Quantitative Control of Solutions to the Axisymmetric Navier-Stokes Equations in Terms of the Weak $L^3$ Norm

Abstract

We are concerned with strong axisymmetric solutions to the 3D incompressible Navier-Stokes equations. We show that if the weak $L^3$ norm of a strong solution u on the time interval [0, T] is bounded by $A \gg 1$ then for each $k\ge 0$ there exists $C_k>1$ such that $\Vert D^k u (t) \Vert _{L^\infty (\mathbb {R}^3)} \le t^{-(1+k)/2}\exp \exp A^{C_k}$ for all $t\in (0,T]$.

1 Introduction

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

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\partial _t u - \Delta u + (u\cdot \nabla ) u + \nabla p =0,\\ &{}\textrm{div}\,u=0 \qquad \text { in } \mathbb {R}^3 \end{array}\right. } \end{aligned}$$

(1)

for $t\in [0,T)$. 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 T provided a regularity criterion holds, such as $\int _0^T \Vert {{\,\textrm{curl}\,}}\, u \Vert _\infty dt <\infty$ (the Beale-Kato-Majda [3] criterion), Lipschitz continuity up to $t=T$ of the direction of vorticity (the Constantin-Fefferman [11] criterion), or if $\int _0^T \Vert u \Vert _p^q dt< \infty$ for any $p\in [3,\infty ]$, $q\in [2,\infty ]$ such that $2/q+3/p\le 1$ (the Ladyzhenskaya-Prodi-Serrin condition), among many others. The non-endpoint case $q<\infty$ of the latter condition was settled in the 1960s [17, 34, 41], while the endpoint case $L^\infty _t L^3_x$ was only settled many years later by Escauriaza, Seregin, and Šverák [12]. The main difficulty of the endpoint case is related to the fact that $L^3$ is 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 $T_0>0$ is a putative blow-up time of (1), then $\Vert u(t) \Vert _3$ must blow-up at least along a sequence of times $t_k \rightarrow T_0^-$. While Seregin [38] showed that the $L^3$ norm must blow-up along any sequence of times converging to $T_0^-$, the question of quantitative control of the strong solution u in terms of the $L^3$ norm remained open until the recent breakthrough work by Tao [44], who showed that

$$\begin{aligned} | \nabla ^j u(x,t) |\le \exp \exp \exp (A^{O(1)}) t^{-\frac{j+1}{2}} \end{aligned}$$

(2)

for all $t\in [0,T]$, $j=0,1$, $x\in \mathbb {R}^3$, whenever

$$\begin{aligned} \Vert u \Vert _{L^\infty ([0,T];L^3 (\mathbb {R}^3))} \le A \end{aligned}$$

for some $A\gg 1$. This result implies in particular a lower bound

$$\begin{aligned} \limsup _{t\rightarrow T_0^-} \frac{\Vert u(t) \Vert _3 }{\left( \log \log \log (T_0-t)^{-1}) \right) ^c} = \infty , \end{aligned}$$

where $c>0$ and $T_0>0$ 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 quantitative estimates, and the second author [31] proved that, in the case of axisymmetric solutions,

$$\begin{aligned} | \nabla ^j u(x,t) |\le \exp \exp (A^{O(1)}) t^{-\frac{j+1}{2}} \end{aligned}$$

for all $t\in [0,T]$, $j=0,1$, $x\in \mathbb {R}^3$, whenever

$$\begin{aligned} \left\| r^{1-\frac{3}{p} }u \right\| _{L^\infty ([0,T];L^p (\mathbb {R}^3))} \le A \end{aligned}$$

for some $A\gg 1$, $p\in (2,3]$. In another work [32] he generalized (2) to higher dimensions ($d\ge 4$), 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 $L^{3,q}$ for $q<\infty$. 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 $q=\infty$.

1.1 Tao’s Stacking Argument and Type I Blow-Up

In order to illustrate the issue at the endpoint space $L^{3,\infty }$, let us recall that the main strategy of Tao [44] is to show that if u concentrates at a particular time, then there exists a widely separated sequence of length scales $(R_k)_{k=1}^K$ and $\alpha =\alpha (A)>0$ such that $\Vert u\Vert _{L^3(\{|x|\sim R_k\})}\ge \alpha$ for all k, which implies that

$$\begin{aligned} \Vert u \Vert _3^3 = \int _{\mathbb {R}^3} |u|^3 \ge \sum _{k} \int _{|x|\sim R_k} |u|^3 \ge \alpha ^3 K. \end{aligned}$$

(3)

The more singularly u concentrates at the origin, the larger one can take K; thus the $L^3$ norm controls the regularity of u. More precisely, if $\Vert u\Vert _3\le A$ and u concentrates at a large frequency N at time T, then one can take $\alpha =\exp (-\exp (A^{O(1)}))$ and $K\sim \log (NT^\frac{1}{2})$, which, by (3), implies that $N\le T^{-\frac{1}{2}}\exp \exp \exp (A^{O(1)})$. 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 $L^3$ situation with that of general Lorentz spaces $L^{3,q}$ with interpolation exponent $q\ge 3$. In that case, $\Vert u\Vert _{L^{3,q}(\{|x|\sim R_k\})}\ge \alpha$ implies

$$\begin{aligned} \Vert u\Vert _{L^{3,q}({\mathbb {R}}^3)} > rsim \big \Vert \Vert u\Vert _{L^{3,q}(\{|x|\sim R_k\})}\big \Vert _{\ell _k^q}\ge \alpha K^{\frac{1}{q}}, \end{aligned}$$

and so one should expect the bounds from the stacking argument (3) used in the Lorentz space $L^{3,q}$ extension [13] to degenerate as $q\rightarrow \infty$. Indeed, if $|u(x)|=|x|^{-1}$ then, for some constant $\alpha >0$, we have $\Vert u \Vert _{L^{3,\infty } (\{|x|\sim R\})} \ge \alpha$ for all $R>0$, yet $\Vert u \Vert _{L^{3,\infty } (\mathbb {R}^3)} \sim 1$ which shows that the first inequality in (3) fails for the $L^{3,\infty }$ norm. For this reason, the approach of Tao [44] (and, for related reasons, of Escauriaza-Seregin-Šverák) to the $L^3$ problem cannot be extended to $L^{3,\infty }$.

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 Nečas, Růžička, and Šverák [26] for finite-energy solutions and by Tsai [45] for locally-finite energy solutions. The latter reference identifies the following as a very natural ansatz for blow-up:

$$\begin{aligned}&u(t,x)=\frac{1}{(T_0-t)^{\frac{1}{2}}}U\left( \frac{x}{(T_0-t)^\frac{1}{2}}\right) ,\nonumber \\&\quad U(y)=a\left( \frac{y}{|y|}\right) \frac{1}{|y|}+o\left( \frac{1}{|y|}\right) \text { as }|y|\rightarrow \infty , \end{aligned}$$

(4)

where $a:S^2\rightarrow {\mathbb {R}}^3$ is 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 $L^3$ such as those in [12, 31, 44] are less effective at controlling such solutions because $|x|^{-1}\notin L^3({\mathbb {R}^3})$, which shows the relevance of the weak norm $L^{3,\infty }$.

Specializing to the case of axial symmetry, it is known, for instance, that certain critical pointwise estimates of u with respect to the distance from the axis imply regularity [6, 7, 33]. Moreover, Koch, Nadirashvili, Seregin, and Šverák [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 $T_0$. 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-Šverák 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

$$\begin{aligned} \partial _\theta u_r=\partial _\theta u_3=\partial _\theta u_\theta =0, \end{aligned}$$

(5)

where $u_r, u_\theta , u_3$ denote (respectively) the radial, angular, and vertical components of u, so that

$$\begin{aligned} u=u_r e_r + u_{\theta }e_\theta + u_3 e_3 \end{aligned}$$

in cylindrical coordinates, where $e_r$, $e_\theta$, $e_3$ denote the cylindrical basis vectors. We assume further that u remains bounded in $L^{3,\infty }$,

$$\begin{aligned} \Vert u\Vert _{L^\infty ([0,T];L^{3,\infty }({\mathbb {R}^3}))}\le A \end{aligned}$$

(6)

for some $A\gg 1$. We prove the following.

Theorem 1.1

(Main result) Suppose u is a classical axisymmetric solution of (1) on $[0,T]\times {\mathbb {R}^3}$ obeying (6). Then

$$\begin{aligned} \Vert \nabla ^ju(t)\Vert _{L_x^\infty ({\mathbb {R}^3})}\le t^{-\frac{1+j}{2}}\exp \exp (A^{O_j(1)}) \end{aligned}$$

for all $j\ge 0$, $t\in [0,T]$.

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 $\Theta :=r u_\theta$ near the axis, as well as localized energy estimates on

$$\begin{aligned} \Phi :=\frac{\omega _r}{r} \quad \text { and }\quad \Gamma :=\frac{\omega _\theta }{r}. \end{aligned}$$

(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 $\Phi$, $\Gamma$,

$$\begin{aligned} \Vert \Phi \Vert _{L^\infty _t L^2_x} + \Vert \Gamma \Vert _{L^\infty _t L^2_x} + \Vert \nabla \Phi \Vert _{L^2_t L^2_x} + \Vert \nabla \Gamma \Vert _{L^2_t L^2_x} , \end{aligned}$$

(8)

controls u via an estimate on $\Vert u_\theta ^2/r \Vert _{L^2}$ (see [8, Lemma 3.1]). They also observed that one can indeed estimate this energy norm as long as the angular velocity $u_\theta$ remains small in any neighbourhood of the axis, namely if

$$\begin{aligned} \Vert r^d u_\theta \Vert _{L^\infty _t ([0,T]; L^{3/(1-d)}(\{ r \le \alpha \} ))} \text { is sufficiently small for some }\alpha >0\text { and }d\in (0,1).\nonumber \\ \end{aligned}$$

(9)

In fact, this can be observed from the PDEs satisfied by $\Phi$, $\Gamma$,

$$\begin{aligned} \begin{aligned}&\left( \partial _t + u \cdot \nabla - \Delta - \frac{2}{r} \partial _r \right) \Gamma + \frac{2}{r^2} u_\theta \omega _r =0,\\&\left( \partial _t + u \cdot \nabla - \Delta - \frac{2}{r} \partial _r \right) \Phi - (\omega _r\partial _r + \omega _3 \partial _3 ) \frac{u_r}{r}=0, \end{aligned} \end{aligned}$$

(10)

which show that, in order to control the energy of $\Gamma$, $\Phi$ one needs to control $u_r/r$, $\omega _r$, $\omega _3$ and $u_\theta$. However, $u_r/r$ can be controlled by $\Gamma$ in the sense that

$$\begin{aligned} \frac{u_r}{r} = \Delta ^{-1} \partial _3 \Gamma - 2 \frac{\partial _r }{r} \Delta ^{-2} \partial _3 \Gamma \end{aligned}$$

(11)

(see [8, p. 1929] for details), which is one of the main properties of function $\Gamma$. In particular, (11) lets us use the Calderón-Zygmund inequality to obtain that

$$\begin{aligned} \left\| D^2 \frac{u_r}{r} \right\| _{L^q} \le \Vert \partial _3 \Gamma \Vert _{L^q} \end{aligned}$$

(12)

for $q\in (1,\infty )$ (see [8, Lemma 2.3] for details). Moreover $\omega _r = r \Phi$, and $\omega _3 = \partial _r (r u_\theta )/r$, which shows that the $L^2$ estimate of $\Phi$, $\Gamma$ relies only on control of $u_\theta$. In fact, away from from the axis, one can easily control $u_\theta$, 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 $u_\theta$ thanks to the weak-$L^3$ bound (6), by utilizing parabolic theory developped by Nazarov and Ural’tseva [24] in the spirit of the Harnack inequality. Namely, noting that the swirl $\Theta :=ru_\theta$ satisfies the autonomous PDE

$$\begin{aligned} \Big (\partial _t+\Big (u+\frac{2}{r}e_r\Big )\cdot \nabla -\Delta \Big )\Theta =0 \end{aligned}$$

(13)

everywhere except for the axis, one can deduce (as observed in [24, Section 4]) Hölder continuity of $\Theta$ 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 Hölder exponent in terms of the weak-$L^3$ norm, and hence we obtain sufficient control of the swirl $\Theta$ in a very small neighbourhood of the axis. As for the outside of the neighbourhood, we obtain pointwise estimates on u and 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 $L^3$ bound (6) and short time control of the dynamics of the energy (8), control of $\Vert \Phi (T) \Vert _{L^2} + \Vert \Gamma (T) \Vert _{L^2}$ can 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 A due to lack of quantitative decay in the $x_3$ direction. The standard approach of propagating $L^2$ control of $\Phi ,\Gamma$ 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 $L^2$ norms that measure $\Phi$ and $\Gamma$ uniformly-locally in $x_3$: namely, we consider

$$\begin{aligned} \Vert \Phi \Vert _{L^\infty _t L^2_{3-{\textrm{uloc}}}} + \Vert \Gamma \Vert _{L^\infty _t L^2_{3-{\textrm{uloc}}}} + \Vert \nabla \Phi \Vert _{L^2_t L^2_{3-{\textrm{uloc}}}} + \Vert \nabla \Gamma \Vert _{L^2_t L^2_{3-{\textrm{uloc}}}}, \end{aligned}$$

(14)

where $\Vert \cdot \Vert _{L^2_{3-{\textrm{uloc}}} }:=\sup _{z\in {\mathbb {R}}}\Vert \cdot \Vert _{L^2({\mathbb {R}}^2\times [z-1,z+1])}$. 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 $x_3$-${\textrm{uloc}}$ control of the solution u itself in terms of (14). We address this difficulty by an $x_3$-${\textrm{uloc}}$ generalization of the $L^4$ estimate on $u_\theta / r^{1/2}$ introduced by [8, Lemma 3.1], together with a $x_3$-${\textrm{uloc}}$ bootstrapping via $\Vert u \Vert _{L^\infty _t L^6_{3-{\textrm{uloc}}}}$, as well as an inductive argument for the norms $\Vert u \Vert _{L^\infty _t W_{{\textrm{uloc}}}^{k-1,6}}$ with respect to $k\ge 1$, where “${\textrm{uloc}}$” refers to the uniformly locally integrable spaces (in all variables, not only $x_3$). We refer the reader to Steps 2–4 in Section 7 for details.

Another challenge is an $x_3$-${\textrm{uloc}}$ estimate on $u_r$ in terms of $\Gamma$. To be more precise, instead of the global estimate (12), we require $L^2_{3-{\textrm{uloc}}}$ control of $u_r/r$, which is much more challenging, particularly considering the bilaplacian term in (11) above. To this end we develop a bilaplacian Poisson-type estimate in $L^2_{3-{\textrm{uloc}}}$ (see Lemma 5.5), which enables us to show that

$$\begin{aligned} \left\| \nabla \partial _r \frac{u_r}{r}\right\| _{L^2_{3-{\textrm{uloc}}}}+ \Big \Vert \nabla \partial _3 \frac{u_r}{r}\Big \Vert _{L^2_{3-{\textrm{uloc}}}}\lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}+\Vert \nabla \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}, \end{aligned}$$

(15)

see Lemma 5.3. Note that this is a $x_3$-${\textrm{uloc}}$ generalization of (12), and also requires the whole gradient on the right-hand side, rather than $\partial _3\Gamma$ only. Such an estimate lets us close the bound in (14), and thus control all subcritical norms of u in terms of $\Vert u \Vert _{L^{3,\infty }}$ (see Section 7 for details).

Having overcome the two difficulties of controlling the energy (14), we deduce (in (73)) that $\Vert \Gamma (t) \Vert _{ L^2_{3-{\textrm{uloc}}}} \le \exp \exp A^{O(1)}$ for all $t\in [1/2,1]$, whenever a solution u satisfies $\Vert u\Vert _{L^\infty ([0,1];L^{3,\infty })}\le A$; see Figure 1 (supposing that $T=1$). This suffices for iteratively improving the quantitative control of u until $t=1$. Indeed, we first deduce a subcritical bound on the swirl-free part of the velocity on the same time interval, namely that $\Vert u_r e_r +u_ze_z\Vert _{L^p_{3-{\textrm{uloc}}}} \lesssim _p \exp \exp A^{O(1)}$ for $p\ge 3$ and $t\in [1/2,1]$. We can then control (in (74)) the time evolution of $\Vert u_\theta r^{-1/2} \Vert _{L^4_{3-{\textrm{uloc}}}}$ over short time intervals, and so, choosing $t_0\in [0,1]$ sufficiently close to 1 (by picking a time of regularity, see Lemma 4.2) we then obtain (in (75)) that $\Vert u_\theta r^{-1/2} \Vert _{L^4_{3-{\textrm{uloc}}}}$ and $\Vert u \Vert _{L^6_{3-{\textrm{uloc}}}}$ are bounded by $\exp \exp A^{O(1)}$ for all $t\in [t_0,1]$, see Figure 1. This subcritical bound allows one to also estimate $\Vert u \Vert _{W^{k,6}_{{\textrm{uloc}}}}\le \exp \exp A^{C_k}$ for every k, on a time interval of the same size (see Step 4 in Section 7), which yields the claim of Theorem 1.1.

Fig. 1

A sketch of the proof of Theorem 1.1

1.3 A Comparison of the Blow-Up Rate

We note that Theorem 1.1, together with the well-known blow-up criterion $\Vert u(t) \Vert _\infty \ge c/(T_0-t)^{1/2}$ (see [30, Corollary 6.25], for example), where $T_0>0$ is a putative blow-up time, immediately implies the following lower bound on the blow-up rate of $\Vert u(t) \Vert _{L^{3,\infty }}$.

Corollary 1.2

(Blow-up rate of the weak-$L^3$ norm) If u is a classical axisymmetric solution of (1) that blows up at $T_0$, then

$$\begin{aligned} \limsup _{t\rightarrow T_0^-}\frac{\Vert u(t)\Vert _{L^{3,\infty }({\mathbb {R}^3})}}{(\log \log (T_0-t)^{-1})^c}=+\infty . \end{aligned}$$

(16)

This corollary is also a consequence of a recent theorem of Chen, Tsai, and Zhang [9], who prove¹

$$\begin{aligned} \limsup _{t\rightarrow T_0^-}\frac{\Vert b(t)\Vert _{\dot{B}^{-1}_{\infty , \infty }(\mathbb {R}^3)}}{\left( \log \log \frac{100}{T_0-t}\right) ^{\frac{1}{48}-}}=+\infty , \end{aligned}$$

where $b:=u_r e_r + u_3 e_3$ denotes the swirl-less part of the velocity field u (see [22, Section 3.3] for the relevant definition of $\dot{B}_{\infty ,\infty }^{-1}$). Thus, since $\dot{B}^{-1}_{\infty , \infty }(\mathbb {R}^3)\supset L^{3,\infty }$, the above blow-up rate implies (16). We conjecture that a variant of Theorem 1.1 holds with the weak-$L^3$ norm replaced by such a critical Besov norm and can be proved using the ideas presented here.

In order to describe the relation of Corollary 1.2 to [9], we note that the argument in [9] proceeds by proving a pointwise estimate of the form

$$\begin{aligned} | r u_\theta | \le C \exp (-c |\log r |^\tau ), \end{aligned}$$

(17)

where $c,C>0$, $\tau \in (0,1)$, for axisymmetric solutions obeying the slightly supercritical bound

$$\begin{aligned} \frac{1}{R^\frac{1}{2}}\Vert u \Vert _{L^\infty ((-R^2 ,0); L^2 (B_R) )}\le K \left( \log \log \frac{100}{R} \right) ^\beta \quad \text { for all }R\in (0,1/4] \end{aligned}$$

for some $\beta \in (0,\frac{1}{8})$ and $K>0$. This is yet another application of Harnack inequality methods to axisymmetric Navier-Stokes equations. Rather than proving Hölder continuity of $\Theta$ under a global control of a critical norm as we do in Proposition 5.1, [9] obtains (17) by an “almost Hölder continuity,”

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{Q_\rho } \Theta \le \exp \left( -c \left( \left( \log \frac{100}{\rho }\right) ^\tau - \left( \log \frac{100}{R} \right) ^\tau \right) \right) \mathop {{{\,\textrm{osc}\,}}}\limits _{Q_R} \Theta \end{aligned}$$

(18)

for $0<\rho < R\le 1/4$, $\tau \in (0,1)$; see [9, Proposition 1.2]. A similar result in the case of $\tau =1/4$ has been obtained independently by Seregin [40, Proposition 1.3]. Note that the case of $\tau =1$ corresponds to Hölder continuity.

