(1)CONCENTRATION OF CRITICAL NORMS NEAR SINGULARITIES TOBIAS BARKER AND CHRISTOPHE PRANGE AbstractThis paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations

CONCENTRATION OF CRITICAL NORMS NEAR SINGULARITIES

TOBIAS BARKER AND CHRISTOPHE PRANGE

AbstractThis paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for lo- cal energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical spaceL³, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and ˇSver´ak, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a con- centration phenomenon near a possible Type I blow-up. Namely, we show if(0, T^∗) is a singular point then

ku(·, t)k_L3(BR(0))≥γ_univ, R=O(√

T^∗−t).

This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namelyL^3,∞and the Besov spaceB˙⁻¹⁺

3

p,∞ p,p∈(3,∞).

Keywords Navier-Stokes equations, Leray-Hopf solutions, local energy solutions, singu- larities, Type I blow-up, concentration,ε-regularity, critical norms, Morrey spaces.

Mathematics Subject Classification (2010) 35A99, 35B44, 35B65, 35D30, 35Q30, 76D05 1. INTRODUCTION

This paper concerns weak Leray-Hopf solutions of the Navier-Stokes system

∂_tu+u· ∇u−∆u+∇p= 0, u(x,0) =u₀(x), ∇ ·u= 0, x∈R³, t >0.

(1)

In particular these solutions satisfy (2) ku(·, t)k²_L2(R³)+ 2

ˆt

0

ˆ

R³

|∇u(x, s)|²dxds≤ ku₀k²_L2(R³)

along with other properties.

Though these solutions were shown to exist for any divergence-free initial data inL²(R³) [34], it is unknown if they are smooth for all positive times. Uniqueness is also still open, although non-uniqueness scenarios were suggested by [25] with supporting numerical evi- dence in [19]. For weaker notions of solutions with bounded kinetic energy, uniqueness can fail as demonstrated by Buckmaster and Vicol in [8]. In investigating the regularity of such solutions, it is natural to ask the following question:

(Q)What type of initial conditions induce smoothing of the associated solutions of the Navier-Stokes equations and can this be described quantatively?

Date: January 9, 2019.

1

The list of contributions to (Q) is vast and we do not attempt to be exhaustive. The first contribution to (Q) was provided by Leray in [34] for u0 ∈ L^p(R³) with p > 3.

By using perturbation methods, further contributions to(Q)were made by Kato [27] for u₀ ∈L³(R³), by Planchon [45] foru₀in critical Besov spaces, by Koch and Tataru [29] for u0 ∈BM O⁻¹(R³), by Maekawa and Terasawa [39] foru0∈L³_uloc(R³)and by Maekawa, Miura and Prange [38] foru₀ ∈L³_uloc(R³+). Here

L^p_uloc(R³) :={u₀∈L^p_loc(R³) : ku₀k_L^p

uloc(R³):= sup

x0∈R³

ku₀k_Lp(B(x0,1) <∞}.

(3)

Recently in [24], Jia and ˇSver´ak made an interesting contribution to(Q)whenu0 ∈L²_uloc andu₀ ∈ L^m(B(0,2))withm > 3. Their result and our first theorem below are in the context of local energy solutions, which were introduced by Lemari´e-Rieusset (see Chapters 32-33 in [33], see also [28]). The definition of local energy solutions is given in Appendix A. Our foremost result is the following theorem.

Theorem 1.For allM ∈(0,∞), there existsS^∗(M)∈(0,¹₄]and an independent universal constant γuniv such that the following holds true. Consider any local energy solutionu, in the sense of Definition 16, to the Navier-Stokes equations (1) with initial data u₀ ∈ L²_uloc,σ(R³)satisfying

ku₀k_L2

uloc(R³)≤M and sup

|¯x|≥R

ku₀k_L2(B1(¯x))

R→∞−→ 0,

u₀ ∈L³(B₂(0)) and ku₀k_L3(B2(0)) ≤γ_univ. Then the above assumptions imply that

u∈L^∞(B¹

3(0)×(β, S^∗(M))), for allβ ∈(0, S^∗(M)).

We also prove an extension of this theorem to: (i) the critical Lorentz-space case in SectionB, i.e.u₀∈L^3,∞(B₂(0)), and (ii) to the critical Besov space case in SectionC, i.e.

u₀ ∈B˙⁻¹⁺

3

p,∞ p(B₂(0)),p ∈ (3,∞). The case of the Besov case in particular requires some technical innovations, which are outlined at the end of the paragraph1.1below.

This result asserts that the regularity of local energy solutions is a somewhat local pro- perty, near initial time. Indeed the solutionu is bounded inB¹

2(0)×(0, S^∗(M)), hence smooth in space, if the initial data is locally in the scale critical spaceL³.

Notice that the result we prove in Section4 is stronger. Indeed, considering the mild solution a associated to an L³ continuous divergence-free extension of the critical data u₀|_B1(0), we prove that¹

(4) u−a∈C_par^0,ν( ¯B¹

3(0)×[0, S^∗(M)]),

for someν ∈ (0,¹₂). We refer to Theorem4in Section3and to Section4for the details, in particular regarding the decomposition of the initial data. This improved regularity for u−arelies heavily on the fact thatu−ahas zero initial data inB1(0).

1In this paper, the parabolic H¨older semi-norm is defined in the following way:

[·]_C0,ν

par(R³×[0,T]):= [·]_C0,ν

t ([0,T];L^∞_x(R³))+ [·]_L∞

t (0,T;C_x^0,2ν(R³)).

