Typicality results for weak solutions of the incompressible Navier--Stokes equations
aa r X i v : . [ m a t h . A P ] F e b TYPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLENAVIER–STOKES EQUATIONS
MARIA COLOMBO, LUIGI DE ROSA AND MASSIMO SORELLA
Abstract.
In the class of L ∞ ((0 , T ); L ( T )) distributional solutions of the incompressible Navier-Stokessystem, the ones which are smooth in some open interval of times are meagre in the sense of Baire category,and the Leray ones are a nowhere dense set. Keywords: incompressible Navier–Stokes equations, nonsmooth distributional solutions, Leray solutions,convex integration, Baire category.
MSC (2010): 35Q30 - 35D30 - 76B03 - 26A21. Introduction
In the last 15 years, the fundamental results of De Lellis and Székelyhidi [11, 12, 14] initiated a researchline which allowed to build nonsmooth distributional solutions of various equations in fluid dynamics withincreasingly many regularity properties. All these results share a common approach called convex integra-tion, which in this context points roughly speaking to build solutions of a nonlinear PDE by an iterativeprocedure, where at each step the constructed functions solve the equation up to a smaller and smallererror, which is corrected each time by means of the nonlinearity of the PDE. This lead to importantresults such as the proof of the Onsager conjecture by Isett [3, 17] and the construction of nonsmoothdistributional solutions to the Navier-Stokes equations by Buckmaster and Vicol [2, 4, 8]. Related recentresults were obtained for the hypodissipative Navier-Stokes equations [9, 15] , the surface-quasigeostrophicequation [6, 7, 18] and the transport equation [1, 20–22] (see also the references quoted therein).A natural question is then “how many” such distributional solutions can be found, compared to thesmooth ones. In this paper we investigate this question in terms of Baire category. We focus on theNavier-Stokes system in the spatial periodic setting T = R / Z (cid:26) ∂ t v + div( v ⊗ v ) + ∇ p − ∆ v = 0div v = 0 in T × [0 , T ] (1.1)where v : T × [0 , T ] → R represents the velocity of an incompressible fluid, p : T × [0 , T ] → R is thehydrodynamic pressure, with the constraint ´ T p dx = 0.We define the following complete metric space D := (cid:8) v ∈ L ∞ ((0 , T ); L ( T )) : v is a distributional solution of (1.1) (cid:9) , endowed with the metric d D ( u, v ) := k u − v k L ∞ t ( L x ) , and its subsets L := { v ∈ D : v is a Leray–Hopf solution of (1.1) }S := (cid:8) v ∈ D : v ∈ C ∞ ( T × I ) for some open interval I ⊂ (0 , T ) (cid:9) . We refer to Section 2.1 for the definitions of distributional and Leray-Hopf solutions. Our main result isthe following
Theorem 1.1.
The set L is nowhere dense in D while the set S is meagre in D . We recall that L is nowhere dense in D if and only if the closure of L has empty interior. In particular, L is meagre in D .A partial answer to the question of “how many” distributional solutions there are, compared to thesmooth ones, was given before by the so called “h-principle”, a term introduced by Gromov in the contextof isometric embeddings. In the context of the Euler equations (see for instance [13, Theorem 6]), it statesthat arbitrarily close in the weak L topology to a (suitably defined) strict subsolution one can build anexact distributional solution. In a slightly different direction, it has been shown in [10] that a dense setof initial data admits infinitely many distributional solutions with the same kinetic energy, and in [2] thatdistributional solutions are nonunique for any initial datum in L for the Navier-Stokes system. Previously,convex integration was also used in [16] to characterize typical energy profiles for the Euler equations interms of Hölder spaces, which requires to introduce a suitable metric space to deal with the right energyregularity. Aknowledgements.
The authors aknowledge the support of the SNF Grant 200021_182565.2.
The iterative proposition and proof of the main theorem
The proof of Theorem 1.1 is based on an iterative proposition, typical of convex integration schemesand analogous to [4, Section 7] and [2, Section 2]; in analogy with the latter, also here we use intermittentjets (see Section 3 below) as the fundamental building blocks. At difference to the previously cited works,we need to keep track of the kinetic energy in some intervals of time along the iteration in such a way tobe able to prescribe it in the limit, and we also need to make sure with a simple use of time cutoffs thatthe support of the perturbation is localized in a converging sequence of enlarging sets. On the contrary,we don’t use the cutoffs to obtain a small set of singular times for our limit, as was done in [2].In turn the proof of Theorem 1.1 follows from the iterative proposition in this way: to show that thesubset L is nowhere dense in the metric space D , we prove that for every v ∈ L there are arbitrarily closeelements which belong to D \ L . In Step 1 of the proof we reduce to such statement, where we chooseelements in
D \ L by imposing locally increasing kinetic energy.The method presented here to prove Theorem 1.1 is quite general in contexts where the convex integra-tion scheme works and should apply also to other contexts.2.1.
Basic notations and definitions.
We recall that a distributional solution of the system (1.1) is avector field v ∈ L ( T × (0 , T ); R ) such that ˆ T ˆ T ( v · ∂ t ϕ + v ⊗ v : ∇ ϕ + v · ∆ ϕ ) dxdt = 0 , for all ϕ ∈ C ∞ c ( T × (0 , T ); R ) such that div ϕ = 0. The pressure does not appear in the distributionalformulation because it can be recovered as the unique 0-average solution of − ∆ p = div div( v ⊗ v ) . (2.1)A Leray Hopf solution of the system (1.1) is a vector field v ∈ L ((0 , T ); H ( T )) ∩ L ∞ ((0 , T ); L ( T ))and for a.e. s ≥ t ∈ [ s, T ] the following inequality holds ˆ T | v ( x, t ) | dx + ˆ ts ˆ T |∇ v ( x, τ ) | dxdτ ≤ ˆ T | v ( x, s ) | dx. (2.2)It is a classical result by Leray that Leray-Hopf solutions are smooth outside a closed set of times ofHausdorff dimension 1 /
2, see for instance [19].
YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 3
The Navier–Stokes–Reynolds system.
In this section, for every integer q ≥ v q , p q , ˚ R q ) to the Navier-Stokes-Reynolds system (cid:26) ∂ t v q + div( v q ⊗ v q ) + ∇ p q − ∆ v q = div ˚ R q div v q = 0 (2.3)where the Reynolds stress ˚ R q is assumed to be a trace-free symmetric matrix valued function. Indeed forany matrix A we will use the notation ˚ A to remark the traceless property.2.3. Parameters.
Define the frequency parameter λ q → + ∞ and the amplitudes parameter δ q → + by λ q = 2 πa ( b q ) ,δ q = λ − βq . The sufficiently large (universal) parameter b is free, and so is the sufficiently small parameter β = β ( b ).The parameter a is chosen to be a sufficiently large multiple of the geometric constant n ∗ . Moreover, wefix another parameter useful to prescribe a precise kinetic energy ǫ := (cid:18) ǫ sup ξ ∈ Λ k γ ξ k C | Λ | C π ) (cid:19) , (2.4)where sup ξ ∈ Λ k γ ξ k C , | Λ | , C are all universal constants independent on q , more precisely: γ ξ are functionsdefined in Lemma 3.1, Λ is the finite set defined in Lemma 3.1, C is the constant given by Lemma A.3, ǫ is a free constant that will be used in the proof of Theorem 1.1.Moreover, we will use the intermittent jets (defined in Section 3) to define the new velocity incrementat step q + 1.2.4. Inductive estimates and iterative proposition.
