Onsager's conjecture for admissible weak solutions

0.68 MB·

by  Tristan Buckmaster

ONSAGER’S CONJECTURE FOR ADMISSIBLE WEAK SOLUTIONS
TRISTAN BUCKMASTER, CAMILLO DE LELLIS, LA´SZLO´ SZE´KELYHIDI JR., AND VLAD VICOL
Abstract. We prove that given any β <1/3, a time interval [0,T], and given any smooth energy
profile e: [0,T] → (0,∞), there exists a weak´solution v of the three-dimensional Euler equations
such that v ∈ Cβ([0,T]×T3), with e(t) = |v(x,t)|2dx for all t ∈ [0,T]. Moreover, we show
T3
7 that a suitable h-principle holds in the regularity class Cβ , for any β < 1/3. The implication of
1 t,x
this is that the dissipative solutions we construct are in a sense typical in the appropriate space of
0
subsolutions as opposed to just isolated examples.
2

n
a
J
0 1. Introduction
3

In this paper we consider the incompressible Euler equations
]
P 
∂ v+v·∇v+∇p = 0
A  t
(1.1)
.
h  divv = 0,
t
a in the periodic setting x ∈ T3 = R3 \Z3, where v is a vector field representing the velocity of the
m
fluid and p is the pressure. We study weak (distributional) solutions v which are H¨older continuous
[
in space, i.e. such that1

1
v |v(x,t)−v(y,t)| ≤ C|x−y|β for all t ∈ [0,T] (1.2)
8
for some constant C which is independent of time t.
7
6
In his famous 1949 note on statistical hydrodynamics Lars Onsager [Ons49] conjectured that the
8
threshold regularity for the validity of the energy conservation of weak solutions of (1.1) is the
0
1. exponent 1/3: in particular he asserted that for larger H¨older exponents any weak solution would
0 conserve the energy, whereas for any smaller exponent there are solutions which do not. The first
7 assertion was fully proved by Constantin, E and Titi in [CET94], after a partial result of Eyink
1 in [Eyi94] (see also [CCFS08] for a sharper criterion in L3-based spaces). Concerning the second
:
v assertion,thefirstproofoftheexistenceofasquaresummableweaksolutionwhichdoesnotpreserve
i
X the energy is due to Scheffer in his pioneering paper [Sch93]. A different proof has been later given
r byShnirelmanin[Shn97]. In[DLS09]thesecondandthirdauthorrealizedthattechniquesfromthe
a
theory of differential inclusions could be applied very efficiently to produce bounded weak solutions
whichviolatetheenergyconservationinseveralforms. Pushedbytheanalogyoftheseconstructions
with the famous C1 solutions of Nash and Kuiper for the isometric embedding problem (cf. [Nas54]
and [Kui55]) they proposed to approach the remaining statement of the Onsager’s conjecture in a
similar way (cf. [DLS12]). Indeed in [DLS13] and [DLS14] they were able to give the first examples
of, respectively, continuous and H¨older continuous solutions which dissipate the energy, reaching
the threshold exponent 1/10. After a series of important partial results improving the threshold and
the techniques from several points of view, cf. [Ise13a, BDLS13, BDLISJ15, Buc15, BDLS16], in
his recent paper Isett [Ise16] has been able to finally reach the Onsager exponent 1/3. The proof of
Date: January 31, 2017.
1ThesmallestconstantC satisfying(1.2)willbedenotedby[v] ,cf. AppendixA.Wewillwritev∈Cβ(T3×[0,T])
β
when v is Ho¨lder continuous in the whole space-time.
1
Isett combines previous ideas with two new important ingredients, one developed by Daneri and
the third author in [DSJ16] (the introduction of Mikado flows, see Section 2.6) and one introduced
by Isett himself the aforementioned paper (the gluing technique, see Section 2.5).
However, the solutions produced in [Ise16] are only shown to be nonconservative and in fact for
those solutions the total kinetic energy fails to be monotonic on any interval of time. Thus Isett’s
theorem left open the question whether it is possible or not to construct solutions which dissipate
the kinetic energy (i.e. with strictly monotonic decreasing energy). In fact the latter is a relevant
point, for at least two reasons: first of all because dissipative solutions satisfy the weak-strong
uniqueness property [Lio96, BDLS11] and secondly because indeed Onsager in his work conjectures
the existence of dissipative solutions. Indeed, in [Ons49] Onsager states:
It is of some interest to note that in principle, turbulent dissipation as described could take place
just as readily without the final assistance by viscosity.