1.1. Comparison of Theorem1to previous literature and novelty of our results. Our theorem can be seen as an extension to the scale-critical case of the pioneering result of Jia and ˇSver´ak [24, Theorem 3.1] for subcriticalu₀ ∈ L^m(B(0,2)),m > 3. The main part of their proof relies upon establishing an anε-regularity criteria for suitable solutions (see Section2Definition1for a definition of suitable solutions) of the perturbed Navier-Stokes equations with subcriticala, i.e.a∈L^∞_t L^m_x(B1×(−1,0)),m >3and∇ ·a= 0:

(5) ∂tv−∆v+v· ∇v+a· ∇v+v· ∇a+∇q= 0, ∇ ·v= 0 in B1(0)×(−1,0).

In particular, they show that if certain scale-critical quantities involvingvandqon the unit cubeB1(0)×(−1,0)are small then one has decay of the oscillation:

1 r⁵

t0

ˆ

t−r²

ˆ

Br(¯x)

v−

ˆt

t−r²

ˆ

Br(¯x)

vdyds

3

dxds⁰ ≤Cr^α

for all(¯x, t)∈B¹

2

×(−¹₄,0)and for someα >0. This implies parabolic H¨older continuity ofvby Campanato’s characterisation. The proof of the decay of the oscillation in Jia and ˇSver´ak’s paper [24] is achieved by contradiction and by compactness arguments. Related arguments were previously used in the context of the Navier-Stokes equations by Lin in [37] and by Ladyzhenskaya and Seregin in [32]. Such arguments applied to the system (5) crucially use that for ain subcritical spaces, we have parabolic H¨older continuity in B¹

2(0)×(−¹₄,0)for the following linear system:

(6)

∂tw−∆w+a·∇w+w·∇a+∇q =∇·G, ∇·a= 0, ∇·w= 0 in B1(0)×(−1,0), for a subcritical forcing termG∈L^5m3 (B1(0)×(−1,0)). Unfortunately, whenu0is critical and hencea, Gbelong to scale invariant spaces with respect to the Navier-Stokes scaling² such asL⁵(B1(0)×(−1,0)), we do not expect solutions of (6) to be H¨older continuous.

Concerning this point, let us emphasize that the regularity result (4) uses in an essential way thatu−ahas zero initial data locally inB1(0). This lack of improvement for the perturbed linear system (6) seems to prevent us from relying on compactness arguments to directly prove the boundedness. Indeed, such compactness arguments are based on H¨older conti- nuity for the linear system. Difficulties with using compactness arguments are also found when provingε- regularity statements for the Navier-Stokes equations in higher dimensions (see [12]-[11]). This is the main difficulty we have to overcome to prove Theorem 1. We handle this difficulty by proving a subcritical Morrey bound thanks to a Caffarelli, Kohn and Nirenberg-type scheme. This point is explained in more details in the paragraph1.2 below.

The extension of our results to the Besov case in SectionCrelies on some ideas which are new as far as we know. In particular, in the Caffarelli, Kohn and Nirenberg-type iteration, we need to exploit the local decay of the kinetic energy near the initial time, because the critical drift is more singular in the Besov case than in theL³ case. Such an insight was used before forglobalestimates by Barker [5] to prove weak-strong uniqueness, in Barker, Seregin and ˇSver´ak’s paper [7] on globalL^3,∞solutions and by Albritton and Barker [2] for global Besov solutions. However, to the best of our knowledge, it is the first time that the decay of the kinetic energy near initial time is used inlocalestimates, such as a Caffarelli, Kohn and Nirenberg-type iteration. We believe this point is of independent interest.

2The Navier-Stokes equations are invariant under the scaling (u^(λ)(x, t), p^(λ)(x, t)) = (λu(λx, λ²t), λ²p(λx, λ²t)), u^(λ)₀ (x) = λu0(λx). We say that a space X ⊂ S⁰(R³) is critical (or scale-invariant) if its norm is invariant under the above rescaling for the initial data. Likewise, we say XT ⊂ S⁰(R³)is critical is its norm is invariant under the rescaling for the velocity field.

1.2. Strategy of Proof of Theorem 1. As is the case in Jia and ˇSver´ak’s paper, the key point is to take advantage of the smallness of the local energy of the perturbationvin the unit ball near initial time, i.e. inB₁(0)×(0, S^∗(M)). There are then two main blocks in the proof.

First we prove a subcritical Morrey bound on the perturbation. Smallness of the local energy together with the smallness ofkak_L5

t,x enables to prove a subcritical Morrey bound onv: forδ ∈(0,3)fixed, for(¯x, t)∈B¹

2(0)×(0, S^∗(M)),

(7) sup

r∈(0,¹₂)

1 r^5−δ

ˆt

t−r²

ˆ

Br(¯x)

|v|³dxds <∞,

withv extended by 0 in negative times. The precise statement is given in Theorem3 in Section2. Estimate (7) is based on a Caffarelli, Kohn, Nirenberg type iteration. The proof requires some technical innovations, in particular concerning the treatment of the pressure and of the perturbation termsa· ∇v and v· ∇a. A major difficulty is that the decay of kak_L5(Br(¯x)×(t−r²,t))does not improve whenr→0. A careful study of the proof in Section 2shows that theL⁵norm forais the critical threshold for iteration to work. Exploiting that ais a solution to the Navier-Stokes equations and the bound (7), one could directly apply ε-regularity away from the initial time to get smoothness of the perturbationv. Instead, we aim at obtaining the boundedness ofvup to the initial time.