We emphasize that the main point of our work is not to improve the blow-up rate but to give an explicit bound on u and its derivatives in terms of only the critical norm—this is a strictly stronger result in the sense that it pertains to all axisymmetric classical solutions, even those not blowing up. A naïve attempt to prove a similar quantitative theorem (e.g., using ideas of estimating axisymmetric vector fields from [21]) would lead to a bound which, compared to Theorem 1.1, would contain more iterated exponentials as well as severe dependence on the time t and subcritical norms of the initial data. Instead, Theorem 1.1 parallels the results in [44] and improves on those in [31] in the sense that the final bound depends only on $\Vert u\Vert _{L_t^\infty L_x^{3,\infty }}$ and a dimensional factor in t. This also leads to additional interesting corollaries: for instance, an explicit rate of convergence for $u(t)\rightarrow 0$ as $t\rightarrow +\infty$, and the non-existence of nontrivial ancient axisymmetric solutions in $L_t^\infty L_x^{3,\infty }$.

A comparison of these results with the work of Chen, Tsai, and Zhang [9] raises the following question: Is it possible to efficiently control (in the sense of Theorem 1.1) u and its derivatives in terms of only b measured in some critical norm? In fact, in our proof of Hölder continuity of $\Theta$ near the axis (Proposition 5.1) one can easily replace (6) with boundedness of $\Vert b(t) \Vert _{L^{3,\infty }}$ in time, since “u” in (13) can be replaced by “b”, due to axisymmetry. However, we do require $L^{3,\infty }$ control of all components of u for other quantitative estimates leading to Theorem 1.1. These include the basic estimates (Lemmas 4.2–4.4), quantitative decay away from the axis (Proposition 5.2), as well as energy estimates on $\Gamma$ and $\Phi$ (Proposition 6.1) and their implementation in the main argument (Section 7).

A related open problem is to explicitly control u in terms of $u_\theta$ only. In fact, despite a number of works [8, 18, 20, 25, 40, 46] on the properties of the swirl $ru_\theta$, its role in the regularity problem of axisymmetric solutions remains unclear.

1.4 An Estimate on the Self-similar Length Scale

One of the remarkable consequences of the quantitative estimate provided by Theorem 1.1 above is that it provides an estimate on the length scale up to which an axisymmetric solution to the NSE (1) can be approximated by a self-similar profile as in (4).

In order to make this precise, we will say that a vector field $b\in L^\infty ({\mathbb {R}^3}; {\mathbb {R}^3})$ is nearly-spherical if there exists $\delta \in (0,1/2)$ such that for every $R>0$, there exists $x_0\in {\mathbb {R}^3}$ with $|x_0|=R$ such that

$$\begin{aligned} |b(x_0)|\ge \frac{\Vert b\Vert _\infty }{2}\quad \text {and}\quad |b(x)-b(x_0)|\le \frac{\Vert b\Vert _\infty }{4}\quad \text {for all }x\in B(x_0,\delta |x_0|). \end{aligned}$$

(19)

Clearly any spherical profile $b(x)=a(x/|x|)$ is nearly-spherical for every $a\in C(\partial B(0,1))$ (in which case the choice of $\delta$ for (19) to hold can be made by a simple continuity argument). Let $\psi \in C_c^\infty ({\mathbb {R}^3}; [0,1])$ be such that $\int \psi =1$, and let $\psi _l (x):=l^{-3} \psi (x/l)$ denote a mollifier at a given length scale $l>0$. We also set $\widetilde{\psi _l} :=\psi _l *\psi _l$.

We note that, letting $R:=2l/\delta$, we can find $x_0\in {\mathbb {R}^3}$ with $|x_0|= 2l/\delta$ and satisfying (19). In particular

$$\begin{aligned} \left| \left( \widetilde{\psi _l} *\frac{b(\cdot )}{|\cdot |}\right) (x_0) \right| =\left| \int _{B(x_0,2l )}\widetilde{\psi _l}(x_0-y)\frac{b(y)}{|y|}\textrm{d}y\right| > rsim \frac{|b(x_0)| - \Vert b \Vert _\infty /4}{(1+\delta )|x_0|} \ge \frac{\delta \Vert b \Vert _\infty }{16 l}, \end{aligned}$$

which shows that

$$\begin{aligned} \left\| \widetilde{\psi _l } *\frac{b(\cdot )}{|\cdot |} \right\| _\infty \ge \frac{\delta \Vert b \Vert _\infty }{16 l} \end{aligned}$$

(20)

for every length scale $l>0$. This simple fact lets us deduce from Theorem 1.1 that, if an axisymmetric solution approximates a self-similar profile b(t, x)/|x| up to length scale l(t), where b is nearly-spherical uniformly on [0, t], then l(t) cannot be smaller than a particular quantitative threshold.

Corollary 1.3

If u is a strong axisymmetric solution u of (1) on [0, T],

$$\begin{aligned} \Big \Vert u(t )-\psi _{l(t )}*\frac{b (t,x)}{|x|}\Big \Vert _{L^{3,\infty }}\le \sigma \Vert b(t)\Vert _\infty \end{aligned}$$

(21)

for $t\in [0,T]$, and $\sigma < c \delta$, where $c>0$ is a sufficiently small constant and b(T) is nearly-spherical with constant $\delta$, then

$$\begin{aligned} l (T) > rsim \delta T^{\frac{1}{2}}\Vert b(T)\Vert _\infty \exp \left( -\exp \left( \Vert b\Vert _{L_{t,x}^\infty ([0,T]\times {\mathbb {R}^3})}^{O(1)}\right) \right) . \end{aligned}$$

Proof

We note that, at time T,

$$\begin{aligned} \Vert u\Vert _\infty& > rsim \Vert \psi _{l}*u\Vert _\infty \\&\ge \left\| \widetilde{\psi }_{l}*\frac{b(\cdot )}{|\cdot |}\right\| _\infty -\left\| \psi _{l}*\left( u-\psi _l*\frac{b(\cdot )}{|\cdot |}\right) \right\| _\infty \\&\ge \frac{\delta \Vert b \Vert _\infty }{16l } - C l^{-1}\left\| u-\psi _l*\frac{b(\cdot )}{|\cdot |} \right\| _{L^{3,\infty }}\\&\ge \left( \frac{\delta }{16} - C\, \sigma \right) \frac{ \left\| b \right\| _\infty }{l}. \end{aligned}$$

Thus $\Vert u (T)\Vert _\infty \ge \delta \Vert b(T) \Vert _\infty / 32l$ if $\sigma \in (0,\delta /32C )$. Since also

$$\begin{aligned} \Vert u (t)\Vert _{L^{3,\infty } } \le \left\| \widetilde{\psi }_{l(t)}*\frac{b(t, \cdot )}{|\cdot |}\right\| _{L^{3,\infty } }+\left\| u(t)-\psi _{l(t)}*\frac{b(t,\cdot )}{|\cdot |}\right\| _{L^{3,\infty } } \le C \Vert b(t,\cdot ) \Vert _{\infty } \end{aligned}$$

for all $t\in [0,T]$, Theorem 1.1 implies that

$$\begin{aligned} \frac{\delta \Vert b (T) \Vert _\infty }{32 \,l(T)} \le \Vert u (T) \Vert _\infty \lesssim T^{-1/2} \exp \exp \left( \Vert b \Vert _{L^\infty ([0,T]\times {\mathbb {R}^3})}^{O(1)}\right) , \end{aligned}$$

from which the claim follows. $\square$

1.5 Organization of the Paper

The structure of the paper is as follows. In the following Section 2 we discuss preliminary concepts related to the Lorentz spaces $L^{p,q}$, the Bogovskiĭ operator, a simple Poisson-type tail estimate that we will later (in Section 5.3) expand to obtain our Poisson-type estimate (15) above, as well as some properties of cylindrical coordinates. In Section 3 we discuss some properties of axisymmetric functions, including an axisymmetric Bernstein inequality (Section 3.1) and a quantified version of Hardy’s inequality (Section 3.2). In Section 4 we present some quantitative estimates of the 3D Navier–Stokes equations, including the Picard iterates (Section 4.1), times of regularity, bounded total speed, and second derivatives estimates (Section 4.2), all of which remain valid without the assumption of axisymmetry. The following section, Section 5, is dedicated to quantitative estimates that are specific to the axisymmetric setting (5) of the equations (1). These include the statement of the Hölder estimate of the swirl $\Theta$ mentioned above (Section 5.1), pointwise estimates away from the axis (Section 5.2), as well as the Poisson-type $x_3$-${\textrm{uloc}}$ estimate on $u_r/r$ (15) (Section 5.3). In Section 6 we prove the energy estimate (14) for $\Gamma$ and $\Phi$ mentioned above, and Section 7 combines the developed methods to prove the main theorem, Theorem 1.1. Finally, Appendix A includes a detailed verification of the Hölder estimate of $\Theta$.

2 Preliminaries

Given $f:\Omega \rightarrow \mathbb {R}$ we let

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{\Omega }\, f :=\sup _{\Omega } f - \inf _{\Omega } f \end{aligned}$$

denote the oscillation of f over $\Omega$. We also denote by the average over $\Omega$.

We use standard definitions of Lebesgue spaces $L^p (\Omega )$, Sobolev spaces $W^{k,p}(\Omega )$, spaces of continuous functions $C(\Omega )$, spaces $C_c(\Omega )$ of continuous functions with compact support. For brevity of notation we often omit “$\Omega$” in the notation if $\Omega = \mathbb {R}^3$; for example $W^{1,\infty } \equiv W^{1,\infty } (\mathbb {R}^3)$. We use the convention $\Vert \cdot \Vert _p :=\Vert \cdot \Vert _{L^p (\mathbb {R}^3 )}$, and we reserve the notation $\Vert \cdot \Vert :=\Vert \cdot \Vert _{2}$ for the $L^2(\mathbb {R}^3)$ norm. We also write $\int :=\int _{\mathbb {R}^3}$. Given $p\in [1,\infty ]$, we define the uniformly local $L^p$ norms,

$$\begin{aligned} \Vert u\Vert _{L_{{\textrm{uloc}}}^p}:=\sup _{x\in {\mathbb {R}^3}}\Vert u\Vert _{L_x^p(B(x,1))}\quad \text { and }\quad \Vert u\Vert _{L_{t,x-{\textrm{uloc}}}^p}:=\big \Vert \Vert u\Vert _{L_{{\textrm{uloc}}}^p}\big \Vert _{L_t^p}, \end{aligned}$$

(22)

as well as the norms that are uniformly local in $x_3$ only,

$$\begin{aligned} \Vert f\Vert _{L^p_{3-{\textrm{uloc}}} ({\mathbb {R}^3})}:=\sup _{z\in {\mathbb {R}}}\Vert f\Vert _{L^p({\mathbb {R}}^2\times [z-1,z+1])}. \end{aligned}$$

(23)

We let $\Psi (x,t) :=(4\pi t)^{-3/2} \textrm{e}^{-x^2/4t}$ denote the heat kernel, which satisfies

$$\begin{aligned} \Vert \nabla ^k \Psi (t) \Vert _p = C_{k,p} t^{-\frac{3}{2} \left( 1-\frac{1}{p} \right) -\frac{k}{2} }. \end{aligned}$$

(24)

We often use the notation $\textrm{e}^{t\Delta }f :=\Psi (t) * f$.

Given $N\in \{ 2^k :k\in \mathbb {N}\}$ we let $P_N$ denote the N-th Littlewood-Paley projection. We recall a localized version of the Bernstein inequality

$$\begin{aligned} \Vert P_N f \Vert _{L^q (\Omega ) } \lesssim _k N^{\frac{3}{p_1}-\frac{3}{q} } \Vert P_N f \Vert _{L^{p_1} (\Omega _{R})} + (RN)^{-k} N^{\frac{3}{p_2}-\frac{3}{q} } \Vert P_N f \Vert _{L^{p_2}}, \end{aligned}$$

(25)

where $\Omega \subset \mathbb {R}^3$ is an open set, $k\ge 1$, $\Omega _{R } :=\{ x\in \mathbb {R}^3 :\textrm{dist}(x,\Omega ) < R \}$, $q\in [1,\infty ]$ and $p_1,p_2 \in [1,q]$; see [44, Lemma 2.1] for a proof.

2.1 Lorentz Spaces

We recall the Lorentz spaces, defined by

$$\begin{aligned} \Vert f \Vert _{L^{p,q}} :=p^{1/q} \Vert \lambda | \{ |f |\ge \lambda \} |^{1/p} \Vert _{L^q (\mathbb {R}_+ , \frac{\textrm{d}\lambda }{\lambda } )} \end{aligned}$$

(26)

for $q<\infty$ and

$$\begin{aligned} \Vert f \Vert _{L^{p,\infty }} :=\Vert \lambda | \{ |f |\ge \lambda \} |^{1/p} \Vert _{L^\infty (\mathbb {R}_+ , \frac{\textrm{d}\lambda }{\lambda } )}. \end{aligned}$$

We recall the Hölder inequality for Lorentz spaces,

$$\begin{aligned} \Vert fg \Vert _{L^{p,q}} \le C_{p_1,p_2,q_1,q_2} \Vert f \Vert _{L^{p_1,q_1}} \Vert g \Vert _{L^{p_2,q_2}}, \end{aligned}$$

(27)

whenever $1/p=1/p_1+1/p_2$, $1/q=1/q_1+1/q_2$, $p_1,p_2,p\in (0,\infty )$, $q_1,q_2,q\in (0,\infty ]$. We refer the reader to [43, Theorem 6.9] for a proof of (27). The Hölder inequality can be very useful when estimating some localized integrals in terms of the $L^{p,\infty }$ norm. For example, if $\phi \in C_0^\infty (\Omega )$ is a smooth cutoff function then we have the simple estimate

$$\begin{aligned} \Vert \phi \Vert _{L^{p,1}} = p\int _0^\infty | \{ |\phi |\ge \lambda \} |^{1/p} \textrm{d}\lambda \le p\int _0^{\Vert \phi \Vert _\infty } | \{ |\phi |\ge \lambda \} |^{1/p} \textrm{d}\lambda \le p |\Omega |^{1/p} \Vert \phi \Vert _\infty , \end{aligned}$$

which shows that, for example

$$\begin{aligned} \int _{\Omega } fg \le \Vert f \Vert _{L^{3,\infty }} \Vert g \Vert _2 |\Omega |^{1/6}. \end{aligned}$$

This simple method allows us to use the weak $L^3$ space to estimate some integrals over a region close to the axis of symmetry.

We also note two Young’s inequalities involving weak $L^p$ spaces

$$\begin{aligned}{} & {} \Vert f*g \Vert _{L^{p,\infty }} \lesssim \Vert f \Vert _1 \Vert g \Vert _{L^{p,\infty }}\qquad \text { for } p\in (1,\infty ), \end{aligned}$$

(28)

$$\begin{aligned}{} & {} \Vert f*g \Vert _{p} \lesssim \Vert f \Vert _r \Vert g \Vert _{L^{q,\infty }} \quad \text { for } p,q,r\in (1,\infty ) \text { with } \frac{1}{p} +1 = \frac{1}{q}+\frac{1}{r} , \end{aligned}$$

(29)

see [22, Proposition 2.4(a)] and [35, Theorem A.16] for details (respectively).

2.2 The Bogovskiĭ Operator

We recall that, given $p\in (1,\infty )$, an open ball $B\subset \mathbb {R}^3$, $b\in W^{1,p}(B)$ such that $\textrm{div}\, b =0$, and $\phi \in C_0^\infty (B; [0,1])$ such that $\phi =1$ on B/2 there exists $\overline{b} \in W^{1,p} (\mathbb {R}^3)$ such that $\overline{b} =0$ outside B and inside B/2,

$$\begin{aligned} \textrm{div}\, \overline{b} = \textrm{div}(\phi b) \quad \text { and } \quad \Vert \overline{b } \Vert _{W^{1,p}} \lesssim _B \Vert b \Vert _{W^{1,p}(B)}, \end{aligned}$$

(30)

due to the Bogovskiĭ lemma (see [4, 5] or [14, Lemma III.3.1], for example). Here we use the non-homogeneous $W^{1,p}$ norm and so the implicit constant in (30) may depend of the size of B. We note that the Bogovskiĭ lemma often assumes that the domain is star-shaped (which is not the case for $B\setminus B/2$), but it can be overcome in this particular setting by applying the partition of identity to $\phi$; see [29, Section 2.3] for example.

2.3 A Poisson-Type Tail Estimate

Here we are concerned with a Poisson equation of the form $-\Delta f = D^2 g$, and we show that any $W^{k,\infty } (B(0,1))$ norm of $\nabla f$ can be bounded by the $L^1_{{\textrm{uloc}}}$ norm of g, if $g=0$ on B(0, 2).

To be more precise, we let $\psi \in C_c^\infty (B(0,1);[0,1])$ be such that $\psi =1$ on B(0, 1/2). Given $y\in \mathbb {R}^3$ we set

$$\begin{aligned} \psi _y (x) :=\psi (x-y). \end{aligned}$$

(31)

and

$$\begin{aligned} \widetilde{\psi } :=\sum _{\begin{array}{c} j\in \mathbb {Z}^3 \\ |j|\le 10 \end{array} } \psi _j. \end{aligned}$$

Lemma 2.1

Suppose that $f = D^2 (-\Delta )^{-1} (g (1-\widetilde{ \psi } ))$ for some $g\in L^2$. Then

$$\begin{aligned} \Vert \psi \nabla f \Vert _{W^{k,\infty }} \lesssim _k \Vert g \Vert _{L^1_{{\textrm{uloc}}}} \qquad \text { for } k\ge 0. \end{aligned}$$

Proof

We note that

$$\begin{aligned} \partial _i f (x) = \int \frac{(x_i-y_i)g(y)(1-{\tilde{\phi }}(y))}{|x-y|^5} \textrm{d}y \end{aligned}$$

for $x\in {{\,\textrm{supp}\,}}\,\phi$, and so

$$\begin{aligned} \begin{aligned} |\nabla f (x) |&\le \int _{\{ |x-y | \ge 5 \} }\frac{|g(y) |}{|x-y|^4}\textrm{d}y \\&\le \sum _{\begin{array}{c} j\in \mathbb {Z}^3\\ |j|\ge 2 \end{array}} \int _{x_1+j_1}^{x_1+j_1+1} \int _{x_2+j_2}^{x_2+j_2+1} \int _{x_3+j_3}^{x_3+j_3+1} \frac{|g(y)|}{|x-y|^4} \textrm{d}y_3\,\textrm{d}y_2 \, \textrm{d}y_1\\&\lesssim \Vert g \Vert _{L^1_{{\textrm{uloc}}} }\sum _{\begin{array}{c} j\in \mathbb {Z}^3\\ |j|\ge 2 \end{array}} |j|^{-4} \lesssim \Vert g \Vert _{L^1_{{\textrm{uloc}}} }, \end{aligned} \end{aligned}$$

as required. An analogous argument applies to higher derivatives of f. $\square$

The above proof demonstrates a simple method of tail estimation which we will later use to obtain a $L^2_{3-{\textrm{uloc}}}$ estimate of $u_r/r$ in terms of $\Gamma$, mentioned in the introduction (recall (15)). In fact, to this end, a similar strategy can be applied in the $x_3$ direction only, and can be extended to the more challenging biLaplacian Poisson equation (see Lemma 5.5 below).

2.4 Cylindrical Coordinates

Given $x\in \mathbb {R}^3$ we denote by $x':=(x_1,x_2)$ the horizontal variables, and $r:=(x_1^2 + x_2^2)^{1/2}$ denotes the radius in the cylindrical coordinates. We often use the notation

$$\begin{aligned} \{ r< r_0 \} :=\{ x\in \mathbb {R}^3 :r<r_0 \} \end{aligned}$$

for a given $r_0>0$.

We recall a version of the Hardy inequality

$$\begin{aligned} \Vert r^{-1} f \Vert _{L^{q} (\Omega )} \lesssim C(\Omega ) \Vert f \Vert _{L^{q} (\Omega )} +\Vert \nabla f \Vert _{L^{q} (\Omega )}, \end{aligned}$$

(32)

where $\Omega$ is a bounded domain and $q\in (1,2]$; see [8, Lemma 2.4] for a proof.

We recall the divergence operator in cylindrical coordinates: if $v= v_r e_r + v_\theta e_\theta + v_3 e_3$ then

$$\begin{aligned} \textrm{div}\, v = \frac{1}{r} \partial _r (rv_r )+\frac{1}{r} \partial _\theta v_\theta + \partial _3 v_3. \end{aligned}$$

(33)

We say that a vector field v is axisymmetric if (5) holds. In such case we have

$$\begin{aligned} |\nabla 'v|^2=(\partial _r v_r)^2+(\partial _rv_\theta )^2+(\partial _rv_3)^2+\frac{1}{r^2}(v_r^2+v_\theta ^2), \end{aligned}$$

(34)

which implies the pointwise bounds

$$\begin{aligned} \frac{|v_r|}{r},\,\frac{|v_\theta |}{r}\le |\nabla 'v|. \end{aligned}$$

Here $\nabla '$ refers to the gradient with respect to the horizontal variables $x'$ only.