In this note we suitably modify the approach of Isett in order to show the following theorem.
Theorem 1.1. Assume e : [0,T] → R is a strictly positive smooth function. Then for any
0 < β < 1/3 there exists a weak solution v ∈ Cβ(T3×[0,T]) to (1.1) such that
ˆ
|v(x,t)|2 dx = e(t).
T3
We are indeed able to prove a stronger statement than Theorem 1.1, namely an h-principle in
the sense of [DSJ16]. Following [DSJ16] we introduce smooth strict subsolutions of the Euler
equations.
Definition 1.2. A smooth strict subsolution of (1.1) on T3×[0,T] is a smooth triple (v¯,p¯,R¯) with
R¯ a symmetric 2-tensor, such that
 ∂ v¯+div(v¯⊗v¯)+∇p¯= −divR¯
 t
(1.3)
 divv¯= 0,
and R¯(x,t) is positive definite for all (x,t).
WethencanprovethatanysmoothstrictsubsolutioncanbesuitablyapproximatedbyCβ solutions
for any β < 1/3. More precisely
Theorem 1.3. Let (v¯,p¯,R¯) be a smooth strict subsolution of the Euler equations on T3 ×[0,T]
and let β < 1/3. Then there exists a sequence (v ,p ) of weak solutions of (1.1) such that v ∈
k k k
Cβ(T3×[0,T]),
v (cid:42)∗ v¯ and v ⊗v (cid:42)∗ v¯⊗v¯+R¯ in L∞
k k k
uniformly in time, and furthermore for all t ∈ [0,T]
ˆ ˆ
|v |2dx = (cid:0)|v¯|2+trR¯(cid:1) dx. (1.4)
k
T3 T3
Theorem1.1canbeconcludedasasimplecorollaryofTheorem1.3. Howeverwegiveanalternative
simplerandself-containedargumentforTheorem1.1: indeedtheproofofTheorem1.3invokessome
results of [DSJ16], whereas the argument for Theorem 1.1 is entirely contained in our note, aside
from technical propositions which are classical statements in the literature, all collected in the
Appendix.
The most important differences in our proof compared to that of [Ise16] rely on the estimates for
the “gluing step” of Isett’s proof (we refer to Section 2.5 for more details) and in a simple remark
2
concerning the regions where the perturbation is added (see Section 2.6). We note that, even
without the extra benefit of imposing the energy profile and achieving the most general h-principle
statement, the proof proposed here is considerably shorter than that of [Ise16].
Acknowledgments. TheworkofT.B.hasbeenpartiallysupportedbytheNationalScienceFoun-
dationgrantDMS-1600868. TheresearchofC.D.L.hasbeensupportedbythegrant200021 159403
of the Swiss National Foundation. L.Sz. gratefully acknowledges the support of the ERC Grant
Agreement No. 277993. V.V. was partially supported by the National Science Foundation grant
DMS-1514771 and by an Alfred P. Sloan Research Fellowship.
2. Outline of the proof
Asalreadymentioned, althoughTheorem1.1canberecoveredasacorollaryofTheorem1.3, inthis
sectionweoutlineanindependentproof,reducingittoasuitableiterativeprocedure,summarizedin
Proposition 2.1 below. The same iteration procedure can be used to prove Theorem 1.3, as shown
in Section 7 at the end of the note, but the corresponding argument we will need some results
from [DSJ16], which we state without proof. In contrast, the proof of Theorem 1.1 is completely
self-contained.
2.1. Inductive proposition. First of all, we impose for the moment that
sup (cid:12)(cid:12)de(t)(cid:12)(cid:12) ≤ 1 (2.1)
dt
t∈[0,T]
(we will see later that this can be done without loosing generality).
Let then q ≥ 0 be a natural number. At a given step q we assume to have a triple (v ,p ,R˚ ) to
q q q
the Euler-Reynolds system (1.3), namely such that
 ∂ v +div(v ⊗v )+∇p = divR˚
 t q q q q q
(2.2)
 divv = 0,
q
to which we add the constraints that
trR˚ = 0 (2.3)
q
and that
ˆ
p (x,t)dx = 0 (2.4)
q
T3
(which uniquely determines the pressure).