Obtaining the boundedness ofu−aand eventually H¨older continuity in the parabolic metric up to initial time is the second main block of the paper. It goes through the use of the Morrey bound (7) to control the nonlinear term in (5) and a bootstrap on the linear equation to get the boundedness. Related arguments were used by Seregin in [49]. This work is done in Section3.

Let us point out that for the subcritical caseu0 ∈L^m(B(0,2))(which corresponds toa belonging to subcritical spaces), Jia and ˇSver´ak prove in [24] that the perturbationv=u−a is H¨older continuous in the parabolic metric up to the initial time. Moreover, in [24] the H¨older exponent degenerates asmapproaches the critical casem= 3. Perhaps at first sight it appears somewhat unexpected that one still obtains H¨older continuity ofvup to the initial time, for the critical initial data case. Our proof for showing this relies upon the structure of estimates for the mild solutiona, in particularsup_s∈(0,S^∗_(M))s¹⁵ka(·, s)k_L5 1and the fact thatvhas zero initial data. Such points allow us to obtain a decay of theL^∞norm ofv near the initial time, which is key from going fromvbeing bounded to H¨older continuous.

1.3. Concentration of norms centered on singularities. In the the second part of the paper, we apply the results of Theorem 1 to obtain certain new concentration results for weak Leray-Hopf solutions which first develop singular points at timeT^∗ > 0. We say that(¯x, t) ∈ R³ ×(0,∞) is a regular point of u, if there exists r ∈ (0,∞) such that u∈L^∞(Br(¯x)×(t−r², t)). A contrario, a point(¯x, t)∈R³×(0,∞)is a singular point, or a blow-up point if it is not regular. A timeT^∗ ∈(0,∞)is called a blow-up time if there existsx¯ ∈ R³ such that(¯x, T^∗) is a singular point. For(¯x, t) ∈ R³ ×R, we define the parabolic cylinder

Q_r(¯x, t) :=B_r(x)×(t−r², t).

Investigation of singular weak Leray-Hopf solutions was first performed by Leray in [34].

In particular, Leray showed that if a weak Leray-Hopf solutionvfirst develops singularities

atT^∗ then

(8) kv(·, t)k_Lp(R³)≥ C(p) (T^∗−t)^12(1−3p)

for 3< p≤ ∞ and 0< t < T^∗.

Behaviour of theL³norm is more subtle. In a breakthrough paper, Escauriaza, Seregin and Sver´ak showed that if(¯x, T^∗)is a singular point then

(9) lim sup

t↑T^∗

kv(·, t)k_L3(Bδ(¯x))=∞ for any fixed δ >0.

See [44] and [13] for local extensions, as well as [16] and [36] for global extensions. Later in [47], Seregin improved (9):

(10) lim

t↑T^∗kv(·, t)k_L3(R³)=∞.

We also refer to [6], [38], [1] and [2] for extensions and refinements (we mention that [6]

and [38] concern the half-space). Recently in [3] Albritton and Barker refined (9) and (10) to show that ifΩis a bounded domain withC² boundary one has

(11) lim

t↑T^∗kv(·, t)k_L3(Bδ(¯x)∩Ω)=∞ for any fixed δ >0.

In this paper we are interested in investigating accumulation behaviour of norms ofv near blow-up timesT^∗on balls whose radius shrinks to zero astapproachesT^∗. We refer to this as‘concentration ofv’. Such phenomenon was investigated for other equations e.g.

nonlinear Schr¨odinger in the wake of the pioneering work [40], see [20], [21]. In cite [35], an interesting concentration result is proven for a weak Leray-Hopf solutionvwhich first blows-up atT^∗ >0. In particular, their results imply that there existst_n↑T^∗andx_n∈R³ such that

(12) kv(·, t_n)k_Lm(B√

C(m)(T∗−tn)(xn))≥ C(m)

(T^∗−t_n)^12(1−m3), 3≤m≤ ∞.

We are not aware of any prior such results of this type for the Navier-Stokes equations.

By using a rescaling argument and an estimate of the existence time of mild solutions in terms of the size of the initial data inL^m_uloc(R³),m >3, Maekawa Miura Prange improved (12). In particular, see [38, Corollary 1.1], they showed that for everyt∈(0, T^∗)(not just a sequencet_n↑T)there existsx(t)∈R³such that

(13) kv(·, t)k_Lp(B√

C(m)(T∗−t)(x(t)))≥ C(m)

(T^∗−t)^12(1−m3) 3≤m≤ ∞.

In (12) or (13) no information is provided on x_n and x(t). It is natural to ask whether the concentration phenomenon occurs on ballsB(x, R) withR = O(√

T^∗−t) and with (x, T^∗)being a singular point. Our second theorem answers this in the affirmative for the L³ for Leray-Hopf solutions which first blow-up at time T^∗ and which satisfy the Type I bound:

sup

x∈¯ R³

sup

r∈(0,r0)

sup

T^∗−r²<t<T^∗

r⁻¹² ˆ

Br(¯x)

|u(x, t)|²dx ¹₂

≤M,

for a fixed radiusr0 ∈(0,∞]andM, T^∗ ∈(0,∞).

