# Blow-up of dynamically restricted critical norms near a potential Navier-
Stokes singularity

Tobias Barker  Department of Mathematical Sciences, University of Bath, Bath
BA2 7AY, UK<EMAIL_ADDRESS>,  Pedro Gabriel Fernández-Dalgo  Cergy
Paris Université, Laboratoire de Mathématiques AGM, UMR CNRS 8088, France
<EMAIL_ADDRESS>and  Christophe Prange  Cergy Paris Université,
Laboratoire de Mathématiques AGM, UMR CNRS 8088, France
<EMAIL_ADDRESS>
###### Abstract.

In this paper we develop new methods to obtain regularity criteria for the
three-dimensional Navier-Stokes equations in terms of dynamically restricted
endpoint critical norms: the critical Lebesgue norm in general or the critical
weak Lebesgue norm in the axisymmetric case. This type of results is inspired
in particular by a work of Neustupa (2014), which handles certain non endpoint
critical norms. Our work enables to have a better understanding of the
nonlocal effect of the pressure on the regularity of the solutions.

Keywords:  Navier-Stokes equations, local regularity, critical norms,
dynamically restricted quantities, potential singularities, time weighted
energy estimates, axisymmetric solutions

AMS classification:  35A99, 35B44, 35B65, 35Q30, 76D05.

##  1\. Introduction

One currently active research direction on the three-dimensional Navier-Stokes
equations

$\partial_{t}v-\Delta v+v\cdot\nabla v+\nabla p=0,\phantom{space}\nabla\cdot
v=0.$ (1.1)

is the investigation of scales that play a role in potential singularity
formation. Many results were proved in the past ten years that: (i) show
dynamically restricted regularity criteria and on the other side of the coin
(ii) show that certain scale-invariant quantities accumulate/concentrate on
specific scales near a potential singularity. Let us cite (non exhaustively)
the following works: [13, 7, 12, 22] on frequency localized results, [21, 16,
3, 18, 8] on spatially localized results. For a more extensive bibliography,
we refer to the recent survey [5].

In this vein, let us mention in particular the results of Nečas and Neustupa
[25] and of Neustupa [23, 24]. The papers [25, 23] lead to [24], where the
author proves a dynamically restricted version of the Ladyženskaja-Prodi-
Serrin regularity criteria, namely if $v$ is a finite-energy weak solution to
(1.1) in $\mathbb{R}^{3}\times(-1,0)$ such that $0$ is a first-time
singularity and such that $(0,0)$ is a singular point,111By definition, the
point $(0,0)$ is a _singular point_ if for all $r\in(0,1)$, $v\notin
L^{\infty}(Q_{(0,0)}(r))$, where $Q_{(0,0)}(r)$ is the parabolic cylinder
$B_{0}(r)\times(-r^{2},0)$. The point $(0,0)$ is called a _regular point_ if
it is not a singular point. then

$\int\limits_{-1}^{0}\Bigg{(}\int\limits_{B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{-at})}|v|^{p}\,dx\Bigg{)}^{\frac{q}{p}}\,ds=\infty$

for critical Lebesgue exponents $\frac{2}{q}+\frac{3}{p}=1$, $3\leq q<\infty$,
$3<p\leq 9$ and a certain $a>0$ (see Footnote 3). In this paper, we address
the endpoint $p=3$ and $q=\infty$. Hence, the result stated below generalizes
in particular the celebrated result of Escauriaza, Seregin and Šverák [14]. In
the case that $v$ is axisymmetric we obtain strengthened results, which
generalize the results of [11, 10, 19, 29]. For an illustration of Neustupa’s
result and our theorem below, we refer to Figure 1.

######  Theorem A.

Let $v$ be a finite-energy weak solution to the Navier-Stokes equations (1.1)
in $\mathbb{R}^{3}\times(-1,0)$ such that $0$ is a first-time
singularity.222In particular, $v\in
C^{\infty}(-1,T;C^{\infty}(\mathbb{R}^{3}))$ for all $T\in(-1,0)$. We
introduce this assumption in order to remove certain technicalities. Notice
that the framework of first-time singularities is relevant for the study of
the global regularity problem for the 3D Navier-Stokes equations. Assume
$(0,0)$ is a singular point.
Then

$\limsup_{t\rightarrow 0^{+}}\|v(\cdot,t)\|_{L^{3}(B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{-at}))}=\infty.$ (1.2)

If in addition, the solution is axisymmetric, then

$\limsup_{t\rightarrow
0^{+}}\|v(\cdot,t)\|_{L^{3,\infty}(B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{-at}))}=\infty.$ (1.3)

Here, and throughout the paper, $a:=\lambda_{S}(B_{0}(1))>\pi$, where
$\lambda_{S}(B_{1})$ is the first eigenvalue of the Dirichlet-Stokes operator
on $B_{0}(1)$.333From our estimates, we can allow
$a\in\big{(}0,\frac{4}{3}\lambda_{S}(B_{0}(1))\big{)}$. We choose
$a:=\lambda_{S}(B_{0}(1))$ in order to fix the ideas.

Figure 1. Illustration for Theorem A

Let us remark the following three facts:

  1. (1)

Theorem A is stated in the global setting for first-time singularities of
Leray-Hopf solutions. In view of the global regularity problem for the three-
dimensional Navier-Stokes equations this is an appropriate setting. That said,
Theorem A can be localized i.e. proved for suitable solutions as is the result
of Neustupa [24, Theorem 1]. Localizing requires minor technical changes in
the proof.

  2. (2)

We believe that Theorem A can be quantified using the strategy developed in
the works [32, 4, 2, 27] and further explained in the recent survey paper [5].
The scale-invariant controls obtained in Section 4 play a key role for this
quantification as do the scale-invariant control $L^{\infty}_{t}L^{3}_{x}$ in
[32, 2] or the Type I assumption in [4].