Moreover,

$$\begin{aligned} |\partial _{rr} f | \lesssim | D^2 f|. \end{aligned}$$

(35)

Indeed, since

$$\begin{aligned} \partial _r = \cos \theta \, \partial _1 +\sin \theta \, \partial _2 = \frac{x_1}{|x'|} \partial _1 + \frac{x_2}{|x'| } \partial _2 , \end{aligned}$$

where $x' :=(x_1,x_2)$ refers to the horizontal variables, we can compute that

$$\begin{aligned} \begin{aligned} \partial _{rr}&= \frac{x_1^2}{|x'|^2} \partial _{11} + 2 \frac{x_1 x_2}{|x'|^2 } \partial _1 \partial _2 + \frac{x_2^2}{|x'|^2 } \partial _{22} , \end{aligned} \end{aligned}$$

from which (35) follows. More generally,

$$\begin{aligned} \begin{aligned} \partial _{rrr}&= \frac{x_1^3}{|x'|^3} \partial _{111} + \frac{3x_1^2 x_2}{|x'|^3} \partial _{11}\partial _2 + \frac{3x_1x_2^2}{|x'|^3 } \partial _1 \partial _{22} + \frac{x_2^3}{|x'|^3} \partial _{222} ,\\ \partial _{rrrr}&= \frac{x_1^4}{|x'|^4} \partial _{1111} + \frac{4x_1^3 x_2}{|x'|^4} \partial _{111}\partial _2 + \frac{6x_1^2x_2^2}{|x'|^4}\partial _{11}\partial _{22} + \frac{4x_1x_2^3}{|x'|^4 } \partial _1 \partial _{222} + \frac{x_2^4}{|x'|^4} \partial _{2222}. \end{aligned} \end{aligned}$$

This shows that

$$\begin{aligned} | D^3_{r,x_3} f | \lesssim | D^3 f|\qquad \text { and } \qquad | D^4_{r,x_3} f | \lesssim | D^4 f| \end{aligned}$$

(36)

for any axisymmetric f (here, for example, $D^4$ refers to all fourth order derivatives with respect to $x_1,x_2,x_3$).

3 Properties of Axisymmetric Functions

Here we discuss some properties of axisymmetric functions, including an axisymmetric Bernstein inequality and a quantified Hardy’s inequality.

3.1 Bernstein Inequalities

Here we discuss a version of the axisymmetric Bernstein inequality provided by [31, Proposition 1] that involves the weak $L^3$ space.

Lemma 3.1

Let $T_m$ be a Fourier multiplier whose symbol m is supported on B(0, N) with $|\nabla ^jm|\le MN^{-j}$ and $1<q< p\le \infty$. If either $-\frac{2}{p}<\alpha <\frac{1}{q}-\frac{1}{p}$ or $p=\infty$ and $\alpha =0$, we have

$$\begin{aligned} \Vert r^\alpha T_mu\Vert _{L^p}\lesssim MN^{\frac{3}{q}-\frac{3}{p}-\alpha }\Vert u\Vert _{L^{q,\infty }} \end{aligned}$$

for all axisymmetric scalar- or vector-valued functions u.

Proof

We normalize $M=N=1$. Under these assumptions on $p,\alpha$, Proposition 1 in [31] implies

$$\begin{aligned} \Vert r^\alpha T_mu\Vert _{L^p}\lesssim \Vert P_{\le 10}u\Vert _{L^{q+\epsilon }} \end{aligned}$$

for $T_mP_{\le 10}=T_m$, since an $\epsilon >0$ sufficiently small depending on $p,q,\alpha$. Let $\psi$ be the kernel such that $P_{\le 10}=\psi *$. Then by the weak Young inequality (29),

$$\begin{aligned} \Vert P_{\le 10}u\Vert _{L^{q+\epsilon }}\lesssim \Vert \psi \Vert _{L^{1+O(\epsilon )}}\Vert u\Vert _{L^{q,\infty }}\lesssim \Vert u\Vert _{L^{q,\infty }}. \end{aligned}$$

$\square$

A useful consequence of the above lemma is the following heat kernel estimate

$$\begin{aligned} \Vert r^\alpha \textrm{e}^{\Delta } \nabla ^jf\Vert _{L^p}&\le \Vert r^\alpha \textrm{e}^{\Delta }\nabla ^jP_{\le 1}f\Vert _{L^p}+\sum _{N>1}\Vert r^\alpha \textrm{e}^{\Delta }\nabla ^jP_Nf\Vert _{L^p}\nonumber \\&\lesssim _{\alpha , p ,q,j}\Vert f\Vert _{L^{q,\infty }}(1+\sum _{N>1}\textrm{e}^{-N^2/100}N^{j+\frac{3}{q}-\frac{3}{p}})\nonumber \\&\lesssim _{ p ,q,j} \Vert f\Vert _{L^{q,\infty }} \end{aligned}$$

(37)

under the same assumptions on the parameters as in Lemma 3.1.

3.2 A Quantified Version of the Hardy Inequality

By the classical Hardy inequality

$$\begin{aligned} \Vert r^{-\frac{3}{p}+\frac{1}{2}}f \Vert _p \lesssim _{p} \left( \Vert f\Vert _{2}+\Vert \nabla f\Vert _{2}\right) \end{aligned}$$

for any axisymmetric f, and $p\in (2,6)$ (see [8, Lemma 2.6], for example). Here we prove a version of this inequality, which is localized in the horizontal variables, “uloc” in $x_3$, and which has a quantified divergence of the constant near $p=2$. Namely we prove the following.

Lemma 3.2

(Quantified Hardy inequality) For $p\in (2,6-\epsilon )$,

$$\begin{aligned} \Vert r^{-\frac{3}{p}+\frac{1}{2}}f\Vert _{L_{3-{\textrm{uloc}}}^p(\{r\le 1\})}&\lesssim _\epsilon (p-2)^{-O(1)}\left( \Vert f\Vert _{L_{3-{\textrm{uloc}}}^2(\{r\le 1\})}+\Vert \nabla f\Vert _{L_{3-{\textrm{uloc}}}^2(\{r\le 1\})}\right) . \end{aligned}$$

Proof

From the Sobolev embedding

$$\begin{aligned} \Vert u\Vert _{L^{2p/(2-p)}({\mathbb {R}}^2)}&\lesssim (2-p)^{-O(1)}\Vert \nabla u\Vert _{L^p({\mathbb {R}}^2)} \end{aligned}$$

for $p<2$, (see, e.g., [42] where the sharp constant is computed), one can prove the two-dimensional Gagliardo-Nirenberg inequality

$$\begin{aligned} \Vert f\Vert _{L^q(B(1))}&\lesssim q\left( \Vert f\Vert _{L^6(B(1))}^{\frac{6}{q}}\Vert \nabla f\Vert _{L^2(B(1))}^{1-\frac{6}{q}}+\Vert f\Vert _{L^p(B(1))}\right) \end{aligned}$$

(38)

for $q>6$. Fix $\epsilon >0$ to be specified. Then

$$\begin{aligned} \Big \Vert \frac{f(\cdot , x_3)}{r^{\frac{3}{q}-\frac{1}{2}}}\Big \Vert _{L_{x'}^q(r\ge \epsilon )}&\le \Vert r^{-\frac{3}{q}+\frac{1}{2}}\Vert _{L_{x'}^{6q/(6-q)}(\{r\ge \epsilon \})}\Vert f(\cdot , x_3)\Vert _{L_{x'}^6({\mathbb {R}}^2)}\\&\lesssim \epsilon ^{-\frac{1}{q}+\frac{1}{6}}\Vert f(\cdot , x_3)\Vert _{L_{x'}^6({\mathbb {R}}^2)}. \end{aligned}$$

Inside, for any $\frac{1}{s}\in (\frac{3}{2p}-\frac{1}{4},\frac{1}{p})$, by (38),

$$\begin{aligned} \Big \Vert \frac{f(\cdot , x_3)}{r^{\frac{3}{p}-\frac{1}{2}}}\Big \Vert _{L_{x'}^p(r\le \min (1,\epsilon ))}&\le \Vert r^{-\frac{3}{p}+\frac{1}{2}}\Vert _{L_{x'}^s(r<\min (1,\epsilon ))}\Vert f(\cdot , x_3)\Vert _{L_{x'}^{ps/(s-p)}(B(1))}\\&\lesssim \Big (\frac{1}{s}-\frac{3}{2p}+\frac{1}{4}\Big )^{-\frac{1}{s}}\Big (\frac{1}{p}-\frac{1}{s}\Big )^{-1}\\&\quad \times \left( \epsilon ^{-\frac{3}{p}+\frac{1}{2}+\frac{2}{s}}\Vert f(\cdot , x_3)\Vert _{L_{x'}^6(B(1))}^{\frac{6}{p}-\frac{6}{s}}\Vert \nabla f(\cdot , x_3)\Vert _{L_{x'}^2(B(1))}^{1-\frac{6}{p}+\frac{6}{s}}\right. \\&\quad \left. +\Vert f(\cdot , x_3)\Vert _{L_{x'}^p(B(1))}\right) . \end{aligned}$$

Upon taking $\epsilon =\Vert f\Vert _6^3/\Vert \nabla f\Vert _2^3$ and $\frac{1}{s}=\frac{4}{3p}-\frac{1}{6}$,

$$\begin{aligned} \Big \Vert \frac{f(\cdot , x_3)}{r^{\frac{3}{p}-\frac{1}{2}}}\Big \Vert _{L_{x'}^p(B(1))}&\lesssim (p-2)^{-O(1)}\left( \Vert f(\cdot , x_3)\Vert _{L_{x'}^6(B(1))}^{\frac{3}{2}-\frac{3}{p}}\Vert \nabla f(\cdot , x_3)\Vert _{L_{x'}^2(B(1))}^{-\frac{1}{2}+\frac{3}{p}}\right. \\&\quad \left. +\Vert f(\cdot , x_3)\Vert _{L_{x'}^p(B(1))}\right) . \end{aligned}$$

Finally by Hölder’s inequality, Sobolev embedding, and Gagliardo-Nirenberg interpolation, we find

$$\begin{aligned} \Big \Vert \frac{f}{r^{\frac{3}{p}-\frac{1}{2}}}\Big \Vert _{L_x^p(B_{{\mathbb {R}}^2}(1)\times B_{{\mathbb {R}}}(z,1))}&\lesssim (p-2)^{-O(1)}\Vert f\Vert _{H_x^1(B_{{\mathbb {R}}^2}(1)\times B_{{\mathbb {R}}}(z,1))}, \end{aligned}$$

as required. $\square$

4 Basic Estimates for the Navier-Stokes Solutions

Here we discuss some estimates for the Navier-Stokes equations without the assumption of axisymmetry.

4.1 The Picard Estimates

We define the flat and sharp Picard iterates

$$\begin{aligned} u^\flat _n(t):=\textrm{e}^{(t-t_n)\Delta }u(t_n)-\int _{t_n}^t \textrm{e}^{(t-t')\Delta }{\mathbb {P}}\textrm{div}( u^\flat _{n-1}\otimes u^\flat _{n-1}(t'))\textrm{d}t',\quad u^\sharp _n :=u-u^\flat _n\nonumber \\ \end{aligned}$$

(39)

for all $n=1,2,\ldots$ and $t\ge t_n$, where $t_n\in [0,\frac{1}{2})$ is an increasing sequence of times, and $u^\flat _0:=0$, $u^\sharp _0:=u$. We have the following.

Lemma 4.1

(Basic Picard estimates) Assume u solves (1) on $[0,1]\times {\mathbb {R}^3}$ with the bound (6). If $p\in (3, \infty ]$ and $-\frac{2}{p}<\alpha <\frac{1}{3}-\frac{1}{p}$ or $p=\infty$ and $\alpha =0$, we have

$$\begin{aligned} \Vert r^\alpha \nabla ^ju^\flat _n\Vert _{L_t^\infty L_x^p([\frac{1}{2},1]\times {\mathbb {R}^3})}&\le A^{O_{n,j,p}(1)}, \end{aligned}$$

(40)

$$\begin{aligned} \Vert u^\sharp _n\Vert _{L_{t}^\infty L_x^q([\frac{1}{2},1]\times {\mathbb {R}^3})}&\le A^{O_{n,q}(1)} \qquad \text { for all }q\in (1,3), \end{aligned}$$

(41)

$$\begin{aligned} \Vert \nabla ^jP_Nu^\flat _n\Vert _{L_{t,x}^\infty ([\frac{1}{2},1]\times {\mathbb {R}^3})}&\le \textrm{e}^{-N^2/O_{n,j}(1)} A^{O_{n,j}(1)}, \end{aligned}$$

(42)

as well as the energy estimate

$$\begin{aligned} \Vert \nabla u^\sharp _n\Vert _{L_{t,x}^2([\frac{1}{2},1]\times {\mathbb {R}^3})}\le A^{O_n(1)}. \end{aligned}$$

(43)

In particular,

$$\begin{aligned} \Vert \nabla u\Vert _{L_{t,x-{\textrm{uloc}}}^2([\frac{1}{2},1]\times {\mathbb {R}^3})}\le A^{O(1)}. \end{aligned}$$

(44)

The proof of (40)–(42) above relies only on the definition (39) as well as basic heat estimates (24), which, together with the weak Young’s inequality (29), can be used in the same way as [44, (3.11)–(3.13)] and [32, Proposition 2.5] to obtain the estimates with $\Vert u \Vert _{L^\infty ([0,1];L^{3,\infty })}\le A$ on the right-hand side.

4.2 Basic Estimates

Here we assume that u satisfies (1) with the weak $L^{3,\infty }$ bound (6) on the time interval [0, T].

Lemma 4.2

(Choice of time of regularity) If u solves (1) on a time interval I and satisfies $\Vert u\Vert _{L_t^\infty L_x^{3,\infty }(I\times {\mathbb {R}^3})}\le A$, then there exists $t_*\in I$ such that

$$\begin{aligned} \Vert \nabla ^ju(t_*)\Vert _{L_{x}^\infty ({\mathbb {R}^3})}&\le |I|^{-\frac{1+j}{2}}A^{O(1)} \end{aligned}$$

for all $j=0,1,2,\ldots ,10$.

Lemma 4.3

(Bounded total speed) We have the bounded total speed estimate

$$\begin{aligned} \Vert u\Vert _{L_t^1L_x^\infty (I/2\times {\mathbb {R}^3})}\le |I|^{\frac{1}{2}}A^{O(1)}. \end{aligned}$$

The two lemmas above follow by the same arguments in [44, Lemma 3.1] and [13, Propositions 3.1–2] using the estimates in Lemma 4.1. In particular, it is straightforward to check that the proofs of Propositions 3.1 and 3.2 in [13] are still valid in Lorentz spaces $L^{p,q}$ with $q =\infty$. Furthermore, we estimate $\nabla ^2 u$ in terms of A.

Lemma 4.4

(2nd order derivatives estimates) If u solves (1) on [0, T] and obeys (6), then

$$\begin{aligned} \Vert \nabla ^2u\Vert _{L_{t,x-{\textrm{uloc}}}^p([\frac{T}{2},T]\times {\mathbb {R}^3})}\lesssim _p A^{O(1)}T^{\frac{5}{2p}-\frac{3}{2}} \end{aligned}$$

for $p\in [1,\frac{4}{3})$, where the “${\textrm{uloc}}$” norm is considered as the supremum of the $L^p$ norms over $B(T^{1/2})\subset \mathbb {R}^3$ (instead of B(1), recall (22)).

Proof

We use an approach due to Constantin [10]. First rescale to make $T=1$. For every $\epsilon \in (0,\frac{1}{2})$, we define the approximation to the function $\langle x\rangle :=(1+|x|^2)^\frac{1}{2}$,

$$\begin{aligned} q_\epsilon (x):=\langle x\rangle -\frac{1}{2(1-\epsilon )}\langle x\rangle ^{1-\epsilon } \end{aligned}$$

which satisfies the properties

$$\begin{aligned} |\nabla q_\epsilon |&\le 1, \end{aligned}$$

(45)

$$\begin{aligned} \xi ^T\nabla ^2q_\epsilon (x)\xi&>\frac{\epsilon }{2}\langle x\rangle ^{-(1+\epsilon )}|\xi |^2, \end{aligned}$$

(46)

$$\begin{aligned} \frac{1-2\epsilon }{2-2\epsilon }\langle x\rangle&\le q_\epsilon (x)\le \langle x\rangle . \end{aligned}$$

(47)

With $\tau$ a time scale to be specified, we define $w:=q_\epsilon (\tau \omega )$ which obeys the equation

$$\begin{aligned} (\partial _t+u\cdot \nabla -\Delta )w=\tau \nabla q_\epsilon (\tau \omega )\cdot (\omega \cdot \nabla u)-\tau ^2{{\,\textrm{tr}\,}}(\nabla \omega ^T\nabla ^2q_\epsilon \nabla \omega ). \end{aligned}$$

Recall that $\omega :=\textrm{curl}\,u$ denotes the vorticity vector. Multiplying by a spatial cutoff at length scale R and integrating over ${\mathbb {R}}^d$,

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int _{\mathbb {R}^3}w\psi \le \int _{\mathbb {R}^3}(u\cdot \nabla \psi +\Delta \psi )w+O(\tau |\nabla u|^2)\psi -\frac{\epsilon }{2}\tau ^2\langle \tau \omega \rangle ^{-(1+\epsilon )}|\nabla \omega |^2\psi . \end{aligned}$$

Let ${\tilde{\psi }}$ be an enlarged cutoff function so that $R|\nabla \psi |+R^2|\Delta \psi |\le 10{\tilde{\psi }}$. We set

$$\begin{aligned} \Vert f \Vert _{L_{{\textrm{uloc}},R}^p} :=\sup _{B(R)\subset \mathbb {R}^3} \Vert f \Vert _{L^p (B(R))}. \end{aligned}$$

Integrating in time starting from a $t_0$ to be specified and taking a supremum over the balls,

$$\begin{aligned}&\Vert w\psi (t)\Vert _{L_{{\textrm{uloc}},R}^1}+\frac{\epsilon }{2}\tau ^2\int _{t_0}^t\int _{\mathbb {R}^3}\langle \tau \omega \rangle ^{-(1+\epsilon )}|\nabla \omega |^2\psi \,d x\textrm{d}t\\&\quad \quad \lesssim \Vert w(t_0)\Vert _{L_{{\textrm{uloc}},R}^1}+\int _{t_0}^t(R^{-2}+R^{-1}\Vert u\Vert _{\infty })\Vert w(t')\Vert _{L_{{\textrm{uloc}},R}^1}\textrm{d}t'+\tau \Vert \nabla u\Vert _{L_{t,x-{\textrm{uloc}},R}^2}^2. \end{aligned}$$

Grönwall’s inequality

$$\begin{aligned} \Vert w(t)\Vert _{L_{{\textrm{uloc}},R}^1}\lesssim \left( \Vert w(t_0)\Vert _{L_{{\textrm{uloc}},R}^1}+\tau R A^{O(1)} \right) \exp (R^{-2}|t-t_0|+R^{-1}A^{O(1)}|t-t_0|^\frac{1}{2}), \end{aligned}$$

where $|t-t_0|^{1/2}$ comes from applying the Cauchy-Schwarz inequality in the time integral and by using the energy bound (44). Setting $R=A^{C_1}$ and $\tau =A^{-2C_1}$ for a sufficiently large $C_1$, we find

$$\begin{aligned} \Vert \langle \tau \omega (t)\rangle \Vert _{L_{{\textrm{uloc}},R}^1}\lesssim \Vert \langle \tau \omega (t_0)\rangle \Vert _{L_{{\textrm{uloc}},R}^1}. \end{aligned}$$

By (44) and Hölder’s inequality, we can find a $t_0\in [1/4,1/2]$ where the right-hand side is bounded by $A^{O(1)}$. Therefore

$$\begin{aligned} \int _{t_0}^t\int _{\mathbb {R}^3}\langle \tau \omega \rangle ^{-(1+\epsilon )}|\nabla \omega |^2\psi \, \textrm{d}x\textrm{d}t&\le \epsilon ^{-1}A^{O(1)}. \end{aligned}$$

We use Hölder’s inequality with the decomposition

$$\begin{aligned} |\nabla \omega |^{\frac{4}{3+\epsilon }}=\big (|\nabla \omega |^{\frac{4}{3+\epsilon }}\langle \tau \omega \rangle ^{-2\frac{1+\epsilon }{3+\epsilon }}\big )\langle \tau \omega \rangle ^{2\frac{1+\epsilon }{3+\epsilon }} \end{aligned}$$

to conclude

$$\begin{aligned} \Vert \nabla \omega \Vert _{L_{t,x-{\textrm{uloc}}}^{4/(3+\epsilon )}([t_0,t]\times {\mathbb {R}^3})}\le \epsilon ^{-O(1)}A^{O(1)}. \end{aligned}$$

To convert this into a bound on $\nabla ^2u$, fix a unit ball $B\subset {\mathbb {R}^3}$ and a cutoff function $\varphi \in C_c^\infty (3B)$ with $\varphi \equiv 1$ in 2B. We decompose $\nabla ^2u=a+b$ where $a=\nabla ^2\Delta ^{-1}{{\,\textrm{curl}\,}}(\varphi \omega )$. Note that $b=\nabla f$ where $f=\nabla \Delta ^{-1}{{\,\textrm{curl}\,}}((1-\varphi )\omega )$ is harmonic in 2B so for any $p\in [1,\frac{4}{3})$,

$$\begin{aligned} \Vert a\Vert _{L_{t,x}^{p}([t_0,t]\times B)}&\lesssim \Vert \nabla \omega \Vert _{L_{t,x}^p([t_0,t]\times 3B)}+\Vert \nabla \varphi \Vert _{L^\infty }\Vert \omega \Vert _{L_{t,x-{\textrm{uloc}}}^2([t_0,t]\times {\mathbb {R}^3})}\le \epsilon ^{-O(1)}A^{O(1)} \end{aligned}$$

and

$$\begin{aligned} \Vert b\Vert _{L_{t,x}^p([t_0,t]\times B)}&\lesssim \Vert \nabla \Delta ^{-1}{{\,\textrm{curl}\,}}((1-\varphi )\omega )\Vert _{L_{t,x}^2([t_0,t]\times 2B)}\\&\lesssim \Vert \omega ^\sharp \Vert _{L_{t,x}^2([t_0,t]\times {\mathbb {R}^3})}+\Vert \omega ^\flat \Vert _{L_{t,x}^\infty ([t_0,t]\times {\mathbb {R}^3})}\le A^{O(1)}, \end{aligned}$$

where we have used (44), Hölder’s inequality, (43), and (40). $\square$

5 Estimates for Axisymmetric Navier-Stokes Solutions

Here we provide some estimates of classical solutions of (1) that are specific to the axisymmetric assumption on the solutions.