We define new “slow” parameters, for all q ≥ s q := (cid:16) s (cid:17) q +1 , (2.5) S q := q X i =0 s i , (2.6)for some fixed parameter s >
0. By choosing a ( s ) sufficiently large, we will guarantee that s − q +1 ≪ λ q , indeed s − q is a slow parameter compared to λ q . Moreover we define the local time interval, for some smallnumber s >
0, for all q ≥ I q := ( t − S q , t + S q ) , (2.7)for some t ∈ (0 ,
1) and s = s ( t ) > B s ( t ) := ( t − s, t + 2 s ) ⊂ [0 , . Observe that I q ⊂ B s ( t ) for all q ≥ B s ( t ), i.e. for example: if v ∈ L ∞ t L px , we denote k v k L p the quantitysup t ∈ B s ( t ) k v ( t, · ) k L px . We use . as an inequality that holds up to a constant independent on q .For q ≥
0, we want to guarantee
M. COLOMBO, L. DE ROSA AND M. SORELLA k v q k L ≤ k v k L − ǫ π δ / q , (2.8a) k ˚ R q k L ≤ λ − ζq δ q +1 , (2.8b) k v q k C x,t ( T × B s ( t )) ≤ λ q , (2.8c)and moreover δ q +1 δ λ ζ/ q ≤ e ( t ) − ˆ T | v q ( x, t ) | dx ≤ δ q +1 ǫ δ , for all t ∈ I , (2.9a)Supp T (˚ R q ) ⊂ I q , (2.9b)Supp T ( v q − v q − ) ⊂ I q , for all q ≥ , (2.9c)which are new with respect to the convex integration scheme proposed by Buckmaster and Vicol in [4,Section 7]. Proposition 2.1 (Iterative Proposition) . Let e : [0 , T ] → (0 , ∞ ) be a strictly positive smooth function.For every ǫ, s > and t ∈ (0 , T ) there exist b > , β ( b ) > , ζ > , a = a ( β, b, ζ, e, ǫ, s ) such thatfor any a ≥ a which is a multiple of the geometric constant n ∗ of Lemma 3.1, the following holds. Let ( v q , p q , ˚ R q ) be a smooth triple solving the Navier-Stokes-Reynolds system (2.3) in T × B s ( t ) satisfyingthe inductive estimates (2.8) - (2.9) .Then there exists a second smooth triple ( v q +1 , p q +1 , ˚ R q +1 ) which solves the Navier-Stokes-Reynoldssystem in T × B s ( t ) (2.3) , satisfies the estimates (2.8) and (2.9) at level q + 1 . In addition, we havethat k v q +1 − v q k L ∞ ( B s ( t ); L ( T )) ≤ ǫδ / π δ / q +1 . (2.10)2.5. Proof of Theorem 1.1.
Step 1. Let v ∈ L ∞ ((0 , T ); L ( T )) be a distributional solution of (1.1) ,such that v ∈ C ∞ ( T × I ) , for some open interval I ⊂ (0 , T ) . Then, we prove the following claim: forevery ǫ > , there exists a distributional solution v ǫ ∈ L ∞ ((0 , T ); L ( T )) of (1.1) such that k v ǫ − v k L ∞ ((0 ,T );( L ( T )) < ǫ (2.11) and the kinetic energy of v ǫ is strictly increasing in a sub-interval of (0 , T ) . Let t ∈ I and choose s > B s ( t ) ⊂ I . Let g ∈ C ∞ ([0 , T ]; [ ǫ , ǫ ]) be such that g ′ ( t ) > sup t ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ddt ˆ T | v ( x, t ) | dx (cid:12)(cid:12)(cid:12)(cid:12) , and consider the kinetic energy (increasing in a neighbourhood of t ) e ( t ) := ˆ T | v ( x, t ) | dx + g ( t ) . (2.12)Since the function v is smooth in T × I we consider the smooth solution p , with zero average, in T × I of (2.1), and define the starting triple ( v , p , R ) := ( v, p, v, p,
0) satisfies the estimates (2.8) and (2.9) at step q = 0, up to enlarge a , thus we can applyProposition 2.1 starting from the triple ( v , p , R ). Hence, we get a sequence { v q } q ∈ N that satisfies (2.8),(2.9) and moreover, from (2.10) we get X q ≥ k v q +1 − v q k L ≤ ǫδ / π X q ≥ δ / q +1 ≤ ǫδ / π X q ≥ ( a − βb ) q +1 ≤ ǫ − a − βb ) < ǫ (2.13) Here Supp T ( u ) denotes the closure of { t ∈ (0 ,
1) : ∃ x ∈ T u ( x, t ) = 0 } . To be precise we considered v − = v . YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 5 where the last holds if a is sufficiently large in order to have a − βb < /
2. Hence, there exists the limit˜ v ǫ := lim q →∞ v q , in L ∞ ( B s ( t ); L ( T )) such that k ˜ v ǫ − v k L ∞ ( B s ( t ); L ( T )) < ǫ and it is a distributionalsolution of the Navier-Stokes equations in B s ( t ) × T , because by (2.8b) we have that lim q →∞ ˚ R q = 0 in L ∞ ( B s ( t ); L ( T )). One can verify that the vector field v ǫ = (cid:26) ˜ v ǫ in B s ( t ) v in [0 , T ] \ B s ( t ) , still solves (1.1) in [0 , T ] × T and satisfies (2.11). Moreover the kinetic energy of v ǫ is increasing in aneighbourhood of t thanks to (2.9a) and (2.12). Step 2. We conclude the proof of Theorem 1.1.
Let v be a distributional solution which is smooth in a subinterval of times and ǫ >
0; for instance,any Leray solution can be taken as v since they are smooth outside a closed set of H / measure 0. Weapply the Step 1 and get a distributional solution of Navier-Stokes v ǫ ∈ L ∞ ((0 , T ); L ( T )) such that k v ǫ − v k L ∞ ((0 ,T ); L ( T )) < ǫ with increasing kinetic energy in a sub-interval of [0 , T ] and therefore suchthat v ǫ ∈ D \ L .Since L is closed with respect to L ∞ L convergence, we deduce that the interior of L which coincideswith the interior of L , is empty.To show that S is a meagre set in D , we rewrite it as S ⊂ [ s ∈ Q + [ t ∈ (0 , ∩ Q { v ∈ D : v ∈ C ∞ (( t − s, t + s ) × T ) } , and we notice that from Step 1 the right-hand side is a countable union of nowhere dense sets, hence it ismeagre. 3. Intermittent jets
In this section we recall from [4] the definition and the main properties of intermittent jets we will usein the convex integration scheme.3.1.
A geometric lemma.
We start with a geometric lemma. A proof of the following version, which isessentially due to De Lellis and Székelyhidi Jr., can be found in [2, Lemma 4.1]. This lemma allows us toreconstruct any symmetric 3 × R in a neighbourhood of the identity as a linear combinationof a particular basis. Lemma 3.1.