  3. (3)

After finishing the paper, we were made aware of the work [20] dealing with
similar hollowed critical conditions in the axisymmetric case. There, De
Giorgi type energy estimates and Moser type iteration are directly used for
the equation for the swirl, which is pressureless, in order to deduce Hölder
continuity.

###  1.1. Heuristics for the proof

The proof is by contraposition, assuming that one of the quantities in (1.2)
or (1.3) is bounded. To fix the ideas, let us assume that

$\operatorname*{ess\,sup}_{s\in(-1,0)}\|v(\cdot,s)\|_{L^{3}(B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{a(-s)}))}\leq M,$ (1.4)

for some $M\geq 1$.

#### Failure of a direct strategy

It seems not possible to directly prove that (1.4) implies the regularity by
adapting the compactness argument of [14]. Indeed, the absence of a global
scale-invariant assumption prevents us from relying on persistence of
singularity type results for obtaining a non-zero solution from a limiting
procedure. It turns out that proving a scale-invariant bound inside the
parabolic is the cornerstone of our strategy.

#### Key difficulty

We work in physical space variables rather than renormalized variables of Giga
and Kohn type as is done by Nečas and Neustupa [25] and Neustupa [23, 24]. Our
study is based on time-weighted scale-invariant local energy controls. The key
point that enables results such as (1.2) or (1.3) to hold is the divergence-
form structure of the non linear term $v\cdot\nabla v=\nabla\cdot(v\otimes v)$
in the Navier-Stokes equations. If the pressure was not there, it would be
easy to obtain a scale-invariant control inside the paraboloid

$\bigcup_{s\in(-1,0)}B_{0}(\theta_{a}(s)),$ (1.5)

because the variation of the local energy localized on the parabolid only
involves a priori control outside the paraboloid, for instance in the region

$\bigcup_{s\in(-1,0)}B_{0}(2\theta_{a}(s))\setminus B_{0}(\theta_{a}(s)).$

The major difficulty we have to face is the pressure term in the local energy
inequality

$\int\limits_{-1}^{0}\int\limits_{\mathbb{R}^{3}}pv\cdot\nabla\Psi^{2}\,dxd\tau.$

Indeed, because of the nonlocality of the pressure, bounding this term
involves estimates of the velocity inside the paraboloid.

#### Key new idea

Our idea to circumvent the problem of the pressure is to introduce a test
function $\Psi_{N}$ that is supported in

$\bigcup_{s\in(-1,0)}B_{0}((N+1)\theta_{a}(s))\setminus
B_{0}(N\theta_{a}(s)),$

for $N\gg 1$ i.e. far away from the region (1.5) where we lack critical
information. Taking $N$ large (in terms of $M$, see (1.4)) enables us to show
that the contribution of the problematic pressure term is negligible compared
to an appropriate localized energy of $v$. This scheme is totally different
from what is done in [24]. There, the fact that non-endpoint critical controls
are considered buys some smallness that allows a control of the pressure term.
Note that in [24] Neustupa eventually applies $\varepsilon$-regularity to
conclude the regularity. Contrary to this, we do not have such smallness in
the endpoint case and do not rely on $\varepsilon$-regularity, see Section 4.

#### Two technical points

First, the constraint on $a$ in Theorem A (see also Footnote 3) comes from the
condition (3.15) that is needed in order to get a damping term in the local
energy inequality.

Second, as mentioned above, we work with scale-invariant weighted local energy
quantities. The weights, see (2) below involve a parameter $\gamma$. In order
for the local energy estimate (2) to be well-defined, we first need to take
$\gamma>0$. This enables us, see Section 3.3, to prove that a certain scale-
invariant kinetic energy is bounded (quantity $f$ in (3.2)), which in turn
enables us to make sense of (2) for $\gamma>-1$ and in particular for
$\gamma=0$. This parameter is used in the final part of the paper, Section 4.

###  1.2. Outline of the paper

Section 2 is concerned with scale-invariant weighted local energy estimates.
The focus is on the control of the pressure term. Section 3 demonstrates the
boundedness of a scale-invariant kinetic energy localized on the paraboloid
(1.5). The final part of the paper, Section 4, is devoted to the proof of
Theorem A. Appendix A surveys some results about Lorentz spaces.

###  1.3. Notations

For $a>0$, we define $\theta_{a}(t)=\sqrt{a(-t)}$. Notice that throughout the
paper $a:=\lambda_{S}(B_{0}(1))$ as stated in Theorem A.

Throughout the paper $C$ is assumed to be a positive universal constant which
may change from line to line. In particular $C$, does not depend on the
parameters $a$, $\gamma$ or $N$ (see Section 2). Note that the inequality
$\ldots\leq C(\ldots)$ is sometimes denoted by $\ldots\lesssim\ldots$. The
notation $C_{b_{1},\ldots\,b_{k}}$ denotes a positive constant depending on
the parameters $b_{1},\ldots\,b_{k}$. Note that the inequality $\ldots\leq
C_{b_{1},\ldots\,b_{k}}(\ldots)$ is sometimes denoted by
$\ldots\lesssim_{b_{1},\ldots\,b_{k}}\ldots$.

##  2\. Weighted energy and pressure estimates

Let $M\geq 1$. In this section we assume

$\operatorname*{ess\,sup}_{s\in(-1,0)}\|v(\cdot,s)\|_{L^{3,\infty}(B_{0}(\sqrt{a})\setminus
B_{0}(\theta_{a}(s)))}\leq M.$ (2.1)

For basic facts about weak Lebesgue and Lorentz spaces we refer to Appendix A.
We mainly rely on the Hölder inequality for Lorentz spaces in the estimates
below, see Proposition A.1. Our objective is to get scale-invariant controls
for the velocity inside the region (1.5) via the local energy inequality and
the use of the critical control (2.1). The final scale-invariant control, see
Proposition 3.1, is obtained in Section 3. Our focus in the present section is
on an appropriate choice of test functions for the local energy inequality and
on the estimates for the pressure.