The size of the approximate solution v and the error R˚ will be measured by a frequency λ and
q q q
an amplitude δ , which are given by
q
λ = 2π(cid:100)a(bq)(cid:101) (2.5)
q
δ = λ−2β (2.6)
q q
where (cid:100)x(cid:101) denotes the smallest integer n ≥ x, a > 1 is a large parameter, b > 1 is close to 1 and
0 < β < 1/3 is the exponent of Theorem 1.1. The parameters a and b are then related to β.
3
We proceed by induction, assuming the estimates
(cid:13) (cid:13)
(cid:13)R˚ (cid:13) ≤ δ λ−3α (2.7)
(cid:13) q(cid:13) q+1 q
0
(cid:107)v (cid:107) ≤ Mδ1/2λ (2.8)
q 1 q q
(cid:107)v (cid:107) ≤ 1−δ1/2 (2.9)
q 0 q ˆ
δ λ−α ≤ e(t)− |v |2 dx ≤ δ (2.10)
q+1 q q q+1
T3
where 0 < α < 1 is a small parameter to be chosen suitably (which will depend upon β), and M is
a universal constant (which is fixed throughout the iteration and whose choice depends on certain
geometric properties of the space of symmetric matrices and on the “squiggling” regions of the
perturbation step, cf. Remark 5.2, Lemma 5.5 and Definition 5.6). We refer to Appendix A for the
definitions of the H¨older norms used above, where we take into account only space regularity.
Proposition2.1. ThereisauniversalconstantM withthefollowingproperty. Assume0 < β < 1/3
and
1−β
1 < b < . (2.11)
2β
Then there exists an α depending on β and b, such that for any 0 < α < α there exists an a
0 0 0
depending on β, b, α and M, such that for any a ≥ a the following holds: Given a strictly positive
0
energy function e : [0,T] → R satisfying (2.1), and a triple (v ,R˚ ,p ) solving (2.2)-(2.4) and
q q q
satisfying the estimates (2.7)–(2.10), then there exists a solution (v ,R˚ ,p ) to (2.2)-(2.4)
q+1 q+1 q+1
satisfying (2.7)–(2.10) with q replaced by q+1. Moreover, we have
1
(cid:107)v −v (cid:107) + (cid:107)v −v (cid:107) ≤ Mδ1/2 . (2.12)
q+1 q 0 λ q+1 q 1 q+1
q+1
The proof of Proposition 2.1 is summarized in the Sections 2.3, 2.4, 2.5 and 2.6, but its details
will occupy most of the paper and will be completed in Section 6 below. We show next that this
proposition immediately implies Theorem 1.1.
2.2. Proof of Theorem 1.1. First of all, we fix any H¨older exponent β < 1/3 and also the
parameters b and α, the first satisfying (2.11) and the second smaller than the threshold given in
Proposition 2.1. Next we show that, without loss of generality, we may further assume the energy
profile satisfies
infe(t) ≥ δ λ−α, supe(t) ≤ δ , and supe(cid:48)(t) ≤ 1, (2.13)
1 0 1
t t t
provided the parameter a is chosen sufficiently large.
To see this, we first note that the Euler equations are invariant under the transformation
v(x,t) (cid:55)→ Γv(x,Γt) and p(x,t) (cid:55)→ Γ2p(x,Γt).
Thus if we choose
(cid:18) δ (cid:19)1/2
1
Γ = ,
sup e(t)
t
then using the scaling invariance, the stated problem reduces to finding a solution with the energy
profile given by
e˜(t) = Γ2e(t),
for which we have
δ inf e(t) (cid:18) δ (cid:19)3/2
infe˜(t) ≥ 1 t , supe˜(t) ≤ δ , and supe˜(cid:48)(t) ≤ 1 supe(cid:48)(t).