Denote by B sym / ( Id ) the closed ball of radius / around the identity matrix in the spaceof symmetric × matrices. There exists a finite set Λ ⊂ S ∩ Q such that there exist C ∞ functions γ ξ : B sym / ( Id ) → R which obey R = X ξ ∈ Λ γ ξ ( R ) ξ ⊗ ξ, for every symmetric matrix R satisfying | R − Id | ≤ / . Moreover for each ξ ∈ Λ , let use define A ξ ∈ S ∩ Q to be an orthogonal vector to ξ . Then for each ξ ∈ Λ we have that { ξ, A ξ , ξ × A ξ } ⊂ S ∩ Q form anorthonormal basis for R . Furthermore, since we will periodize functions, let n ∗ be the l.c.m. of thedenominators of the rational numbers ξ, A ξ and ξ × A ξ , such that { n ∗ ξ, n ∗ A ξ , n ∗ ξ × A ξ } ⊂ Z . M. COLOMBO, L. DE ROSA AND M. SORELLA
Vector fields.
Let Φ : R → R be a smooth function with support contained in a ball of radius 1.We normalize Φ such that φ = − ∆Φ obeys14 π ˆ R φ ( x , x ) dx dx = 1 . (3.1)We remark that by definition φ has zero average. Define ψ : R → R to be a smooth, zero averagefunction with support in the ball of radius 1 satisfying T ψ ( x ) dx = 12 π ˆ R ψ ( x ) dx = 1 . We define the parameters r ⊥ , r || and µ as follows r ⊥ := r ⊥ ,q +1 := λ − / q +1 (2 π ) − / , (3.2a) r || := r || ,q +1 := λ − / q +1 , (3.2b) µ := µ q +1 := λ / q +1 (2 π ) / . (3.2c)We define φ r ⊥ , Φ r ⊥ , and ψ r || to be the rescaled cut-off functions φ r ⊥ ( x , x ) := 1 r ⊥ φ (cid:18) x r ⊥ , x r ⊥ (cid:19) , Φ r ⊥ ( x , x ) := 1 r ⊥ Φ (cid:18) x r ⊥ , x r ⊥ (cid:19) ,ψ r || ( x ) := (cid:18) r || (cid:19) / ψ (cid:18) x r || (cid:19) . With this rescaling we have φ r ⊥ = − r ⊥ ∆Φ r ⊥ . Moreover the functions φ r ⊥ and Φ r ⊥ are supported inthe ball of radius r ⊥ in R , ψ r || is supported in the ball of radius r || in R and we keep the normalizations k φ r ⊥ k L = 4 π and k ψ r || k L = 2 π. We then periodize the previous functions φ r ⊥ ( x + 2 πn, x + 2 πm ) = φ r ⊥ ( x , x ) , Φ r ⊥ ( x + 2 πn, x + 2 πm ) = Φ r ⊥ ( x , x ) ,ψ r || ( x + 2 πn ) = ψ r || ( x ) . For every ξ ∈ Λ (recalling the notations in Lemma 3.1), we introduce the functions defined on T × R ψ ξ ( x, t ) := ψ r || ( n ∗ r ⊥ λ q +1 ( x · ξ + µt )) , (3.3a)Φ ξ ( x ) := Φ r ⊥ ( n ∗ r ⊥ λ q +1 ( x − α ξ ) · A ξ , n ∗ r ⊥ λ q +1 ( x − α ξ ) · ( ξ × A ξ )) , (3.3b) φ ξ ( x ) := φ r ⊥ ( n ∗ r ⊥ λ q +1 ( x − α ξ ) · A ξ , n ∗ r ⊥ λ q +1 ( x − α ξ ) · ( ξ × A ξ )) , (3.3c)where α ξ are shifts which ensure that the functions { Φ ξ } have mutually disjoint support.In order for such shifts α ξ to exist, it is sufficient to assume that r ⊥ is smaller than a universal constant,which depends only on the geometry of the finite set Λ.It is important to note that the function ψ ξ oscillates at frequency proportional to r ⊥ r − || λ q +1 , whereas φ ξ and Φ ξ oscillate at frequency proportional to λ q +1 . Definition 3.2.
The intermittent jets are vector fields W ξ : T × R → R defined as W ξ ( x, t ) := ξψ ξ ( x, t ) φ ξ ( x ) . YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 7 If σ := r ⊥ n ∗ ∈ N , thanks to the choice of n ∗ in Lemma 3.1 we have that W ξ has zero average in T andis (cid:0) T σ (cid:1) periodic. Moreover, by our choice of α ξ , we have that W ξ ⊗ W ξ ′ ≡ , whenever ξ = ξ ′ ∈ Λ, i.e. { W ξ } ξ ∈ Λ have mutually disjoint support. The essential identities obeyed by theintermittent jets are k W ξ k pL p ( T ) = 18 π k ψ ξ k pL p ( T ) k φ ξ k pL p ( T ) div( W ξ ⊗ W ξ ) = 2( W ξ · ∇ ψ ξ ) φ ξ ξ = 1 µ ∂ t ( φ ξ ψ ξ ξ ) (3.4) T W ξ ⊗ W ξ = ξ ⊗ ξ, where the last identity will be useful to apply Lemma 3.1.We denote by P =0 the operator which projects a function onto its non-zero frequencies P =0 f = f − ffl T f ,and by P H we will denote the usual Helmholtz projector onto divergence-free vector fields, P H f = f −∇ (∆ − div f ). Motivated by (3.4), we define W ( t ) ξ ( x, t ) := − µ P H P =0 φ ξ ( x ) ψ ξ ( x, t ) ξ. (3.5)Lastly, we note that the intermittent jets W ξ are not divergence free, then we introduce the followingtwo functions W ( c ) ξ , V ξ : T × R → R V ξ ( x, t ) := 1 n ∗ λ q +1 ξψ ξ ( x, t )Φ ξ ( x ) ,W ( c ) ξ ( x, t ) := 1 n ∗ λ q +1 ∇ ψ ξ ( x, t ) × ( ∇ × Φ ξ ( x ) ξ ) . Using ∆Φ ξ = − λ q +1 n ∗ φ ξ we compute the intermittent jets in terms of V ξ λ q +1 n ∗ W ξ = λ q +1 n ∗ ξφ ξ ψ ξ = − ∆Φ ξ ψ ξ ξ = ∇ × ( ψ ξ ∇ × (Φ ξ ξ )) − ∇ ψ ξ × ( ∇ × Φ ξ ξ )= ∇ × ∇ × ( ψ ξ Φ ξ ξ ) − ∇ × ( ∇ ψ ξ × Φ ξ ξ ) − ∇ ψ ξ × ( ∇ × Φ ξ ξ )= ∇ × ∇ × ( ψ ξ Φ ξ ξ ) − ∇ ψ ξ × ( ∇ × Φ ξ ξ )= λ q +1 n ∗ (cid:16) ∇ × ∇ × V ξ − W ( c ) ξ (cid:17) , (3.7)from which we deduce div( W ξ + W ( c ) ξ ) ≡ . Moreover, since r ⊥ ≪ r || , the correction W cξ is comparatively small in L with respect to W ξ , moreprecisely we state the following lemma (see [5, Section 7.4]). Lemma 3.3.