Let444Later on, $N$ will be taken large depending in particular on $M$, see
(3.15). $N\geq 4$ and $\gamma>0$.555The restriction $\gamma>0$ is needed for
the validity of (2) for all $t\in(s,0]$. We will subsequently show, see (4.2),
that the second term in the left hand side of (2) also makes sense for
$\gamma>-1$. This fact will be used in Section 4. Let $\varphi_{N}\in
C^{\infty}_{c}(\mathbb{R}^{3})$ with $\varphi_{N}$ positive,
$\varphi_{N}(x)=1$ on $B_{0}(N)$, $\operatorname*{supp}\varphi_{N}\subset
B_{0}(N+1)$ and $\|\nabla\varphi_{N}\|_{L^{\infty}}\lesssim 1$. We then test
the Navier-Stokes equations (1.1) with

$\theta_{a}^{\gamma}(t)\Psi_{N}^{2}(x,t),\qquad t<0,$ (2.2)

where

$\Psi_{N}(x,t):=\varphi_{N}\Big{(}\frac{x}{\theta_{a}(t)}\Big{)},$ (2.3)

and integrate over $\mathbb{R}^{3}\times(s,t)$ with $t\in(s,0]$.666For $t=0$,
we integrate over $\mathbb{R}^{3}\times(s,\delta)$ and let $\delta\rightarrow
0^{-}$. This yields for all $s\in(-1,0)$ and for all $t\in(s,0]$,

$\displaystyle\theta_{a}(t)^{\gamma}\|\Psi_{N}v(\cdot,t)\|_{2}^{2}+\frac{a\gamma}{2}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-2}\|\Psi_{N}v\|_{2}^{2}\,d\tau+2\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma}\|\Psi_{N}\nabla
v\|_{2}^{2}\,d\tau$
$\displaystyle\leq\theta_{a}(s)^{\gamma}\|\Psi_{N}v(\cdot,s)\|_{2}^{2}+\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma}\int\limits_{\mathbb{R}^{3}}\big{[}-\nabla|v|^{2}\cdot\nabla\Psi_{N}^{2}+(|v|^{2}+2p)(v\cdot\nabla\Psi_{N}^{2})\big{]}\,d\tau$
$\displaystyle\qquad+\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-2}\int\limits_{\mathbb{R}^{3}}\Big{(}\frac{1}{2}ax\cdot\nabla\Psi_{N}^{2}(x,\tau)\Big{)}|v(x,\tau)|^{2}\,d\tau.$
(2.4)

Notice that

$\|ax\cdot\nabla\Psi_{N}^{2}\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim aN,$ (2.5)

which will be used below, see (2.16). Now, observe that the identity

$\Psi_{N}^{2}|\nabla
v|^{2}=|\nabla(\Psi_{N}v)|^{2}-|\nabla\Psi_{N}|^{2}|v|^{2}-\frac{1}{2}\nabla\Psi_{N}^{2}\cdot\nabla|v|^{2}$
(2.6)

allows $\nabla(\Psi_{N}v)$ to appear. Hence, using the identity (2.6) and
multiplying (2) by $\theta_{a}(s)^{-\gamma-1}$, and for all $s\in(-1,0)$,
$t\in(s,0]$,

$\displaystyle\frac{\theta_{a}(t)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v(\cdot,t)\|_{2}^{2}+\frac{a\gamma}{2}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}\,d\tau+2\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau$
$\displaystyle\qquad\leq\frac{1}{\theta_{a}(s)}\|\Psi_{N}v(\cdot,s)\|_{2}^{2}+\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\mathbb{R}^{3}}\big{[}2|\nabla\Psi_{N}|^{2}|v|^{2}+(|v|^{2}+2p)(v\cdot\nabla\Psi_{N}^{2})\big{]}\,dx\,d\tau$
$\displaystyle\qquad\quad+\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\mathbb{R}^{3}}\frac{1}{2}ax\cdot\nabla\Psi_{N}^{2}|v|^{2}\,dx\,d\tau.$
(2.7)

We now split the right hand side of (2) into the pressureless terms

$\displaystyle\begin{split}K^{I}(s,t)=&\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\mathbb{R}^{3}}\big{[}2|\nabla\Psi_{N}|^{2}|v|^{2}+|v|^{2}v\cdot\nabla\Psi_{N}^{2}\big{]}\,dx\,d\tau\\\
&+\frac{1}{2}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\mathbb{R}^{3}}ax\cdot\nabla\Psi_{N}^{2}(x,\tau)|v|^{2}\,dx\,d\tau\end{split}$
(2.8)

and the pressure term

$K^{II}(s,t)=\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\mathbb{R}^{3}}2p(v\cdot\nabla\Psi_{N}^{2})\,dx\,d\tau.$
(2.9)

The quantity $K^{I}(s,t)$ is easy to bound using the a priori critical control
(2.1). Indeed, all the terms in the right hand side of (2.8) involve a
derivative of the cut-off function $\Psi_{N}$, hence, these integrals are all
supported in the complement of (1.5). The quantity $K^{II}(s,t)$ concentrates
the difficulties, because of the nonlocality of the pressure. We handle this
issue first.

