DISSIPATIVE CONTINUOUS EULER FLOWS 1. Introduction In what ...

DISSIPATIVE CONTINUOUS EULER FLOWS ´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR. Abstract. We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.

1. Introduction T3

In what follows denotes the 3-dimensional torus, i.e. T3 = S1 ×S1 ×S1 . In this note we prove the following theorem. Theorem 1.1. Assume e : [0, 1] → R is a positive smooth function. Then there is a continuous vector field v : T3 × [0, 1] → R3 and a continuous scalar field p : T3 × [0, 1] → R 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)

Obviously, if we choose a strictly decreasing function e(t), Theorem 1.1 yields continuous solutions of the incompressible Euler equations which “dis´ sipate” the total kinetic energy 12 |v|2 (x, t) dx. This is not possible for C 1 solutions: in that case one can multiply the first equation in (1) by v to derive   2  |v|2 |u| + div u +p = 0. ∂t 2 2 Integrating this last identity in x we then conclude ˆ d |v|2 (x, t) dx = 0 . dt T3 2

(3)

Theorem 1.1 shows therefore that this formal computation cannot be justified for distributional solutions, even if they are continuous. The pair (v, p) in Theorem 1.1 solves (1) in the following sense: ˆ 1ˆ (∂t ϕ · v + ∇ϕ : v ⊗ v + p div ϕ) dxdt = 0 (4) 0

T3

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´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

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for all ϕ ∈ Cc∞ (T3 × (0, 1); R3 ) and ˆ 1ˆ v · ∇ψ dxdt = 0 0

T3

for all ψ ∈ Cc∞ (T3 × (0, 1)).

Remark 1. In the usual definition of weak solution, (4) is replaced by the same condition for divergence free test fields: therefore p disappears from the identity. With this alternative definition, for every weak solution v which belongs to L2 a corresponding pressure field can then be recovered using ∆p = −div div (v ⊗ v) .

(5)

p is then determined up to an arbitrary function of t: this arbitrariness can ffl be overcome by imposing, for instance, p(x, t) dx = 0. However, as it is well-known, the equation (5) and the continuity of v does not guarantee the continuity of p. 1.1. Onsager’s Conjecture. The possibility that weak solutions might dissipate the total kinetic energy has been considered for a rather long time in the fluid dynamics literature: this phenomenon goes under the name of “anomalous dissipation”. In fact, to our knowledge, the existence of dissipative solutions was considered for the first time by Lars Onsager in his famous 1949 note about statistical hydrodynamics, see [18]. In that paper Onsager conjectured that (a) C 0,α solutions are energy conservative when α > 31 ; (b) There exist dissipative solutions with C 0,α regularity for any α
0 and λ0 > 1 with the following property. There exist pairwise disjoint subsets Λj ⊂ {k ∈ Z3 : |k| = λ0 }

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) R= γk (R) Id − ⊗ ∀R ∈ Br0 (Id) . 2 |k| |k|

(22)

k∈Λj

Remark 2. Though it will not be used in the sequel, the cardinality of each set Λj constructed in the proof of the Lemma is indeed bounded a priori independently of all the other parameters. A close inspection of the proof shows that it gives sets with cardinality at most 98. This seems however far from optimal: one should be able to find sets Λj with cardinality 14. The proof of the Geometric Lemma is based on the following well-known fact. Proposition 3.3. The set Q3 ∩ S2 is dense in S2 .

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´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

Proof. Let s : R2 → S2 be the inverse of the stereographic projection:   2v 2u u2 + v 2 − 1 s(u, v) := . , , u2 + v 2 + 1 u2 + v 2 + 1 u2 + v 2 + 1 It is obvious that s(Q2 ) ⊂ Q3 . Since Q2 is dense in R2 and s is a diffeomorphism onto S2 \ (0, 0, 1), the proposition follows trivially.  Indeed, much more can be proved: n1 Z3 ∩ S2 , distributes uniformly on the sphere for n ∈ N large whenever n ≡ 1, 2, 3, 4, 5, 6 (mod 8). This problem was raised by Linnik (see [16]), who proved a first result in its direction, and solved thanks to a breakthrough of Iwaniec [14] in the theory of modular forms of half-integral weight (see, for instance, [9] and [19]). Proof of Lemma 3.2. For each vector v ∈ R3 \ {0}, we denote by Mv the 3 × 3 symmetric matrix given by v v ⊗ . Mv = Id − |v| |v| With this notation the identity (22) reads as 2 1 X  (j) R= γk (R) Mk . 2