For any
N, M ≥ and p ∈ [1 , ∞ ] the following inequalities hold k∇ N ∂ Mt ψ ξ k L p . r /p − / || (cid:18) r ⊥ λ q +1 r || (cid:19) N (cid:18) r ⊥ λ q +1 µr || (cid:19) M (3.8a) k∇ N φ ξ k L p + k∇ N Φ ξ k L p . r /p − ⊥ λ Nq +1 (3.8b) k∇ N ∂ Mt W ξ k L p . r /p − ⊥ r /p − / || λ Nq +1 (cid:18) r ⊥ λ q +1 µr || (cid:19) M (3.8c) M. COLOMBO, L. DE ROSA AND M. SORELLA r || r ⊥ k∇ N ∂ Mt W ( c ) ξ k L p . r /p − ⊥ r /p − / || λ Nq +1 (cid:18) r ⊥ λ q +1 µr || (cid:19) M (3.8d) λ q +1 k∇ N ∂ Mt V ξ k L p . r /p − ⊥ r /p − / || λ Nq +1 (cid:18) r ⊥ λ q +1 µr || (cid:19) M . (3.8e) The implicit constants are independent of λ q +1 , r ⊥ , r || , µ . Proof of the iterative proposition
Given ( v q , p q , ˚ R q ) a triple solving the Navier-Stokes-Reynolds system (2.3) in T × B s ( t ) satisfyingthe inductive estimates (2.8) and (2.9) at step q , we have to construct ( v q +1 , p q +1 , ˚ R q +1 ) which still solvesthe Navier-Stokes-Reynolds system (2.3) in T × B s ( t ) and satisfies the estimates (2.8) and (2.9) at step q + 1 and the estimate (2.10) holds.4.1. Mollification.
In order to avoid a loss of derivatives in the iterative scheme, we replace v q by amollified velocity field ˜ v ℓ . For this purpose we choose a small parameter ℓ ∈ (0 ,
1) which lies between λ − q and λ − q +1 and that satisfies ℓλ q ≤ λ − αq +1 ℓ − ≤ λ αq +1 , where 0 < α ≪
1. This can be done since αb > ℓ as the geometric mean of the two bounds imposed before ℓ = λ − α/ q +1 λ − q . With this choice we also have that ℓ ≪ s q +1 . Let { θ ℓ } ℓ> and { ϕ ℓ } ℓ> be two standard families ofFriedrichs mollifiers on R (space) and R (time) respectively. We define the mollification of v q and ˚ R q inspace and time, at length scale ℓ by v ℓ := ( v q ∗ x θ ℓ ) ∗ t ϕ ℓ , ˚ R ℓ := (˚ R q ∗ x θ ℓ ) ∗ t ϕ ℓ , where we possibly extend to 0 the definition of v q outside B s ( t ). We have that v ℓ solves ( ∂ t v ℓ + div( v ℓ ⊗ v ℓ ) + ∇ p ℓ − ν ∆ v ℓ = div(˚ R ℓ + ˚ R com )div v ℓ = 0 , (4.2)where ˚ R com is defined by ˚ R com = ( v ℓ ˚ ⊗ v ℓ ) − (( v q ˚ ⊗ v q ) ∗ x θ ℓ ) ∗ t ϕ ℓ . We introduce the following notations y + I q := ( t − S q − y, t + S q + y ) and ˜ I q := s q +1 + I q . Let η ∈ C ∞ c ( ˜ I q ; R + ) such that η ( t ) ≡ t ∈ I q , k η k C N ≤ C (cid:18) s (cid:19) Nq , Moreover, we define ˜ v ℓ = ηv ℓ + (1 − η ) v q . Note that ˜ v ℓ satisfies Supp T (˜ v ℓ − v q ) ⊂ ˜ I q ⊂ I q +1 , YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 9 that will be crucial in order to guarantee (2.9c) at step q + 1.Moreover, using (4.2) and that ( v q , p q , ˚ R q ) is a Navier–Stokes–Reynolds solution, we have that ˜ v ℓ satisfies ∂ t ˜ v ℓ + div(˜ v ℓ ⊗ ˜ v ℓ ) − ∆˜ v ℓ = ( v ℓ − v q ) ∂ t η + η (1 − η )div( v ℓ ˚ ⊗ ( v q − v ℓ ))+ η (1 − η )div( v q ˚ ⊗ ( v ℓ − v q ))+ η div( R ℓ + R com ) + (1 − η )div(˚ R q ) − ∇ π ℓ , for some pressure π ℓ .We recall the inverse divergence operator from [12]. Definition 4.1.
We define the Reynolds operator R : C ∞ ( T ; R ) → C ∞ ( T ; R ) as R v := 14 ( ∇ P H ∆ − v + ( ∇ P H ∆ − v ) T ) + 34 ( ∇ ∆ − v + ( ∇ ∆ − v ) T ) −
12 div (∆ − v Id) , for every smooth v with zero average. If v ∈ C ∞ ( T ; R ) we define R v := R ( v − ffl T v ) . We have the following
Proposition 4.2 ( R = div − ) . For any v ∈ C ∞ ( T ; R ) with zero average we have(1) R v ( x ) is a symmetric traceless matrix, for each x ∈ T ,(2) div R v = v − ffl T v ,(3) R can be extended to a continuous operator from L p to L p ,(4) R∇ can be extended to a continuous operator from L p to L p . Using (2.9b) and that η ( t ) ≡ I q , we have(1 − η )div(˚ R q ) ≡ . Thus ˜ v ℓ solves ∂ t ˜ v ℓ + div(˜ v ℓ ⊗ ˜ v ℓ ) − ∆˜ v ℓ + ∇ π ℓ = div( R ℓ + R com + R loc ) , where R ℓ = η ˚ R ℓ , R com = η ˚ R com and R loc := η (1 − η ) v ℓ ˚ ⊗ ( v q − v ℓ ) + η (1 − η ) v q ˚ ⊗ ( v ℓ − v q ) + R (( v ℓ − v q ) ∂ t η ) . A simple bound on v ℓ − v q on L ∞ t L is given by k v ℓ − v q k L . ℓ k v q k C ≤ ℓλ q ≪ λ − ζq +1 δ q +2 , where the last holds if 4 ζ + 2 βb < α. Then using the previous bound, (2.8a) and that kRk L → L . k R com k L + k R loc k L ≪ λ − ζq +1 δ q +2 , where we used that λ ζq +1 ≫ C (cid:0) s (cid:1) q , unless to possibly enlarge a ( s, ζ ). Note that we also have the propertyon the compact supports of the errorsSupp( R ℓ ) ∪ Supp( R com ) ∪ Supp( R loc ) ⊂ ˜ I q ⊂ I q +1 . The mollified functions satisfy k ˜ v ℓ k C Nx,t ( T × B s ( t )) . λ q ℓ − N +1 . λ − αq ℓ − N , N ≥ , (4.3a) k ˜ v ℓ k L ≤ k v q k L + k v q − v ℓ k L ≤ k v k L − δ / q + λ − αq , (4.3b) k ˜ v ℓ − v q k L . ℓλ q ≤ λ − αq +1 , (4.3c) k R ℓ k L ≤ λ − ζq δ q +1 , (4.3d) k R ℓ k C Nx,t . λ − ζq δ q +1 ℓ − − N , N ≥ . (4.3e)We are now ready to go to the perturbation step, in which we will add a small perturbation to ˜ v ℓ in orderto cancel the bigger error R ℓ proving (2.8b), (2.9b) and satisfying all the other estimates (2.8), (2.9) and(2.10).4.2. Amplitudes.