###  2.1. Pressure estimates

To treat the quantity $K^{II}(s,t)$, observe that

$\nabla\Psi_{N}^{2}(x,\tau)=\frac{2}{\theta_{a}(\tau)}\Psi_{N}(x,t)\nabla\varphi_{N}\Big{(}\frac{x}{\theta_{a}(\tau)}\Big{)},$

thus,

$\big{|}K^{II}(s,t)\big{|}\leq
C_{0}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-1}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}|pv\Psi_{N}|\,dx\,d\tau.$

Then we decompose the pressure term. Consider a cut-off function $\eta\in
C^{\infty}_{c}(\mathbb{R}^{3})$ such that:

$\eta(x)=\left\\{\begin{array}[]{lcc}1&\text{if}&|x|<\frac{1}{2}\sqrt{a},\\\
\in[0,1]&\text{if}&\frac{1}{2}\sqrt{a}<|x|<\frac{3}{4}\sqrt{a},\\\
0&\text{if}&\frac{3}{4}\sqrt{a}<|x|,\end{array}\right.$

and satisfying $|\nabla\eta|<\frac{C}{\sqrt{a}}$ and
$|\nabla^{2}\eta|<\frac{C}{a}$. Let us also write
$\eta(x)=\eta_{1}(x,t)+\eta_{2}(x,t)$, where $\eta_{1}$ and $\eta_{2}$ are
$C^{\infty}_{0}$ with values in $[0,1]$, and such that $\eta_{1}(\cdot,t)$ is
equal to $1$ on $B_{0}(\theta_{a}(t))$ and $\eta_{1}(\cdot,t)$ is supported on
$B_{0}(2\theta_{a}(t))$.777Notice that this decomposition of $\eta$ is only
needed for $t\in(-\frac{1}{4},0)$. This decomposition is classical, see for
instance [9], except for the additional localization on the paraboloid via
$\eta_{1}$. We summarize the decomposition in the following lemma.

######  Lemma 2.1.

Let $v$ be a finite-energy weak solution to the Navier-Stokes equations (1.1)
in $\mathbb{R}^{3}\times(-1,0)$ such that $v\in
C^{\infty}(-1,T;C^{\infty}(\mathbb{R}^{3}))$ for all $T\in(-1,0)$. There exist
a universal constant $C\in(0,\infty)$ such that for all
$\tau\in(-\frac{1}{16(N+1)^{2}},\,0)$ and for all $x\in
B_{0}((N+1)\theta_{a}(\tau))\setminus B_{0}(N\theta_{a}(\tau))$,

$\displaystyle|p(x,\tau)|=|\eta(x)p(x,\tau)|\leq$
$\displaystyle\frac{C}{N^{3}\theta_{a}(\tau)^{3}}\int\limits_{B_{0}(\theta_{a}(\tau))}|v(y,\tau)|^{2}\,dy$
$\displaystyle+\frac{C}{N^{3}\theta_{a}(\tau)^{3}}\int\limits_{B_{0}(2\theta_{a}(\tau))\setminus
B_{0}(\theta_{a}(\tau))}|v(y,\tau)|^{2}\,dy$
$\displaystyle+\Bigg{|}\frac{1}{4\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(2\theta_{a}(\tau))}\frac{\partial^{2}}{\partial_{y_{i}}\partial_{y_{j}}}\left(\frac{1}{|x-y|}\right)[\eta_{2}v_{i}v_{j}](y,\tau)\,dy\,\Bigg{|}$
$\displaystyle+\frac{C}{(\sqrt{a})^{3}}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}[|v|^{2}+|p|](y,\tau)\,dy$ $\displaystyle=:$
$\displaystyle\bar{p}_{1}(\tau)+\bar{p}_{2}(\tau)+\bar{p}_{3}(x,\tau)+\bar{p}_{4}(\tau).$
(2.10)

###### Proof.

Using the identity

$\eta(x)p(x,\tau)=-\frac{1}{4\pi}\int\limits_{\mathbb{R}^{3}}\frac{1}{|x-y|}[\Delta(\eta
p)](y,\tau)\,dy$

and the fact that $\Delta p=-\partial_{i}\partial_{j}(v_{i}v_{j})$, we obtain

$\eta(x)p(x,\tau):=p_{1}(x,\tau)+p_{2}(x,\tau)+p_{3}(x,\tau)+p_{4}(x,\tau),$

where

$\displaystyle p_{1}(x,\tau):=$
$\displaystyle\frac{1}{4\pi}\int\limits_{B_{0}(\theta_{a}(\tau))}\frac{\partial^{2}}{\partial_{y_{i}}\partial_{y_{j}}}\left(\frac{1}{|x-y|}\right)[\eta_{1}v_{i}v_{j}](y,\tau)\,dy,$
$\displaystyle p_{2}(x,\tau):=$
$\displaystyle\frac{1}{4\pi}\int\limits_{B_{0}(2\theta_{a}(\tau))\setminus
B_{0}(\theta_{a}(\tau))}\frac{\partial^{2}}{\partial_{y_{i}}\partial_{y_{j}}}\left(\frac{1}{|x-y|}\right)[\eta
v_{i}v_{j}](y,\tau)\,dy,$ $\displaystyle p_{3}(x,\tau):=$
$\displaystyle\frac{1}{4\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(2\theta_{a}(\tau))}\frac{\partial^{2}}{\partial_{y_{i}}\partial_{y_{j}}}\left(\frac{1}{|x-y|}\right)[\eta_{2}v_{i}v_{j}](y,\tau)\,dy$

and

$\displaystyle p_{4}(x,\tau):=$
$\displaystyle\frac{1}{2\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}\frac{x_{i}-y_{i}}{|x-y|^{3}}\left(\frac{\partial\eta}{\partial_{y_{j}}}v_{i}v_{j}\right)(y,\tau)\,dy$
$\displaystyle+\frac{1}{4\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}\frac{1}{|x-y|}\left(\frac{\partial^{2}\eta}{\partial_{y_{i}}\partial_{y_{j}}}v_{i}v_{j}\right)(y,\tau)\,dy$
$\displaystyle+\frac{1}{2\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}\frac{x_{i}-y_{i}}{|x-y|^{3}}\left(\frac{\partial\eta}{\partial_{y_{j}}}\,p\right)(y,\tau)\,dy$
$\displaystyle+\frac{1}{4\pi}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}\frac{1}{|x-y|}\left(\Delta\eta\,p\right)(y,\tau)\,dy.$