(23)

k∈Λj

Step 1 Fix a λ0 > 1 and for each set F ⊂ {k ∈ Z3 : |k| = λ0 } we consider the set c(F ) which is the interior of the convex hull, in S 3×3 , of {Mk : k ∈ F }. We claim in this step that it suffices to find a λ0 and N disjoint subsets Fj ⊂ {k ∈ Z3 : |k| = λ0 } such that (d) −Fj = Fj ; (e) c(Fj ) contains a positive multiple of the identity. Indeed, we will show below that, if Fj satisfies (d) and (e), then we can find a (j) r0 > 0, a subset Γj ⊂ Fj and positive smooth functions λk ∈ C ∞ (B2r0 (Id)) such that X (j) R= λk (R)Mk . k∈Γj (j)

We then find Λj and the functions γk by • defining Λj := Γj ∪ −Γj ; (j) • setting λk = 0 if k ∈ Λj \ Γj ; • defining q (j)

γk :=

(j)

(j)

λk + λ−k

for every k ∈ Λj . Observe that the functions and the sets satisfy both (a) and (b). Moreover, (j) (j) since at least one of the λ±k is positive on B2r0 (Id), γk is smooth in Br0 (Id). We now come to the existence of the set Γj . For simplicity we drop the subscripts. The open set c(F ) contains an element αId with α > 0. Then

DISSIPATIVE CONTINUOUS EULER FLOWS

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there are seven matrices A1 , . . . , A7 in c(F ) such that αId belongs to the interior of their convex hull, which is an open convex simplex S. We choose ˜ of center αId and radius ϑ is contained in S. Then ϑ so that the unit ball U ˜ each point R ∈ U can be written in a unique way as a convex combination of the elements Ai : R=

7 X

βi (R)Ai

i=1

˜. and the functions βi are positive and smooth on U P By Caratheodory’s Theorem, each Ai is the convex combination λi,n Mvi,n of at most 7 Mvi,n with vi,n ∈ F , where we require that each λi,n is positive (observe that Caratheodory’s Theorem guarantees the existence of 7 points Mvi,n such that Ai belongs to the closed convex hull of them; if we insist on the property that the corresponding coefficients are all positive, then we might be obliged to choose a number smaller than 7). ϑ Set r0 := 2α . Then, R=

X1 βi (αR)λi,n Mvi,n α

∀R ∈ B2r0 (Id) .

i,n

and each coefficient 1 βi (αR)λi,n α is positive for every R ∈ B2r0 (Id). The set Γj is then given by {vi,n }. Note that we might have vi,n = vl,m for two distinct pairs (i, n) and (l, m). Therefore, for k ∈ Γj , the function λk will be defined as X 1 βi (αR)λi,n . λk (R) = α (i,n):k=vi,n

Step 2 By Step 1, in order to prove the lemma, it suffices to find a number λ0 and N disjoint families F1 , . . . , FN ⊂ λ0 S2 ∩ Z3 such that the sets c(Fi ) contain all a positive multiple of the identity. By Proposition 3.3 there is a sequence λk ↑ ∞ such that the sets S2 ∩ λ1k Z3 converge, in the Hausdorff sense, to the entire sphere S2 . Given this sequence {λk }, we can easily partition each λk S2 ∩ Z3 into N disjoint symmetric families {Fjk }j=1,...,N in such a way that, for each fixed j, the corresponding sequence of sets { λ1k Fjk }k converges in the Hausdorff sense to S2 . Hence, any point of c(S2 ) is contained in c( λ1k Fjk ) provided k is large enough. On the other hand it is easy to see that c(S2 ) contains a multiple of the identity αId (for instance one can adapt the argument of Lemma 4.2 in [6]). By Step 1, this concludes the proof. 

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´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

˚1 and p1 4. The maps v1 , R ˚1 and p1 of Proposition We have now all the tools to define the maps v1 , R 2.2. Recalling (17) and the discussion therein, w := v1 − v is the sum of two maps, wo and wc . wo is a highly oscillatory function based on “patching Beltrami flows” and it will be defined first, in Section 4.1. wc will then be added so as to ensure that v1 is divergence free: in order to achieve this we will use the classical Leray projector, see Section 4.3 for the precise definition. p1 is related to wo by a simple formula, given in Section 4.4. ˚1 . Essentially, this last matrix field Finally, in Section 4.5 we will define R can also be thought of as a “corrector term”, analogous to wc . In fact, if we consider the point of view of [6], the Euler-Reynolds system can be stated equivalently as the fact that the 4 × 4 matrix   ˚1 v1 v1 ⊗ v1 + p1 Id − R U := v1 0 ˚1 has therefore the same flavor as wc is a divergence-free in space-time. R and is also defined through a suitable (elliptic) operator, cp. with Definition 4.2. 4.1. The perturbation wo . We start by defining a partition of unity on the space of √ velocities, i.e. the state space. Choose two constants c1 and c2 such that 23 < c1 < c2 < 1 and we fix a function ϕ ∈ Cc∞ (Bc2 (0)) which is nonnegative and identically 1 on the ball Bc1 (0). We next consider the lattice Z3 ⊂ R3 and its quotient by (2Z)3 , i.e. we define the equivalence relation (k1 , k2 , k3 ) ∼ (`1 , `2 , `3 )