(14)

Let us now state our concentration result.

Theorem 2. Letγ_univ, M ∈ (0,∞) andS^∗(M)given by Theorem 1. LetT^∗ ∈ (0,∞) andr0 ∈(0,∞]be fixed. There existst∗(T^∗, M, r0)∈[0,∞)such that the following holds true. Letube a Leray-Hopf solution to(1)inR³×(0,∞)satisfying the type I bound(14).

Furthermore, supposeu first blows-up at T^∗ and has a singular point at the space-time point(0, T^∗). Then the above assumptions imply that

(15) ku(·, t)k

L³ |·|≤2q_T∗−t

S∗(M)

> γuniv, for allt∈(t∗(M, r0), T^∗).

We further extend this result. Indeed, we also prove: (i) in SectionBthe concentration of the criticalL^3,∞norm, and (ii) in SectionCthe concentration of the critical Besov space normB˙⁻¹⁺

3

p,∞p,p∈(3,∞).

By translation invariance of the Navier-Stokes equations and of the type I condition (14), the concentration result (15) holds at any blow-up point(¯x, T^∗). Notice also that ifr₀=∞, t∗= 0.

LetM⁰ ∈ (0,∞). It is clear that the type I condition (14) is satisfied by Leray-Hopf solutions blowing-up at timeT^∗ >0and such that

|u(x, t)| ≤ M⁰

|x|, for all(x, t)∈R³×(0, T^∗).

More generally, it is also satisfied for Leray-Hopf solutionsu blowing-up at timeT^∗ > 0 and satisfying a scale-critical Morrey-type bound, i.e.

ku(·, t)k_M˙2,3 := sup

¯ x∈R³

sup

r∈(0,∞)

r⁻¹² ˆ

Br(0)

|u(x, t)|²dx¹₂

≤M⁰

for allt∈(0, T^∗). This condition corresponds to (14) withr0 =∞andM =M⁰. Hence the concentration in Theorem2holds for anyt∈(0, T^∗). It is less obvious to see that type I blow-ups satisfying the bound

(16) √

T^∗−t|u(x, t)| ≤M⁰ or

(17) √

T^∗−t^θ|x|^1−θ|u(x, t)| ≤M⁰

for someθ∈ (0,1)also enter the framework of Theorem2. Yet, (16) and (17) imply that there existsr0∈(0,∞)andM(M⁰, u0, r)∈(0,∞)such that (14) holds. This is proved in [50] (see also p. 844-849 of [48]). We note that forr₀ =∞this implication fails for the case of the half-space with Dirichlet boundary condition. This is demonstrated by Giga in [18, Theorem 3.1] by using shear flows.

1.4. Strategy of proof. The strategy to prove Theorem2is to rescale the solution appro- priately and then reduce to Theorem1. The Type I condition (14) ensures that the rescaled solution has initial data that can be controlled inL²_uloc(R³). We refer to Section4for more details.

1.5. Final discussion. As a corollary to Theorem1we see that ifuis a weak Leray-Hopf solution (with initial datau₀ ∈L²(R³)) which first blows up atT^∗ >0and has a singular point(0, T^∗), then the following holds true. Namely, there existst∗(T^∗,ku₀k_L2(R³))such that fort∈[t∗, T^∗)we have:

(18) ku(·, t)k_L3(B1(0)) > γuniv.

Although the result in [3] shows thatlimt↑T^∗ku(·, t)k_L3(B1(0)) = ∞, it does not provide quantitative information on at which moment in timeku(·, t)k_L3(B1(0))begins to grow.

Soon after the present work was submitted to arXiv, Kang, Miura and Tsai uploaded to arXiv an independent work [26] with a different proof of Theorem 1. Their proof of

the subcritical Morrey bounds (Theorem3) is completely different to ours and relies upon compactness arguments as opposed to a Caffarelli, Kohn and Nirenberg type iteration. The H¨older continuity of Theorem4and the extension of Theorem1to wider critical spaces is not present in [26].

Outline of the paper. Section2 is devoted to the proof of a Morrey bound such as (7).

The main result in this section is Theorem3, which is proved using a Caffarelli, Kohn and Nirenberg type iteration. Section3handles the bootstrap arguments on the perturbed linear system in order to prove the boundedness and the H¨older continuity of the perturbation u−aup to initial time. The main result in this section is Theorem4. Section4is concerned with the proof of Theorem1and its application to the concentration of theL³norm near a potential singularity, Theorem2. In AppendixAwe recall well-known results about mild solutions and local energy solutions, and we give pressure formulas. AppendixBis devoted to the extension of our results to the Lorentz spaceL^3,∞. AppendixCis concerned with the extension of our results to the Besov spaceB˙⁻¹⁺

3

p,∞ p forp∈(3,∞), which requires some new ideas.

Throughout the paper the constantC∈(0,∞)denotes a universal constant, unless stated otherwise.

2. PROPAGATION OF A MORREY-TYPE BOUND

The goal of this section is to prove a Morrey-type bound for local suitable solutions of

∂_tv−∆v+∇q =−v· ∇v−a· ∇v− ∇ ·(a⊗v), ∇ ·v = 0, (19)

by using a Caffarelli, Kohn and Nirenberg [9] type iteration.