Notice that

$\operatorname*{dist}\big{(}B_{0}(2\theta_{a}(\tau)),B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))\big{)}\geq(N-2)\theta_{a}(\tau),$

and

$\operatorname*{dist}\big{(}B_{0}(\tfrac{1}{2}\sqrt{a}),B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))\big{)}\geq\frac{1}{4}\sqrt{a}$

for $\tau\in(-\frac{1}{16(N+1)^{2}},0)$. Combining the above two estimates
with the above pressure decomposition concludes the proof of the lemma.∎

By Lemma 2.1, we have for all $s\in(-\frac{1}{16(N+1)^{2}},\,0)$ and for all
$t\in(s,0]$,

$\big{|}K^{II}(s)\big{|}\\\
\leq\frac{C}{\theta_{a}(s)^{\gamma+1}}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\big{(}\bar{p}_{1}(\tau)+\bar{p}_{2}(\tau)+\bar{p}_{3}(x,\tau)+\bar{p}_{4}(\tau)\big{)}|v\Psi_{N}|\,dx\,d\tau,$
(2.11)

where $\bar{p}_{i}$, $i=1,\ldots\,4$ are defined in (2.10). We now turn to the
estimates of each term in the right hand side of (2.11).

#### Term involving $\bar{p}_{1}$

The term involving $\bar{p}_{1}$ is the main source of difficulties in this
paper. Indeed, this term concentrates the nonlocal part of the pressure that
cannot be directly controlled via the a priori critical assumption. Here we
control this term via the kinetic energy in $B_{0}(\theta_{a}(\tau))$. In
Section 3.2 we will then be able to control the scale-invariant local energy
via a Gronwall type argument.

From Hölder’s inequality for Lorentz spaces, see Proposition A.1, the fact
that

$\|\mathbf{1}_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\|_{L^{\frac{3}{2},1}(\mathbb{R}^{3})}\leq
CN^{2}\theta_{a}(\tau)^{2},$

and by (2.1), we obtain

$\displaystyle\begin{split}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}|v(x,\tau)|\,dx\leq&CN^{2}\theta_{a}(\tau)^{2}\|v(\cdot,\tau)\|_{L^{3,\infty}(B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau)))}\\\ \leq&CN^{2}M\theta_{a}(\tau)^{2}.\end{split}$
(2.12)

Therefore

$\displaystyle\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\bar{p}_{1}(\tau)|v(x,\tau)\Psi_{N}|\,dx\,d\tau$
$\displaystyle\leq\,\frac{C}{N^{3}}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-4}\Bigg{(}\int\limits_{B_{0}(\theta_{a}(\tau))}|v(x,\tau)|^{2}\,dx\Bigg{)}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}|v(x,\tau)|\,dx\,d\tau$
$\displaystyle\leq\,\frac{C_{*}M}{N}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-2}\int\limits_{B_{0}(\theta_{a}(\tau))}|v(x,\tau)|^{2}\,dx\,d\tau,$
(2.13)

where $C_{*}\in(0,\infty)$ is a universal constant.

#### Term involving $\bar{p}_{2}$

We estimate this term using Hölder’s inequality for Lorentz spaces, (2.1) and
(2.12),

$\displaystyle\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\bar{p}_{2}(\tau)|v(x,\tau)\Psi_{N}|\,dx\,d\tau$
$\displaystyle\leq\,\frac{C}{N^{3}}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-4}\Bigg{(}\int\limits_{B_{0}(2\theta_{a}(\tau))\setminus
B_{0}(\theta_{a}(\tau))}|v(x,\tau)|^{2}\,dy\Bigg{)}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}|v(x,\tau)|\,dx\,d\tau$
$\displaystyle\leq\,\frac{CM^{3}}{N}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\,d\tau$
$\displaystyle\leq\frac{CM^{3}}{a(\gamma+1)N}\theta_{a}(s)^{\gamma+1}.$

#### Term involving $\bar{p}_{3}$

Observe that for this term, the singularity of the kernel is seen. Calderón-
Zygmund’s theorem gives for $\tau\in(-\frac{1}{16(N+1)^{2}},\,0)$,

$\|\bar{p}_{3}(\cdot,\tau)\|_{L^{\frac{3}{2},\infty}(B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau)))}\leq
C\|v(\cdot,\tau)\|_{L^{3,\infty}(B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\theta_{a}(\tau)))}^{2}\leq CM^{2}.$ (2.14)

Then, by Hölder’s inequality for Lorentz spaces, interpolation of $L^{3,1}$
between $L^{2}$ and $L^{6}$ [6, Theorem 5.3.1], the Sobolev inequality and
Young’s inequality,

$\displaystyle\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\bar{p}_{3}(x,\tau)|v(x,\tau)\Psi_{N}|\,dx\,d\tau$
$\displaystyle\leq\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\|\bar{p}_{3}(\cdot,\tau)\|_{L^{\frac{3}{2},\infty}(B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau)))}\|v(\cdot,\tau)\Psi_{N}\|_{L^{3,1}(\mathbb{R}^{3})}\,d\tau$
$\displaystyle\leq
C_{\bar{p}_{3}}M^{2}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-\frac{3}{2}}\|\Psi_{N}v(\cdot,\tau)\|_{L^{2}(\mathbb{R}^{3})}\,d\tau+C_{\bar{p}_{3}}M^{2}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-\frac{1}{2}}\|\nabla(\Psi_{N}v(\cdot,\tau))\|_{L^{2}(\mathbb{R}^{3})}\,d\tau,$

where $C_{\bar{p}_{3}}\in(0,\infty)$ is a universal constant. Now, by the
Cauchy-Schwarz and Young’s inequalities we have on the one hand