1
t supte(t) t t supte(t) t
4
If a is chosen sufficiently large then we can ensure
(cid:18) δ (cid:19)3/2 inf e(t)
supe˜(cid:48)(t) ≤ 1 supe(cid:48)(t) ≤ 1, and t ≥ λ−α.
sup e(t) sup e(t) 0
t t t t
Now we apply Proposition 2.1 iteratively with (v ,R ,p ) = (0,0,0). Indeed the pair (v ,R )
0 0 0 0 0
trivially satisfies (2.7)–(2.9), whereas the estimate (2.10) and (2.1) follows as a consequence of
(2.13). Notice that by (2.12) v converges uniformly to some continuous v. Moreover, we recall
q
that the pressure is determined by
∆p = divdiv(−v ⊗v +R˚ ) (2.14)
q q q q
and (2.4) and thus p is also converging to some pressure p (for the moment only in Lr for every
q
r < ∞). Since R˚ → 0 uniformly, the pair (v,p) solves the Euler equations.
q
Observe that using (2.12) we also infer2
(cid:88)∞ (cid:107)v −v (cid:107) (cid:46) (cid:88)∞ (cid:107)v −v (cid:107)1−β(cid:48)(cid:107)v −v (cid:107)β(cid:48) (cid:46) (cid:88)∞ δ1−2β(cid:48) (cid:16)δ1/2 λ (cid:17)β(cid:48) (cid:46) (cid:88)∞ λβ(cid:48)−β
q+1 q β(cid:48) q+1 q 0 q+1 q 1 q+1 q+1 q q
q=0 q=0 q=0 q=0
and hence that v is uniformly bounded in C0Cβ(cid:48) for all β(cid:48) < β. To recover the time regularity, we
q t x
could use the Euler equations and the general result in [Ise13b]. Nevertheless, we believe that the
followingshortandself-containedproofofthetime-regularitymaybeofindependentinterest:
Fix a smooth standard mollifier ψ in space, let q ∈ N, and consider v˜ := v∗ψ , where ψ (x) =
q 2−q (cid:96)
(cid:96)−3ψ(x(cid:96)−1). From standard mollification estimates we have
(cid:107)v˜ −v(cid:107) (cid:46) (cid:107)v(cid:107) 2−qβ(cid:48), (2.15)
q 0 β(cid:48)
and thus v˜ −v → 0 uniformly as q → ∞. Moreover, v˜ obeys the following equation
q q
∂ v˜ +div(v⊗v)∗ψ +∇p∗ψ = 0.
t q 2−q 2−q
Next, since
−∆p∗ψ = divdiv(v⊗v)∗ψ ,
2−q 2−q
using Schauder’s estimates, for any fixed ε > 0 we get
(cid:107)∇p∗ψ (cid:107) ≤ (cid:107)∇p∗ψ (cid:107) (cid:46) (cid:107)v⊗v(cid:107) 2q(1+ε−β(cid:48)) (cid:46) (cid:107)v(cid:107)2 2q(1+ε−β(cid:48)),
2−q 0 2−q ε β(cid:48) β(cid:48)
(where the constant in the estimate depends on ε but not on q). Similarly,
(cid:107)(v⊗v)∗ψ (cid:107) (cid:46) (cid:107)v⊗v(cid:107) 2q(1−β(cid:48)) (cid:46) (cid:107)v(cid:107)2 2q(1−β(cid:48)).
2−q 1 β(cid:48) β(cid:48)
Thus the above estimates yield
(cid:107)∂ v˜ (cid:107) (cid:46) (cid:107)v(cid:107)2 2q(1+ε−β(cid:48)). (2.16)
t q 0 β(cid:48)
Next, for β(cid:48)(cid:48) < β(cid:48) we conclude from (2.15) and (2.16) that
(cid:107)v˜ −v˜ (cid:107) (cid:46) (cid:0)(cid:107)v˜ −v(cid:107) +(cid:107)v˜ −v(cid:107) (cid:1)1−β(cid:48)(cid:48)(cid:0)(cid:107)∂ v˜ (cid:107) +(cid:107)∂ v˜ (cid:107) (cid:1)β(cid:48)(cid:48)
q q+1 C0Cβ(cid:48)(cid:48) q 0 q+1 0 t q 0 t q+1 0
x t
(cid:46) (cid:107)v(cid:107)1+β(cid:48)(cid:48)2−qβ(cid:48)(1−β(cid:48)(cid:48))2qβ(cid:48)(cid:48)(1+ε−β(cid:48)) = (cid:107)v(cid:107)1+β(cid:48)(cid:48)2−q(β(cid:48)−(1+ε)β(cid:48)(cid:48))
β(cid:48) β(cid:48)
(cid:46) (cid:107)v(cid:107)1+β(cid:48)(cid:48)2−qε
β(cid:48)
2Throughout the manuscript we use the the notation x (cid:46) y to denote x ≤ Cy, for a sufficiently large constant
C >0, which is independent of a,b, and q, but may change from line to line.