Here we define the amplitudes of the perturbation, namely the functions needed toapply Lemma 3.1 and cancel the Reynolds error R ℓ . We define χ : R + → R + , a smooth function such that χ ( z ) := ( if ≤ z ≤ z if z ≥ z ≤ χ ( z ) ≤ z for z ∈ (1 ,
2) and χ ( z ) ≥ z ∈ [0 , ∞ ). We define for all t ∈ I = [ t − s , t + s ] ρ ( t ) := 13 ´ T χ (cid:16) | R ℓ ( x,t ) | λ ζq δ δ q +1 (cid:17) dx (cid:18) e ( t ) − ˆ T | ˜ v ℓ ( x, t ) | dx − δ q +2 (cid:19) (4.4)and with a little abuse of notation we define ρ ( t ) := ρ (cid:16) t + s (cid:17) for all t > t + s ,ρ ( t ) := ρ (cid:16) t − s (cid:17) for all t < t − s . Now, we consider another local cut-off in time ˜ η ∈ C ∞ c ( I q +1 ; R + ) such that˜ η ( t ) ≡ t ∈ ˜ I q , k ˜ η k C N ≤ C (cid:18) s (cid:19) Nq , and we define ρ ( x, t ) := ˜ η ( t ) ρ ( t ) χ | R ℓ ( x, t ) | λ ζq δ δ q +1 ! . (4.5) Lemma 4.3.
The following estimates hold δ q +1 δ λ ζq ≤ ρ ( t ) ≤ ǫ δ q +1 δ , (4.6) (cid:12)(cid:12)(cid:12)(cid:12) R ℓ ( x, t ) ρ ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (4.7) k ρ k L ≤ π ǫ δ q +1 δ . (4.8) Proof.
Note that (cid:12)(cid:12) k v q k L − k ˜ v ℓ k L (cid:12)(cid:12) ≤ k v q − ˜ v ℓ k L k v q + ˜ v ℓ k L . ℓ k v q k C k v q k L . ℓλ q ≤ λ − ζq δ q +1 , (4.9)where in the last inequality we used that 2 β + ζb < α . Moreover, thanks to the construction of χ and (2.8b)we have (2 π ) ≤ ˆ T χ | R ℓ ( x, t ) | λ ζq δ δ q +1 ! dx ≤ π ) . (4.10)Thus, thanks to (2.9a), (4.9) and (4.10) we get ρ ( t ) ≤ · (2 π ) (cid:18) e ( t ) − ˆ T | v q ( x, t ) | dx (cid:19) + 13 · (2 π ) (cid:18) ˆ T | v q ( x, t ) | dx − ˆ T | ˜ v ℓ ( x, t ) | dx − δ q +2 (cid:19) YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 11 ≤ · (2 π ) (cid:18) δ q +1 ǫ δ (cid:19) ≤ ǫ δ q +1 δ . and similarly ρ ( t ) ≥ · (2 π ) (cid:18) e ( t ) − ˆ T | v q ( x, t ) | dx (cid:19) + 16 · (2 π ) (cid:18) ˆ T | v q ( x, t ) | dx − ˆ T | ˜ v ℓ ( x, t ) | dx − δ q +2 (cid:19) ≥ · (2 π ) δ q +1 δ λ ζ/ q − δ q +1 λ ζq − δ q +2 ! ≥ δ q +1 δ λ ζq , where the last holds if we choose a ( ζ ) sufficiently large. Thus (4.6) holds.The proof of (4.7) follows from the following computation, observing that Supp T ( R ℓ ) ⊂ ˜ I q , ˜ η ( t ) ≡ t ∈ ˜ I q and that χ ( z ) ≥ z/ z ≥ (cid:12)(cid:12)(cid:12)(cid:12) R ℓ ( x, t ) ρ ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | R ℓ ( x, t ) | ρ ( t ) | R ℓ ( x,t ) | δ q +1 λ ζq δ = δ q +1 ρ ( t ) λ ζq δ ≤ / . We conclude the proof by estimating ˆ T | ρ ( x, t ) | dx ≤ ˆ | Rℓ ( x,t ) | λζqδ δq +1 < | ρ ( x, t ) | dx + ˆ | Rℓ ( x,t ) | λζqδ δq +1 ≥ | ρ ( x, t ) | dx ≤ π (cid:18) δ q +1 ǫ δ (cid:19) + ˆ T | λ ζq ǫ R ℓ | dx ≤ π (cid:18) δ q +1 ǫ δ (cid:19) + 8 ǫ λ ζq k R ℓ k L ≤ π ǫ (cid:18) δ + λ − ζq (cid:19) δ q +1 ≤ π ǫ δ q +1 δ . (cid:3) We can now define the amplitudes functions a ξ : T × (0 , T ) → R as a ξ ( x, t ) := a ξ,q +1 ( x, t ) := ρ / ( x, t ) γ ξ (cid:18) Id − R ℓ ( x, t ) ρ ( x, t ) (cid:19) , (4.11)where γ ξ are defined in Lemma 3.1, hence we also get the identity ρ ( x, t ) Id − R ℓ ( x, t ) = X ξ ∈ Λ a ξ ( x, t ) ξ ⊗ ξ. (4.12) Lemma 4.4.
The following estimates hold k a ξ k L ≤ δ / q +1 C | Λ | ǫ πδ / , (4.13) k a ξ k C Nx,t . ℓ − − N , (4.14) where C is the universal constant for which Lemma A.3 holds.Proof. We define ρ ( x, t ) := ρ ( t ) χ | R ℓ ( x, t ) | λ ζq δ δ q +1 ! ,a ξ ( x, t ) := ρ / ( x, t ) γ ξ (cid:18) Id − R ℓ ( x, t ) ρ ( x, t ) (cid:19) , a ξ ( x, t ) = ˜ η ( t ) a ξ ( x, t ) . The first estimate follows from (4.8) and the definition of ǫ k a ξ k L ≤ k ρ k / L k γ ξ k C k ˜ η k C ≤ (cid:18) π δ q +1 ǫ δ (cid:19) / k γ ξ k C ≤ δ / q +1 C | Λ | ǫ πδ / . We prove the second estimate. We introduce the notation ˜ γ ξ ( x, t ) := γ ξ (cid:16) Id − R ℓ ( x,t ) ρ ( x,t ) (cid:17) and thanks toProposition A.2 we have k a ξ k C Nx,t . k ρ / k C N k ˜ γ k C + k ρ / k C k ˜ γ k C N . We now estimate every piece. Using Proposition A.1 and (2.9a) k ρ k C Nt . ℓ − N . Thanks to the previous inequality, Proposition A.1 and Proposition A.2 we get k ρ k C Nx,t . ℓ − − N . (4.15)Using Proposition A.1, estimate (4.3e), the previous estimate and that ρ is bounded from below by δ q +1 δ λ ζq ,we have k ˜ γ k C N . (cid:13)(cid:13)(cid:13)(cid:13) R ℓ ρ (cid:13)(cid:13)(cid:13)(cid:13) C N . ℓ − − N and using also that δ q +1 δ λ ζq ≥ ℓ (choosing ζ = ζ ( α ) sufficiently small), we have k ρ / k C Nx,t . ℓ − − N . Hence k a ξ k C Nx,t . ℓ − − N . Moreover, by applying Proposition A.2 we get k a ξ k C Nx,t . k a ξ k C Nx,t k ˜ η k C + k ˜ η k C N k a ξ k C x,t . k a ξ k C Nx,t , since s − q +1 ≪ λ q ≪ ℓ − , up to enlarge a ( s, α ). (cid:3) Principal part of the perturbation, incompressibility and temporal correctors.