⇐⇒

ki − `i is even ∀i.

We then denote by Cj , j = 1, . . . , 8 the 8 equivalence classes of Z3 / ∼. For each k ∈ Z3 denote by ϕk the function ϕk (x) := ϕ(x − k) . Observe that, if k 6= ` ∈ Ci , then |k − `| ≥ 2 > 2c2 . Hence ϕk and ϕ` have disjoint supports. On the other hand, the function X ψ := ϕ2k k∈Z3

is smooth, bounded and bounded away from zero. We then define ϕk (v) αk (v) := p ψ(v) and (j)

φk (v, τ ) :=

X l∈Cj

−i(k· µl )τ

αl (µv)e

.

DISSIPATIVE CONTINUOUS EULER FLOWS

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Since αl and α˜l have disjoint supports for l 6= ˜l ∈ Cj , it follows that for all v, τ, j X (j) |φk (v, τ )|2 = αl (µv)2 , (24) l∈Cj (j) 2 j=1 |φk (v, τ )|

P8

and in particular = 1. Furthermore, for the same reason there exist for any m = 0, 1, 2, . . . constants C = C(m) such that (j)

sup |Dvm φk (v, τ )| ≤ C(m)µm .

(25)

v,τ

Fix next any (v, τ ) and j. Observe that there is at most one l ∈ Cj with the property that αl (µv) 6= 0 and this l has the property that |µv − l| < 1. Thus, in a neighborhood of (v, τ ) we will have   l (j) (j) (j) ∂τ φk + i(k · v)φk = ik · v − φk , (26) µ Combining (25) and (26), for any m = 0, 1, 2, . . . we find constants C = C(m, |k|) such that (j)

(j)

sup |Dvm (∂τ φk + i(k · v)φk | ≤ C(m, |k|)µm−1 .

(27)

v,τ

We apply Lemma 3.2 with N = 8 to obtain λ0 > 1, r0 > 0 and pairwise (j) disjoint families Λj together with corresponding functions γk ∈ C ∞ (Br0 (Id)). Next, set ρ(t) :=

1 3(2π)3

   ˆ  δ e(t) 1 − − |v|2 (x, t) dx 2 T3

and ˚ t) , R(x, t) := ρ(t)Id − R(x, and define   8 X X p (j) R(x, t) (j) wo (x, t) := ρ(t) γk φk (v(x, t), λt) Bk eiλk·x . ρ(t)

(28)

j=1 k∈Λj

4.2. The constants η and M . Note that wo is well-defined only if Rρ ∈ Br0 (Id) where r0 is given in Lemma 3.2. This is ensured by an appropriate choice of η. Indeed, ρ(t) ≥

1 δ e(t) ≥ cδ min e(t) =: cδm , 3(2π)3 4 t∈[0,1]

where c is a dimensional (positive) constant and m > 0 by assumption. Then

R

1 ˚ η

ρ(t) − Id ≤ cδm kRk ≤ cm .

´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

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Thus, it suffices to choose η := 12 cmr0 =

r0 min e(t) . 24(2π)3 t∈[0,1]

(29)

Observe that this choice is independent of δ > 0. Notice next that, by our choice of ρ(t) and by (7), ρ(t) ≤ δe(t). Thus there exists a constant M > 1 depending only on e (in particular independent of δ) so that √ Mδ kwo k0 ≤ . (30) 2 This fixes the choice of the constant M in Proposition 2.2 4.3. The correction wc . We next define the Leray projector onto divergencefree vectorfields with zero average. Definition 4.1. Let v ∈ C ∞ (T3 , R3 ) be a smooth vector field. Let Qv := ∇φ +

v, T3

where φ ∈ C ∞ (T3 ) is the solution of ∆φ = div v in T3

ffl

with T3 φ = 0. Furthermore, let P = I − Q be the Leray projection onto divergence-free fields with zero average. The vector field v1 is then the sum of v with the Leray projection w of wo , namely v1 (x, t) := v(x, t) + Pwo (x, t) =: v(x, t) + w(x, t) and hence wc (x, t) := −Qwo (x, t) = w(x, t) − wo (x, t). 4.4. The pressure p1 . We define p1 := p −

|wo |2 . 2

(31)