5
Here we have chosen ε > 0 sufficiently small (in terms of β(cid:48) and β(cid:48)(cid:48)) so that that β(cid:48)−(1+ε)β(cid:48)(cid:48) ≥ ε.
Thus, the series
(cid:88)
v = v˜ + (v˜ −v˜ )
0 q+1 q
q≥0
converges in C0Cβ(cid:48)(cid:48). Since we already know v ∈ C0Cβ(cid:48), we obtain that v ∈ Cβ(cid:48)(cid:48)([0,T]×T3) as
x t t x
desired, with β(cid:48)(cid:48) < β(cid:48) < β < 1/3 arbitrary.
Finally, since δ → 0 as q → ∞, from (2.10) we have
q+1 ˆ
|v|2 dx = e(t),
T3
which completes the proof of the theorem.
2.3. Stages. Except for Section 7, the rest of the paper is devoted to the proof of Proposition 2.1.
It will be useful to make the assumption that α is small enough so to have
(cid:18) δ (cid:19)3/2 λ
λ3α ≤ q ≤ q+1 , (2.17)
q δ λ
q+1 q
which also require that a is large enough to absorb any constant appearing from the ratio λ /a(bq),
q
for which we have the elementary bounds
λ
q
2π ≤ ≤ 4π. (2.18)
abq
The proof consists of three stages, in each of which we modify v . Roughly speaking, the stages
q
are as follows:
• Mollification: (v ,R˚ ) (cid:55)→ (v ,R˚ );
q q (cid:96) (cid:96)
• Gluing: (v ,R˚ ) (cid:55)→ (v¯ ,R˚ );
(cid:96) (cid:96) q q
• Perturbation: (v¯ ,R˚ ) (cid:55)→ (v ,R˚ ).
q q q+1 q+1
2.4. Mollification step. The first stage is mollification: we mollify v at length scale (cid:96) in order
q
to handle the loss of derivative problem, typical of convex integration schemes. To this aim, we fix
a standard mollification kernel ψ in space and introduce the mollification parameter
δ1/2
q+1
(cid:96) := , (2.19)
δ1/2λ1+3α/2
q q
and define
v :=v ∗ψ
(cid:96) q (cid:96)
R˚ :=R˚ ∗ψ +(v ⊗˚v )∗ψ −v ⊗˚v
(cid:96) q (cid:96) q q (cid:96) (cid:96) (cid:96)
where f⊗˚g is the traceless part of the tensor f ⊗g. These functions obey the equation
 ∂ v +div(v ⊗v )+∇p = divR˚
 t (cid:96) (cid:96) (cid:96) (cid:96) (cid:96)
(2.20)
 divv = 0,
(cid:96)
in view of (2.2).
Observe, again choosing α sufficiently small and a sufficiently large we can assume
λ−3/2 ≤ (cid:96) ≤ λ−1, (2.21)
q q
6
which will be applied repeatedly in order to simplify the statements of several estimates.
From(2.21),standardmollificationestimatesandPropositionA.2weobtainthefollowingbounds3
Proposition 2.2.
(cid:107)v −v (cid:107) (cid:46) δ1/2 λ−α, (2.22)
(cid:96) q 0 q+1 q
(cid:107)v (cid:107) (cid:46) δ1/2λ (cid:96)−N ∀N ≥ 0, (2.23)
(cid:96) N+1 q q
(cid:13) (cid:13)
(cid:13)R˚ (cid:13) (cid:46) δ (cid:96)−N+α ∀N ≥ 0. (2.24)
(cid:13) (cid:96)(cid:13) q+1
ˆ N+α
(cid:12) (cid:12)
(cid:12)(cid:12) |vq|2−|v(cid:96)|2 dx(cid:12)(cid:12) (cid:46) δq+1(cid:96)α. (2.25)
(cid:12) T3 (cid:12)
Proof of Proposition 2.2. The bounds (2.22) and (2.23) follow from the obvious estimates
(cid:107)v −v (cid:107) ≤ (cid:107)v (cid:107) (cid:96) (cid:46) δ1/2λ (cid:96) (cid:46) δ1/2 λ−α
(cid:96) q 0 q 1 q q q+1 q
and
(cid:107)v (cid:107) ≤ (cid:107)v (cid:107) (cid:96)−N (cid:46) δ1/2λ (cid:96)−N .