We first note that $u_\theta$ satisfies

$$\begin{aligned} \left( \partial _t + u \cdot \nabla - \Delta + \frac{1}{r^2} \right) u_\theta +\frac{u_r}{r} u_\theta =0, \end{aligned}$$

(48)

which in particular gives that the swirl $\Theta :=ru_\theta$ satisfies

$$\begin{aligned} \Big (\partial _t+\Big (u+\frac{2}{r}e_r\Big )\cdot \nabla -\Delta \Big )\Theta =0 \end{aligned}$$

(49)

in $(\mathbb {R}^3 \setminus \{ r=0 \}) \times (0,T)$. It then follows that, at each time, $(r,x_3)\mapsto u_\theta (r,x_3,t)$ is a continuous function on $\overline{\mathbb {R}_+} \times \mathbb {R}$ with $u_\theta (0,x_3)=0$ for all $x_3$ (see [23, Lemma 1] for details). In particular

$$\begin{aligned} \Theta (0,x_3,t) =0\qquad \text { for all } x_3\in \mathbb {R}, t\in (0,T). \end{aligned}$$

(50)

Moreover, since $\omega = \omega _r e_r+ \omega _\theta e_\theta + \omega _3 e_3$ is a smooth vector field we see (also by [23, Lemma 1]) that $\Phi = \frac{\omega _r}{r}$, $\Gamma :=\frac{\omega _\theta }{r}$ (recall (7)) satisfy

$$\begin{aligned} | \Phi (r,x_3 ,t ) |, | \Gamma (r,x_3 ,t ) | \lesssim C (x_3,t) \end{aligned}$$

(51)

for $r\in [0,1]$.

5.1 Hölder Continuity Near the Axis

Here we consider the parabolic equation

$$\begin{aligned} \mathcal {M} V :=\partial _t V - \Delta V + b \cdot \nabla V =0 \end{aligned}$$

(52)

in a space-time cylinder

$$\begin{aligned} Q_R (x_0,t_0):=B( x_0, R)\times (t_0 - R^2 ,t_0). \end{aligned}$$

We assume that at each point of $Q_R:=Q_R (0,0)$

$$\begin{aligned} \text {either } \textrm{div}\, b=0 \quad \text { or } \quad V=0. \end{aligned}$$

(53)

We also assume that

$$\begin{aligned} {\mathcal {N}}(R ):=2+\sup _{R'\le R }(R')^{-\alpha }\Vert b\Vert _{L_t^{\ell }L_x^q(Q_{R'})}<\infty , \end{aligned}$$

(54)

where $\alpha :=\frac{3}{q}+\frac{2}{\ell }-1\in [0,1)$. In such setting [24, Corollary 3.6] observed that V must be Hölder continuous in the interior of $Q_{R}$, and in the proposition below we state a version of their result in which we quantify the dependence of the Hölder exponent in terms of $\mathcal {N}$.

Proposition 5.1

If V is a Lipschitz solution of (52) on $Q_{2R}$ then

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{B(r)}V(0)\lesssim \left( \frac{r}{R}\right) ^\gamma \mathop {{{\,\textrm{osc}\,}}}\limits _{Q_R}V \end{aligned}$$

for all $r\le R$, where $\gamma =\exp (-{{\mathcal {N}}}^{O(1)})$.

Proof

See Appendix 1. $\square$

We note that the swirl $\Theta$ satisfies (52) with $b:=u +2e_r /r$ (recall (49) above). Moreover $\textrm{div}\, b = 0$ everywhere except for the axis, since $\textrm{div}\, u =0$, $\textrm{div}(e_r /r )=0$ (recall (33)) there. Furthermore, $\Theta =0$ on the axis (recall (50)), and so the assumption (53) holds. Thus Proposition 5.1 shows that $\Theta$ is Hölder continuous in a neighborhood of the axis. We explore this in more detail in the proof of Theorem 1.1 below, where we quantify $\mathcal {N}$ in terms of the weak-$L^3$ bound A (see Step 1 in Section 6 below).

5.2 Pointwise Estimates Away from the Axis

The following is a more precise version of Proposition 8 in [31].

Proposition 5.2

(Pointwise bounds away from the axis) Let u solve (1) on [0, 1] satisfying (5) and (6). Then for every $\epsilon \in (0,4/15 )$, we have

$$\begin{aligned} |\nabla ^ju|\le \left( r^{-1-j}+r^{-\frac{1}{3}+\epsilon } \right) A^{O_{\epsilon ,j}(1)} \end{aligned}$$

for each $t\in [1/2,1]$. We also have

$$\begin{aligned} \Vert u\Vert _{L^p(\{r\ge 1\})}\le A^{O_p(1)} \end{aligned}$$

for each such t, and $p\in (3,\infty ]$.

Proof

We first pick any $\alpha \in (1/3-\epsilon /2,1/3)$ and $c =c(j)>0$ sufficiently small so that

$$\begin{aligned} (1-\alpha + j ) c< \epsilon /2 \quad \text { and } \quad c< \alpha / (1-\alpha ). \end{aligned}$$

(55)

We also pick $n=n(j)\in \mathbb {N}$ sufficiently large so that

$$\begin{aligned} n\ge (2+j )\left( 1 + \frac{1}{c} \right) . \end{aligned}$$

(56)

We set $t_k :=1/2 - (1/2)^k$ and we define a sequence of regions $\{x\in {\mathbb {R}^3}:r\ge R/2\}=\Omega _1\supset \Omega _2\supset \cdots \supset \Omega _n=\{x\in {\mathbb {R}^3}:r\ge R\}$ such that ${{\,\textrm{dist}\,}}(\Omega _i^c,\Omega _{i+1})\ge R/2n$.

Given such a sequence of times we now consider the corresponding Picard iterates $u^\flat _k$, $u^\sharp _k$, for $k\in \{ 0, 1,\ldots , n\}$.

Step 1. We show that

$$\begin{aligned} \Vert P_N u^\flat _k (t) \Vert _{L^\infty (\{ r \ge R /2 \} )},\Vert P_N u^\sharp _k (t) \Vert _{L^\infty (\{ r \ge R /2 \} )}\lesssim R^{-\alpha }N^{1-\alpha }A^{O_k(1)} \end{aligned}$$

(57)

for all $\alpha \in [0,\frac{1}{3})$, $R>0$ and $t\in [t_k,1]$, $k\ge 0$.

In fact, we first observe that Lemma 3.1 gives that

$$\begin{aligned} \Vert r^\alpha P_Nu (t) \Vert _{\infty }\lesssim N^{1-\alpha }\Vert u(t) \Vert _{L^{3,\infty }} \lesssim N^{1-\alpha } A^{O(1)}. \end{aligned}$$

(58)

Thus, since the first inequality above is valid for any axisymmetric function, it remains to note that the second inequality is also valid for each $u^\flat _k$, $u^\sharp _k$, on $[t_k,1]$, $k\ge 0$. Indeed, the case $k=0$ follows trivially, while the inductive step follows by applying Young’s inequality (28) for weak $L^p$ spaces, and Hölder’s inequality (27) for Lorentz spaces

$$\begin{aligned} \begin{aligned} \Vert u^\flat _k (t) \Vert _{L^{3,\infty }}&\lesssim \Vert \Psi (t- t_k ) \Vert _1 \Vert u(t_k) \Vert _{L^{3,\infty }}\\ {}&\quad + \int _{t_k}^t \Vert \nabla \Psi (t-t')\Vert _1 \Vert (u^\flat _{k-1} \otimes u^\flat _{k-1} )(t')\Vert _{L^{3/2,\infty }} \textrm{d}t'\\&\le C_k A + C_k \Vert u^\flat _{k-1} \Vert ^2_{L^\infty ([t_{k-1},1 ]; L^{3,\infty })} \int _{t_k}^t (t-t')^{-\frac{1}{2} } \textrm{d}t' \le A^{O_k(1)} \end{aligned} \end{aligned}$$

for $t\in [t_k ,1 ]$, as required, where we also used the heat kernel bounds (24).

Step 2. We show that the inequality from Step 1 can be improved for $u^\sharp _k$ for large k, namely

$$\begin{aligned} \Vert P_Nu^\sharp _k\Vert _{L^\infty ([\frac{1}{2},1]\times \{r\ge R\})}&\le NA^{O_k(1)}((RN)^{-(k-1)\alpha }+N^{-(k-1)}) \end{aligned}$$

(59)

for every $k\ge 1$ and $N\in 2^{\mathbb {N}}\cap [100^k\max (1,R^{-1}),\infty )$.

We will show that,

$$\begin{aligned} X_{k,N}\le N^{-\frac{4}{5}}A^{O_k(1)}((RN)^{-(k-1)\alpha }+N^{-(k-1)}), \end{aligned}$$

(60)

for $k\ge 1$ and $N\ge 100^k\max (1,R^{-1})$, using induction with respect to k, where

$$\begin{aligned} X_{k,N}:=\Vert P_Nu^\sharp _k\Vert _{L^\infty ([t_{k+1},1]; L^{5/3} ( \Omega _k ))}. \end{aligned}$$

Then (59) follows by the local Bernstein inequality (25).

As for the base case $k=1$ we note that (37) gives that

$$\begin{aligned} \Vert P_Nu^\sharp _1(t)\Vert _{{5/3}}&\lesssim \int _{t_1}^t\Vert P_N\textrm{e}^{(t-t')\Delta }{\mathbb {P}}\textrm{div}(u\otimes u)(t')\Vert _{{5/3}}\textrm{d}t'\\&\lesssim \int _{t_1}^t\textrm{e}^{-(t-t')N^2/O(1)}N^{\frac{6}{5}}\Vert (u\otimes u)(t')\Vert _{L^{\frac{3}{2},\infty }}\textrm{d}t'\\&\lesssim N^\frac{6}{5}\Vert \textrm{e}^{-tN^2/O(1)}\Vert _{L^1(t_1,1)}\Vert u\Vert _{L^{3,\infty }}^2 \end{aligned}$$

for $t\in [t_1,1]$. Thus

$$\begin{aligned} X_{1,N}\le \Vert P_Nu^\sharp _1\Vert _{L^\infty ([t_2,1]; L^{5/3} ) }\le N^{-\frac{4}{5}}A^{O(1)}, \end{aligned}$$

(61)

due to Hölder’s inequality for Lorentz spaces (27).

As for the inductive step, we use the Duhamel formula for $u^\sharp _k$ (recall (39)), and the local Bernstein inequality (25) to obtain

$$\begin{aligned} \begin{aligned}&\Vert P_N u^\sharp _k (t) \Vert _{L^{5/3} (\Omega _{k})}\\&\quad \lesssim \int _{t_k}^t \Vert P_N \textrm{e}^{(t-t')\Delta } \mathbb {P} \textrm{div}(u\otimes u - u^\flat _{k-1} \otimes u^\flat _{k-1} ) \Vert _{L^{5/3} (\Omega _k )} \textrm{d}t' \\&\quad \le \int _{t_k}^t N \textrm{e}^{-(t-t')N^2/O(1)} \textrm{d}t' \left( \Vert P_N ( u\otimes u - u^\flat _{k-1} \otimes u^\flat _{k-1} ) \Vert _{L^\infty ([t_{k},1]; L^{5/3} (\Omega _{k-1}) )} \right. \\&\qquad \left. +(NR)^{-(k-1)\alpha } \Vert P_N ( u\otimes u - u^\flat _{k-1} \otimes u^\flat _{k-1} ) \Vert _{L^\infty ([t_k , 1]; L^{5/3})} \right) \\&\quad \lesssim N^{-1} \left( \Vert P_N ( u\otimes u - u^\flat _{k-1} \otimes u^\flat _{k-1} ) \Vert _{L^\infty ([t_{k},1]; L^{5/3} (\Omega _{k-1}) )}\right. \\&\qquad \left. + N^{\frac{1}{5}} (NR)^{-(k-1)\alpha } A^{O(1)} \right) , \end{aligned} \end{aligned}$$

where we used the weak $L^3$ bound (6) and Lemma 3.1 for the $u\otimes u$ term and (40) for the $u^\flat _{k-1} \otimes u^\flat _{k-1}$ term. Thus we can use the paraproduct decomposition in the first term on the right-hand side to obtain

$$\begin{aligned} X_{k,N}&\lesssim N^{-1}\Vert Y_1+\cdots +Y_5\Vert _{L^\infty ([t_{k},1]; L^{5/3} (\Omega _{k-1}))}+N^{-\frac{4}{5}}( NR)^{-(k-1)\alpha }A^{O_k(1)}, \end{aligned}$$

(62)

where

$$\begin{aligned} Y_1&:=2\sum _{N'\sim N}P_{N'}u^\sharp _{k-1}\odot P_{\le N/100}u^\sharp _{k-1},\\ Y_2&:=\sum _{N_1\sim N_2 > rsim N}P_{N_1}u^\sharp _{k-1}\otimes P_{N_2}u^\sharp _{k-1},\\ Y_3&:=\sum _{N_1\sim N_2 > rsim N}P_{N_1}u^\flat _{k-1}\otimes P_{N_2}u^\sharp _{k-1},\\ Y_4&:=2\sum _{N'\sim N}P_{N'}u^\flat _{k-1}\odot P_{\le N/100}u^\sharp _{k-1},\\ Y_5&:=2\sum _{N'\sim N}P_{\le N/100}u^\flat _{k-1}\odot P_{N'}u^\sharp _{k-1}, \end{aligned}$$

where we use the notation $a\odot b:=a\otimes b + b\otimes a$. Using (57),

$$\begin{aligned} \Vert Y_1\Vert _{L^\infty ([t_k,1];L^{5/3}(\Omega _{k-1}))}&\lesssim \sum _{N'\sim N}X_{k-1,N'}\sum _{N'\lesssim N} R^{-\alpha }(N')^{1-\alpha } A^{O_k(1)}\\&\lesssim R^{-\alpha }N^{1-\alpha } A^{O_k(1)}\sum _{N'\sim N}X_{k-1,N'} \end{aligned}$$

and

$$\begin{aligned} \Vert Y_2\Vert _{L^\infty ([t_k,1];L^{5/3}(\Omega _{k-1}))}&\lesssim R^{-\alpha }A^{O_k(1)}\sum _{N' > rsim N} (N')^{1-\alpha } X_{k-1,N'}. \end{aligned}$$

Moreover, the frequency-localized bounds (42) for $u^\flat _{k-1}$ give that

$$\begin{aligned} \Vert Y_3\Vert _{L^\infty ([t_k,1]; L^{5/3}(\Omega _{k-1}))}&\lesssim A^{O_k(1)}\sum _{N' > rsim N}\textrm{e}^{-(N')^2/O_k(1)}N'X_{k-1,N'}, \end{aligned}$$

and (41), as well as boundedness of $P_{\le N/100}$ on $L^{5/3}$ give that

$$\begin{aligned} \Vert Y_4\Vert _{L^\infty ([t_k,1];L^{5/3}(\Omega _{k-1}))}&\lesssim A^{O_k(1)}\sum _{N'\sim N}\textrm{e}^{-(N')^2/O_k(1)}N'\lesssim \textrm{e}^{-N^2/O_k(1)}A^{O_k(1)}. \end{aligned}$$

Finally, using boundedness of $P_{\le N/100}$ on $L^\infty$ and (40) we obtain

$$\begin{aligned} \Vert Y_5\Vert _{L^\infty ([t_k,1]; L^{5/3}(\Omega _{k-1}))}&\lesssim A^{O_k(1)}\sum _{N'\sim N}X_{k-1,N'}. \end{aligned}$$

Combining these estimates into (62), we have shown

$$\begin{aligned} \begin{aligned} X_{k,N}&\le A^{O_k(1)}\left( ((RN)^{-\alpha }+N^{-1})\sum _{N'\sim N} X_{k-1,N'} + N^{-1} R^{-\alpha } \sum _{N' > rsim N } (N')^{1-\alpha } X_{k-1,N'}\right. \\&\quad \left. +N^{-1}\sum _{N' > rsim N}\textrm{e}^{-(N')^2/O_k(1)}N' X_{k-1,N'}+N^{-\frac{4}{5}}(NR)^{-(k-1)\alpha }+N^{-1} \textrm{e}^{-N^2/O_k(1)}\right) . \end{aligned} \end{aligned}$$

(63)

Since the upper bounds on $X_{k-1,N'}$ provided by the inductive assumption (60) are comparable for all $N'\sim N$, up to constants depending only on k, we thus obtain that

$$\begin{aligned} \begin{aligned} \sum _{N'\sim N} X_{k-1,N'}&\le A^{O_{k}(1)}N^{-\frac{4}{5}} \left( (RN)^{-\alpha ({k-2}) } + N^{-{k-2}} \right) ,\\&R^{-\alpha }\sum _{N' > rsim N } (N')^{1-\alpha } X_{k-1,N'}\\&\le A^{O_{k}(1)} R^{-\alpha }\sum _{N' > rsim N}(N')^{1-\alpha -\frac{4}{5}} \left( (RN')^{-\alpha ({k-2}) } + (N')^{-({k-2})} \right) \\&\le A^{O_{k}(1)} N^{\frac{1}{5}} \left( (RN)^{-\alpha ({k-1}) } + N^{-({k-1})} \right) , \end{aligned} \end{aligned}$$

where, in the last line we used the fact that $(k-1)(1-\alpha ) - 4/5 <0$ for any $k\ge 2$. A similar estimate for $\sum _{N' > rsim N}\textrm{e}^{-(N')^2/O_k(1)}N' X_{k-1,N'}$ now allows us to deduce from (63) that

$$\begin{aligned} \begin{aligned} X_{k,N}&\le N^{-\frac{4}{5} }A^{O_k(1)}((RN)^{-(k-1)\alpha }+N^{-(k-1)}), \end{aligned} \end{aligned}$$

as required.

Step 3. We prove the claim.