Definition 1(Local suitable solution). Leta∈ L⁵(Q1). The pair(v, q)is a local suitable solution inQ₁to the perturbed Navier-Stokes equations (19) if

vbelongs toL^∞(−1,0;L²(B1(0)))∩L²(−1,0;H¹(B1(0))), qbelongs toL³²(Q1), vis inCw([−1,0];L²(B1(0))),(v, q)is a solution to (19) inQ1in the sense of distributions and for all0≤φ∈C_c^∞(Q₁), we have the following local energy equality

ˆ

B1(0)

|v(x, t)|²φ(x, t)dx+ 2 ˆt

−1

ˆ

B1(0)

|∇v|²φdxds

≤ ˆt

−1

ˆ

B1(0)

|v|²(∂tφ+ ∆φ)dxds+ ˆt

−1

ˆ

B1(0)

(|v|²+ 2q)v· ∇φdxds

− ˆt

−1

ˆ

B1(0)

(a· ∇v)·vφdxds+ ˆt

−1

ˆ

B1(0)

(a⊗v) : (∇vφ+v⊗ ∇φ)dxds (20)

for allt∈(−1,0].

The following theorem is a generalization to scale-critical drifts of theε-regularity result for subcritical drifts proved in the paper by Jia and ˇSver´ak [24].

Theorem 3. For allδ ∈ (0,3), , there existsC∗(δ) ∈ (0,∞), for all E ∈ (0,∞), there existsε∗(δ, E) ∈ (0,∞), for alla ∈ L⁵(Q₁) and all local suitable solutionvto (19) in

Q₁(0,0)such that

(21) sup

−1<s<0

ˆ

B1(0)

|v(x, s)|²dx+ ˆ

Q1(0,0)

|∇v|²dxds≤E,

the conditions

(22) kak_L5(Q1(0,0)) ≤ε∗

and (23)

ˆ

Q1(0,0)

|v|³+|q|³²dxds≤ε∗

imply that for all(¯x, t)∈Q¯_1/2(0,0), for allr ∈(0,¹₄],

(24) −

ˆ

Qr(¯x,t)

|v|³dxds≤C∗ε

2

∗3r^−δ.

Let (¯x, t) ∈ Q¯_1/2(0,0) be fixed for the rest of this section. For all n ∈ N, we let rn:= 2⁻ⁿ. We actually prove that for alln≥2,

(25) −

ˆ

Qrn(¯x,t)

|v|³dxds≤ε

2

∗3r^−δ_n .

In this estimate, contrary to (24), there is no constant C∗(δ) in the right hand side. The constantC∗(δ)comes from going from scalesr_nto allr ∈(0,¹₄].

The proof is by iteration following the scheme of Caffarelli, Kohn and Nirenberg [9] (see also [46]). Our aim is to propagate fork≥2the following two bounds

1 r_k²

ˆ

Q_rk(¯x,t)

|v|³dxds+ 1 r

1+δ 2

k

ˆ

Q_rk(¯x,t)

|q−(q)_rk(s)|³²dxds≤ε

2

∗3r_k^3−δ, (A_k)

sup

t−r_k²<s<t

ˆ

B_rk(¯x)

|v(x, s)|²dx+ ˆ

Q_rk(¯x,t)

|∇v|²dxds≤C_Bε

2

∗3r³⁻

2 3δ

(B_k) k

where

(q)r_k(s) :=− ˆ

B_rk(¯x)

q(x, s)dx, for constantsε∗(δ, E), C_B(δ)∈(0,∞)to be chosen such that

(26) CB = 10C₁^{2 211}

1−2^−23δ + 2⁶ and

(27) max D1(δ)ε

1

∗3, D2(1 +E)ε

1

∗3,5D3(δ)ε∗, D4(δ)ε

7

∗9,2C3C

3 2

Bε

1

∗3, 12D5(δ)ε

1

∗3,12D6(δ)ε

2

∗15,12D7(δ)ε

1

∗3,12D8(δ)ε

4

∗3

≤1, where the constantsD1, . . . are defined in the course of the proof of Theorem3. Notice that (A_k) is a bound on a scale-invariant quantity forv. However the quantity on the left hand side of (A_k) related to the pressureqis not scale-critical. Indeed, we allow for some room in the rate of decay inr_kfor the oscillation of the pressure, which gives more flexibility in the argument. The bound (B_k) is a bound on the local energy.

Remark2. The reason one cannot takeδ = 0appears clearly in the proof of Theorem 3.

Indeed, the lack of improvement of decay ofkak_L5(Qr)combined withδ= 0would lead to a linear growth inkwhen controlling some terms, in particularI₃andI₄ below. We would then not be able to offset this growth by takingε∗ small uniformly ink. Notice that if one knows for some reason thatahas more integrability, then one can then take δ = 0. This would be the case for instance ifasolves the Navier-Stokes equations so that one can apply a Serrin’s type criteria. However, in view of proving Theorem1, we need Theorem3for a generalasmall inL⁵_t,x, not necessarily a solution to the Navier-Stokes equations.

The proof of (Bk) for k = n+ 1 ≥ 3 relies on the local energy inequality (20), the bounds (A_k) and (B_k) for all2 ≤k≤n. The idea is to use test functions(φ_n)n≥2, which are almost solutions to the backward heat equation. The following lemma is taken directly from [46, Lemma 15.11], see also [9]. We give a statement for the sake of completeness.