$\displaystyle
C_{\bar{p}_{3}}M^{2}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-\frac{3}{2}}\,\|\Psi_{N}v(\cdot,\tau)\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq
C_{a,\gamma}M^{4}\theta_{a}(s)^{\gamma+1}+\frac{a\gamma}{16}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-2}\|\Psi_{N}v(\cdot,\tau)\|_{L^{2}}^{2}\,d\tau$

and on the other hand

$\displaystyle
C_{\bar{p}_{3}}M^{2}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-\frac{1}{2}}\,\|\nabla(\Psi_{N}v(\cdot,\tau))\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq
C_{a,\gamma}M^{4}\theta_{a}(s)^{\gamma+1}+\frac{1}{4}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma}\|\nabla(\Psi_{N}v(\cdot,\tau))\|_{L^{2}}^{2}\,d\tau.$

#### Term involving $\bar{p}_{4}$

We have by Hölder’s inequality for Lorentz spaces and (2.12)

$\displaystyle\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-1}\int\limits_{B_{0}((N+1)\theta_{a}(\tau))\setminus
B_{0}(N\theta_{a}(\tau))}\bar{p}_{4}(\tau)|v(x,\tau)\Psi_{N}|\,dx\,d\tau$
$\displaystyle\leq\,\frac{CN^{2}M}{(\sqrt{a})^{3}}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma+1}\Bigg{(}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}\left(\frac{1}{2}\sqrt{a}\right)}\big{[}|v|^{2}+|p|\big{]}\,dx\Bigg{)}\,d\tau$
$\displaystyle\leq\,\frac{CSMN^{2}}{(\sqrt{a})^{3}}\theta_{a}(s)^{\gamma+1},$

where

$S=\int\limits_{-1}^{0}\Bigg{(}\int\limits_{B_{0}(\frac{3}{4}\sqrt{a})\setminus
B_{0}(\frac{1}{2}\sqrt{a})}\big{[}|v|^{3}+|p|^{\frac{3}{2}}\big{]}\,dx\Bigg{)}^{\frac{2}{3}}<\infty.$

###  2.2. Local energy estimates under a priori critical control outside of
the paraboloid

We recall that $M\geq 1$ and $N\geq 4$. Summarizing the pressure estimates
above we get, for all $s\in(-\frac{1}{16(N+1)^{2}},\,0)$ and for all
$t\in(s,0]$, first $\theta_{a}(s)\leq\frac{\sqrt{a}}{4(N+1)}$ and then

$\big{|}K^{II}(s)\big{|}\leq\,\frac{CM}{N}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}\,d\tau+\frac{a\gamma}{16}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v(\cdot,\tau)\|_{L^{2}}^{2}\,d\tau\\\
+\frac{1}{4}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v(\cdot,\tau))\|_{L^{2}}^{2}\,d\tau+C_{a,\gamma}(M^{4}+SMN^{2}).$
(2.15)

Similarly, for $K^{I}(s)$, using in particular (2.5), we get

$\big{|}K^{I}(s)\big{|}\leq\frac{a\gamma}{16}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v(\cdot,\tau)\|_{L^{2}}^{2}\,d\tau+\frac{1}{4}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v(\cdot,\tau))\|_{L^{2}}^{2}\,d\tau\\\
+C_{a,\gamma}(M^{4}+M^{2}N^{2}).$ (2.16)

Then, from (2) combined with (2.16) and (2.15), we find for
$s\in(-\frac{1}{16(N+1)^{2}},0)$, and $t\in(s,0]$,

$\displaystyle\frac{\theta_{a}(t)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}$
$\displaystyle\|\Psi_{N}v(\cdot,t)\|_{2}^{2}+\frac{a\gamma}{4}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}\,d\tau+\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau$
$\displaystyle\leq$
$\displaystyle\frac{1}{\theta_{a}(s)}\|\Psi_{N}v(\cdot,s)\|_{2}^{2}+\frac{C_{*}M}{N}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}\,d\tau$
$\displaystyle+C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2}).$ (2.17)

where $C_{*}\in(0,\infty)$ is the universal constant in (2.13).

Let us emphasize that the gain of the factor $N^{-1}$ in the term
$\frac{C_{*}M}{N}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}$
plays a crucial role below. Taking $N$ large makes it possible to make
$\frac{C_{*}M}{N}$ small; see (3.15).

##  3\. Scale-invariant bounds on the interior of the paraboloid

The objective of this section is to prove the following result.

######  Proposition 3.1.

Let $M\geq 1$. Let $v$ be a finite-energy weak solution to the Navier-Stokes
equations (1.1) in $\mathbb{R}^{3}\times(-1,0)$ such that $v\in
C^{\infty}(-1,T;C^{\infty}(\mathbb{R}^{3}))$ for all $T\in(-1,0)$.
Assume that

$\operatorname*{ess\,sup}_{s\in(-1,0)}\|v(\cdot,s)\|_{L^{3,\infty}(B_{0}(\sqrt{a})\setminus
B_{0}(\theta_{a}(s)))}\leq M.$

Then,

$\operatorname*{ess\,sup}_{s\in\left(-1,0\right)}\frac{1}{\theta_{a}(s)}\int\limits_{B_{0}(\theta_{a}(s))}|v(x,\tau)|^{2}\,d\tau<+\infty.$
(3.1)

We shall connect information between critical quantities in order to rewrite
the time weighted energy estimates as a differential inequality. Let us denote

$f(s):=\frac{1}{\theta_{a}(s)}\|\Psi_{N}v(\cdot,s)\|_{2}^{2}$ (3.2)

and for all $\gamma\in\mathbb{R}$ we define888Notice that $f$ and $g_{\gamma}$
are dimensionally critical in the sense of Caffarelli, Kohn and Nirenberg [9].