The prin-cipal part of w q +1 is defined as w ( p ) q +1 := X ξ ∈ Λ a ξ W ξ . (4.16)The incompressibility corrector w ( c ) q +1 , that we define in order to have the incompressibility of w q +1 , isdefined as w ( c ) q +1 := X ξ ∈ Λ curl( ∇ a ξ × V ξ ) + ∇ a ξ × curl V ξ + a ξ W ( c ) ξ . Note that w ( p ) q +1 + w ( c ) q +1 = X ξ ∈ Λ ∇ × ∇ × ( a ξ V ξ ) , div( w ( p ) q +1 + w ( c ) q +1 ) = 0 , where the first equation follows from a direct computation similar to (3.7) with amplitudes functions a ξ W ξ = a ξ ∇ × ∇ × V ξ − a ξ W ( c ) ξ YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 13 = ∇ × ( a ξ ∇ × V ξ ) − ∇ a ξ × ( ∇ × V ξ ) − a ξ W ( c ) ξ = ∇ × ∇ × ( a ξ V ξ ) − ∇ × ( ∇ a ξ × V ξ ) − ∇ a ξ × ( ∇ × V ξ ) − a ξ W ( c ) ξ . Moreover, we introduce a temporal corrector similar to (3.5) with amplitude functions w ( t ) q +1 := − µ X ξ ∈ Λ P H P =0 (cid:0) a ξ φ ξ ψ ξ ξ (cid:1) . (4.17)Note that w ( t ) q +1 satisfies ∂ t w ( t ) q +1 + X ξ ∈ Λ P =0 (cid:0) a ξ div( W ξ ⊗ W ξ ) (cid:1) = − µ X ξ ∈ Λ P H P =0 ∂ t (cid:0) a ξ φ ξ ψ ξ ξ (cid:1) + 1 µ X ξ ∈ Λ P =0 (cid:0) a ξ ∂ t (cid:0) φ ξ ψ ξ ξ (cid:1)(cid:1) = (Id − P H ) 1 µ X ξ ∈ Λ P =0 ∂ t (cid:0) a ξ φ ξ ψ ξ ξ (cid:1)| {z } =: ∇ P q +1 − µ X ξ ∈ Λ P =0 (cid:0) ∂ t a ξ (cid:0) φ ξ ψ ξ ξ (cid:1)(cid:1) . From this computation and the identity (4.12), it follows thatdiv( w ( p ) q +1 ⊗ w ( p ) q +1 + R ℓ ) + ∂ t w ( t ) q +1 = X ξ ∈ Λ div (cid:0) a ξ P =0 ( W ξ ⊗ W ξ ) (cid:1) + ∇ ρ + ∂ t w ( t ) q +1 = X ξ ∈ Λ P =0 (cid:0) ∇ a ξ P =0 ( W ξ ⊗ W ξ ) (cid:1) + ∇ ρ + X ξ ∈ Λ P =0 (cid:0) a ξ div ( W ξ ⊗ W ξ ) (cid:1) + ∂ t w ( t ) q +1 = X ξ ∈ Λ P =0 (cid:0) ∇ a ξ P =0 ( W ξ ⊗ W ξ ) (cid:1) + ∇ ρ + ∇ P q +1 − µ X ξ ∈ Λ P =0 (cid:0) ∂ t a ξ (cid:0) φ ξ ψ ξ ξ (cid:1)(cid:1) . (4.18)4.4. The velocity increment and proof of the inductive estimates.
We now define the total incre-ment w q +1 := w ( p ) q +1 + w ( c ) q +1 + w ( t ) q +1 (4.19)and the new vector field is then given by v q +1 := ˜ v ℓ + w q +1 . (4.20)In this section we verify that the inductive estimates (2.8) hold with q replaced by q + 1, and that (2.10)is satisfied.4.4.1. Proof of (2.10) . We want to apply Lemma A.3 in L with f = a ξ and g σ = W ξ , which is byconstruction (cid:0) T σ (cid:1) − periodic with σ ∼ λ q +1 r ⊥ , where ∼ means up to a constant depending only on n ∗ and ξ ∈ Λ. For this purpose, note that by (4.4) we get k D j a ξ k L ≤ δ / q +1 C | Λ | ǫ πδ / ℓ − j , and thus we can take C f = δ / q +1 C | Λ | ǫ πδ / . By conditions on ℓ we have ℓ − ≤ λ αq +1 , whereas by (3.2) wehave that λ q +1 r ⊥ = (cid:16) λ q +1 π (cid:17) / . Thus, since α < · and a is huge, Lemma A.3 is applicable. Combining the resulting estimate with the normalization k W ξ k L = 1 we obtain k w ( p ) q +1 k L ≤ X ξ ∈ Λ C δ / q +1 C | Λ | ǫ πδ / k W ξ k L ≤ ǫ πδ / δ / q +1 . (4.21)For the correctors w ( c ) q +1 and w ( t ) q +1 we can use rougher estimates since they are considerably smaller than w ( p ) q +1 . The following estimates are consequence of Proposition 4.2, estimates (3.2), (3.8) and Lemma 4.4 k w ( p ) q +1 k L p . X ξ ∈ Λ k a ξ k C k W ξ k L p . ℓ − r /p − ⊥ r /p − / || (4.22a) k w ( c ) q +1 k L p . X ξ ∈ Λ k a ξ k C k V ξ k W ,p + k a ξ k C k W ( c ) ξ k L p . ℓ − r /p − ⊥ r /p − / || λ − q +1 + ℓ − r /p − ⊥ r /p − / || r ⊥ r || . ℓ − r /p − ⊥ r /p − / || λ − / q +1 (4.22b) k w ( t ) q +1 k L p . µ − X ξ ∈ Λ k a ξ k C k φ ξ k L p k ψ ξ k L p . ℓ − r /p − ⊥ r /p − || µ − . ℓ − r /p − ⊥ r /p − / || λ − / q +1 , (4.22c)where in the last inequality we used also the continuity of P H in L p (for any 1 < p < ∞ ) and the fact that k φ ξ ψ ξ k L p = k φ ξ k L p k ψ ξ k L p , thanks to Fubini.Combining (4.21), with the last two estimates of (4.22) for p = 2, and using (3.2), we obtain for aconstant C > q ) that k w q +1 k L ≤ ǫ πδ / δ / q +1 + Cℓ − r ⊥ r || + Cℓ − λ − / q +1 ! ≤ ǫ πδ / δ / q +1 Cλ α − / q +1 + Cλ α − / q +1 ! ≤ ǫ πδ / δ / q +1 . Moreover from (4.3), by choosing a sufficiently large we get k v q +1 − v q k L ≤ k w q +1 k L + k ˜ v ℓ − v q k L ≤ ǫ πδ / δ / q +1 , thus (2.10) is satisfied.4.4.2. Proof of (2.8a) . The bound (2.8a) follows easily from and the previous estimates (if q = 0) k v q +1 k L = k v q +1 − v q + v q k L ≤ k v q k L + k v q +1 − v q k L ≤ k v k L − ǫ π δ / q + ǫδ / π δ / q +1 ≤ k v k L − ǫ π δ / q +1 , where in the last inequality we have used that a is taken sufficiently large and b ≫
1. If q = 0, then (2.8a)is trivial. In the last inequality, we have implicitly used that α < / (7 ·
74) and a be sufficiently large. YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 15
Proof of (2.9c) . The property (2.9c) is verified since v q +1 − v q = ˜ v ℓ − v q + w q +1 and Supp T (˜ v ℓ − v q ) ⊂ Supp T η ⊂ I q +1 , Supp T w q +1 ⊂ Supp T a ξ ⊂ Supp T ˜ η ⊂ I q +1 .4.4.4. Proof of (2.8c) . Taking either a spatial or a temporal derivative, using Lemma 3.3, Lemma 4.4, (3.2)and (4.1), we have k w ( p ) q +1 k C x,t . k a ξ k C x,t k W ξ k C x,t + k a ξ k C x,t k W ξ k C x,t . ℓ − r − ⊥ r − / || + ℓ − r − ⊥ r − / || λ q +1 . λ / αq +1 , k w ( p ) q +1 k C x,t . k a ξ k C x,t k V ξ k C x,t + k a ξ k C x,t k W ( c ) ξ k C x,t , . ℓ r − ⊥ r − / || λ − q +1 λ q +1 + ℓ − r ⊥ r || r − ⊥ r − / || λ q +1 . λ / αq +1 , k w ( t ) q +1 k C x,t . k w ( t ) q +1 k C ,αx,t . µ k a ξ φ ξ ψ ξ k C ,αx,t . µ k a ξ k C x,t k φ ξ k C x,t k ψ ξ k C ,αt . µ ℓ − r − ⊥ r − / || λ q +1 λ αq +1 . λ − / αq +1 . In the latter inequality we have used that P H is continuous on Hölder spaces. Therefore, using that α < /
40, that a is sufficiently large and thanks to estimate (4.3a), we have k v q +1 k C x,t ( B s ( t ) × T ) ≤ k ˜ v ℓ k C x,t ( B s ( t ) × T ) + k w q +1 k C x,t ≤ λ q +1 . The new Reynolds stress.