Lemma 3(Construction of test functionsφ_n). There exists a constantC₁ ∈ (0,∞)such that for all(¯x, t)∈R³×(0,∞), for alln∈N,n≥2, there exists

φn∈C_c^∞ B¹

2(¯x)×(t−¹₉, t+ ^r₂ⁿ²)

such that

C₁⁻¹r⁻¹_n ≤φ_n≤C₁r_n⁻¹ and |∇φ_n| ≤C₁r⁻²_n on Q_rn(¯x, t), (28)

φ_n≤C₁r²_nr⁻³_k and |∇φ_n| ≤C₁r_n²r_k⁻⁴ onQ_rk−1(¯x, t)\Q_rk(¯x, t), 2≤k≤n, (29)

(suppφ_n)∩Q₁(¯x, t)⊂Q¹

3(¯x, t), (30)

|∂_tφn+ ∆φn| ≤C1r²_n on R³×(−∞,0], (31)

wherer_n= 2⁻ⁿ.

Assuming (B_k) for2≤k≤n, we easily get the bound (A_k) fork=nonvby interpo- lation. To prove the pressure bound, we need a representation formula for the pressure.

Lemma 4(Pressure estimate). There exists a constantC2 ∈ (0,∞)such that for allρ ∈ (0,∞), for alla∈L⁵(Q_ρ(0,0)), for all weak solutionq ∈L³²(Q_ρ(0,0))to

−∆q =∇ · ∇ ·(v⊗v) +∇ · ∇ ·(a⊗v) +∇ · ∇ ·(v⊗a) in Q_ρ(0,0),

we have

r^−1+δ2 ˆ

Qr(0,0)

|q−(q)r(s)|³²dxds

≤C₂r^−1+δ2 ˆ

Q2r(0,0)

|v|³dxds+C₂r^1−δ2 ˆ

Q2r(0,0)

|v|³dxds

!¹₂ ˆ

Q2r(0,0)

|a|⁵dxds

!₁₀³

+C2r^6−2δ sup

−r²<s<0

ˆ

2r<|x|<ρ

|v(x, s)|²

|x|⁴ dx

!³₂

+C₂r^4−2δ ˆ0

−r²

ˆ

2r<|x|<ρ

|v(x, s)a(x, s)|

|x|⁴ dx

!³₂ ds

+C₂r^4−2δρ⁻⁹² ˆ

Qρ(0,0)

|v|³+|q|³²dxds

+C2r^44−5δ10 ρ⁻³⁹¹⁰ ˆ

Qρ(0,0)

|v|³dxds

!¹₂ ˆ

Qρ(0,0)

|a|⁵dxds

!₁₀³ , (32)

for all0< r≤ρ/2.

Proof of Lemma4. The proof follows the lines of [46, Lemma 15.12]. We first adapt the decomposition of the pressure given in Lemma18. We take a cut-off functionϕsuch that ϕ= 1onB(0,³₄ρ),suppϕ⊂B(0, ρ)andρ|∇ϕ| ≤K2andρ²|∇²ϕ| ≤K₂⁰ for universal constantsK₂, K₂⁰ ∈(0,∞). We have for allx∈R³and almost everys∈(−ρ²,0),

ϕq(x, s) = − ∇²N ∗(ϕv⊗v)− ∇²N∗(ϕa⊗v)− ∇²N ∗(ϕv⊗a)

−2∇N ∗(∇ϕ·(v⊗v))−2∇N ∗(∇ϕ·(a⊗v))−2∇N ∗(∇ϕ·(v⊗a))

−N ∗(∇²ϕ: (v⊗v))−N ∗(∇²ϕ: (a⊗v))−N ∗(∇²ϕ: (v⊗a))

−2∇N ∗((∇ϕ)q)−N ∗(q∆ϕ)

=:q1,1(x.s) +q1,2(x.s) +q1,3(x.s) +q2,1(x, s) +q2,2(x, s) +q2,3(x, s) +. . . +q₄(x, s) +q₅(x, s),

where N(x) = −_4π|x|¹ . We focus on the terms which involve a. The other terms are estimated exactly as in [46]. Forq1split between a local part and a nonlocal part as follows:

for allx∈R³ and almost everys∈(−ρ²,0) q_1,1(x, s) = −

ˆ

B2r(0)

∇²N(x−y) : (a⊗v)ϕ dy− ˆ

R³\B2r(0)

∇²N(x−y) : (a⊗v)ϕ dy

=q1,1,loc(x, s) +q1,1,nonloc(x, s).

For the local part, we use Calder´on-Zygmund estimates and obtain ˆ

Qr(0,0)

|q_1,1,loc−(q_1,1,loc)_r(s)|³²dxds≤Cr ˆ

Qr(0,0)

|q_1,1,loc−(q_1,1,loc)_r(s)|¹⁵⁸ dxds

!⁴₅

≤Crkak

3 2

L⁵(Q1(0,0))kvk

3 2

L³(Q2r(0,0)),

which yields the second term in the right hand side of (32). As for the nonlocal pressure, we estimate its gradient as follows forx∈B_r(0)ands∈(−1,0)

|∇q_1,1,nonloc(x, s)| ≤C ˆ

Bρ(0)\B2r(0)

|av|

|y|⁴dy, so that

kq_1,1,nonloc−(q1,1,nonloc)r(s)k

L³²(Br(0))≤Cr³ ˆ

Bρ(0)\B2r(0)

|av|

|y|⁴dy

which gives the fourth term in the right hand side of (32) by integrating in time. It remains to see howq2,j andq3,j, j = 2,3, lead to the last term in the right hand side of (32). We have