(cid:96) N+1 q 1 q q
Next, applying Proposition A.2,
(cid:13) (cid:13)
(cid:13)R˚ (cid:13) (cid:46)(cid:107)R˚ (cid:107) (cid:96)−N−α+(cid:107)v (cid:107)2(cid:96)2−N−α (cid:46) δ λ−3α(cid:96)−N−α+δ λ2(cid:96)2(cid:96)−N−α (cid:46) δ λ−3α(cid:96)−N−α,
(cid:13) (cid:96)(cid:13) q 0 q 1 q+1 q q q q+1 q
N+α
on the other hand, by (2.21) λ−3α ≤ (cid:96)2α, from which (2.24) follows. Similarly, by Proposition A.2,
q
ˆ ˆ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:13) (cid:13)
(cid:12)(cid:12)(cid:12) T3|vq|2−|v(cid:96)|2 dx(cid:12)(cid:12)(cid:12) =(cid:12)(cid:12)(cid:12) T3(|vq|2)(cid:96)−|v(cid:96)|2 dx(cid:12)(cid:12)(cid:12) (cid:46) (cid:13)(cid:13)(|vq|2)(cid:96)−|v(cid:96)|2(cid:13)(cid:13)0 (cid:46) (cid:107)vq(cid:107)21(cid:96)2,
which implies (2.25). (cid:3)
2.5. Gluing step. In the second stage we encounter the new crucial ingredient introduced by Isett
in [Ise16]: we glue together exact solutions to the Euler equations in order to produce a new v ,
q
closetov , whoseassociatedReynoldsstresserrorhassupportinpairwisedisjointtemporalregions
q
of length τ in time, where
q
(cid:96)2α
τ = . (2.26)
q
δ1/2λ
q q
Theparameterτ shouldbecomparedtotheparameterµ−1 usedinthepaper[BDLISJ15]. Indeed,
q
τ−1 satisfies precisely the same parameter inequalities that µ satisfies in Section 2 of [BDLISJ15].
q
We note in particular that like in [BDLISJ15] we have the CFL-like condition
(2.23)
τ (cid:107)v (cid:107) (cid:46) τ δ1/2λ (cid:96)−α (cid:46) (cid:96)α (cid:28) 1 (2.27)
q (cid:96) 1+α q q q
as long as a is sufficiently large.
˚
More precisely, we aim to construct a new triple (v ,R ,p ) solving the Euler Reynolds equation
q q q
˚
(2.2) such that the temporal support of R is contained in pairwise disjoint intervals I of length
q i
∼ τ and such that the gaps between neighbouring intervals is also of length ∼ τ . More precisely,
q q
for any n ∈ Z let
t = nτ , I = [t + 1τ ,t + 2τ ]∩[0,T], J = (t − 1τ ,t + 1τ )∩[0,T].
n q n n 3 q n 3 q n n 3 q n 3 q
3Inthefollowing,whenconsideringhigherordernorms(cid:107)·(cid:107) or(cid:107)·(cid:107) ,thesymbol(cid:46)willimplythattheconstant
N N+1
in the inequality might also depend on N.
7
We require
suppR˚ ⊂ (cid:91) I ×T3. (2.28)
q n
n∈N
˚
Moreover, (v ,R ) will satisfy the following estimates for any N ≥ 0
q q
(cid:107)v −v (cid:107) (cid:46) δ1/2 (cid:96)α (2.29)
q (cid:96) 0 q+1
(cid:107)v (cid:107) (cid:46) δ1/2λ (cid:96)−N (2.30)
q 1+N q q
(cid:13) (cid:13)
(cid:13)R˚(cid:13) (cid:46) δ (cid:96)−N+α (2.31)
(cid:13) q(cid:13) q+1
N+α
(cid:13) (cid:13)
(cid:13)∂ R˚ +(v ·∇)R˚(cid:13) (cid:46) δ δ1/2λ (cid:96)−N−α (2.32)
(cid:13) t q q q(cid:13) q+1 q q
ˆ N+α
(cid:12) (cid:12)
(cid:12)(cid:12) |v¯q|2−|v(cid:96)|2dx(cid:12)(cid:12) (cid:46) δq+1(cid:96)α (2.33)
(cid:12) T3 (cid:12)
where the implicit constants depend only on M,α, and N, cf. Propositions 4.2, 4.3 and 4.4.