˚1 . In order to specify the choice of R ˚1 we 4.5. The Reynolds stress R introduce a new operator. Definition 4.2. 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 ∞ 3 3 where u ∈ C (T , R ) is the solution of ∆u = v − with

T3

ffl T3

u = 0.

v in T3

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Lemma 4.3 (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. Proof. It is obvious by inspection that Rv is symmetric. Since Pv is divergencefree, we obtain for the trace 3 3 tr(Rv) = (2div u) − div u = 0. 4 2 Similarly, we have 1 3 1 div (Rv) = ∆(Pu) + (∇div u + ∆u) − ∇div u. (32) 4 4 2 ffl On the other hand recall that Pu = u − ∇φ − u = u − ∇φ, where ∆φ = div u. Therefore ∆(Pu) = ∆u − ∇div u. Plugging this identity into (32), we obtain div (Rv) = ∆u ffl and since u solves ∆u = v − v, (b) follows readily.  Then we set ˚1 := R (∂t v1 + div (v1 ⊗ v1 ) + ∇p1 ) . R Note that [∂t v1 + div (v1 ⊗ v1 ) + ∇p1 ] has average zero. Indeed: • the vector field div (v1 ⊗ v1 ) + ∇p1 has average 0 because it is the divergence of the matrix field v1 ⊗ v1 + p1 Id; ˚ shows • for the same reason, the identity ∂t v = −div (v ⊗ v + pId − R) that ∂t v has average 0; on the other hand w = Pwo has average 0 because of the definition of P; this implies that ∂t w has also average zero and thus we conclude as well that ∂t v1 = ∂t v + ∂t w has average zero. ˚1 (x, t) is symmetric and traceTherefore from Lemma 4.3 it follows that R free and that the identity ˚1 ∂t v1 + div (v1 ⊗ v1 ) + ∇p1 = div R holds. ˚1 ) satThe rest of this note is devoted to prove that the triple (v1 , p1 , R isfies the estimates (9), (10), (11) and (12). This will be achieved by an appropriate choice of the parameters µ and λ. In particular, we will show that the estimates hold provided µ is sufficiently large and λ much larger than µ. 5. Schauder Estimates In the following m = 0, 1, 2, . . . , α ∈ (0, 1), and β is a multiindex. We introduce the usual (spatial) H¨older norms as follows. First of all, the supremum norm is denoted by kf k0 := supT3 |f |. We define the H¨older seminorms

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´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

as [f ]m = max kDβ f k0 , |β|=m

[f ]m+α = max sup |β|=m x6=y

|Dβ f (x) − Dβ f (y)| . |x − y|α

The H¨ older norms are then given by kf km =

m X

[f ]j

j=0

kf km+α = kf km + [f ]m+α . Recall the following elementary inequalities: [f ]s ≤ C εr−s [f ]r + ε−s kf k0




(33)

for r ≥ s ≥ 0, and [f g]r ≤ C [f ]r kgk0 + kf k0 [g]r




(34)

for any 1 ≥ r ≥ 0. Finally, we recall the classical Schauder estimates for the Laplace operator and the corresponding estimates which we can infer for the various operators involved in our construction. Proposition 5.1. 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 .

(35)

Moreover we have the estimates kQvkm+α ≤ C(m, α)kvkm+α kPvkm+α ≤ C(m, α)kvkm+α kRvkm+1+α ≤ C(m, α)kvkm+α kR(div A)km+α ≤ C(m, α)kAkm+α kRQ(div A)km+α ≤ C(m, α)kAkm+α .

