QUANTITATIVE CONTROL OF SOLUTIONS TO THE AXISYMMETRIC NAVIER-STOKES EQUATIONS IN TERMS OF THE WEAK L3NORM W. S. O ZANSKI S. PALASEK

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QUANTITATIVE CONTROL OF SOLUTIONS TO THE AXISYMMETRIC

NAVIER-STOKES EQUATIONS IN TERMS OF THE WEAK L3NORM

W. S. O ˙

ZA ´

NSKI, S. PALASEK

Abstract. We are concerned with strong axisymmetric solutions to the 3D incompressible

Navier-Stokes equations. We show that if the weak L3norm of a strong solution uon the

time interval [0, T ] is bounded by A≫1 then for each k≥0 there exists Ck>1 such that

∥Dku(t)∥L∞(R3)≤t−(1+k)/2exp exp ACkfor all t∈(0, T ].

1. Introduction

We are concerned with the 3D incompressible Navier-Stokes equations,

(∂tu−∆u+ (u· ∇)u+∇p= 0,

div u= 0 in R3(1)

for t∈[0, T ). While the question of global well-posedness of the equations remains open,

it is well-known that the unique strong solution on a time interval [0, T ) can be continued

past Tprovided a regularity criterion holds, such as ´T

0∥curl u∥∞dt < ∞(the Beale-Kato-

Majda [3] criterion), Lipschitz continuity up to t=Tof the direction of vorticity (the

Constantin-Feﬀerman [11] criterion), or if ´T

0∥u∥q

pdt < ∞for any p∈[3,∞], q∈[2,∞]

such that 2/q + 3/p ≤1 (the Ladyzhenskaya-Prodi-Serrin condition), among many others.

The non-endpoint case q < ∞of the latter condition was settled in the 1960s [17, 41, 34],

while the endpoint case L∞

tL3

xwas only settled many years later by Escauriaza, Seregin, and

Sver´ak [12]. The main diﬃculty of the endpoint case is related to the fact that L3is a critical

space for 3D Navier–Stokes, and [12] settled it with an argument by contradiction using a

blow-up procedure and new unique continuation results. This result implies that if T0>0 is

a putative blow-up time of (1), then ∥u(t)∥3must blow-up at least along a sequence of times

tk→T−

0. While Seregin [38] showed that the L3norm must blow-up along any sequence

of times converging to T−

0, the question of quantitative control of the strong solution uin

terms of the L3norm remained open until the recent breakthrough work by Tao [44], who

showed that

|∇ju(x, t)| ≤ exp exp exp(AO(1))t−j+1

2(2)

for all t∈[0, T ], j= 0,1, x∈R3, whenever

∥u∥L∞([0,T ];L3(R3)) ≤A

for some A≫1. This result implies in particular a lower bound

lim sup

t→T−

∥u(t)∥3

(log log log(T0−t)−1))c=∞,

where c > 0 and T0>0 is the putative blow-up time, and has subsequently been improved in

some settings. For example, Barker and Prange [2] and Barker [1] provided remarkable local

arXiv:2210.10030v3 [math.AP] 12 Jul 2023

2 W. S. O ˙

ZA ´

NSKI, S. PALASEK

quantitative estimates, and the second author [31] proved that, in the case of axisymmetric

solutions,

|∇ju(x, t)| ≤ exp exp(AO(1))t−j+1

for all t∈[0, T ], j= 0,1, x∈R3, whenever



r1−3

pu



L∞([0,T ];Lp(R3)) ≤A

for some A≫1, p∈(2,3]. In another work [32] he generalized (2) to higher dimensions

(d≥4), where, due to an issue related to the lack of Leray’s intervals of regularity, one

obtains an analogue of (2) with four exponential functions. Recently Feng, He, and Wang

[13] extended (2) to the non-endpoint Lorentz spaces L3,q for q < ∞. We emphasize that

all these generalizations rely on the same stacking argument by Tao [44]. In particular, the

argument breaks down for the endpoint case q=∞.