|∇q_i,j(x, s)| ≤Cρ⁻⁴ ˆ

Bρ(0)\Bρ 2(0)

|a||v|dx

≤Cρ⁻¹³⁵ ka(·, s)k_L5(B1(0))kv(·, s)k_L3(Bρ(0)), fori, j = 2,3. Hence,

kq_i,j−(qi,j)r(s)k

3 2

L³²(Qr(0,0)) ≤Cr⁹² ˆ0

−r²

k∇q_i,j(·, s)k

3 2

L^∞(Br(0))ds

≤Cr⁹²ρ⁻³⁹¹⁰ ˆ0

−r²

ka(·, s)k

3 2

L⁵(B1(0))kv(·, s)k

3 2

L³(Bρ(0))ds

≤Cr⁴⁹¹⁰ρ⁻³⁹¹⁰kak

3 2

L⁵(Q1(0,0))kvk

3 2

L³(Qρ(0,0))

which concludes the proof.

With these two lemmas, we can now proceed with the proof of Morrey bound.

Proof of Theorem3. Step 1:(A_k)for k = 2. This is a direct consequence of assumption (23). Indeed,r2 = 2⁻²so that

1 r²₂

ˆ

Qr2(¯x,t)

|v|³dxds+ 1 r

1+δ 2

2

ˆ

Qr2(¯x,t)

|q−(q)r2(s)|³²dxds

≤16 ˆ

Q1/2(¯x,t)

|v|³dxds+ 16 ˆ

Q1/2(¯x,t)

|q|³²dxds

≤16ε∗ = 2^7−δε

1

∗3ε

2

∗3r^3−δ₁ =D₁(δ)ε

1

∗3ε

2

∗3r^3−δ₁ ≤ε

2

∗3r^3−δ₁ , by our choice ofε∗, see (27).

Step 2:(A_k) for k = 2and (21) implies(B_k)for k = 2. We take φ = φ₂ in the local energy inequality (20) and bound every term in the right hand side. Then

C₁⁻¹r⁻¹₂ sup

t−r²₂<s<t

ˆ

Br2(¯x)

|v(x, s)|²dx+C₁⁻¹r⁻¹₂ ˆ

Qr2(¯x,t)

|∇v|²dxds

≤C₁r²₂ ˆ

Q_1/2(¯x,t)

|v|²dxds+ ˆ

Q_1/2(¯x,t)

|v|³|∇φ₂|dxds+ 2 ˆ

Q_1/2(¯x,t)

|v||q||∇φ₂|dxds

+ ˆ

Q_1/2(¯x,t)

|a||v||∇v||φ₂|dxds+ ˆ

Q_1/2(¯x,t)

|a||v|(|∇v||φ₂|+|v||∇φ₂|)dxds

≤C1r²₂2⁻⁵³ε

2

∗3 + 3C12⁴ε∗+C12³kak_L5E+C12⁴⁻²³kak_L5E

≤C1(1 + 2¹⁰ε

1

∗3 + 2⁹Eε

1

∗3)ε

2

∗3r²⁻

2 3δ 2

≤C1(1 +D2(1 +E)ε

1

∗3)ε

2

∗3r²⁻

2 3δ

2 ≤2C1ε

2

∗3r²⁻

2 3δ

2 ≤CBε

2

∗3r²⁻

2 3δ

2 ,

where the last line follows from the choice ofε∗(see (26)) andC_B(see (27)).

Letn≥2.

Step 3:(A_k)and(B_k)for all2 ≤k ≤nimplies(B_k)fork= n+ 1. Let us first notice that assuming (B_k) for2≤k < nin this argument is only needed in the casea6= 0. Step 3 relies on the local energy inequality (20) and the use of the test functionφ_nconstructed in Lemma3. We have

C₁⁻¹r_n⁻¹ sup

t−r_n²<s<t

ˆ

Brn(¯x)

|v(x, s)|²dx+C₁⁻¹r⁻¹_n ˆ

Qrn(¯x,t)

|∇v|²dxds

≤C1r²_n ˆ

Q1/2(¯x,t)

|v|²dxds+ ˆ

Q1/2(¯x,t)

|v|³|∇φ_n|dxds+ 2

ˆ

Q1/2(¯x,t)

v· ∇φ_nqdxds + 2

ˆ

Q1/2(¯x,t)

|a||v||∇v||φ_n|dxds+ ˆ

Q1/2(¯x,t)

|a||v|²|∇φ_n|dxds

=I1+. . . I5.

It immediately follows from the smallness hypothesis (23) that

I1 ≤C1ε

2

∗3r²⁻

2 3δ

n ≤ CB

10C1

ε

2

∗3r²⁻

2 3δ

n .

For the other terms, one decomposesQ_1/2(¯x, t)into the union ofQ_rn(¯x, t)and the annuli Q_rk−1(¯x, t)\Q_rk(¯x, t)for2≤k≤n. The second and third terms are treated in a standard

way. We have using (A_k) for2≤k≤n, I2≤C1r_n⁻²

ˆ

Qrn(¯x,t)

|v|³dxds+C1r²_n

n

X

k=2

r⁻⁴_k

ˆ

Q_rk−1(¯x,t)\Q_rk(¯x,t)

|v|³dxds

≤C1ε

2

∗3r^3−δ_n +C1ε

2

∗3r²_n

n

X

k=3

r⁻⁴_k r^5−δ_k−1+C1r²_nr⁻⁴₂ ˆ

Q1/2(¯x,t)

|v|³dxds

≤C₁

1 + 2^2(1+δ)

1−2^−(1−δ) + 2⁸ ε

2

∗3r²⁻

2 3δ

n ≤ C_B

10C1

ε

2

∗3r²⁻

2 3δ

n .