(36) (37) (38) (39) (40)

Proof. The estimates (35) are the usual Schauder estimates, see for instance [12, Chapter 4]. The meticulous reader will notice that the estimates in [12] are stated in Rn for the potential-theoretic solution of the Laplace operator. The periodic case is however an easy corollary. Take for instance φ and f and consider them as periodic functions defined on R3 . Consider g = f χ, where χ is a cut-off function supported in B6π (0) and identically 1 on B4π (0). Let φ˜ be the potential-theoretic solution in R3 of ∆φ˜ = g. For φ˜ we can invoke

DISSIPATIVE CONTINUOUS EULER FLOWS

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the Schauder estimates as in [12, Chapter 4]. Moreover φ − φ˜ is an harmonic function in ffl B4pi (0). Obviously kφkL2 (B4π (0)) can be easily bounded using ∆φ = f , φ = 0 and the Parseval identity. Thus, standard properties of ˜ C m,α ([2π]3 ) ≤ C(m, α)kf k0 . harmonic functions give kφ − φk The estimates (36), (37), (38) and (39) are easy consequences of (35) and the definitions of the operators. The estimate (40) requires a little more care. Let u : T3 → R3 be the unique solution of X 2 ∆∆ui = ∂i ∂jn Ajn with

ffl

j,n

u = 0. Then kukm+1+α ≤ C(m, α)kAkm+α .

(41)

First of all, with the argument above, one can reduce this estimate to a corresponding one for the potential-theoretic solution of the biLaplace operator in R3 . For this case we can then invoke general estimates for elliptic khomogeneous constant coefficients operators (see for instance [24, Theorem 1]) or use the same arguments of [12, Chapter 4] replacing the fundamental solution of the Laplacian with that of the biLaplacian. Finally, (40) follows from the identity
3
1 1 RQ(div A) = ∇Pu + (∇Pu)T + ∇u + (∇u)T − (div u)Id 4 4 2 and the estimates (41) and (37).  In what follows we will use the convention that greek subscripts of H¨older norms denote always exponents in the open interval (0, 1). Proposition 5.2. Let k ∈ Z3 \ {0} and λ ≥ 1 be fixed. (i) For any a ∈ C ∞ (T3 ) and m ∈ N we have ˆ [a]m iλk·x dx ≤ m . 3 a(x)e λ T

(42)

(ii) Let φλ ∈ C ∞ (T3 ) be the solution of

with

∆φλ = fλ in T3

´ T3

φλ = 0, where fλ (x) := a(x)eiλk·x −

a(y)eiλk·y dy. T3

Then for any α ∈ (0, 1) and m ∈ N we have the estimate k∇φλ kα ≤ where C = C(α, m).

C C C kak0 + m−α [a]m + m [a]m+α , λ1−α λ λ

(43)

´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

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Proof. For j = 0, 1, . . . define "

#  j k k Aj (y, ξ) := −i i 2 · ∇ a(y) eik·ξ , |k|2 |k| " # j k Fj (y, ξ) := i 2 · ∇ a(y) eik·ξ . |k| Direct calculation shows that   1 1 Fj (x, λx) = div Aj (x, λx) + Fj+1 (x, λx). λ λ In particular for any m ∈ N a(x)eiλk·x = F0 (x, λx) =

m−1   1 X 1 1 div Aj (x, λx) + m Fm (x, λx) j λ λ λ j=0

Integrating this over T3 and using that |k| ≥ 1 we obtain (42). Next, using (33) and (34) we have for any j ≤ m − 1 kAj (·, λ·)kα ≤ C (λα [a]j + [a]j+α ) ≤ Cλj+α λ−m [a]m + kak0




and similarly kFm (·, λ·)kα ≤ C (λα [a]m + [a]m+α ) . Moreover, according to standard (35),  m−1 1 1 X 1 kAj (·, λ·)kα + m kFm (·, λ·)kα + k∇φkα ≤ C j λ λ λ j=0

T

 F0 (x, λx) dx , 3

hence, using (42) for the last term, k∇φkα ≤

C C C kak0 + m−α [a]m + m [a]m+α λ1−α λ λ

as required.



Corollary 5.3. Let k ∈ Z3 \ {0} be fixed. For a smooth vectorfield 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). Proof. This is an immediate consequence of the definition of R, the Schauder estimate (37) for P and Proposition 5.2 above. 

DISSIPATIVE CONTINUOUS EULER FLOWS

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6. Estimates on the corrector and the energy In all subsequent estimates, unless otherwise stated, C denotes a generic ˚ as well as constant that can vary from line to line, and depends on e, v, R on λ0 , α and δ, but is independent of λ and µ. Smallness of the respective quantities will be achieved by an appropriate choice of λ, µ in Section 8. Moreover, all estimates will implicitly assume (18), in particular that 1 ≤ µ ≤ λ. Our aim is to estimate the space-time sup-norm of v1 − v = w = wo + wc ˚1 . Since wc and R ˚1 are defined in terms of the singular integral and R operators P, Q and R, which act in space, instead of obtaining directly estimates of the C 0 norm, we will use Schauder estimates to obtain bounds on spatial H¨ older norms. Thus, in the sequel the H¨older norms will denote spatial norms, and are understood to be uniform in time t ∈ [0, 1]. Moreover, if the H¨ older exponent is denoted by a greek letter, then it is a number in the open interval (0, 1). It will be convenient to write wo as wo (x, t) = W (x, t, λt, λx), where W (y, s, τ, ξ) :=