1.1. Tao’s stacking argument and Type I blow-up. In order to illustrate the issue at

the endpoint space L3,∞, let us recall that the main strategy of Tao [44] is to show that if

uconcentrates at a particular time, then there exists a widely separated sequence of length

scales (Rk)K

k=1 and α=α(A)>0 such that ∥u∥L3({|x|∼Rk})≥αfor all k, which implies that

∥u∥3

3=ˆR3

|u|3≥X

kˆ|x|∼Rk

|u|3≥α3K. (3)

The more singularly uconcentrates at the origin, the larger one can take K; thus the L3

norm controls the regularity of u. More precisely, if ∥u∥3≤Aand uconcentrates at a large

frequency Nat time T, then one can take α= exp(−exp(AO(1))) and K∼log(NT 1

2), which,

by (3), implies that N≤T−1

2exp exp exp(AO(1)). This controls the solution in the sense that

higher frequencies do not admit concentrations, and so a simple argument [44, Section 6]

implies the conclusion (2).

Let us contrast this L3situation with that of general Lorentz spaces L3,q with interpolation

exponent q≥3. In that case, ∥u∥L3,q({|x|∼Rk})≥αimplies

∥u∥L3,q(R3)≳

∥u∥L3,q({|x|∼Rk})

ℓq

≥αK 1

and so one should expect the bounds from the stacking argument (3) used in the Lorentz

space L3,q extension [13] to degenerate as q→ ∞. Indeed, if |u(x)|=|x|−1then, for some

constant α > 0, we have ∥u∥L3,∞({|x|∼R})≥αfor all R > 0, yet ∥u∥L3,∞(R3)∼1 which shows

that the ﬁrst inequality in (3) fails for the L3,∞norm. For this reason, the approach of Tao

[44] (and, for related reasons, of Escauriaza-Seregin-ˇ

Sver´ak) to the L3problem cannot be

extended to L3,∞.

This issue is in fact closely related to the study of Type 1 blow-ups and approximately

self-similar solutions to (1). Leray famously conjectured the existence of backwards self-

similar solutions that blow up in ﬁnite time, a possibility later ruled out by Neˇcas, R˚uˇziˇcka,

and ˇ

Sver´ak [26] for ﬁnite-energy solutions and by Tsai [45] for locally-ﬁnite energy solutions.

QUANTITATIVE ESTIMATES FOR AXISYMMETRIC NSE 3

The latter reference identiﬁes the following as a very natural ansatz for blow-up:

u(t, x) = 1

(T0−t)1

U x

(T0−t)1

2!, U(y) = ay

|y|1

|y|+o1

|y|as |y|→∞,(4)

where a:S2→R3is smooth. While Tsai [45] shows that there are no solutions exactly

of this form, solutions that approximate this proﬁle or attain it in a discretely self-similar

way are promising candidates for singularity formation, as demonstrated by, for example, the

Scheﬀer constructions [27, 28, 36, 37], and the recent numerical evidence of an approximately

self-similar singularity for the axisymmetric system due to Hou [15]. Unfortunately, criteria

pertaining to L3such as those in [12, 44, 31] are less eﬀective at controlling such solutions

because |x|−1/∈L3(R3), which shows the relevance of the weak norm L3,∞.

Specializing to the case of axial symmetry, it is known, for instance, that certain critical

pointwise estimates of uwith respect to the distance from the axis imply regularity [6, 7, 33].

Moreover, Koch, Nadirashvili, Seregin, and ˇ

Sver´ak [16] proved a Liouville-type theorem for

ancient axisymmetric solutions. Furthermore, Seregin [39] proved that ﬁnite-time blow-up

cannot be of Type I. Thus, roughly speaking, no axisymmetric solution can approximate the

proﬁle (4) all the way up to a putative blow-up time T0. However, this regularity is only

qualitative (indeed, the proof uses an argument by contradiction based on a “zooming in”

procedure), and so explicit bounds on the solution have not been available.

The main purpose of this work is to make this regularity quantitative, in a similar sense

in which Tao [44] quantiﬁed the Escauriaza-Seregin-ˇ

Sver´ak theorem [12]. This allows us to

not only to rule out Type I singularies, but also to control how singular they can possibly

become. For example, it lets us estimate the length scale up to which a solution can be

approximated by a self-similar proﬁle, see Corollary 1.3 for details.