The termI₃ requires more care. The idea is to write it as a telescoping series. Indeed, one needs to substract the mean of the pressure onQrk−1(¯x, t). In order to do this, we introduce cut-off functionsχk ∈ C_c^∞(R³×(−∞, s))fork ≥1such that0 ≤χk ≤ 1,χk ≡ 1on Q_7rk/8(¯x, t)andsuppχ_k∩Q¯₁(¯x, t)⊂Q¯_rk(¯x, t). We then have

I3= 2

n

X

k=2

ˆ

Q_rk−1

(q−(q)rk−1(s))v· ∇((χ_k−1−χk)φn)dxds

+ 2 ˆ

Qrn

(q−(q)rn(s))v· ∇(χ_nφn)dxds

≤2¹¹C1r_n²

n

X

k=2

r⁻⁴_k−1 ˆ

Q_rk−1

|q−(q)rk−1(s)||v|dxds+ 2⁶C1r⁻²_n ˆ

Qrn

|q−(q)rn(s)||v|dxds

≤2¹¹C1ε

2

∗3r²_n

n

X

k=2

r⁻

2 3δ

k−1 + 2⁶C1ε

2

∗3r²⁻

2 3δ n

≤C1

2¹¹

1−2^−23δ + 2⁶

ε

2

∗3r²⁻

2 3δ

n ≤ CB

10C1

ε

2

∗3r²⁻

2 3δ

n .

In the above calculation we have used the bounds:

k∇((χ_k−χ_k+1)φ_n)k_L∞(Qrn)≤2¹⁰C₁r_n²r_k⁻⁴ and k∇(χ_nφ_n))k_L∞(Qrn)≤17C₁r_n⁻². We refer the reader to p.295 of [46]. As forI₄andI₅we have

I4 ≤2C1r²_n

n

X

k=2

r⁻³_k

ˆ

Q_rk−1(¯x,t)\Q_rk(¯x,t)

|a||v||∇v|dxds

+ 2C1r⁻¹_n ˆ

Qrn(¯x,t)

|a||v||∇v|dxds

≤2C₁r²_nkak_L5(Q1) n

X

k=1

r⁻³_k+1 ˆ

Q_rk(¯x,t)

|v|¹⁰³ dxds

!₁₀³ ˆ

Q_rk(¯x,t)

|∇v|²dxds

!¹₂

≤2C1CBε

2 3+1

∗ r_n²

n

X

k=1

r_k+1⁻³ r³⁻

2 3δ k

≤ 2⁴

1−2^−23δC1CBε

2 3+1

∗ r²⁻

2 3δ

n =D3(δ)ε∗CBε

2

∗3r²⁻

2 3δ

n ≤ C_B

10C1

ε

2

∗3r²⁻

2 3δ n

and

I5 ≤C1r²_nkak_L5(Q1(0,0)) n

X

k=1

r⁻⁴_k+1r

2 3

k

ˆ

Q_rk(¯x,t)

|v|³dxds

!²₃

≤C₁ε¹⁺

4

∗ 9r²_n

n

X

k=1

r⁻⁴_k+1r⁴⁻

2 3δ k

≤ 2⁴ 1−2^−23δ

C₁ε¹⁺

4

∗ 9r²⁻

2 3δ

n =D₄(δ)ε

7

∗9ε

2

∗3r²⁻

2 3δ

n ≤ C_B

10C₁ε

2

∗3r²⁻

2 3δ

n .

These estimates imply (B_k) for k = n+ 1. Indeed, using the lower bound for φ_n on Qrn(¯x, t)and the fact thatQrn+1(¯x, t)⊂Qrn(¯x, t), we obtain

r⁻¹_n+1 sup

t−r²_n+1<s<t

ˆ

B_rn+1(¯x)

|v(x, s)|²dx+r⁻¹_n+1 ˆ

Q_rn+1(¯x,t)

|∇v|²dxds

≤2r_n⁻¹ sup

t−r_n²<s<t

ˆ

B_rn(¯x)

|v(x, s)|²dx+ 2r_n⁻¹ ˆ

Q_rn(¯x,t)

|∇v|²dxds

≤2C1(I1+. . . I5)

≤C_Bε

2

∗3r²⁻

2 3δ n+1 ,

which is the result.

Step 4:(Bk)for2 ≤ k≤ n+ 1implies(Ak)fork = n+ 1. First, by interpolation, we easily get that there exists a universal constantC₃∈(0,∞)such that

r_n+1⁻² ˆ

Q_rn(¯x,t)

|v|³dxds

≤C3 r_n+1⁻¹ sup

t−r²_n+1<s<t

ˆ

B_rn+1(¯x)

|v(x, s)|²dx+r⁻¹_n+1 ˆ

Q_rn+1(¯x,t)

|∇v|²dxds

!³

2

.

Therefore, by (Ak) fork=n+ 1,

r_n+1⁻² ˆ

Q_rn+1(¯x,t)

|v|³dxds≤C₃C

3 2

Bε∗r_n+1^3−δ ≤ 1 2ε

2

∗3r_n+1^3−δ.