X

ak (y, s, τ )Bk eik·ξ

(44)

|k|=λ0

  8 X X p (j) R(y, s) (j) γk φk (v(y, s), τ ) Bk eik·ξ . (45) = ρ(s) ρ(s) j=1 k∈Λj

We summarize the main properties of the coefficients W : Proposition 6.1. (i) Let ak ∈ C ∞ (T3 × [0, 1] × R) be given by (44). Then for any r ≥ 0 kak (·, s, τ )kr ≤ Cµr , k∂s ak (·, s, τ )kr ≤ Cµr+1 , k∂τ ak (·, s, τ )kr ≤ Cµr , k(∂τ ak + i(k · v)ak )(·, s, τ )kr ≤ Cµr−1 . (ii) The matrix-function W ⊗ W can be written as X (W ⊗ W )(y, s, τ, ξ) = R(y, s) + Uk (y, s, τ )eik·ξ ,

(46)

1≤|k|≤2λ0

where the coefficients Uk ∈ C ∞ (T3 × [0, 1] × R; S 3×3 ) satisfy 1 Uk k = (tr Uk )k 2

(47)

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´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

and for any r ≥ 0 kUkµ (·, s, τ )kr ≤ Cµr , k∂s Ukµ (·, s, τ )kr ≤ Cµr+1 , k∂τ Uk (·, s, τ )kr ≤ Cµr , k(∂τ Ukµ + i(k · v)Ukµ )(·, s, τ )kr ≤ Cµr−1 , ˚ but is indeIn all these estimates the constant C depends on r and e, v, R pendent of (s, τ ) and µ. Proof. The estimates for ak are a consequence of (25) and (27). Indeed, ˚ one since the constants in the estimates are allowed to depend on e, v, R, (j) only needs to keep track of the number of derivatives of φk with respect to v. Then the estimates on ak , ∂s ak and ∂τ ak + i(k · v)ak immediately follow. From the triangle inequality we can then also conclude the estimate on ∂τ ak . Next, consider the expansion of ξ 7→ W ⊗ W into a Fourier series in ξ, i.e. X (W ⊗ W )(y, s, τ, ξ) = U0 (y, s, τ ) + Uk (y, s, τ )eik·ξ . 1≤|k|≤2λ0

Since each Uk is the sum of finitely many terms of the form ak0 ak00 , the estimates for Uk follow from those for ak . Next, since U0 is given by the average (in ξ), in order to obtain (46) we need to show that W ⊗ W (y, s, τ, ξ) dξ = R(y, s). T3

To this end we calculate:    2 k k (21) ρ X X (j) −1 (j) 2 γk (ρ R) |φk (v, τ )| Id − ⊗ W ⊗ W dξ = 2 |k| |k| T3 j k∈Λj    2 k k (24) ρ X X X (j) −1 2 = γk (ρ R) (αl (v)) Id − ⊗ 2 |k| |k| j

(22)

= R

j

=R

k∈Λj l∈Cj

XX

X

(αl (v))2

l∈Cj

(αl (v))2 = R.

l∈Z3

Finally, (47) is a direct consequence of Proposition 3.1, in particular (20).  After this preparation we are ready to estimate all the terms in the perturbation scheme. First of all we verify that the corrector term wc is indeed much smaller than the main perturbation term wo :

DISSIPATIVE CONTINUOUS EULER FLOWS

21

Lemma 6.2 (Estimate on the corrector). kwc kα ≤ C

µ

(48)

λ1−α

Proof. We start with the observation that, since k · Bk = 0,   X k × Bk iλx·k  1 −iak (x, t, λt) wo (x, t) = ∇ ×  e + λ |k|2 |k|=λ0

k × Bk iλx·k 1 X i∇ak (x, t, λt) × + e . λ |k|2 |k|=λ0

Hence wc (x, t) =

1 Quc (x, t), λ

where uc (x, t) =

X

i∇ak (x, t, λt) ×

|k|=λ0

(49)

k × Bk iλx·k e . |k|2

The estimate (48) then follows from the Schauder estimate (36) for Q combined with kuc kα ≤ Cµλα .  Next, we verify the estimate on the energy (9). Lemma 6.3 (Estimate on the energy). ˆ µ 2 1 e(t)(1 − δ) − |v1 | dx ≤ C 1−α . 2 λ 3