We first consider the case $R\ge 100^{n/c}$, and we note that, by (57)

$$\begin{aligned} \Vert P_{N\le R^{c}}\nabla ^ju^\sharp _n\Vert _{L_{t,x}^\infty ([\frac{1}{2},1]\times \{r\ge R\})}&\le \sum _{N\le R^c}A^{O_n(1)}N^{1-\alpha +j}R^{-\alpha }\\&\le A^{O_n(1)}R^{-\alpha +(1-\alpha +j)c}\le A^{O_{n}(1) }R^{-\frac{1}{3} +\varepsilon }, \end{aligned}$$

where we used the choice of $\alpha >1/3-\epsilon /2$ and the first property of our choice (55) of c in the last inequality. On the other hand for $N>R^c$ we can use (59) with $k= n$ to obtain arbitrarily fast decay in N. Comparing the terms on the right-hand side of (59) we see that $N^{-(n-2)}$ dominates $(RN)^{-(n-2)\alpha }$ if and only if $N\le R^{\alpha /(1-\alpha )}$, which allows us to apply the decomposition

$$\begin{aligned} \Vert P_{N>R^{c}}\nabla ^ju^\sharp _n\Vert _{L_{t,x}^\infty ([\frac{1}{2},1]\times \{r\ge R\})}&\le \sum _{R^c<N\le R^{\alpha /(1-\alpha )}}A^{O_n(1)}N^{-n+2+j}\\&\quad +\sum _{N>R^{\alpha /(1-\alpha )}}A^{O_n(1)}N^{1+j}(RN)^{-(n-1)\alpha }\\&\le A^{O_n (1)} R^{c(-n+2+j)}\\&\le A^{O_n(1)}R^{-1-j}, \end{aligned}$$

where we used the second property of our choice (55) of c in the second inequality, and the choice (56) of n in the last inequality.

We now suppose that $R\le 100^{n/c}$. The low frequencies can be estimated directly from the weak $L^3$ bound (6),

$$\begin{aligned} \Vert P_{\le 100^{2n/c}R^{-1}}\nabla ^ju\Vert _{L_{t,x}^\infty ([\frac{1}{2},1]\times \{r\ge R\})}&\lesssim _{n,c} A^{O(1)}R^{-1-j}. \end{aligned}$$

On the other hand, for $N> 100^{2n/c}R^{-1}$ we have in particular $N>R^{\alpha /(1-\alpha )}$, which shows that the dominant term on the right-hand side of (59) is $(RN)^{-(n-2)\alpha }$, and so

$$\begin{aligned}&\Vert P_{>100^{2n/c}R^{-1}}\nabla ^ju^\sharp _n (t) \Vert _{L^\infty (\{r\ge R\})}\\&\le \sum _{N>100^{2n/c}R^{-1}}N^{1+j}A^{O_n(1)}(RN)^{-(n-1)\alpha }\le A^{O_n(1)}R^{-1-j} \end{aligned}$$

for every $t\in [1/2,1]$, as desired. As for the estimate for $u^\flat$ we use (40) to obtain

$$\begin{aligned} \Vert \nabla ^j u^\flat _n \Vert _{L^\infty (\{ r\ge R \} )} \le R^{-1/3+\epsilon } \Vert r^{1/3-\epsilon } \nabla ^j u^\flat _n \Vert _{\infty } \lesssim _\epsilon R^{-1/3+\epsilon } A^{O_{\epsilon , j} (1)}, \end{aligned}$$

as needed.

The estimate for $\Vert u \Vert _{L^p (\{ r \ge 1\} )}$ follows by an $L^p$ analogue of Step 1, as well as applying the $X_{k,N}$ estimates (60) in the $L^p$ variant of Step 3. $\square$

5.3 A Poisson-Type Estimate on $u_r/r$

Here we discuss how derivatives of $u_r/r$ can be controlled by $\Gamma$ using the representation (11),

$$\begin{aligned} \frac{u_r}{r} = \Delta ^{-1} \partial _3 \Gamma - 2 \frac{\partial _r }{r} \Delta ^{-2} \partial _3 \Gamma , \end{aligned}$$

(64)

see [8, p. 1929], which will be an essential part of our $x_3$-${\textrm{uloc}}$ energy estimates for $\Phi$ and $\Gamma$ (see Proposition 6.1 below).

Lemma 5.3

(The $L^2_{3-{\textrm{uloc}}}$ estimate on $u_r/r$)

$$\begin{aligned} \Big \Vert \nabla \partial _r \frac{u_r}{r}\Big \Vert _{L_{3-{\textrm{uloc}}}^2}+ \Big \Vert \nabla \partial _3 \frac{u_r}{r}\Big \Vert _{L_{3-{\textrm{uloc}}}^2}\lesssim \Vert \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}+\Vert \nabla \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}. \end{aligned}$$

(65)

A version of the above estimate without the localization in $x_3$ has appeared in [8, Lemma 2.3]. As mentioned in the introduction, the localization makes the estimate much more challenging, particularly due to the bilaplacian term in (64).

In order to prove Lemma 5.3 we note that, since

$$\begin{aligned} \frac{\partial _r }{r} = \Delta ' - \partial _{rr}, \end{aligned}$$

(64) gives that

$$\begin{aligned} \frac{u_r}{r} = -\Delta ^{-1} \partial _3 \Gamma + 2(\partial _{rr} -\Delta ' ) \Delta ^{-2} \partial _3 \Gamma . \end{aligned}$$

(66)

Thus, since $| \nabla \partial _3 \frac{u_r}{r} | = |(\partial _r \partial _3 \frac{u_r}{r} , \partial _3 \partial _3 \frac{u_r}{r})|$ (and similarly for $|\nabla \partial _r \frac{u_r}{r} |$), we can use (35) and (36) to observe that

$$\begin{aligned} \begin{aligned} \left| \nabla \partial _3 \frac{u_r}{r} \right| + \left| \nabla \partial _r \frac{u_r}{r} \right|&\lesssim |D^2_{r,x_3} \Delta ^{-1} \partial _3 \Gamma | + |D^2_{r,x_3} (\partial _{rr} - \Delta ')\Delta ^{-2} \partial _3 \Gamma |\\&\lesssim |\nabla \Gamma | + | D^2 \Delta ^{-1} \nabla ' \Gamma | + | D^4 \Delta ^{-2} \nabla ' \Gamma |, \end{aligned} \end{aligned}$$

where we used $\partial _{33} = \Delta - \Delta '$ in the last line. In particular, each of the terms on the right-hand side involves at least one derivative in the horizontal variables. Thus, in order to estimate the left-hand side of (65) it suffices to find suitable bounds on the last two terms, which we achieve in Lemmas 5.4–5.5 below. Their claims give us (65), as required.

Lemma 5.4

If $f= \Delta ^{-1} \nabla ' \Gamma$ then

$$\begin{aligned} \Vert D^2 f \Vert _{L^2_{3-{\textrm{uloc}}}} \le \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} + \Vert \nabla ' \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} . \end{aligned}$$

Proof

Let I(x) denote the kernel matrix of $D^2 (-\Delta )^{-1}$. We have that

$$\begin{aligned} |\nabla ^j I(x) | \le \frac{C}{|x|^{3+j}} \qquad \text { for } j=0,1, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} D^2 f(x)&= \mathrm {p.v.}\int _{\mathbb {R}^3} I( x-y ) \nabla ' \Gamma (y) \textrm{d}y \\&= \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (y) {\tilde{\phi }} (y_3) I(x-y ) \textrm{d}y + \mathrm {p.v.} \int _{\mathbb {R}^3} \Gamma (y) (1-{\tilde{\phi }} (y_3) ) \nabla ' I( x-y ) \textrm{d}y \\&=: f_1 (x) + f_2 (x). \end{aligned} \end{aligned}$$

The Calderón-Zygmund inequality (see [35, Theorem B.5], for example) gives that

$$\begin{aligned} \Vert f_1 \Vert _{L^2_{3-{\textrm{uloc}}}} \le \Vert f_1 \Vert _{L^2} \lesssim \Vert \nabla ' \Gamma \, {\tilde{\phi }} \Vert _{L^2} \lesssim \Vert \nabla ' \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}. \end{aligned}$$

Moreover, noting that $\int _{\mathbb {R}^2 } \frac{ \textrm{d}x_1 \, \textrm{d}x_2}{(a^2 +x_1^2 +x_2^2)^2} = C a^{-2}$, we can use Young’s inequality for convolutions to obtain

$$\begin{aligned} \begin{aligned} \Vert f_2 (\cdot , x_3 ) \Vert _{L^2}&\le \int _{\mathbb {R}} \frac{ \Vert \Gamma (\cdot , y_3 ) \Vert _{L^2} (1-{\tilde{\phi }} (y_3) )}{|x_3-y_3|^2} \textrm{d}y_3 \\&\le \sum _{j\ge 1} \int _{\{ | x_3 - y_3 | \in (j,j+1) } \frac{ \Vert \Gamma (\cdot , y_3 ) \Vert _{L^2} (1-{\tilde{\phi }} (y_3) )}{|x_3-y_3|^2} \textrm{d}y_3 \\&\le \sum _{j\ge 1} j^{-2} \int _{\{ | x_3 - y_3 | \in (j,j+1) } \Vert \Gamma (\cdot , y_3 ) \Vert _{L^2} \textrm{d}y_3 \\&\le \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}. \end{aligned} \end{aligned}$$

Integration in $x_3$ over ${{\,\textrm{supp}\,}}\, \phi$ finishes the proof. $\square$

For the bilaplacian term in (66) one needs to work harder:

Lemma 5.5

Let $f=D^4 \Delta ^{-2} \nabla ' \Gamma$. Then

$$\begin{aligned} \Vert f \Vert _{L^2_{3-{\textrm{uloc}}}} \le \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} + \Vert \nabla \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} . \end{aligned}$$

Proof

We have that

$$\begin{aligned} f(x) = \mathrm {p.v.}\int _{\mathbb {R}^3} \mathrm {p.v.}\int _{\mathbb {R}^3} \nabla ' \Gamma (z) I(x-y) I(y-z) \textrm{d}z \, \textrm{d}y. \end{aligned}$$

Recalling that ${\tilde{\phi }} = \sum _{|j|\le 10 } \phi _j$, and $\tilde{{\tilde{\phi }} } = \sum _{|j|\le 20 } \phi _j$ we use the partition of unity,

$$\begin{aligned} \begin{aligned} 1&= \tilde{{\tilde{\phi }} } (z_3) + (1- \tilde{{\tilde{\phi }} } (z_3 ) ) {{\tilde{\phi }} } (y_3) + \sum _{\begin{array}{c} |j|>10 \\ |k|>20 \end{array} } \phi _j (y_3) \phi _k (z_3) \\&= \tilde{{\tilde{\phi }} } (z_3) + (1- \tilde{{\tilde{\phi }} } (z_3 ) ) {{\tilde{\phi }} } (y_3)\\&\quad + \sum _{|j|> 10 } \phi _j (y_3) \left( \sum _{\begin{array}{c} |k|>20\\ |k-j |\le 10 \end{array}}\phi _k (z_3) + \sum _{\begin{array}{c} |k|>20\\ |k-j|>10 \\ k\le j/2 \end{array} }\phi _k (z_3)+ \sum _{\begin{array}{c} |k|>20\\ |k-j|>10 \\ j/2<k\le 2j \end{array} }\phi _k (z_3)+ \sum _{\begin{array}{c} |k|>20\\ |k-j|>10 \\ k> 2j \end{array} }\phi _k (z_3) \right) , \end{aligned} \end{aligned}$$

to decompose f accordingly,

$$\begin{aligned} \begin{aligned} f(x)&= \mathrm {p.v.} \int _{\mathbb {R}^3} \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) \tilde{{\tilde{\phi }}} (z_3) I( x-y) I(y-z) \textrm{d}y \, \textrm{d}z \\&\quad +\mathrm {p.v.}\int _{\mathbb {R}^3}I(x-y){\tilde{\phi }} (y_3) \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) (1-\tilde{{\tilde{\phi }}} (z_3)) I(y-z) \textrm{d}z \, \textrm{d}y \\&\quad + \mathrm {p.v.} \int _{\mathbb {R}^3}I(x-y)\sum _{|j|>10 } \phi _j (y_3) \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j | \le 10 \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \, \textrm{d}y \\&\quad + \mathrm {p.v.} \int _{\mathbb {R}^3}I(x-y)\sum _{|j|>10 } \phi _j (y_3) \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j |> 10\\ k\le j/2 \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \, \textrm{d}y \\&\quad + \mathrm {p.v.}\int _{\mathbb {R}^3}I(x-y)\sum _{|j|>10 } \phi _j (y_3) \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j |> 10 \\ j/2 < k \le 2j \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \, \textrm{d}y \\&\quad + \mathrm {p.v.}\int _{\mathbb {R}^3}I(x-y)\sum _{|j|>10 } \phi _j (y_3) \mathrm {p.v.} \int _{\mathbb {R}^3} \nabla ' \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j |> 10 \\ k >2j \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \, \textrm{d}y \\&=: f_1 (x) + f_2 (x) + f_3 (x) + f_4 (x) + f_5 (x) + f_6 (x) . \end{aligned} \end{aligned}$$

Clearly $f_1$ involves localization of $\nabla ' \Gamma$ in $z_3$, and so we can use the Calderón-Zygmund inequality twice to obtain

$$\begin{aligned} \Vert f_1 \Vert _{L^2} \lesssim \Vert \nabla \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}. \end{aligned}$$

As for $f_2$ we integrate by parts in the z-integral (note that this does not conflict with the principal value, as the singularity has been cut off, and the far field has sufficient decay) and apply the Calderón-Zygmund estimate in x to obtain

$$\begin{aligned} \begin{aligned} \Vert f_2 \Vert _{L^2}&\lesssim \left\| {\tilde{\phi }} (y_3 ) \int _{\mathbb {R}^3} \frac{ |\Gamma (z)| (1-\tilde{{\tilde{\phi }} } (z_3))}{ | y-z |^4 } \textrm{d}z \right\| _{L^2} \lesssim \sup _{y_3 \in {{\,\textrm{supp}\,}}\, {\tilde{\phi }} } \left\| \int _{\mathbb {R}^3} \frac{ |\Gamma (z)| (1-\tilde{{\tilde{\phi }} } (z_3))}{ | y-z |^4 } \textrm{d}z \right\| _{L^2_{y'}}\\&\lesssim \sup _{y_3 \in {{\,\textrm{supp}\,}}\, {\tilde{\phi }} } \int _{\mathbb {R}} \frac{ \Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }} } (z_3))}{ | y_3-z_3 |^2 } \textrm{d}z_3 \\&\lesssim \sup _{y_3 \in {{\,\textrm{supp}\,}}\, {\tilde{\phi }} } \sum _{j \ge 1} j^{-2} \int _{|z_3 - y_3 |\in (j,j+1)} \Vert \Gamma (\cdot , z_3 ) \Vert _{L^2_{z'}} \textrm{d}z_3 \lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}, \end{aligned} \end{aligned}$$

where we used Young’s inequality in the second line (as in the lemma above).

As for $f_3$, we integrate by parts in z and then in y to obtain

$$\begin{aligned} |f_3 (x) | \lesssim \sum _{|j|>10} \int _{\mathbb {R}^3} \frac{ \phi _j (y_3) }{|x-y|^4} \left| \mathrm {p.v.} \int _{\mathbb {R}^3} \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j | \le 10 \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \right| \textrm{d}y . \end{aligned}$$

We note that the integration by parts is justified as

$$\begin{aligned} f_3 = D^2 (-\Delta )^{-1} \left( (1-\sum _{|j|\le 10 } \phi _j (y_3) ) D^2 (-\Delta )^{-1} \left( \nabla ' \Gamma (1-\sum _{k\in I} \phi _k (z_3) \right) \right) , \end{aligned}$$

where $I:=\{ -20 ,\ldots , 20 \} \cup \{ j-10 , \ldots , j+10\}$ is a finite index set. Thus, the operation of integration by parts above is equivalent to moving $\nabla '$ outside of the outer brackets, which in turn holds since the sums do not depend on $x'$ and $\nabla '$ commutes with other differential symbols.

Thus, using Young’s inequality in $x'$

$$\begin{aligned} \begin{aligned} \Vert f_3 (\cdot , x_3) \Vert _{L^2_{x'}}&\lesssim \sum _{|j|>10} \int _{\mathbb {R}} \frac{ \phi _j (y_3) }{|x_3-y_3|^2} \left\| \mathrm {p.v.} \int _{\mathbb {R}^3} \Gamma (z) \sum _{\begin{array}{c} |k|>6 \\ |k-j | \le 2 \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \right\| _{L^2_{y'}} \textrm{d}y_3 \\&\lesssim \sum _{|j|>2} j^{-2} \left\| \mathrm {p.v.} \int _{\mathbb {R}^3} \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j | \le 10 \end{array}} \phi _k (z_3 ) I(y-z) \textrm{d}z \right\| _{L^2_{y}} \\&\lesssim \sum _{|j|>10} j^{-2} \left\| \Gamma (z) \sum _{\begin{array}{c} |k|>20 \\ |k-j | \le 10 \end{array}} \phi _k (z_3 ) \right\| _{L^2} \lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} \end{aligned} \end{aligned}$$

for each $x_3 \in {{\,\textrm{supp}\,}}\, \phi$, where we applied the Cauchy-Schwarz inequality (in $y_3$) in the second line.

As for $f_4$ we note that

$$\begin{aligned} |y_3-z_3 |\ge & {} |y_3| - |z_3| \ge (j-1) - (k+1) \ge \frac{j}{2}\\{} & {} -2 \ge (j+2)/4 \ge (|y_3 | +1 ) /4 \ge |y_3 - x_3|/4. \end{aligned}$$

Thus, we can integrate by parts in z to obtain

$$\begin{aligned} |f_4 (x) | \le \int _{\mathbb {R}^3} \int _{\mathbb {R}^3 \cap \{ |y_3 -z_3 | \ge |x_3 - y_3 |/4 \} } \frac{|\Gamma (z) | (1-\tilde{\phi } (y_3 )) (1-\tilde{\phi }(y_3-z_3 ))}{|x-y|^3 |y-z|^4 } \textrm{d}z \, \textrm{d}y. \end{aligned}$$

Hence, applying Young’s inequality in $x'$ and then in $y'$ we obtain

$$\begin{aligned} \begin{aligned} \Vert f_{4} (\cdot , x_3) \Vert _{L^2}&\le \int _{\mathbb {R}} \left\| \int _{\mathbb {R}^3\cap \{ |y_3 -z_3 | \ge |x_3 - y_3 |/4 \}} \frac{ \Gamma (z) (1-{\tilde{\phi }} (y_3) ) (1-{\tilde{\phi }} (y_3 - z_3 )) }{|y-z|^4} \textrm{d}z \right\| _{L^2_{y'}} \\&\quad \cdot \underbrace{ \int _{\mathbb {R}^2} \frac{\textrm{d}x_1 \, \textrm{d}x_2}{\left( |x_3 - y_3 |^2 + x_1^2 +x_2^2 \right) ^{3/2} } }_{=C|x_3-y_3 |^{-1}}\textrm{d}y_3\\&\lesssim \int _{\mathbb {R}} \int _{\mathbb {R}\cap \{ |y_3 -z_3 | \ge |x_3 - y_3 |/4 \}}\\&\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-{\tilde{\phi }} (y_3) ) (1-{\tilde{\phi }} (y_3 - z_3 )) }{|x_3 - y_3 | \, |y_3 - z_3 |^2} \textrm{d}z_3 \, \textrm{d}y_3 . \end{aligned} \end{aligned}$$

(67)

Hence

$$\begin{aligned} \begin{aligned} \Vert f_4 (\cdot , x_3 )\Vert _{L^2}&\le \int _{\mathbb {R}} \frac{1-{\tilde{\phi }} (y_3 ) }{|x_3 - y_3|^{3/2}} \left( \sum _{j\ge 1 } \int _{\{|y_3 - z_3 | \in (j,j+1) \} } \frac{ \Vert \Gamma (\cdot , z_3 ) \Vert _{L^2}}{ | y_3 - z_3 |^{3/2}} \textrm{d}z_3 \, \right) \textrm{d}y_3 \\&\lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} \int _{\mathbb {R}} \frac{1-{\tilde{\phi }} (y_3 ) }{|x_3 - y_3|^{3/2}} \textrm{d}y_3 \lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}. \end{aligned} \end{aligned}$$

As for $f_5$ we have

$$\begin{aligned} \frac{1}{4} \le \frac{|x_3 -y_3 |}{|x_3 - z_3 | } \le 4 , \end{aligned}$$

since

$$\begin{aligned} |x_3 - y_3 | \le |y_3 | + |x_3 | \le j+2 \le 2j-8 \le 4k - 8 \le 4 (|z_3 | - |x_3 | )\le 4 | x_3 - z_3 | \end{aligned}$$

and

$$\begin{aligned} |x_3 - z_3 | \le |z_3| + |x_3| \le k+2 \le 2j+2 \le 4(j-2) \le 4 (| y_3 | - |x_3 | ) \le 4 |x_3 - y_3 |. \end{aligned}$$