1.2. The main regularity theorem. We suppose that a strong solution to (1) on the time

interval [0, T ] is axisymmetric, namely that

∂θur=∂θu3=∂θuθ= 0,(5)

where ur, uθ, u3denote (respectively) the radial, angular, and vertical components of u, so

that

u=urer+uθeθ+u3e3

in cylindrical coordinates, where er,eθ,e3denote the cylindrical basis vectors. We assume

further that uremains bounded in L3,∞,

∥u∥L∞([0,T ];L3,∞(R3)) ≤A(6)

for some A≫1. We prove the following.

Theorem 1.1 (Main result).Suppose uis a classical axisymmetric solution of (1) on [0, T ]×

R3obeying (6). Then

∥∇ju(t)∥L∞

x(R3)≤t−1+j

2exp exp(AOj(1))

for all j≥0,t∈[0, T ].

4 W. S. O ˙

ZA ´

NSKI, S. PALASEK

We note that, although our proof of the above theorem does use some of the basic a priori

estimates (see Section 4.2) pointed out by Tao [44], it follows a completely diﬀerent scheme.

Our main ingredients are parabolic methods applied to the swirl Θ :=ruθnear the axis, as

well as localized energy estimates on

Φ:=ωr

rand Γ :=ωθ

r.(7)

In a sense, we use those estimates to replace the Carleman inequalities appearing in Tao’s

[44] approach.

To be more precise, our proof builds on the work of Chen, Fang, and Zhang [8], who

showed that the energy norm of Φ, Γ,

∥Φ∥L∞

tL2

x+∥Γ∥L∞

tL2

x+∥∇Φ∥L2

tL2

x+∥∇Γ∥L2

tL2

x,(8)

controls uvia an estimate on ∥u2

θ/r∥L2(see [8, Lemma 3.1]). They also observed that one

can indeed estimate this energy norm as long as the angular velocity uθremains small in

any neighbourhood of the axis, namely if

∥rduθ∥L∞

t([0,T ];L3/(1−d)({r≤α})) is suﬃciently small for some α > 0 and d∈(0,1).(9)

In fact, this can be observed from the PDEs satisﬁed by Φ, Γ,

∂t+u·∇−∆−2

r∂rΓ + 2

r2uθωr= 0,

∂t+u·∇−∆−2

r∂rΦ−(ωr∂r+ω3∂3)ur

r= 0,

(10)

which show that, in order to control the energy of Γ, Φ one needs to control ur/r,ωr,ω3

and uθ. However, ur/r can be controlled by Γ in the sense that

r= ∆−1∂3Γ−2∂r

r∆−2∂3Γ (11)

(see [8, p. 1929] for details), which is one of the main properties of function Γ. In particular,

(11) lets us use the Calder´on-Zygmund inequality to obtain that



D2ur

r



Lq≤ ∥∂3Γ∥Lq(12)

for q∈(1,∞) (see [8, Lemma 2.3] for details). Moreover ωr=rΦ, and ω3=∂r(ruθ)/r, which

shows that the L2estimate of Φ, Γ relies only on control of uθ. In fact, away from from the

axis, one can easily control uθ, while near the axis the smallness condition (9) is required

in an absorption argument by the dissipative part of the energy, see [8, (3.11)–(3.14)] for

details.

In this work we obtain such control of uθthanks to the weak-L3bound (6), by utilizing

parabolic theory developped by Nazarov and Ural’tseva [24] in the spirit of the Harnack

inequality. Namely, noting that the swirl Θ :=ruθsatisﬁes the autonomous PDE

∂t+u+2

rer·∇−∆Θ = 0 (13)

everywhere except for the axis, one can deduce (as observed in [24, Section 4]) H¨older con-

tinuity of Θ near the axis. A similar observation, but in a case of limited regularity of u

was used by Seregin [39] in his proof of no Type I blow-ups for axisymmetric solutions.

We quantify this approach (see Proposition 5.1 below) to obtain an estimate on the H¨older

QUANTITATIVE ESTIMATES FOR AXISYMMETRIC NSE 5

exponent in terms of the weak-L3norm, and hence we obtain suﬃcient control of the swirl

Θ in a very small neighbourhood of the axis. As for the outside of the neighbourhood, we

obtain pointwise estimates on uand all its derivatives, which are quantiﬁed with respect to

A, and which improve the second author’s estimates [31, Proposition 8]. This would enable

one to close the energy estimates for the quantities in (8) if there exist suﬃciently many

starting times where the energy norms are ﬁnite. Indeed, given a weak L3bound (6) and

short time control of the dynamics of the energy (8), control of ∥Φ(T)∥L2+∥Γ(T)∥L2can be

propagated from an initial time very close to t=T. Unfortunately, there are no times when

we can explicitly control these energies in terms of Adue to lack of quantitative decay in

the x3direction. The standard approach of propagating L2control of Φ,Γ from the initial

data at t= 0 (for instance, as in [8]) would lead to additional exponentials in Theorem 1.1.

