (again all depending only on k).
DISSIPATIVE EULER FLOWS
33
We use then the identity (125) to write − [b, R](F ) = R(bF ) − bR(F ) = S (baeiλk·x ) − bS (aeiλk·x ) iλk·x iλk·x + R bF − div S (bae ) − bR F − div S (ae ) B1 (ab) iλk·x B2 (ab) iλk·x aA(b) e + e = 2 eiλk·x + R λ λ λ2 B1 (a) iλk·x B2 (a) iλk·x − bR e + e . (127) λ λ2 Using the Leibniz rule we can write B1 (ab) = B1 (a)b + aB1 (b) and B2 (ab) = B2 (a)b+aB2 (b)+C1 (a)C1 (b), where C1 is an homogeneous operator of order 1. We can then reorder all terms to write aA(b) iλk·x e −[b, R](F ) = λ2 aB2 (b) + C1 (a)C1 (b) iλk·x aB1 (b) iλk·x e e +R +R λ λ2 1 1 − [b, R] (B1 (a))eiλk·x − 2 [b, R] b(B2 (a))eiλk·x . (128) λ λ 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 kak2 kbkN + kakN kbk2 kak0 kbk1 +C 2−α λ λN −α kak2+α kbkN +α + kakN +α kbk2+α +C λN
1 1
iλk·x + [b, R] b(B1 (a))e
+ 2 [b, R] b(B2 (a))eiλk·x . (129) λ| λ α α {z }
k[b, R](F )kα ≤ C
II
(Indeed the above estimate is sub-optimal as we get, for instance, terms of type kak0 kbkN + kak1 kbkN −1 + kak2 kbkN −2 instead of kak2 kbkN ; 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 kak0 kbk1 kak2 kbkN + kakN kbk2 k[b, R](F )kα ≤ C +C 2−α λ λN −α 1 kak2+α kbkN +α + kakN +α kbk2+α
′ iλk·x + R] b(B (a))e +C
.
[b, 2 λN λ2 α
where B2′ = B2 + B1 ◦ B1 is second order and this time we have applied Proposition E.1 with m = N −2 and m = N −3 to handle the corresponding
´ ´ SZEKELYHIDI ´ 34 TRISTAN BUCKMASTER, CAMILLO DE LELLIS, AND LASZL O JR.
two terms of order λ−2 and λ−3 arising from II. Proceeding now inductively, we end up with kak2 kbkN + kakN kbk2 kak0 kbk1 +C 2 λ λN −α kak2+α kbkN +α + kakN +α kbk2+α +C λN
1
′ ′ + N R b(BN (130) (a)eiλk·x . (a))eiλk·x − bR BN λ α Finally, applying Proposition D.1 to the final term we reach the desired estimate. kR(bF ) − bR(F )k0 ≤
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[18] Robert, R. Statistical hydrodynamics (Onsager revisited). In Handbook of mathematical fluid dynamics, Vol. II. North-Holland, Amsterdam, 2003, pp. 1–54. [19] Scheffer, V. An inviscid flow with compact support in space-time. J. Geom. Anal. 3, 4 (1993), 343–401. [20] Shnirelman, A. Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys. 210, 3 (2000), 541–603. ¨ r Mathematik, Universita ¨ t Leipzig, D-04103 Leipzig Institut fu E-mail address:
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