In particular, the triangle inequality gives that

$$\begin{aligned} |y_3 - z_3 | \le 5|x_3 -z_3 |. \end{aligned}$$

Thus we can integrate by parts twice (in z and then in y, so that the derivative falls on $I(x-y)$), and then use Young’s inequality twice (as in (67) above) and Tonelli’s Theorem to obtain

$$\begin{aligned} \begin{aligned} \Vert f_5 (\cdot , x_3 ) \Vert _{L^2}&\le \int _\mathbb {R}\int _{\{ |x_3 - y_3 | /4\le |x_3-z_3 | \le 4 | x_3 - y_3 | \} }\\&\frac{ \Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1- {\tilde{\phi }} (y_3 - z_3 )) (1-\tilde{{\tilde{\phi }}} (z_3 )) }{|x_3-y_3 |^2 | y_3 - z_3 |} \textrm{d}z_3 \, \textrm{d}y_3 \\&\le \int _\mathbb {R}\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }}} (z_3 ))}{|x_3 - z_3 |^2 } \int _{\{ |y_3-z_3 | \le 5 |x_3 - x_3 | \} } \frac{ 1- {\tilde{\phi }} (y_3 - z_3 ) }{ | y_3 - z_3 |} \textrm{d}y_3 \, \textrm{d}z_3 \\&\lesssim \int _\mathbb {R}\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }}} (z_3 ))}{|x_3 - z_3 |^2 } \log (5|x_3-z_3|)\textrm{d}z_3 \\&\lesssim \sum _{j\ge 1 } \int _{|z_3-x_3| \in (j,j+1)} \frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} }{|x_3-z_3|^2 } \log (5|x_3-z_3|) \textrm{d}z_3 \\&\lesssim \sum _{j\ge 1 } j^{-2}\log (5j) \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} \lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}}. \end{aligned} \end{aligned}$$

Finally, for $f_6$ we observe that

$$\begin{aligned} \frac{1}{4} \le \frac{ |x_3 - z_3 |}{|y_3 - z_3 |} \le 4, \end{aligned}$$

since

$$\begin{aligned} |y_3-z_3| \ge |z_3| - |y_3| \ge k - j -2 > \frac{k-8 }{2}\ge \frac{k+2}{4} \ge \frac{|x_3| + |z_3| }{4} \ge \frac{|x_3- z_3|}{4} \end{aligned}$$

and

$$\begin{aligned} |y_3-z_3 |\le & {} | y_3| + |z_3| \le j+k +2\\\le & {} \frac{3k+4}{2} \le 4(k-2) \le 4(|z_3| - |x_3| )\le 4|x_3-z_3|. \end{aligned}$$

In particular, the triangle inequality gives that

$$\begin{aligned} | x_3 - y_3 | \le 5 |x_3 -z_3|. \end{aligned}$$

Thus, similarly to the case of $f_5$ (although without integrating by parts in y), we apply Young’s inequality twice, and Tonelli’s Theorem to obtain

$$\begin{aligned} \begin{aligned} \Vert f_6 (\cdot , x_3 ) \Vert _{L^2}&\le \int _\mathbb {R}\int _{ \mathbb {R}} \frac{ \Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-{\tilde{\phi }} (y_3)) (1-\tilde{{\tilde{\phi }}} (z_3 )) }{|x_3-y_3 | | y_3 - z_3 |^2} \textrm{d}z_3 \, \textrm{d}y_3 \\&\le \int _\mathbb {R}\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }}} (z_3 )) }{|x_3-z_3|^2 } \\&\qquad \int _{\{ \frac{1}{4} |z_3 - x_3 |\le |y_3 - z_3 | \le 4 |z_3 - x_3 | \} } \frac{ 1-{\tilde{\phi }} (y_3) }{|x_3-y_3 | } \textrm{d}y_3 \, \textrm{d}z_3 \\&\le \int _\mathbb {R}\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }}} (z_3 )) }{|x_3-z_3|^2 } \int _{\{ 1\le |x_3-y_3 | \le 5 |x_3-z_3 | \} } \frac{ 1 }{|x_3-y_3 | } \textrm{d}y_3 \, \textrm{d}z_3 \\&\lesssim \int _\mathbb {R}\frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2} (1-\tilde{{\tilde{\phi }}} (z_3 )) }{|x_3-z_3|^2 } \log (5|x_3-z_3|) \textrm{d}z_3 \\&\lesssim \sum _{j\ge 1 } \int _{|z_3-x_3| \in (j,j+1)} \frac{\Vert \Gamma (\cdot , z_3 ) \Vert _{L^2}\log (5|x_3-z_3|) }{|x_3-z_3|^2 } \textrm{d}z_3 \\&\lesssim \sum _{j\ge 1 } \log (5j )j^{-2}\Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} \lesssim \Vert \Gamma \Vert _{L^2_{3-{\textrm{uloc}}}} \end{aligned} \end{aligned}$$

for $x_3 \in {{\,\textrm{supp}\,}}\, \phi$. Integration of the squares of the above estimates for $f_3,f_4,f_5,f_6$ gives the claim. $\square$

6 Energy Estimates for $\omega /r$

In this section, we assume the weak $L^3$ bound (6) on the time interval [0, 1] and prove an energy bound for $\Phi ^2 + \Gamma ^2$ at time 1, that is we prove the following.

Proposition 6.1

(An $L^2_{3-{\textrm{uloc}}}$ energy estimate for $\Phi$ and $\Gamma$) Let u be a classical solution of (1) satisfying the weak $L^3$ bound (6) on [0, 1]. Then

$$\begin{aligned} \Vert \Phi (1)\Vert _{L_{3-{\textrm{uloc}}}^2({\mathbb {R}^3})} + \Vert \Gamma (1)\Vert _{L_{3-{\textrm{uloc}}}^2({\mathbb {R}^3})} \le \exp \exp A^{O(1)}. \end{aligned}$$

(68)

Recall (23) that $\Vert \cdot \Vert _{L^p_{3-{\textrm{uloc}}} }:=\sup _{z\in {\mathbb {R}}}\Vert \cdot \Vert _{L^p({\mathbb {R}}^2\times [z-1,z+1])}$. We note that we will only use (in (73) below) the bound on $\Gamma$.

Proof

We fix a cutoff function $\phi \in C_c^\infty ((-1,1);[0,1])$ such that $\phi \equiv 1$ in $[-1/2,1/2]$, and we define the translate

$$\begin{aligned} \phi _z(y):=\phi (y-z). \end{aligned}$$

Clearly, we have the pointwise inequality

$$\begin{aligned} \phi _z',\phi _z''\lesssim \sum _{i=-2}^2\phi _{z+i}. \end{aligned}$$

We will consider the energies

$$\begin{aligned}&E(t):=\sup _{z\in {\mathbb {R}}}E_z(t),\quad E_z(t):=\frac{1}{2}\int _{\mathbb {R}^3}(\Phi (t,x)^2+\Gamma (t,x)^2)\phi _z(x_3)\textrm{d}x,\\&F(t):=\sup _{z\in {\mathbb {R}}}F_z(t),\quad F_z(t):=\int _{t_0}^t\int _{\mathbb {R}^3}(\nabla \Phi (s,x)^2+\nabla \Gamma (s,x)^2)\phi _z(x_3)\textrm{d}x\, \textrm{d}s \end{aligned}$$

for $t\in [t_0,1]$, where $t_0\in [0,1]$ will be chosen in Step 3 below. Given $z\in {\mathbb {R}}$, we multiply the equations (10) by $\phi _z \Gamma$ and $\phi _z \Phi$, respectively, and integrate to obtain, at a given time t,

$$\begin{aligned} \begin{aligned} E_z'&\le \int _{\mathbb {R}^3}\Big (-(|\nabla \Phi |^2+|\nabla \Gamma |^2)\phi _z+\frac{1}{2}(\Phi ^2+\Gamma ^2)(u_z\phi _z'+\phi _z'')\\&\quad +(\omega _r\partial _r+\omega _3\partial _3)\frac{u_r}{r}\Phi \phi _z-2r^{-1}u_\theta \Phi \Gamma \phi _z\Big )\textrm{d}x\\&=:-F_z'(t) + I_1+I_2+I_3. \end{aligned} \end{aligned}$$

(69)

The second term on the right hand side can be bounded directly,

$$\begin{aligned} I_1\lesssim (1+\Vert u_z\Vert _{L_x^\infty ({\mathbb {R}^3})})E(t). \end{aligned}$$

(70)

The remaining terms $I_2,I_3$ are more challenging. In order to estimate them, as well as choose $t_0$ and deduce the claim (68), we follow the steps below.

Step 1. We use the Hölder estimate (Proposition 5.1) to show that $|\Theta |\le r^\gamma A^{O(1)}$ whenever $r\le \frac{1}{2}$ and $t\in [3/4,1]$, where $\gamma =\exp (-A^{O(1)})$.

To this end we note that, due to incompressibility, $\textrm{div}(u+\frac{2}{r}e_r)=4\pi \delta _{\{x'=0\}}$, which enables us to apply Proposition 5.1 to the equation for the swirl $\Theta$ (recall (13)).

Moreover, in the notation of Proposition 5.1, for every $R<\frac{1}{2}$, $t_0\in [\frac{1}{2},1]$ and $x_0\in (0,0)\times {\mathbb {R}}$ (i.e., on the $x_3$-axis),

$$\begin{aligned} R^{-\frac{4}{5}}\Vert u+\frac{e_r}{r}\Vert _{L_t^\infty L_x^\frac{5}{3}(Q((t_0,x_0),R))}&\lesssim R^{-\frac{1}{2}}\Vert u\Vert _{L_t^\infty L_{{\textrm{uloc}}}^2([t_0-R^2,t_0]\times {\mathbb {R}^3})}+1\le A^{O(1)}, \end{aligned}$$

by Hölder’s inequality and (44) applied on the timescale $R^2$. (In particular note that each scale R leads to a different decomposition $u=u^\flat _n+u^\sharp _n$, but they all obey the same bounds up to being suitably rescaled.) Thus, for every $r\in (0,1/2)$, $\mathop {{{\,\textrm{osc}\,}}}\limits _{B(x_0,r)}\Theta (t_0)\lesssim r^\gamma \mathop {{{\,\textrm{osc}\,}}}\limits _{Q(1/2)}\Theta$ for $r\in (0,1/2)$, which implies the claim.

Step 2. We show that

$$\begin{aligned} \int _{t_0}^t \left| I_2 + I_3 \right| \lesssim \frac{1}{2} F(t) + r_0^{-10} + \int _{t_0}^t G E \end{aligned}$$

for each $t_0\in [3/4,1]$ and $t\in [t_0,1]$, where

$$\begin{aligned} r_0 :=\textrm{e}^{-\gamma ^{-2}}, \end{aligned}$$

(71)

$\gamma = \exp ( -A^{O(1)} )$ is given by Step 1, and

$$\begin{aligned} G :=r_0^{-3} + \Vert u\Vert _\infty + \Vert D^2 u \Vert _{L^{5/4}_{{\textrm{uloc}}}} + \Vert \nabla u \Vert _{L^2_{{\textrm{uloc}}}} \end{aligned}$$

at each $t'\in [t_0,t]$.

To this end, we proceed similarly to [8]. Using integration by parts, we compute

$$\begin{aligned} I_2&=2\pi \int _{\mathbb {R}}\int _0^\infty (-\partial _3u_\theta \partial _r\frac{u_r}{r}\Phi +\frac{\partial _r(ru_\theta )}{r}\partial _3\frac{u_r}{r}\Phi )\phi _z(x_3)r\, \textrm{d}r \,\textrm{d}x_3\\&=\int _{\mathbb {R}^3}u_\theta (\partial _r\frac{u_r}{r}\partial _3\Phi \phi _z-\partial _3\frac{u_r}{r}\partial _r\Phi \phi _z+\partial _r\frac{u_r}{r}\Phi \phi _z')\\&=:I_{2,1}+I_{2,2}+I_{2,3}. \end{aligned}$$

Let us further decompose $I_{2,i}=I_{2,i,{\textrm{in}}}+I_{2,i,{\textrm{out}}}$ ($i=1,2,3$) by writing

$$\begin{aligned} \int = \int _{\{ r < r_0 \} } + \int _{\{ r\ge r_0 \} }. \end{aligned}$$

We decompose

$$\begin{aligned} I_{2,1,{\textrm{in}}}&=I_{2,1,{\textrm{in}},1}+I_{2,1,{\textrm{in}},2}, \end{aligned}$$

where

and $\Omega :=\{ x' :r <1 \} \times {{\,\textrm{supp}\,}}\phi _z$. We compute using Hölder’s inequality and Sobolev embedding

$$\begin{aligned} \bigg |\int _\Omega \partial _r\frac{u_r}{r}\bigg |&\le \Vert r^{-1}\partial _ru_r\Vert _{L^1(\Omega )}+\Vert r^{-2}u_r\Vert _{L^1(\Omega )}\\&\lesssim \Vert r^{-1}\Vert _{L^{15/8} (\Omega )}\Vert \nabla u\Vert _{L^{15/7}(\Omega )}\lesssim \Vert \nabla ^2u\Vert _{L^{5/4}(\Omega )}+\Vert \nabla u\Vert _{L^2(\Omega )} \lesssim G. \end{aligned}$$

Thus, integrating by parts, and applying Hölder’s inequality in Lorentz spaces (27), and Young’s inequality, we obtain

$$\begin{aligned} |I_{2,1,{\textrm{in}},1}|&\le G \int _{B(r_0)\times {\mathbb {R}}} \left( r\Phi ^2\phi _z+|u_\theta \Phi \phi _z'| \right) \textrm{d}x\\&\lesssim G (r_0E+\Vert u_\theta \Vert _{L_x^{3,\infty }({\mathbb {R}^3})}\Vert \Phi \Vert _{L_x^2(\Omega )}|\Omega |^\frac{1}{6})\\&\lesssim G(E+A^{O(1)}). \end{aligned}$$

As for $I_{2,1,{\textrm{in}},2}$ we note that $p=2(1-\gamma )/(1-2\gamma )$ is such that $p-2=2\gamma /(1-2\gamma ) \ge \gamma$ and so we can use the quantified Hardy inequality (Lemma 3.2) to obtain, for we estimate for $t\in [\frac{1}{2},1]$,

where we have also applied Poincaré’s inequality and our choice (71) of $r_0$. Thus

$$\begin{aligned} \int _{t_0}^tI_{2,1,{\textrm{in}},2}\le \frac{1}{20}F(t)+\int _{t_0}^tE. \end{aligned}$$

An analogous argument, in which “$\partial _r$” and “$\partial _3$” are switch, gives us the same bound for $I_{2,2,{\textrm{in}},2}$. As for $I_{2,2,{\textrm{in}},1}$, we integrate by parts, and apply Hölder’s inequality for Lorentz spaces (27), and Young’s inequality, to obtain

which, thanks to the smallness of $r_0 = \exp (-\exp (A^{O(1)}))$ (recall (71)), gives that

$$\begin{aligned} \int _{t_0}^t|I_{2,2,{\textrm{in}},1}|\le \frac{1}{20}F(t)+(t-t_0). \end{aligned}$$

We similarly decompose $I_{2,3,{\textrm{in}}}=I_{2,3,{\textrm{in}},1}+I_{2,3,{\textrm{in}},2}$ to find

where we have used Lemma 3.2 and change of variables, the pointwise estimate $|u_r/r|\le |\nabla u|$, and Hölder’s inequality to bound

where we used (34) in the third line, and the Hardy inequality (32) in the last line. Next

where we have used the Hardy inequality (Lemma 3.2). Thus Lemma 5.3 and Young’s inequality imply that

$$\begin{aligned} \int _{t_0}^t|I_{2,3,{\textrm{in}},2}|\le \frac{1}{20}F(t)+\int _{t_0}^tE. \end{aligned}$$

Next let us consider the contributions to $I_2$ from outside $B(r_0)$. Using Hölder’s inequality, we obtain that

$$\begin{aligned} |I_{2,1,{\textrm{out}}}|&=\bigg |\int _{\{r>r_0\}}u_\theta \partial _r\frac{u_r}{r}\partial _3\Phi \phi _z\,\textrm{d}x\bigg |\\&\le \Vert u_\theta \Vert _{L_{3-{\textrm{uloc}}}^6(\{r>r_0\})}\Vert r^{-1}\partial _ru_r-r^{-2}u_r\Vert _{L_{3-{\textrm{uloc}}}^3(\{r>r_0\})}\Vert \nabla \Phi \Vert _{L_{3-{\textrm{uloc}}}^2({\mathbb {R}^3})}. \end{aligned}$$

Hence, since Proposition 5.2 shows that $|u|\le A^{O(1)} (r^{-1} + r^{-1/4})$ and $|\partial _r u_r | \le A^{O(1)} (r^{-2} + r^{1/4})$, we see that the first two norms on the right hand side are finite and bounded by, say, $r_0^{-10}$. Thus, an application of Young’s inequality gives that

$$\begin{aligned} \int _{t_0}^t|I_{2,1,{\textrm{out}}}|\le \frac{1}{20}F(t)+r_0^{-10}(t-t_0). \end{aligned}$$

The remaining outer parts of $I_2$, i.e. $I_{2,2,{\textrm{out}}}$ and $I_{2,3,{\textrm{out}}}$ can be estimated in a similar way, with the latter bounded by, say, $E+r_0^{-10}$.

Finally let us consider $I_3$. Taking p such that, for example, $\frac{1}{p}=\frac{1}{2}-\frac{\gamma }{4}$, we have $p-2= 2\gamma /(2-\gamma ) \ge \gamma$, and so our quantified Hardy’s inequality (Lemma 3.2) shows that

$$\begin{aligned} |I_{3,{\textrm{in}}}|&\le \left\| r^{-2+\frac{6}{p}}u_\theta \right\| _{L^{\left( 1-\frac{2}{p} \right) ^{-1}}(\{r\le r_0\})}\Vert r^{-\frac{3}{p}+\frac{1}{2}}\Phi \Vert _{L_{3-{\textrm{uloc}}}^p}\Vert r^{-\frac{3}{p}+\frac{1}{2}}\Gamma \Vert _{L_{3-{\textrm{uloc}}}^p}\\&\lesssim \gamma ^{-O(1)}r_0^{\gamma /2}\left( \Vert \Phi \Vert _{L_{3-{\textrm{uloc}}}^2}+\Vert \nabla \Phi \Vert _{L_{3-{\textrm{uloc}}}^2}\right) \left( \Vert \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}+\Vert \nabla \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}\right) , \end{aligned}$$

which gives that $\int _{t_0}^t|I_{3,{\textrm{in}}}|\le \frac{1}{20}F(t)+\int _{t_0}^t E$. On the other hand, for $r\ge r_0$ we have the simple bound

$$\begin{aligned} |I_{3,{\textrm{out}}}|&\le 2 \Vert r^{-1} u_\theta \Vert _{L_x^\infty (\{r\ge r_0\})}\Vert \Phi \Vert _{L_{3-{\textrm{uloc}}}^2}\Vert \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}\le r_0^{-5/4}E, \end{aligned}$$

as required.

Step 3. Given $\tau >0$ we use the choice of time of regularity (Lemma 4.2) to find $t_0\in [1-\tau ,1]$ such that $E(t_0) \lesssim A^{O(1)} \tau ^{-3}$.

Indeed, Lemma 4.2 lets us choose $t_0\in [1-\tau , 1]$ such that

$$\begin{aligned} \Vert \nabla ^2 u(t_0)\Vert _{\infty }\le A^{O(1)}\tau ^{-\frac{3}{2}}. \end{aligned}$$

It follows from the axial symmetry and (34) that $|\Phi | + |\Gamma | \le | \nabla \omega |$, and so

$$\begin{aligned} \Vert \Phi (t_0) \phi _z^{1/2} \Vert _{L^2 (\{ r\le 1 \} )} + \Vert \Gamma (t_0) \phi _z^{1/2} \Vert _{L^2 (\{ r\le 1 \} )} \lesssim \Vert \nabla \omega (t_0)\Vert _{L^\infty (B(1)\times {\mathbb {R}})}\le A^{O(1)}\tau ^{-\frac{3}{2}}\nonumber \\ \end{aligned}$$

(72)

for every $z\in \mathbb {R}$. Using the decomposition $\omega =\omega _1^\sharp +\omega _1^\flat$ on the interval [0, 1], by (44), (40), and Hölder’s inequality,

$$\begin{aligned}&\Vert \Phi (t_0)\phi _z^{1/2} \Vert _{L^2( \{ r>1 \} ) }+\Vert \Gamma (t_0)\phi _z^{1/2}\Vert _{L^2( \{ r>1 \} ) }\\&\lesssim \Vert \omega _1^\sharp \Vert _{L^2({\mathbb {R}^3})}+\Vert r^{-1}\omega _1^\flat \Vert _{L^2 (\{ r>1 \} \cap {{\,\textrm{supp}\,}}\, \phi _z ) }\\&\lesssim \Vert \nabla u^\sharp _1\Vert _{L^2({\mathbb {R}^3})}+\Vert r^{-1}\Vert _{L_{x'}^4(B(1)^c)}\Vert \omega _1^\flat \Vert _{L^4({\mathbb {R}^3})}\\&\le A^{O(1)}. \end{aligned}$$

This and (72) proves the claim of this step.