To avoid this issue and prove eﬃcient bounds, we replace (8) with L2norms that measure

Φ and Γ uniformly-locally in x3: namely, we consider

∥Φ∥L∞

tL2

3−uloc +∥Γ∥L∞

tL2

3−uloc +∥∇Φ∥L2

tL2

3−uloc +∥∇Γ∥L2

tL2

3−uloc ,(14)

where ∥·∥L2

3−uloc := supz∈R∥·∥L2(R2×[z−1,z+1]). See Proposition 6.1 below for an estimate of

such energy norm. This approach gives rise to two further challenges.

One of them is the x3-uloc control of the solution uitself in terms of (14). We address

this diﬃculty by an x3-uloc generalization of the L4estimate on uθ/r1/2introduced by [8,

Lemma 3.1], together with a x3-uloc bootstrapping via ∥u∥L∞

tL6

3−uloc , as well as an inductive

argument for the norms ∥u∥L∞

tWk−1,6

uloc with respect to k≥1, where “uloc” refers to the

uniformly locally integrable spaces (in all variables, not only x3). We refer the reader to

Steps 2–4 in Section 7 for details.

Another challenge is an x3-uloc estimate on urin terms of Γ. To be more precise, instead

of the global estimate (12), we require L2

3−uloc control of ur/r, which is much more challeng-

ing, particularly considering the bilaplacian term in (11) above. To this end we develop a

bilaplacian Poisson-type estimate in L2

3−uloc (see Lemma 5.5), which enables us to show that



∇∂r

r



L2

3−uloc

+



∇∂3

r



L2

3−uloc

≲∥Γ∥L2

3−uloc +∥∇Γ∥L2

3−uloc ,(15)

see Lemma 5.3. Note that this is a x3-uloc generalization of (12), and also requires the

whole gradient on the right-hand side, rather than ∂3Γ only. Such an estimate lets us close

the estimate of (14), and thus control all subcritical norms of uin terms of ∥u∥L3,∞(see

Section 7 for details).

Having overcome the two diﬃculties of controlling the energy (14), we deduce (in (73)) that

∥Γ(t)∥L2

3−uloc ≤exp exp AO(1) for all t∈[1/2,1], whenever a solution usatisﬁes ∥u∥L∞([0,1];L3,∞)≤

A; see Figure 1 (supposing that T= 1). This suﬃces for iteratively improving the quantita-

tive control of uuntil t= 1. Indeed, we ﬁrst deduce a subcritical bound on the swirl-free part

of the velocity on the same time interval, namely that ∥urer+uzez∥Lp

3−uloc ≲pexp exp AO(1)

for p≥3 and t∈[1/2,1]. We can then control (in (74)) the time evolution of ∥uθr−1/2∥L4

3−uloc

over short time intervals, and so, choosing t0∈[0,1] suﬃciently close to 1 (by picking a time

of regularity, see Lemma 4.2) we then obtain (in (75)) that ∥uθr−1/2∥L4

3−uloc and ∥u∥L6

3−uloc

are bounded by exp exp AO(1) for all t∈[t0,1], see Figure 1. This subcritical bound allows

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QUANTITATIVECONTROLOFSOLUTIONSTOTHEAXISYMMETRICNAVIER-STOKESEQUATIONSINTERMSOFTHEWEAKL3NORMW.S.O˙ZA´NSKI,S.PALASEKAbstract.Weareconcernedwithstrongaxisymmetricsolutionstothe3DincompressibleNavier-Stokesequations.WeshowthatiftheweakL3normofastrongsolutionuonthetimeinterval[0,T]isboundedbyA≫1thenforea...

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QUANTITATIVE CONTROL OF SOLUTIONS TO THE AXISYMMETRIC NAVIER-STOKES EQUATIONS IN TERMS OF THE WEAK L3NORM W. S. O ZANSKI S. PALASEK

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