$g_{\gamma}(s,t):=\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-1}}{\theta_{a}(s)^{\gamma+1}}f(\tau)\,d\tau.$
(3.3)

We compute the $s$ derivative of $g_{\gamma}(\cdot,t)$,

$\displaystyle\frac{\partial g_{\gamma}}{\partial s}(s,t)$
$\displaystyle=\frac{a(\gamma+1)}{2}\theta_{a}(s)^{-(\gamma+3)}\int\limits_{s}^{t}\theta_{a}(\tau)^{\gamma-2}\|\Psi_{N}v(\cdot,\tau)\|_{2}^{2}\,d\tau-\theta_{a}(s)^{-(\gamma+1)}\theta_{a}(s)^{\gamma-1}f(s)$
$\displaystyle=\theta_{a}(s)^{-2}\left[\frac{a(\gamma+1)}{2}g_{\gamma}(s,t)-f(s)\right].$
(3.4)

Now, let us rewrite (2.2) so as to make the functions $f$ and $g_{\gamma}$
appear,

$\displaystyle\begin{split}\frac{\theta_{a}(t)^{\gamma+1}}{\theta_{a}(s)^{\gamma+1}}&f(t)+\frac{a\gamma}{4}g_{\gamma}(s,t)+\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\\\
\leq&\,f(s)+\frac{C_{*}M}{N}g_{\gamma}(s,t)+C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2}).\end{split}$
(3.5)

Finally, we replace the identity (3) in (3.5) to get for
$s\in(-\frac{1}{16(N+1)^{2}},0)$, and $t\in(s,0]$,

$\displaystyle\begin{split}\frac{\theta_{a}(t)^{\gamma+1}}{\theta_{a}(s)^{\gamma+1}}f(t)\,+\,&\theta_{a}(s)^{2}\frac{\partial
g_{\gamma}}{\partial
s}(s,t)-\Big{(}\frac{a}{2}\Big{(}\frac{\gamma}{2}+1\Big{)}+\frac{C_{*}M}{N}\Big{)}g_{\gamma}(s,t)+\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\\\
\leq&C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2}).\end{split}$ (3.6)

where $C_{*}\in(0,\infty)$ is the universal constant in (2.13).

###  3.1. Gaining control through a Friedrich’s type estimate

We demonstrate the following Poincaré-type inequality

$\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\geq\,\Lambda
g_{\gamma}(s,t)-C_{\Lambda}NM^{2}$

for some $\Lambda>\frac{a}{2}$ and a constant $C_{\Lambda}\in(0,\infty)$
depending on $\Lambda$; see (3.13) below. For that, we adapt the computations
made in [24] by Neustupa to the weighted energy norms used in our work.

#### Step 1: a partition of the test function $\varphi_{N}$

Consider $\xi\in(0,1)$ and a test function $\varphi_{1}^{\xi}$ defined as

$\varphi_{1}^{\xi}(x)=\left\\{\begin{array}[]{lcc}1&\text{if}&|x|<1+\frac{1}{4}\xi\\\
\in[0,1]&\text{if}&1+\frac{1}{4}\xi<|x|<1+\frac{3}{4}\xi\\\
0&\text{if}&1+\frac{3}{4}\xi<|x|\end{array}\right.$

and satisfying

$|\nabla\varphi_{1}^{\xi}|\leq 4\xi^{-1}.$ (3.7)

Let us define $\varphi_{N,2}^{\xi}:=\varphi_{N}-\varphi_{1}^{\xi}$. Moreover,
we denote $\Psi_{1}^{\xi}(x,t):=\varphi_{1}^{\xi}(\frac{x}{\theta_{a}(t)})$
and $\Psi_{N,2}^{\xi}:=\varphi_{N,2}^{\xi}(\frac{x}{\theta_{a}(t)})$. Thus,
$\Psi_{N}=\Psi_{1}^{\xi}+\Psi_{N,2}^{\xi}$ in view of (2.3).

Observe that

$\displaystyle\|\nabla(\Psi_{1}^{\xi}v)\|_{2}^{2}$
$\displaystyle=\|\Psi_{1}^{\xi}\nabla v\|_{2}^{2}+2\big{(}\Psi_{1}^{\xi}\nabla
v,\nabla(\Psi_{1}^{\xi})\otimes
v\big{)}_{L^{2}}+\|\nabla(\Psi_{1}^{\xi})\otimes v\|_{2}^{2}$
$\displaystyle\leq\|\nabla
v\|_{L^{2}(\operatorname*{supp}(\Psi_{1}^{\xi}))}^{2}+\int\limits_{\mathbb{R}^{3}}\partial_{i}\big{(}({\Psi_{1}^{\xi}})^{2}\big{)}v_{j}\partial_{i}v_{j}\,dx+\|\nabla(\Psi_{1}^{\xi})\otimes
v\|_{2}^{2}$
$\displaystyle\leq\|\nabla(\Psi_{N}v)\|_{2}^{2}-\frac{1}{2}\int\limits_{\mathbb{R}^{3}}\Delta(\Psi_{1}^{\xi})|v|^{2}\,dx+\|\nabla(\Psi_{1}^{\xi})\otimes
v\|_{2}^{2}$