Step 4. We prove the claim.

Integration in time of the energy inequality (69) from initial time $t_0$ chosen in Step 3 above, taking $\sup _{z\in {\mathbb {R}}}$, and applying the estimate (70) for $I_1$ and Step 2 for $I_2$, $I_3$ we find that

$$\begin{aligned}&E(t)+\frac{1}{2}F(t)\le \underbrace{E(t_0)}_{\le A^{O(1)}\tau ^{-3} }+r_0^{-10}\\&\quad +\int _{t_0}^tO(r_0^{-3}+\Vert u\Vert _{\infty }+\Vert \nabla ^2u\Vert _{L_{{\textrm{uloc}}}^{5/4}}+\Vert \nabla u\Vert _{L_{{\textrm{uloc}}}^2})E(t')\textrm{d}t'. \end{aligned}$$

for $t\in [t_0,1]$. Thus, by Grönwall’s inequality,

$$\begin{aligned} E(1)&\le (A^{O(1)}\tau ^{-3}+r_0^{-10})\exp \left( O\left( r_0^{-3}(t-t_0)+A^{O(1)}(t-t_0)^\frac{1}{5}\right) \right) . \end{aligned}$$

Setting $\tau :=r_0^4$, we see that the last exponential function is O(1), and the prefactor gives the required estimate (68). $\square$

7 Proof of Theorem 1.1

In this section we prove Theorem 1.1. Namely, given the $L^{3,\infty }$ bound (6) on the time interval [0, 1], we show that $|\nabla ^j u |\le \exp \exp A^{O_j (1)}$ at time 1.

Step 1. We show that $\Vert b \Vert _{L^p_{3-{\textrm{uloc}}} (\mathbb {R}^3)} \le C_p \exp \exp A^{O(1)}$ for each $p\in [3,\infty )$, $t\in [1/2,1]$, where $b:=u_re_r+u_ze_z$ denotes the swirl-free part of the velocity field.

To this end we apply Proposition 6.1 to find

$$\begin{aligned} \Vert \Gamma \Vert _{L_t^\infty L_{3-{\textrm{uloc}}}^2([\frac{1}{2},1]\times {\mathbb {R}^3})}\le \exp \exp A^{O(1)}. \end{aligned}$$

(73)

On the other hand Proposition 5.2 shows that

$$\begin{aligned} \Vert r^2\omega \Vert _{L_x^\infty (\{r\le 10\})}\le A^{O(1)}. \end{aligned}$$

Interpolating between this inequality and (73) we obtain

$$\begin{aligned} \Vert \omega _\theta \Vert _{L_{3-{\textrm{uloc}}}^p(\{r\le 10\})}&=\Vert \Gamma ^{\frac{2}{3}}(r^2\omega _\theta )^{\frac{1}{3}}\Vert _{L_{3-{\textrm{uloc}}}^p(\{r\le 10\})}\\&\lesssim \Vert \Gamma \Vert _{L_{3-{\textrm{uloc}}}^2}^\frac{2}{3}\Vert r^2\omega _\theta \Vert _{L_x^\infty (\{r\le 10\})}^\frac{1}{3}\le \exp \exp A^{O(1)} \end{aligned}$$

for all $p\le 3$. Noting that

$$\begin{aligned} {{\,\textrm{curl}\,}}b=\omega _\theta e_\theta ,\quad \textrm{div}b=0 \end{aligned}$$

almost everywhere, and that $\textrm{div}\, b =0$ we now localize b to obtain an $L^p$ estimate near the axis. Namely, for any unit ball $B\subset \{r\le 10\}$, let $\phi \in C_c^\infty (B)$ such that $\phi \equiv 1$ on B/2. Observe that for all $p\in [1,3)$ we can use Hölder’s inequality for Lorentz spaces (27) to obtain

$$\begin{aligned} \Vert \textrm{div}(\phi b)\Vert _{L^p({\mathbb {R}^3})}=\Vert b\cdot \nabla \phi \Vert _{p}\lesssim \Vert b\Vert _{L^{3,\infty }}\Vert \nabla \phi \Vert _{L^{3p/(3-p) ,1}}\lesssim A. \end{aligned}$$

Applying the Bogovskiĭ operator (30) to $\textrm{div}(\phi b)$ on the domain $B\setminus (B/2)$, we find ${\tilde{b}} \in W^{1,p}$ such that $\textrm{div}{\tilde{b}}=0$, $\Vert b-{\tilde{b}}\Vert _{W^{1,p}(B)}\le A^{O(1)}$, ${\tilde{b}}\equiv b$ in B/2, and ${\tilde{b}}\equiv 0$ outside B. Then for any $p\in (1,3)$,

$$\begin{aligned} \Vert b\Vert _{L^{3p/(3-p)}(B/2)}&\le \Vert {\tilde{b}}\Vert _{{3p/(3-p)}}\lesssim \Vert \nabla {\tilde{b}}\Vert _{p}\lesssim \Vert \hspace{-.025in}{{\,\textrm{curl}\,}}{\tilde{b}}\Vert _{L^p(B)}\\&\le \Vert \omega _\theta \Vert _{L^p(B)}+\Vert b-{\tilde{b}}\Vert _{W^{1,p}(B)}\\&\le \exp \exp A^{O(1)}, \end{aligned}$$

which is our desired localized estimate. Here we have used the boundedness of the operator $\nabla f \mapsto {{\,\textrm{curl}\,}}\, f$ in $L^p$ (which is a consequence of the identity ${{\,\textrm{curl}\,}}\,{{\,\textrm{curl}\,}}\, f = \nabla (\textrm{div}\, f ) - \Delta f$, which in turn implies that $\nabla f = \nabla (-\Delta )^{-1} {{\,\textrm{curl}\,}}({{\,\textrm{curl}\,}}f)$ for divergence-free f). Combining this with the pointwise estimates away from the axis (Proposition 5.2) gives the claim of this step.

Step 2. We show that there exists $C_0>1$ such that

$$\begin{aligned} \Big \Vert \frac{u_\theta (t)}{r^{\frac{1}{2}}}\Big \Vert _{L_{3-{\textrm{uloc}}}^4}^4\le \Big \Vert \frac{u_\theta (t_0)}{r^{\frac{1}{2}}}\Big \Vert _{L_{3-{\textrm{uloc}}}^4}^4+1+\exp \exp A^{C_0}\int _{t_0}^t\left\| \frac{u_{\theta }}{r^{\frac{1}{2}}} \right\| _{L^4_{3-{\textrm{uloc}}}}^4 \end{aligned}$$

(74)

for each $t_0 \in [1/2,1]$ and $t\in [t_0,1]$.

To this end we provide a localization of the estimate of $u_\theta /r^{1/2}$ in the spirit of [8, Lemma 3.1]. Indeed, one can calculate from the equation (48) for $u_\theta$ that for a smooth cutoff $\psi =\psi (x_3)$,

$$\begin{aligned}&\frac{1}{4}\frac{\textrm{d}}{\textrm{d}t}\int _{\mathbb {R}^3}\frac{u_\theta ^4}{r^2}\psi +\frac{3}{4}\int _{\mathbb {R}^3}\Big |\nabla \frac{u_\theta ^2}{r}\Big |^2\psi +\frac{3}{4}\int _{\mathbb {R}^3}\frac{u_\theta ^4}{r^4}\psi _z\\&\quad =-\frac{3}{2}\int _{\mathbb {R}^3}\frac{1}{r^3}u_ru_\theta ^4\psi +\frac{1}{8}\int _{\mathbb {R}^3}\frac{1}{r^2}u_\theta ^2(2u_\theta ^2u_z-\partial _z(u_\theta ^2))\psi ' =:I_1+I_2+I_3. \end{aligned}$$

As before, we choose $\psi \in C_c^\infty ((-2,2))$ with $\psi \equiv 1$ in $[-1,1]$ and define the translates $\psi _z(x):=\psi (x-z)$ for all $z\in {\mathbb {R}}$. We consider the energies

$$\begin{aligned} E_z(t):=\frac{1}{4}\int _{\mathbb {R}^3}\frac{u_\theta ^4}{r^2}\psi _z ,\qquad&F_z(t):=\frac{3}{4}\int _{t_0}^t\int _{\mathbb {R}^3}\Big |\nabla \frac{u_\theta ^2}{r}\Big |^2\psi _z,\\ E(t):=\sup _{z\in {\mathbb {R}}}E_z(t),\qquad&F(t):=\sup _{z\in {\mathbb {R}}}F_z(t). \end{aligned}$$

By Step 1 and Sobolev embedding,

$$\begin{aligned} |I_1|&\lesssim \Vert u_r\Vert _{L_{3-{\textrm{uloc}}}^6}\Big \Vert r^{-\frac{1}{2}}\frac{u_\theta ^2}{r}\Big \Vert _{L^{12/5}(\Omega )}^2\\&\le \exp \exp A^{O(1)}\left( \Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}^\frac{1}{2}\Big \Vert \nabla \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}^\frac{3}{2}+\Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}^\frac{1}{2}\right) , \end{aligned}$$

where $\Omega :=\mathbb {R}^2 \times {{\,\textrm{supp}\,}}\, \psi$. It follows that

$$\begin{aligned} \int _{t_0}^t|I_1|\le \frac{1}{20}F(t)+\exp \exp A^{O(1)}\int _{t_0}^tE+(t-t_0). \end{aligned}$$

Similarly,

$$\begin{aligned} |I_2|&\lesssim \Vert u_z\Vert _{L^6_{3-{\textrm{uloc}}}}\Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^2_{3-{\textrm{uloc}}}}\Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^3(\Omega )}\\&\le \exp \exp A^{O(1)}E^\frac{1}{2}\left( \Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}^\frac{1}{2}\Big \Vert \nabla \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}^\frac{1}{2}+\Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )} \right) , \end{aligned}$$

which yields the same bound as $I_1$. Finally,

$$\begin{aligned} |I_3|&=\frac{1}{8}\bigg |\int _{\mathbb {R}^3}\frac{u_\theta ^2}{r}\partial _3\frac{u_\theta ^2}{r}\psi '\bigg |\lesssim \Big \Vert \frac{u_\theta ^2}{r}\Big \Vert _{L_{3-{\textrm{uloc}}}^2}\Big \Vert \nabla \frac{u_\theta ^2}{r}\Big \Vert _{L^2(\Omega )}, \end{aligned}$$

so we have

$$\begin{aligned} \int _{t_0}^t|I_3|\le \frac{1}{20}F(t)+\int _{t_0}^tO(E). \end{aligned}$$

Summing and taking the supremum over $z\in {\mathbb {R}}$ gives the claim of this step.

Step 3. We deduce that

$$\begin{aligned} \Vert u\Vert _{L_t^\infty L_{3-{\textrm{uloc}}}^6([t_0,1]\times {\mathbb {R}^3})}\le \exp \exp A^{O(1)}, \end{aligned}$$

(75)

where

$$\begin{aligned} t_0 :=1-\exp (-\exp A^{O(1)}). \end{aligned}$$

Indeed, Lemma 4.2 and Proposition 5.2 give a $t_0\in [1-\exp (- \exp A^{C_0}),1]$ such that $\Vert r^{-\frac{1}{2}}u_\theta (t_0)\Vert _{L_x^4({\mathbb {R}^3})}\le \exp \exp A^{2C_0}$. Therefore, applying Grönwall’s inequality to the claim of the previous step,

$$\begin{aligned} \left\| \frac{u_\theta }{r^{\frac{1}{2}}}\right\| _{L_t^\infty L_{3-{\textrm{uloc}}}^4([t_0,1]\times {\mathbb {R}^3})}\le \exp \exp A^{O(1)}. \end{aligned}$$

Combining this with Proposition 5.2 and Hölder’s inequality,

$$\begin{aligned}&\Vert u_\theta \Vert _{L_t^\infty L_{3-{\textrm{uloc}}}^6([t_0,1]\times {\mathbb {R}^3})}\\&\le \Vert ru_\theta \Vert _{L_x^\infty (\{r\le 1\})}^\frac{1}{3}\Vert r^{-\frac{1}{2}}u_\theta \Vert _{L_t^\infty L_{3-{\textrm{uloc}}}^4([t_0,1]\times {\mathbb {R}^3})}^\frac{2}{3}+\Vert u\Vert _{L_t^\infty L_x^6([t_0,1]\times \{r>1\})}\\&\le \exp \exp A^{O(1)}, \end{aligned}$$

which, together with Step 1, implies (75).

We note that Step 3 already provides a subcritical local regularity condition of the type of Ladyzhenskaya-Prodi-Serrin, which guarantees local boundedness of all spatial derivatives of u, and can be proved by employing the vorticity equation for example (see [35, Theorem 13.7]). In the last step below we use a robust tail estimate of the pressure function (recall Lemma 2.1) to provide a simpler justification of pointwise bounds by $\exp \exp A^{O(1)}$.

Step 4. We prove that, if $\Vert u \Vert _{L^\infty ([1-t_1 ,1 ] ; W^{k-1,6}_{{\textrm{uloc}}} )} \lesssim \exp \exp A^{O(1)}$ for some $k\ge 1$ and $t_1=\exp (-\exp A^{O(1)})$, then the same is true for k (with some other $t_1$ of the same order).

Let $I =[a,b] \subset [t_1,1]$, and let $\chi \in C^\infty (\mathbb {R})$ be such that $\chi (t)=0$ for $t<a+(b-a)/8$ and $\chi (t)=1$ for $t>(a+b)/2$. We set $\phi \in C_c^\infty (B(0,2);[0,1])$ such that $\phi =1$ on B(0, 1/2) and $\sum _{j\in \mathbb {Z}^3} \phi _j =1$, where $\phi _j :=\phi (\cdot - j)$ for each $j\in \mathbb {R}^3$.

Letting $v:=\chi \phi \nabla ^k u$ we see that $v(t_1)=0$, and

$$\begin{aligned} \begin{aligned} v_t -\Delta v&= \underbrace{- \chi ' \phi \nabla ^k u - 2\chi \nabla \phi \cdot \nabla (\nabla ^k u) - \chi \Delta \phi (\nabla ^k u)}_{=: f_1}-\chi \phi \textrm{div}(1+T )\nabla ^k (u\otimes u ) \\&= f_1 - \phi \textrm{div}(1+T ) ((\chi \nabla ^k u \otimes u + u\otimes \chi \nabla ^k u ){\tilde{\phi }} )\\&\quad -\chi \phi \textrm{div}(1+T) \sum _{\begin{array}{c} |\alpha |+|\beta |+|\gamma |=k \\ |\alpha |,|\beta |< k \end{array}} C_{\alpha ,\beta , \gamma } (D^\alpha u\otimes D^\beta u D^\gamma \tilde{ \phi } ) \\&\quad -\chi \phi \textrm{div}T \nabla ^k (u\otimes u (1-\tilde{\phi } ) ) \\&=: f_1 + f_2 + f_3 + f_4. \end{aligned} \end{aligned}$$

We can now estimate $\Vert v(t) \Vert _6$, by extracting the same norm on the right-hand side and ensuring that the length of the interval is sufficiently small, so that the norm can be absorbed. Namely,

$$\begin{aligned} \Vert v(t) \Vert _{6}&= \left\| \int _{a}^t\textrm{e}^{(t-t')\Delta } f_1 (t')\textrm{d}t' + \int _{a}^t\textrm{e}^{(t-t')\Delta }f_2(t' ) \textrm{d}t'+ \int _{a}^t\textrm{e}^{(t-t')\Delta } f_3 (t')\textrm{d}t' \right. \\&\quad \left. + \int _{a}^t\textrm{e}^{(t-t')\Delta } f_4 (t')\textrm{d}t' \right\| _6 \\&\le \left( \Vert \chi \nabla ^k u {\tilde{\phi }} \Vert _{L^\infty ([a,t]; L^{6} )} + \Vert \chi ' \nabla ^{k-1} u {\tilde{\phi }} \Vert _{L^\infty ([a,1]; L^{6} )} \right) \int _{a}^t \Vert \Psi (t-t' ) \Vert _{W^{1,1}} \textrm{d}t' \\&\quad + \Vert \chi \nabla ^k u \tilde{ \phi }^{1/2} \Vert _{L^\infty ([a,t];L^6 )} \Vert u \tilde{ \phi }^{1/2}\Vert _{L^\infty ([a,t ]; L^6) } \int _{a}^t \Vert \Psi (t-t') \Vert _{W^{1,6/5}} \textrm{d}t' \\&\quad + \Vert u \Vert _{L^\infty ([a,1]; W^{k-1,6}_{{\textrm{uloc}}})}^2 \int _{a}^t \Vert \Psi (t-t' ) \Vert _{W^{1,6/5}} \textrm{d}t' \\&\quad + \Vert \textrm{div}\, T (u\otimes u (1-{\tilde{\phi }} ) )\Vert _{L^\infty ([a,1]; W^{k,6}(B(0,2 )) )} \int _{a}^t \Vert \Psi (t-t' ) \Vert _{1} \textrm{d}t' \\&\le \Vert \chi \nabla ^k u \Vert _{L^\infty ([a,t];L^6_{{\textrm{uloc}}} )} \left( (b-a)^{1/2} + \exp \exp A^{O(1)} (b-a)^{1/4} \right) + \exp \exp A^{O(1)} \end{aligned}$$

for each $t\in (a,b)$, where we used Young’s inequality, heat estimates (24) and the Calderón-Zygmund inequality. By replacing $\phi$ (in the definition of v) by $\phi _z$ for any $z\in \mathbb {R}^3$, we obtain the same bound, and so

$$\begin{aligned}{} & {} \Vert \chi \nabla ^k u \Vert _{L^\infty ([a,b];L^6_{{\textrm{uloc}}} )}\\{} & {} \quad \le \Vert \chi \nabla ^k u \Vert _{L^\infty ([a,b];L^6_{{\textrm{uloc}}} )} (b-a)^{1/4} \exp \exp A^{O(1)} + \exp \exp A^{O(1)}. \end{aligned}$$

Thus, for any b, a such that $t_1\le a<b\le 1$ and $(b-a)^{1/4} \le \exp \exp A^{O(1)}/2$ we can absorb the first term on the right-hand side by the left-hand side to obtain

$$\begin{aligned} \Vert \nabla ^k u \Vert _{L^\infty ([(a+b)/2,b]; L^6_{{\textrm{uloc}}} )} \le \exp \exp A^{O(1)}. \end{aligned}$$

Since the upper bound is independent of the location of $[a,b]\subset [t_1, 1]$, we obtain the claim.

Appendix A: Quantitative Parabolic Theory

Here we prove Proposition 5.1. Namely, we consider parabolic cylinders

$$\begin{aligned} Q_R^{\lambda ,\theta }(t_0,x_0):=[t_0-\theta R^2,t_0]\times B(x_0,\lambda R),\quad Q_R^{\lambda ,\theta }:=Q_R^{\lambda ,\theta }(0,0),\quad Q_R :=Q_{R}^{1,1} \end{aligned}$$

and we consider Lipschitz solutions V of $\mathcal {M}V = 0$ on $Q_R^{\lambda , \theta }$, namely we suppose that

$$\begin{aligned} \int _{\mathbb {R}} \int \left( \partial _t V \phi +\nabla V \cdot \nabla \phi + b\cdot \nabla V \phi \right) =0 \end{aligned}$$

(76)

for all $\phi \in C_c^\infty (Q_R^{\lambda ,\theta } )$, where the (distributional) supports of $\textrm{div}b$ and V are disjoint. Moreover we assume that (54) holds, namely

$$\begin{aligned} {\mathcal {N}}(R ):=2+\sup _{{R'}\le 2R }(R')^{-\alpha }\Vert b\Vert _{L_t^{\ell }L_x^q(Q_{R'})}<\infty , \end{aligned}$$

where $\alpha :=\frac{n}{q}+\frac{2}{\ell }-1\in [0,1)$. We also say that V is a subsolution (or supersolution) of $\mathcal {M}V=0$, i.e. $\mathcal {M}V \le 0$ (or $\mathcal {M}V \ge 0$), if (76) holds with “$=$” replaced by “$\le$” (or “$\ge$”) for all nonnegative test functions.

We will show that

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{B(r)}V(0)\lesssim \left( \frac{r}{R}\right) ^\gamma \mathop {{{\,\textrm{osc}\,}}}\limits _{Q(R)}V \end{aligned}$$

(77)

for all $r\le R$, where $\gamma =\exp (-{{\mathcal {N}}}^{O(1)})$.