The gluing procedure will be broken up into two parts: first, we construct a sequence of exact
solutions to the Euler equations with appropriate stability estimates in Section 3 and then we glue
the solutions together in Section 4 with a partition of unity in order to construct v satisfying the
q
properties mentioned above. This is indeed the key idea of Isett in [Ise16]. The main difference
˚
with [Ise16] is in the construction of the tensor R : in this paper we use the usual elliptic operators
q
introduced in [DLS13]. This has the advantage that our Reynolds stress remains trace free, in
contrast to the one of [Ise16], and in turn this is crucial to control the energy in the perturbation
˚
step below. It should be noticed that in [Ise16] the author resorts to a different definition of R
q
because he is not able to find efficient estimates. Our main technical improvement is that this
difficulty can be overcome employing suitable commutator estimates on the advective derivative of
differential operators of negative order, cf. the proof of Proposition 3.4 and Proposition D.1. This
remarkallowsusnotonlytokeepabettercontrolontheenergyandatrace-freeReynoldsstresswith
the desired estimate, but it also shortens the arguments considerably compared to [Ise16].
2.6. Perturbation and proof of Proposition 2.1. The gluing procedure can be used to localize
˚
the Reynolds stress error R to small disjoint temporal regions, but it cannot be used to completely
q
eliminate the error.
First of all note that as a corollary of (2.10), (2.25) and (2.33), by choosing a sufficiently large we
can ensure that ˆ
δ
q+1 ≤ e(t)− |v |2 dx ≤ 2δ . (2.34)
2λα q q+1
q T3
˚
Starting with the solution (v ,p ,R ) satisfying (2.28) and the estimates (2.29)-(2.34), we then
q q q
produce a new solution (v ,p ,R˚ ) of the Euler-Reynolds system (2.2) with estimates
q+1 q+1 q+1
M
(cid:107)v −v (cid:107) +λ−1 (cid:107)v −v (cid:107) ≤ δ1/2 (2.35)
q+1 q 0 q+1 q+1 q 1 2 q+1
δ1/2 δ1/2λ
(cid:107)R˚ (cid:107) (cid:46) q+1 q q . (2.36)
q+1 α λ1−4α
q+1
ˆ
(cid:12)(cid:12)(cid:12)e(t)− |vq+1|2 dx− δq+2(cid:12)(cid:12)(cid:12) (cid:46) δq1/2δq1/+21λ1q+2α , (2.37)
(cid:12) T3 2 (cid:12) λq+1
cf. Corollary 5.8 and Propositions 6.1 and 6.2.
8
As in previous papers [DLS14, Ise13a, BDLS13, BDLISJ15, BDLS16] the key idea, introduced in
[DLS13], for reducing the size of the error is to add a highly oscillatory perturbation w to v .
q+1 q
PreviousschemesheavilyreliedonBeltramiflows, buttheseseemedinsufficienttopushthemethod
beyond H¨older exponent 1/5. A new set of flows, called Mikado flows, with much better properties
wereintroducedin[DSJ16]andindeed, akeyelementintheproofofIsett[Ise16]istheobservation,
already used in [DSJ16], that Mikado flows behave better under advection by a mean flow.
˚
An important point is that the Mikado flows will not only be used to “cancel” the error R ,
q
but also to “improve the energy” in areas where the error vanishes identically. In particular, the
perturbation will be added in spacetime regions which are disjoint and contained in time-slabs of
thickness 2τ , but with the property that their projections on the time axis is a covering of the
q
interval [0,T].