(50)

T

Proof. Taking the trace of identity (46) in Proposition 6.1 we have X |W (y, s, τ, ξ)|2 = tr R(y, s) + ck (y, s, τ )eik·ξ 1≤|k|≤2λ0

for some coefficients ck ∈

C ∞ (T3

× [0, 1] × R), which satisfy the estimates

kck (·, s, τ )kr ≤ Cµr . From part (i) of Proposition 5.2 with m = 1 we deduce ˆ 2 ≤ Cµ |w | − tr R dx o 3 λ T

and

ˆ

T

µ v · wo dx ≤ C . λ 3

Hence, combining with (48) we see that ˆ µ 2 2 2 3 |v1 | − |v| − |wo | dx ≤ C λ1−α . T

´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

22

Recalling that tr R = 3ρ =

1 (2π)3



ˆ e(t)(1 − 21 δ) −

 |v|2 dx ,

T3

we conclude (50).



7. Estimates on the Reynolds stress Rewrite ∂t v1 +div (v1 ⊗ v1 ) + ∇p1 = [∂t wo + v · ∇wo ] + h i ˚ + div (wo ⊗ wo − 12 |wo |2 Id + R)

(51)

+ [∂t wc + div (v1 ⊗ wc + wc ⊗ v1 − wc ⊗ wc + v ⊗ wo )] In other words we split the Reynolds stress into the three parts on the right hand side. We will refer to them as the transport part, the oscillation part, and the error. In the following we will estimate each term separately. Lemma 7.1 (The transport part).  kR(∂t wo + v · ∇wo )kα ≤ C

µ2 λα + 1−α µ λ

 (52)

Proof. Observe that 

 X

R(∂t wo + v · ∇wo ) = λR 

(∂τ ak + i(k · v)ak )(x, t, λt)Bk eiλk·x  +

|k|=λ0

 + R

 X

(∂s ak + v · ∇y ak )(x, t, λt)Bk eiλk·x  .

|k|=λ0

For the first term Corollary 5.3 with m = 2 implies the bound µ µ1+α λα + 1−α + , µ λ λ whereas for the second term Corollary 5.3 with m = 1 implies the bound µ

+

λ1−α Since 1 ≤ µ ≤ λ, we obtain (52).

µ2 µ2+α + . λ1−α λ 

Lemma 7.2 (The oscillation part). 2 ˚ α≤C µ kR(div (wo ⊗ wo − 12 |wo |2 Id + R)k λ1−α

(53)

DISSIPATIVE CONTINUOUS EULER FLOWS

23

Proof. Recall the formula (46) from Proposition 6.1. Since ρ is a function of t only, we can write the oscillation part in (51) as ˚ div (wo ⊗ wo − 21 (|wo |2 − ρ)Id + R)
= div wo ⊗ wo − R − 12 (|wo |2 − tr R)Id   X = div  (Uk − 21 (tr Uk )Id)(x, t, λt)eiλk·x  1≤|k|≤2λ0

X

=

div y [Uk − 12 (tr Uk )Id](x, t, λt)eiλk·x

1≤|k|≤2λ0

Corollary 5.3 with m = 1 then implies (53).



Concerning the error, we are going to treat three terms separately, as follows. Lemma 7.3 (Estimate on the error I). kR(∂t wc )kα ≤ C

µ2 λ1−α

(54)

Proof. Recall from (49) that wc = λ1 Quc . Now X k × Bk iλx·k ∂t uc (x, t) =λ e + i(∇∂τ ak )(x, t, λt) × |k|2 |k|=λ0

+

X

i(∇∂s ak )(x, t, λt) ×

|k|=λ0

Moreover, for any ck ∈

k × Bk iλx·k e . |k|2

C ∞ (T3 ; R3 )

we have   1 k iλx·k iλx·k ck (x, t, λt)e = div ck (x, t, λt) ⊗ 2 e − iλ |k|   1 k · ∇ ck (x, t, λt)eiλx·k − iλ |k|2 Therefore ∂t uc can be written as ∂t uc = div Uc + u ˜c ,

where kUc kα ≤ Cµλα , k˜ uc kα ≤ Cµ2 λα . Therefore we have R∂t wc = λ1 (RQdiv Uc + RQ˜ uc ). From the Schauder estimate (40) for the operator RQdiv , we conclude that 1 kR∂t wc kα ≤ (kRQdiv Uc kα + kRQ˜ uc kα ) λ µ2 C uc kα ) ≤ C 1−α ≤ (kUc kα + k˜ λ λ 

24

´ ´ SZEKELYHIDI ´ CAMILLO DE LELLIS AND LASZL O JR.