To this end we first prove the Harnack inequality for Lipschitz subsolutions of $\mathcal {M} V =0$.

Lemma A.1

(based on Lemma 3.1 in [24]) Let V be a Lipschitz solution of ${\mathcal {M}}{\mathcal {V}}\le 0$ in $Q_R^{\lambda ,\theta }$ where $\lambda \in (1,2]$ and $\theta \in (0,1]$. Then

Proof

We first note that, for any r, a satisfying

$$\begin{aligned} \frac{3}{r}+\frac{2}{a} \in \left[ \frac{3}{2},\frac{5}{2} \right] , \end{aligned}$$

we have the interpolation inequality

$$\begin{aligned} \Vert \zeta U\Vert _{L_t^aL_x^r(Q_R^{\lambda ,\theta })}&\lesssim _{\lambda ,\theta }R^{\frac{3}{r}+\frac{2}{a}-\frac{3}{2}}\Vert \zeta U\Vert _{{\mathcal {V}}(Q_R^{\lambda ,\theta })}, \end{aligned}$$

(78)

by [19, (3.4) in Chapter II], where ${\mathcal {V}}$ is the energy space $L_t^\infty L_x^2\cap L_{t}^2\dot{H}^1_x$.

Since V is a subsolution, we have, for a non-negative test function $\eta$,

$$\begin{aligned} \int _{Q_R^{\lambda ,\theta }}(\partial _tV\eta +\nabla V\cdot \nabla \eta +b\cdot \nabla V\eta )\le 0. \end{aligned}$$

We let $\eta :=\varphi '(V)\xi$ where $\xi$ is a cutoff function vanishing on a neighborhood of the boundary of $Q_R^{\lambda ,\theta }$, and $\varphi$ is a convex function vanishing on ${\mathbb {R}}_-$. Taking $U:=\varphi (V)$ we obtain

$$\begin{aligned} \int _{Q_R^{\lambda ,\theta }\cap \{V>0\}}\left( \partial _tU\xi +\nabla U\cdot \nabla \xi +\frac{\varphi ''(V)}{\varphi '(V)^2}|\nabla U|^2\xi +b\cdot \nabla U\xi \right) \le 0. \end{aligned}$$

We now take

$$\begin{aligned} \varphi (\tau ):=\tau _+^p\quad (p>1)\qquad \text { and }\qquad \xi :=\chi _{\{t<{\overline{t}}\}}U\zeta ^2, \end{aligned}$$

where $\zeta$ is a smooth cutoff function in $Q_R^{\lambda ,\theta }$ and ${\overline{t}}\in (-\theta R^2,0)$,

$$\begin{aligned} \int _{B_{\lambda R}}(\zeta U)^2({\overline{t}})dx+ & {} \int _{Q_R^{\lambda ,\theta }\cap \{t<{\overline{t}}\}}(2-p^{-1})|\nabla U|^2\zeta ^2+U\nabla U\cdot \nabla (\zeta ^2)\nonumber \\{} & {} +\frac{1}{2}b\cdot \nabla (U^2) \zeta ^2-\partial _t(\zeta ^2)U^2\le 0. \end{aligned}$$

(79)

Using integration by parts and recalling the assumption $\textrm{div}\, b \ge 0$, we can apply Hölder’s inequality to obtain

$$\begin{aligned} \int _{Q_R^{\lambda ,\theta }\cap \{t<{\overline{t}}\}}b\cdot \nabla (U^2)\zeta ^2&\ge -\int _{Q_R^{\lambda ,\theta }\cap -\{t<{\overline{t}}\}}b\cdot \nabla (\zeta ^2)U^2\\&\hspace{-1in}\ge -\Vert b\Vert _{L_t^\ell L_x^q(Q_R^{\lambda ,\theta })}\Vert |U|^\frac{1}{s}\zeta ^{\frac{1}{s}-1}\nabla \zeta \Vert _{L_{t,x}^{2s}(Q)}\Vert (\zeta |U|)^{2-\frac{1}{s}}\Vert _{L_t^{(1-\frac{1}{2s}-\frac{1}{\ell })^{-1}}L_x^{(1-\frac{1}{2s}-\frac{1}{q})^{-1}}(Q)}\\&\hspace{-1in}=-\Vert b\Vert _{L_t^\ell L_x^q(Q_R^{\lambda ,\theta })}\Vert U\zeta ^{1-s}|\nabla \zeta |^s\Vert _{L_{t,x}^2(Q_R^{\lambda ,\theta })}^\frac{1}{s}\Vert \zeta U\Vert _{L_t^aL_x^r(Q_R^{\lambda ,\theta })}^{2-\frac{1}{s}}, \end{aligned}$$

where $s>2$ and r and a are defined by

$$\begin{aligned} \frac{1}{2s}+\frac{1}{q}+\frac{1}{r}\Big (2-\frac{1}{s}\Big )=1,\quad \frac{1}{2s}+\frac{1}{\ell }+\frac{1}{a}\Big (2-\frac{1}{s}\Big )=1. \end{aligned}$$

Applying Young’s inequality to separate the last term, and utilizing the interpolation inequality (78) (which is valid since

$$\begin{aligned} \frac{3}{r}+\frac{2}{a}=\frac{3}{2}+1-2\left( 1+2/\left( \frac{3}{q}+\frac{2}{\ell }\right) \right) ^{-1} \in (3/2,11/6), \end{aligned}$$

as needed) we obtain, after plugging into the local energy inequality (79),

$$\begin{aligned} \sup _{t\in [-\theta R^2,0]}\int _{B_{\lambda R}}(\zeta U)^2dx+\int _{Q_R^{\lambda ,\theta }\cap \{t<{\overline{t}}\}}(2-p^{-1})|\nabla U|^2\zeta ^2+U\nabla U\cdot \nabla (\zeta ^2)-\partial _t(\zeta ^2)U^2\\ -O\left( R^2\Vert b\Vert _{L_t^\ell L_x^q(Q_R^{\lambda ,\theta })}^{2s}\Vert U\zeta ^{1-s}|\nabla \zeta |^s\Vert _{L_{t,x}^2(Q_R^{\lambda ,\theta })}^2\right) -\frac{1}{10}\Vert \zeta U\Vert _{{\mathcal {V}}(Q_R^{\lambda ,\theta })}^2\le 0. \end{aligned}$$

Absorbing $\nabla U$ from the term on the third term on the left-hand side by the second term we obtain

$$\begin{aligned} \Vert \zeta U\Vert _{{\mathcal {V}}(Q_R^{\lambda ,\theta })}^2\lesssim \int _{Q_R^{\lambda ,\theta }}\left( |\nabla \zeta |^2+\zeta |\partial _t\zeta |+R^2\Vert b\Vert _{L_t^\ell L_x^q(Q_R^{\lambda ,\theta })}^{2s}\zeta ^{2-2s}|\nabla \zeta |^{2s}\right) U^2. \end{aligned}$$

We now set

$$\begin{aligned} \lambda _m:=1+2^{-m}(\lambda -1)\qquad \text { and }\qquad \theta _m:=\frac{1}{2}\theta (1+4^{-m}), \end{aligned}$$

and we substitute $\zeta$ with $\zeta _m$ such that

$$\begin{aligned} \zeta _m\equiv 1\text { in }Q_R^{\lambda _{m+1},\theta _{m+1}},\quad \zeta _m\equiv 0\text { outside }Q_R^{\lambda _m,\theta _m},\quad |\partial _t\zeta _m|\le \frac{4^mC}{\theta R^2},\quad \frac{|\nabla \zeta _m|}{\zeta _m^{1-\frac{1}{s}}}\le \frac{2^mC }{R}, \end{aligned}$$

where C may depend on $\lambda$. Then the energy estimate and (78), taken with $r=l=10/3$, yield

$$\begin{aligned} \Vert \zeta _mU\Vert _{L_{t,x}^{10/3}(Q_R^{\lambda ,\theta })}&\lesssim \Vert \zeta _mU\Vert _{{\mathcal {V}}(Q_R^{\lambda ,\theta })}\le CR^{-1}(\theta ^{-\frac{1}{2}}+2^m+{\mathcal {N}}^s)2^{ms}\Vert U\Vert _{L_{t,x}^2(Q_R^{\lambda ,\theta })}. \end{aligned}$$

Recalling the definition of U and replacing p with $p_m:=(5/3)^m$, Hölder’s inequality implies

Iterating, we have

and we conclude by taking $m\rightarrow \infty$. $\square$

In the next three lemmas we focus on nonnegative solutions to $\mathcal {M}V\le 0$ and we find lower bounds on the mass distribution of such solutions. We first show that if $V\ge k$ in $Q_R$, except for a small (quantified) “portion of $Q_R$”, then in fact $V\ge k/2$ everywhere in a smaller cylinder.

Lemma A.2

(based on part 2 of Corollary 3.1 in [24]) If V is a non-negative solution of ${\mathcal {M}}{\mathcal {V}}\ge 0$ in $Q_R^{\lambda ,\theta }$ and

$$\begin{aligned} |\{V<k\}\cap Q_R^{\lambda ,\theta }|\le ({\mathcal {N}}/\theta )^{-5C}|Q_R^{\lambda ,\theta }|, \end{aligned}$$

then

$$\begin{aligned} V\ge \frac{k}{2}\quad \text {in }Q_R^{1,\theta /2}. \end{aligned}$$

Proof

We apply Lemma A.1 to $k-V$ to find

which implies the result. $\square$

We now show that, if the cylinder $Q_R^{1,\theta }$ is flat enough, then a lower bound on the bottom lid of $Q_R^{1,\theta }$ (i.e. at $t=-\theta R^2$) implies a similar lower bound at every t.

Lemma A.3

(based on Lemma 3.2 in [24]) Suppose V is non-negative with ${\mathcal {M}}{\mathcal {V}}\ge 0$ in a neighbuorhood of $Q_R^{1,\theta _0}$ and

$$\begin{aligned} |\{V(-\theta _0R^2)\ge k\}\cap B_R|\ge \delta _0|B_R| \end{aligned}$$

for some $\delta _0>0$ and $\theta _0\le C^{-1}\delta _0^6{\mathcal {N}}^{-1}$. Then

$$\begin{aligned} |\{V({\overline{t}})\ge \frac{1}{3}\delta _0k\}\cap B_R|\ge \frac{1}{3}\delta _0|B_R| \end{aligned}$$

for all ${\overline{t}}\in [-\theta _0R^2,0]$.

Proof

By the calculations in [24], with $\zeta$ a smooth cutoff function supported in $B_R$,

$$\begin{aligned}&\int _{B_R}(V({\overline{t}})-k)_-^2\zeta ^2+\int _{Q_R^{1,\theta _0}}\chi _{\{t<{\overline{t}}\}}|\nabla (V-k)_-|^2\zeta ^2\le \int _{B_R}(V(-\theta _0R^2)-K)_-^2\zeta ^2 \end{aligned}$$

(80)

$$\begin{aligned}&\quad \quad +\int _{Q_R^{1,\theta _0}}\chi _{\{t<{\overline{t}}\}}(V-k)_-^2\big (O(|\nabla \zeta |^2)+b\cdot \nabla (\zeta ^2)+(\textrm{div}b)\zeta ^2\big ). \end{aligned}$$

(81)

We choose $\zeta$ such that $\zeta \equiv 1$ in $B_{(1-\sigma )R}$ and $|\nabla \zeta |\le \frac{2}{\sigma R}$ where $\sigma <1$ is to be specified. Note that, due to (53),

$$\begin{aligned} \int _{Q_R^{1,\theta _0}}\chi _{\{t<{\overline{t}}\}}(V-k)_-^2(\textrm{div}b)\zeta ^2&\le k^2\int _{Q_R^{1,\theta _0}}\chi _{\{t<{\overline{t}}\}}(\textrm{div}b)\zeta ^2\\&=-k^2\int _{Q_R^{1,\theta _0}}\chi _{\{t<{\overline{t}}\}}b\cdot \nabla (\zeta ^2). \end{aligned}$$

Then the right-hand side of (80) is bounded by

$$\begin{aligned} k^2\Big ((1-\delta _0)|B_R|+O(\theta _0\sigma ^{-2}|B_R|)+\frac{4}{\sigma R}\Vert b\Vert _{L_t^\ell L_x^q(Q_R)}\Vert 1\Vert _{L_t^{\ell '}L_x^{q'}(Q_R^{1,\theta _0})}\Big ). \end{aligned}$$

From here one can proceed with the argument exactly as in [24] to arrive at

$$\begin{aligned} \left| \left\{ V({\overline{t}})<\frac{1}{3}\delta _0k\right\} \cap B_R\right| \le \left( 1-\frac{1}{3}\delta _0\right) ^{-2}(1-\delta _0+O(\sigma +\sigma ^{-2}\theta _0+\sigma ^{-1}\theta _0^{2/\ell '}{\mathcal {N}})). \end{aligned}$$

Setting $\sigma =C^{-1/5}\delta _0^2$ and $\theta _0$ as above proves the claimed bound. $\square$

We now show that for any given “portion of $Q^{1,\theta }_R$” (in the sense of a set with the measure arbitrarily close to $|Q^{1 , \theta }|$) V is greater or equal a constant multiple of some lower bound, if, for each t, the lower bound occurs at least on some “portion of $B_R$”. Although this enables us to obtain a lower bound on almost the entire cylinder, we lose an exponential in the process.

Lemma A.4

(based on Lemma 3.3 in [24]) Let $V\ge 0$ be a solution of ${\mathcal {M}}{\mathcal {V}}\ge 0$ in $Q_R^{\lambda ,\theta }$ satisfying

$$\begin{aligned} |\{V(t)\ge k_0\}\cap B_R|\ge \delta _1|B_R|\text { for all }t\in [-\theta R^2,0] \end{aligned}$$

for some $k_0>0$, $\delta _1>0$. Then for any $\mu >0$ and $s>C({\mathcal {N}}+\theta ^{-1}) /(\delta _1\mu )^2$,

$$\begin{aligned} |\{V<2^{-s}k_0\}\cap Q_R^{1,\theta }|\le \mu |Q_R^{1,\theta }|. \end{aligned}$$

Proof

With $k_m=2^{-m}k_0$, we define

$$\begin{aligned} {\mathcal {E}}_m(t):=\{x\in B_R :k_{m+1}\le V(x,t)<k_m\};\quad {\mathcal {E}}_m:=\{(t,x)\in Q_R^{1,\theta } :x\in {\mathcal {E}}_m(t)\}. \end{aligned}$$

Integrating the inequality ${\mathcal {M}}{\mathcal {V}}\ge 0$ against the test function $\eta =(V-k_m)_-\xi (x)^2$ where $\xi$ is a smooth cutoff vanishing in a neighborhood of $\partial B_{\lambda R}$ and satisfying $\xi \equiv 1$ in $B_R$,

$$\begin{aligned} \int _{Q^{\lambda ,\theta }_{ R} \cap \{ V< k_m \} } | \nabla V |^2 \xi ^2&\le \int _{Q^{\lambda , \theta }_R } | \nabla (V-k_m )_- |^2 \xi ^2 \lesssim \left. \int _{B_{\lambda R}\cap \{ V< k_m \} } (V-k_m)_-^2 \xi ^2 \right| _{t=-\theta R^2 } \nonumber \\&\quad +\int _{-\theta R^2 }^0 \int _{B_{\lambda R}\cap \{ V< k_m \}}(V-k_m)_-^2 |\nabla \xi |^2 + 2 (V-k_m)_-^2 \xi b \cdot \nabla \xi \nonumber \\&\lesssim k_m^2R^n(1+\theta {\mathcal {N}}), \end{aligned}$$

(82)

by Hölder’s inequality and the trivial bound $0\le (V-k_m)_-\le k_m$. From De Giorgi’s inequality [19, (5.6) in Chapter II],

$$\begin{aligned} (k_m-k_{m+1})|\{V(t)<k_{m+1}\}\cap B_R|\lesssim \frac{R}{\delta _1}\int _{{\mathcal {E}}_m(t)}|\nabla V(t)| \end{aligned}$$

for all $t\in [-\theta R^2,0]$. Integrating in time, squaring, and applying Cauchy-Schwarz gives

$$\begin{aligned} k_{m+1}^2\left| \{V<k_{m+1}\}\cap Q_R^{1,\theta }\right| ^2\lesssim \frac{R^2}{\delta _1^2}\int _{{\mathcal {E}}_m}|\nabla V|^2\textrm{d}x\textrm{d}t|{\mathcal {E}}_m|. \end{aligned}$$

Combined with (82), this gives

$$\begin{aligned} \left| \{V<k_{m+1}\}\cap Q_R^{1,\theta }\right| ^2\lesssim \delta _1^{-2}R^{n+2}(1+\theta {\mathcal {N}})|{\mathcal {E}}_m|. \end{aligned}$$

We conclude

$$\begin{aligned} s\left| \{V<k_s\}\cap Q_R^{1,\theta }\right| ^2&\le \sum _{m=0}^{s-1}\left| \{V<k_{m+1}\}\cap Q_R^{1,\theta }\right| ^2\\&\lesssim \delta _1^{-2}R^{n+2}(1+\theta {\mathcal {N}})\sum _{m=0}^{s-1}|{\mathcal {E}}_m|\\&\lesssim \delta _1^{-2}(\theta ^{-1}+{\mathcal {N}})|Q_R^{1,\theta }|^2. \\ \end{aligned}$$

$\square$

We can now combine Lemmas A.2–A.4 to obtain a pointwise lower bound for V in the interior of a cylinder, with an exponential dependence on $\mathcal {N}$.

Lemma A.5

(based on part 1 of Corollary 3.2 in [24]) If V is a non-negative solution of ${\mathcal {M}}{\mathcal {V}}\ge 0$ in $Q_R^{2,1}$ and

$$\begin{aligned} |\{V(-\Theta R^2)\ge k\}\cap B_R|\ge \delta |B_R| \end{aligned}$$

for some $k>0$ and $\Theta \le C^{-1}\delta ^6{\mathcal {N}}^{-1}$, then

$$\begin{aligned} V\ge \exp (-\delta ^{-2}({\mathcal {N}} /\Theta )^{20C})k\quad \text {in }Q_R^{1,\Theta /2}. \end{aligned}$$

Proof

This is a straightforward application of Lemmas A.3, A.4, and A.2 in sequence, with the latter two applied with $R\rightarrow \frac{3}{2}R$ to compensate for the shrinking domain in Lemma A.2. $\square$

By considering $V-\inf V$ and $\sup V - V$ the above lemma now allows us to estimate oscillations of solutions to $\mathcal {M} V=0$ with no sign restrictions.

Lemma A.6

(based on Lemma 3.5 of [24]) If V solves ${\mathcal {M}} V=0$ in $Q_R^{2,1}$ then

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{Q^{(1)}}V\le (1-\exp (-{\mathcal {N}}^{50C}))\mathop {{{\,\textrm{osc}\,}}}\limits _{Q^{(2)}}V \end{aligned}$$

where $Q^{(1)}=Q_R^{1,\Theta /2}$, $Q^{(2)}=Q_R^{2,1}$, and $\Theta =C^{-2}{\mathcal {N}}^{-1}$.

Proof

Consider the positive supersolutions $V_1=V-\inf _{Q^{(2)}}V$ and $V_2=\sup _{Q^{(2)}}V-V$. With $k=\mathop {{{\,\textrm{osc}\,}}}\limits _{Q^{(2)}}V$, clearly we must have $|\{V_i(-\Theta R^2)\ge k\}\cap B_{2R}|\ge |B_{2R}|/2$ for either $i=1$ or $i=2$. Fix this i, so $V_i$ obeys the hypotheses of Lemma A.5. Let us assume for concreteness that $i=1$; the other case is analogous. Then by the lemma,

$$\begin{aligned} \inf _{Q^{(2)}}V+\exp (-{\mathcal {N}}^{50C})\mathop {{{\,\textrm{osc}\,}}}\limits _{Q^{(2)}}V\le V\le \sup _{Q^{(2)}}V \end{aligned}$$

for all $(t,x)\in Q^{(1)}$, which immediately implies the result. $\square$

Finally, iterating Lemma A.6 we obtain the required Hölder continuity (77), i.e. we can prove Proposition 5.1.

Proof of Proposition5.1

Iterating Lemma A.6, we have

$$\begin{aligned} \mathop {{{\,\textrm{osc}\,}}}\limits _{Q^{2,1}_{(\Theta /2)^{k/2}R/2}}V\le (1-\exp (-{\mathcal {N}}^{50C}))^k\mathop {{{\,\textrm{osc}\,}}}\limits _{Q_{R/2}^{2,1}}V. \end{aligned}$$

We conclude upon taking $k=\lfloor \log \frac{R}{r}(\log \frac{2}{\Theta })^{-1}\rfloor$. $\square$