Proof of Proposition 2.1. The estimate (2.12) is a consequence of (2.22), (2.23), (2.29), (2.30) and
(2.35):
M
(cid:107)v −v (cid:107) +λ−1 (cid:107)v −v (cid:107) ≤ δ1/2 λ +Cδ1/2 (cid:96)α+Cδ1/2λ λ−1 ,
q+1 q 0 q+1 q+1 1 1 2 q+1 q+1 q+1 q q q+1
where the constant C depends on α,β,M, but not on a,b and q. In particular, for every fixed b
(2.12) holds if a is large enough. For (2.8), we use the induction assumption to get
M
(cid:107)v (cid:107) ≤ Mδ1/2λ + δ1/2 λ +Cδ1/2 (cid:96)α+Cδ1/2λ λ−1
q+1 1 q q 2 q+1 q+1 q+1 q q q+1
and again a sufficiently large choice of a will guarantee (cid:107)v(cid:107) ≤ Mδ1/2 λ . Similarly for (2.9),
q+1 q+1 q+1
which will follow from
(cid:107)v (cid:107) ≤ (cid:107)v (cid:107) +(cid:107)v −v (cid:107) ≤ 1−δ1/2+Mδ1/2 .
q+1 0 q 0 q+1 q 0 q q+1
From (2.36) and (2.37), the inequalities (2.7) and (2.10) follow as a consequence of the parameter
inequality
δq1/2δq1/+21λq δq+2
≤ . (2.38)
λ λ8α
q+1 q+1
To see this, one divides by the right hand side, takes logarithms and divides by logλ , to obtain
q
(cid:18) (cid:19)
1
−β −βb+1−b+2b2β+8bα+O ≤ 0,
logλ
q
(cid:16) (cid:17)
where the error term O 1 is due to the constants in (2.18). From the relation (2.11), if α is
logλq
sufficiently small we obtain
−β−βb+1−b+2b2β +8bα < 0. (2.39)
Hence fixing b to satisfy (2.11), choosing subsequently α sufficiently small and then a sufficiently
large, we obtain (2.38).
Finally, an entirely analogous argument shows (2.10) from (2.37). (cid:3)
9
3. Stability estimates for classical exact solutions
3.1. Classical solutions. For each i, let t = iτ , and consider smooth solutions of the Euler
i q
equations

∂ v +div(v ⊗v )+∇p = 0
 t i i i i




divv = 0 (3.1)
i




 v (·,t ) = v (·,t ).
i i (cid:96) i
defined over their own maximal interval of existence. Next, recall the following
Proposition 3.1. For any α > 0 there exists a constant c = c(α) > 0 with the following property.
Given any initial data u ∈ C∞, and T ≤ c(cid:107)u (cid:107) , there exists a unique solution u : R3 ×
0 0 1+α
[−T,T] → R3 to the Euler equation

∂ u+div(u⊗u)+∇p = 0
 t




divu = 0,




 u(·,0) = u
0
Moreover, u obeys the bounds
(cid:107)u(cid:107) (cid:46)(cid:107)u (cid:107) . (3.2)
N+α 0 N+α
for all N ≥ 1, where the implicit constant depends on N and α > 0.
Proof of Proposition 3.1. Theproofoftheexistenceofauniquesolutionisstandard(seee.g.[MB02,
Chapter 4]), and follows from the restriction T ≤ c(cid:107)u (cid:107) . The higher-order bounds (3.2) are
0 1+α
also standard, and can be obtained as follows: For any multi-index θ with |θ| = N we have
∂ ∂θv+v·∇∂θv+[∂θ,v·∇]v+∇∂θp = 0.
t
Using the equation for the pressure −∆p = ∇v·∇v and Schauder estimates we obtain
(cid:107)∇∂θp(cid:107) (cid:46) (cid:107)∇v·∇v(cid:107) (cid:46) (cid:107)v(cid:107) (cid:107)v(cid:107) .
α N−1+α 1+α N+α
Therefore
(cid:107)(∂ +v·∇)∂θv(cid:107) (cid:46) (cid:107)v(cid:107) (cid:107)v(cid:107) ,
t α 1+α N+α
and (3.2) follows by applying (B.3) and Gr¨onwall’s inequality. (cid:3)
An immediate consequence is:
Corollary 3.2. If a is sufficiently large, for |t−t | ≤ τ , we have
i q
(cid:107)v (cid:107) (cid:46) δ1/2λ (cid:96)1−N−α (cid:46) τ−1(cid:96)1−N+α for any N ≥ 1. (3.3)
i N+α q q q
Proof of Corollary 3.2. We apply Proposition 3.1 and use the estimate (2.27) to obtain
(cid:107)v (cid:107) (cid:46) (cid:107)v(t )(cid:107)
i N+α i N+α
for any N ≥ 1. From (2.23) we then deduce the estimate (3.3). (cid:3)
10