Here we will define the new Reynolds stress ˚ R q +1 . By definitions, ˜ v q +1 solves div ˚ R q +1 − ∇ p q +1 = ∂ t (˜ v ℓ + w q +1 ) + div((˜ v ℓ + w q +1 ) ⊗ (˜ v ℓ + w q +1 )) − ∆(˜ v ℓ + w q +1 )= − ∆ w q +1 + ∂ t ( w ( p ) q +1 + w ( c ) q +1 ) + div(˜ v ℓ ⊗ w q +1 + w q +1 ⊗ ˜ v ℓ ) | {z } div( R lin )+ ∇ p lin + div (cid:16) ( w ( c ) q +1 + w ( t ) q +1 ) ⊗ w q +1 + w ( p ) q +1 ⊗ ( w ( c ) q +1 + w ( t ) q +1 ) (cid:17)| {z } div( R cor )+ ∇ p cor + div( w ( p ) q +1 ⊗ w ( p ) q +1 + R ℓ ) + ∂ t w ( t ) q +1 | {z } div( R osc )+ ∇ p osc +div( R com ) − ∇ p ℓ . More precisely R lin := −R ∆ w q +1 + R ∂ t ( w ( p ) q +1 + w ( c ) q +1 ) + ˜ v ℓ ˚ ⊗ w q +1 + w q +1 ˚ ⊗ ˜ v ℓ ,R cor := (cid:16) w ( c ) q +1 + w ( t ) q +1 (cid:17) ˚ ⊗ w q +1 + w ( p ) q +1 ˚ ⊗ (cid:16) w ( c ) q +1 + w ( t ) q +1 (cid:17) ,R osc := X ξ ∈ Λ R (cid:0) ∇ a ξ P =0 ( W ξ ⊗ W ξ ) (cid:1) − µ X ξ ∈ Λ R (cid:0) ∂ t a ξ ( φ ξ ψ ξ ξ ) (cid:1) ,p lin := 2˜ v ℓ · w q +1 , p cor := | w q +1 | − | w ( p ) q +1 | ,p osc := ρ + P q +1 , where the definitions of p osc and R osc are justified by the previous computation (4.18). Hence we define p q +1 := p ℓ − p cor − p lin − p osc and ˚ R q +1 := R lin + R cor + R osc + R com + R loc , where the last two were defined during the mollification step. We observe that the new Reynolds-stress˚ R q +1 is traceless, this property will be crucial in the energy estimates.4.6. Estimates for the new Reynolds stress.
We need to estimate the new stress ˚ R q +1 in L . How-ever, since the Calderón-Zygmund operator ∇R fails to be bounded on L , we introduce an integrabilityparameter, p ∈ (1 ,
2] such that p − ≪ . Recalling the parameters choice (3.2), we fix p to obey r /p − ⊥ r /p − || ≤ (2 π ) / λ p − / (7 p ) q +1 ≤ λ αq +1 , (4.23)where we recall that 0 < α < · . For instance, we take p = − α .4.6.1. Linear error Reynolds stress.
By using Proposition 4.2 we get that k R lin k L p . kR ∆ w q +1 k L p + k ˜ v ℓ ˚ ⊗ w q +1 + w q +1 ˚ ⊗ ˜ v ℓ k L p + kR ∂ t ( w ( p ) q +1 + w ( c ) q +1 ) k L p . k∇ w q +1 k L p + k ˜ v ℓ k L ∞ k w q +1 k L p + X ξ ∈ Λ k ∂ t curl( a ξ V ξ ) k L p . X ξ ∈ Λ k a ξ k C k W ξ k W ,p + k ˜ v ℓ k C X ξ ∈ Λ k a ξ k C k W ξ k W ,p + X ξ ∈ Λ ( k a ξ k C k ∂ t V ξ k W ,p + k ∂ t a ξ k C k V ξ k W ,p ) . Thus, by appealing to Lemma 3.3, Lemma 4.4, estimates (4.3) and to the choice of p = − α , we conclude k R lin k L p . ℓ − r p − ⊥ r p − / || λ q +1 + ℓ − r p − ⊥ r p − || λ q +1 + ℓ − λ − q +1 r p − ⊥ r p − || . ℓ − λ αq +1 λ q +1 r ⊥ r / || . λ α − q +1 ≪ λ − ζq +1 δ q +2 , where for the last inequality we used that α < · and 2 βb + 3 ζ < .4.6.2. Corrector error.
The estimate on the corrector error is a consequence of (4.22) and our choice of p k R cor k L p ≤ k w ( c ) q +1 + w ( t ) q +1 k L p k w q +1 k L p + k w ( p ) q +1 k L p k w ( c ) q +1 + w ( t ) q +1 k L p ≤ k w ( c ) q +1 + w ( t ) q +1 k L p k w q +1 k L p . ℓ − r /p − ⊥ r p − || λ − / q +1 . λ α + α − / q +1 ≪ λ − ζq +1 δ q +2 , where the last inequality is justified as before. YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 17
Oscillation error.