Lemma 7.4 (Estimate on the error II). kR (div (v1 ⊗ wc + wc ⊗ v1 − wc ⊗ wc )) kα ≤ C

µ λ1−2α

.

(55)

Proof. We first estimate kv1 ⊗ wc +wc ⊗ v1 − wc ⊗ wc kα ≤ ≤ C(kv1 k0 kwc kα + kv1 kα kwc k0 + kwc k0 kwc kα ) (48)

≤ C ≤C

µ λ1−α

(kv1 kα + kwc kα )

µ

(kvkα + kwc kα + kwo kα )  µ  µ ≤ C 1−α C + C 1−α + Cλα . λ λ λ1−α

Recall that 1 ≤ µ ≤ λ and hence 1 ≤

µ λ1−α

≤ λα . Thus we conclude

kv1 ⊗ wc + wc ⊗ v1 − wc ⊗ wc kα ≤ C

µ λ1−2α

.

(55) follows from the latter inequality and the Schauder estimate (39).



Lemma 7.5 (Estimate on the error III). kR (div (v ⊗ wo )) kα ≤ C

µ2 . λ1−α

(56)

Proof. Since Bk · k = 0, we can write div (v ⊗ wo ) = wo · ∇v + (div wo )v X = [ak (Bk · ∇)v + v(Bk · ∇ak )] eiλk·x |k|=λ0

The claim follows from Corollary 5.3 with m = 1.



8. Conclusion: Proof of Proposition 2.2 In this section we collect the estimates from the preceding sections. For simplicity we set µ = λβ . It should be noted, however, that due to the requirement (18) we can only ensure µ ∼ λβ for large λ. We claim that for an appropriate choice of α and β, the estimates (9),(10), (11) and (12) will be satisfied for sufficiently large λ. First of all recall that, by the choice of M we have √ Mδ (57) kwo k0 ≤ 2

DISSIPATIVE CONTINUOUS EULER FLOWS

25

(cp. with (30)) and M > 1. Therefore (11) follows from the estimate kwc kα ≤ Cµλα−1 (cp. with (48)) if, for instance, we can prescribe √ µ δ α+β−1 C 1−α = Cλ ≤ . λ 2 On the other hand (12) follows easily from (57) and the identity p1 − p = − 12 |wo |2 . Also, from (50) it follows that (9) is satisfied provided 1 Cλα+β−1 ≤ δ min e(t). 8 t∈[0,1] Finally, (52),(53) as well as the estimates on the error (54)-(56) imply that   ˚1 kα ≤ C λα−β + λα+2β−1 + λ2α+β−1 . kR Therefore, any choice of α, β such that α < β,

α + 2β < 1

(58)

will ensure that (9),(10), (11) and (12) will be valid for sufficiently large λ. This completes the proof. As a side remark observe that (58) requires α < 13 . References [1] Chiodaroli, E. A counterexample to well-posedness of entropy solutions to the incompressible Euler system. Preprint (2012). [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, 1 (1994), 207–209. [3] Constantin, P., and Majda, A. The Beltrami spectrum for incompressible fluid flows. Comm. Math. Phys. 115, 3 (1988), 435–456. ´kelyhidi, Jr., L. h-principle and rigidity for [4] Conti, S., De Lellis, C., and Sze C 1,α isometric embeddings. To appear in the Proceedings of the Abel Symposium 2010 (2011). [5] Cordoba, D., Faraco, D., and Gancedo, F. Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 200, 3 (2011), 725–746. ´kelyhidi, Jr., L. The Euler equations as a differential [6] De Lellis, C., and Sze inclusion. Ann. of Math. (2) 170, 3 (2009), 1417–1436. ´kelyhidi, Jr., L. On admissibility criteria for weak solutions [7] De Lellis, C., and Sze of the Euler equations. Arch. Ration. Mech. Anal. 195, 1 (2010), 225–260. ´kelyhidi, Jr., L. The h-principle and the equations of fluid [8] De Lellis, C., and Sze dynamics. Preprint. (2011). [9] Duke, W. An introduction to the Linnik problems. In Equidistribution in number theory, an introduction, vol. 237 of NATO Sci. Ser. II Math. Phys. Chem. Springer, Dordrecht, 2007, pp. 197–216. [10] Eliashberg, Y., and Mishachev, N. Introduction to the h-principle, vol. 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002. [11] Eyink, G. L. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78, 3-4 (1994), 222–240.

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