By using the boundedness on L p of the Reynolds operator R , Lemma 3.3, Lemma4.4, (3.2), Fubini (to separate φ ξ and ψ ξ ) and the choice of p we can estimate the second summand in thedefinition of R osc as (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ X ξ ∈ Λ R (cid:0) ∂ t a ξ ( φ ξ ψ ξ ξ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ µ − X ξ ∈ Λ k a ξ k C k φ ξ k L p k ψ ξ k L p . µ − ℓ − λ αq +1 ≪ λ − ζq +1 δ q +2 . To estimate the remaining summand we will use Lemma A.4. We apply it with a = ∇ a ξ , κ = σ = λ q +1 r ⊥ and P ≥ σ ( f ) = P =0 ( W ξ ⊗ W ξ ), that is a T σ − periodic function. Then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X ξ ∈ Λ R (cid:0) ∇ a ξ P =0 ( W ξ ⊗ W ξ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . ( λ q +1 r ⊥ ) − k P =0 ( W ξ ⊗ W ξ ) k L p k∇ a ξ k C . ℓ − λ − / q +1 k W ξ k L p . ℓ − λ − / q +1 r p − ⊥ r p − || . λ α + α − / q +1 ≪ λ − ζq +1 δ q +2 . Then (2.8b) at step q + 1 follows easily using also the previous estimates for R com and R loc k ˚ R q +1 k L ≤ k R lin k L + k R cor k L + k R osc k L + k R com k L + k R loc k L ≤ λ − ζq +1 δ q +2 + 13 λ − ζq +1 δ q +2 ≤ λ − ζq +1 δ q +2 , where in the last inequality we have used that 2 βb + 3 ζ < α . Finally, since Supp T w q +1 ⊂ I q +1 , then also(2.9b) holds at step q + 1.4.7. Energy estimate.
In order to complete the proof of Proposition 2.1 we only need to prove the energyestimate (2.9a) at step q + 1. Lemma 4.5.
The following estimate holds for all t ∈ I δ q +2 λ ζ/ q +1 ≤ e ( t ) − ˆ T | v q +1 ( x, t ) | dx ≤ δ q +2 ǫ δ . (4.24) Proof.
Recalling (4.16) and the mutually disjoint supports of { W ξ } ξ ∈ Λ we notice that | w ( p ) q +1 | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ξ ∈ Λ a ξ W ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X ξ ∈ Λ Tr( a ξ W ξ ⊗ a ξ W ξ )= X ξ ∈ Λ a ξ Tr (cid:18) T W ξ ⊗ W ξ (cid:19) + X ξ ∈ Λ a ξ Tr (cid:18) W ξ ⊗ W ξ − T W ξ ⊗ W ξ (cid:19) = 3 ρ + X ξ ∈ Λ a ξ Tr (cid:18) W ξ ⊗ W ξ − T W ξ ⊗ W ξ (cid:19) , (4.25)where in the last equation we used the traceless property of R ℓ and (4.12).Applying Lemma A.5 with f replaced by a ξ (which oscillates at frequency ∼ ℓ − ), the constant C f ∼ ℓ − (thanks to the estimate of Lemma 4.4) and g σ replaced with W ξ ⊗ W ξ − ffl T W ξ ⊗ W ξ (where σ = λ q +1 r ⊥ ), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T X ξ ∈ Λ a ξ Tr (cid:18) W ξ ⊗ W ξ − T W ξ ⊗ W ξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ℓ − λ q +1 r ⊥ ≪ δ q +2 , (4.26) where in the last inequality we used that α < · and 2 βb < . We write the identity e ( t ) − ˆ T | v q +1 | = e ( t ) − (cid:18) ˆ T | ˜ v ℓ | + ˆ T | w ( p ) q +1 | (cid:19) − (cid:18) ˆ T | w ( c ) q +1 + w ( t ) q +1 | + 2 ˆ T ˜ v ℓ · w q +1 (cid:19) − (cid:18) ˆ T w ( p ) q +1 · ( w ( c ) q +1 + w ( t ) q +1 ) (cid:19) (4.27)and thanks to (4.25), (4.26) and to the definition of ρ (4.5), using also that ˜ η ≡ I , we have δ q +2 λ ζ/ q +1 ≤ e ( t ) − (cid:18) ˆ T | ˜ v ℓ | + ˆ T | w ( p ) q +1 | (cid:19) ≤ δ q +2 , for all t ∈ I , up to possibly enlarge a ( ζ ). Moreover, by using (4.3) and (4.22) we can estimate (cid:12)(cid:12)(cid:12)(cid:12) ˆ T | w ( c ) q +1 + w ( t ) q +1 | + 2 ˆ T ˜ v ℓ · w q +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ q +2 λ ζ/ q +1 , (cid:12)(cid:12)(cid:12)(cid:12) ˆ T w ( p ) q +1 · ( w ( c ) q +1 + w ( t ) q +1 ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ q +2 λ ζ/ q +1 , from which (4.24) follows. (cid:3) Appendix A. Useful tools
In this section we state some useful results needed in the convex integration scheme.
Proposition A.1.
Let
Ψ : Ω → R and u : R n → Ω be two smooth functions, with Ω ⊂ R N . Then, forevery m ∈ N + , there exists a constant C > (depending only on m, N, n ) such that [Ψ ◦ u ] m ≤ C ([Ψ] [ u ] m + k D Ψ k C m − k u k m − C [ u ] m ) , [Ψ ◦ u ] m ≤ C ([Ψ] [ u ] m + k D Ψ k C m − k u k mC ) , where [ f ] m = max | β = m | k D β f k . Proposition A.2.
Let f, g : T → R be two smooth real value functions. For any integer r ≥ thereexists a constant C > , depending only on r such that [ f g ] r ≤ C ([ f ] r k g k C + k f k C [ g ] r ) , where [ f ] m = max | β = m | k D β f k . The following lemma is essentially Lemma 3.7 in [4].
Lemma A.3.
Fix integers
N, σ ≥ and let ζ > such that π √ ζσ ≤ and ζ (2 π √ ζ ) N σ N ≤ . (A.1) Let p ∈ { , } and let f, g ∈ C ∞ ( T ; R ) . Suppose that there exists a constant C f > such that k∇ j f k L p ≤ C f ζ j , holds for all ≤ j ≤ N + 4 . Then we have that k f g σ k L p ≤ C C f k g σ k L p , where C is a universal constant. The following lemma is essentially Lemma B.1 in [4].
YPICALITY RESULTS FOR WEAK SOLUTIONS OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS 19
Lemma A.4.
Fix κ ≥ , p ∈ (1 , , and a sufficiently large L ∈ N . Let a ∈ C L ( T ) be such that thereexists ≤ λ ≤ κ , C a > with k D j a k L ∞ ≤ C a λ j , for all ≤ j ≤ L . Assume furthermore that ´ T a ( x ) P ≥ κ f ( x ) dx = 0 . Then we have k|∇| − ( a P ≥ κ ( f )) k L p . C a (cid:18) λ L κ L − (cid:19) k f k L p κ for any f ∈ L p ( T ) , where the implicit constant depends on p and L . Lemma A.5.
Let g : T → R such that T g ( x ) dx = 0 , and let g σ : T → R : g σ ( x ) := g ( σx ) . Let f : T → R such that k∇ f k C ≤ C f ζ, then we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ T g σ ( x ) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) . C f ζσ k g σ k L ( T ) , where . means up to a universal constant. References [1] E. Brué, M. Colombo, and C. De Lellis,
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Email address : [email protected] Luigi De RosaÉcole Polytechnique Fédérale de Lausanne, Institute of Mathematics, Station 8, CH-1015 Lausanne, Switzer-land.
Email address : [email protected] Massimo SorellaÉcole Polytechnique Fédérale de Lausanne, Institute of Mathematics, Station 8, CH-1015 Lausanne, Switzer-land.
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