Here we used that $\Psi_{N}=1$ on the support of $\Psi_{1}^{\xi}$. Thus,

$\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\geq\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{1}^{\xi}v)\|_{2}^{2}\,d\tau-
c_{1}(s,t,a,\xi),$ (3.8)

where

$c_{1}(s,t,a,\xi):=\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\Bigg{(}-\frac{1}{2}\int\limits_{\operatorname*{supp}(\nabla\Psi_{1}^{\xi})}\Delta(\Psi_{1}^{\xi})|v|^{2}\,dx+\|\nabla(\Psi_{1}^{\xi})\otimes
v\|_{2}^{2}\Bigg{)}\,d\tau.$

Using the fact that

$\nabla\Psi_{1}^{\xi}=\frac{1}{\theta_{a}(t)}\nabla\varphi_{1}^{\xi}\Big{(}\frac{x}{\theta_{a}(t)}\Big{)},$

we get

$|c_{1}(s,t,a,\xi)|\leq\frac{C}{\xi^{2}}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\int\limits_{\operatorname*{supp}(\nabla\Psi_{1}^{\xi})}|v(x,\tau)|^{2}\,dx\,d\tau,$

and then using Hölder’s inequality for Lorentz spaces, we obtain

$-c_{1}(s,t,\xi)\geq-C_{a,a,\gamma,\xi}M^{2},$

where $C_{a,\gamma,\xi}\in(0,\infty)$ is a constant depending on $a$, $\gamma$
and $\xi$.

#### Step 2: a Poincaré inequality

Now, we need to manipulate the quantity
$\|\nabla(\Psi_{1}^{\xi}v)(\cdot,\tau)\|_{2}^{2}$. Consider
$\tau\in(-\frac{1}{16(N+1)^{2}},0)$. Since

$\int\limits_{B_{0}((1+\xi)\theta_{a}(\tau))}\nabla(\Psi_{1}^{\xi})\cdot v=0,$

we can use a right inverse of the divergence999Such an operator is sometimes
called a Bogovskii operator, see [15]. to infer that there exists a function

$w^{\xi}(\cdot,\tau)\in
W_{0}^{1,2}\big{(}B_{0}((1+\xi)\theta_{a}(\tau))\big{)}$

such that

$\nabla\cdot w^{\xi}(\cdot,\tau)=\nabla(\Psi_{1}^{\xi})\cdot v(\cdot,\tau)$

and

$\|\nabla w^{\xi}\|_{L^{2}\left(B_{0}((1+\xi)\theta_{a}(\tau))\right)}\leq
C\|\nabla(\Psi_{1}^{\xi})\cdot v\|_{2},$

where $C\in(0,\infty)$ does not depend on the parameters $\xi,\tau$. Hence
$\Psi_{1}^{\xi}v-w^{\xi}$ is divergence free in
$B_{0}{((1+\xi)\theta_{a}(\tau))}$ with zero trace on the boundary. By
Poincaré’s inequality for trace-free divergence-free functions, we have

$\|\Psi_{1}^{\xi}v-w^{\xi}\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})}\leq\frac{1}{\sqrt{\lambda_{S}(B_{0}{((1+\xi)\theta_{a}(\tau))})}}\|\nabla(\Psi_{1}^{\xi}v-w^{\xi})\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})},$
(3.9)

where $\lambda_{S}(B_{0}{(r)})$ is the first eigenvalue of the Dirichlet-
Stokes operator on $B_{0}(r)$.101010It is well known that
$\lambda_{S}(B_{0}(1))$ is greater than $\pi^{2}$, the principal eigenvalue of
the Dirichlet-Laplace operator. By homogeneity we find

$\frac{1}{\sqrt{\lambda_{S}(B_{0}((1+\xi)\theta_{a}(\tau)))}}=\frac{(1+\xi)\theta_{a}(\tau)}{\sqrt{\lambda_{S}(B_{0}{(1)})}}.$
(3.10)

Thus, using Poincaré’s inequality (3.9), the scaling property (3.10) and (3.7)
we find

$\displaystyle\|\Psi_{1}^{\xi}v\|_{2}$
$\displaystyle\leq\|\Psi_{1}^{\xi}v-w^{\xi}\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})}+\|w^{\xi}\|_{2}$
$\displaystyle\leq\frac{(1+\xi)\theta_{a}(\tau)}{\sqrt{\lambda_{S}(B_{0}(1))}}\|\nabla(\Psi_{1}^{\xi}v-w^{\xi})\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})}+\frac{(1+\xi)\theta_{a}(\tau)}{\pi}\|\nabla
w^{\xi}\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})}$
$\displaystyle\leq\frac{(1+\xi)\theta_{a}(\tau)}{\sqrt{\lambda_{S}(B_{0}(1))}}\|\nabla(\Psi_{1}^{\xi}v)\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})}+\frac{C(1+\xi)}{\xi\pi}\|v\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})\setminus
B_{0}{(\theta_{a}(\tau))})}.$

As we want to manipulate the square of
$\|\nabla(\Psi_{1}^{\xi}v)\|_{L^{2}(B_{0}{(1+\xi)\theta_{a}(\tau)})}$, we
introduce a parameter $\kappa>0$ to get

$\|\Psi_{1}^{\xi}v\|_{2}^{2}\leq\frac{(1+\xi)^{2}(1+\kappa)\theta_{a}(\tau)^{2}}{\lambda_{S}(B_{0}(1))}\|\nabla(\Psi_{1}^{\xi}v)\|^{2}_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau)})}+C_{\xi,\kappa}\|v\|_{L^{2}(B_{0}{((1+\xi)\theta_{a}(\tau))})\setminus
B_{0}{(\theta_{a}(\tau))})}^{2},$

where $C_{\xi,\kappa}\in(0,\infty)$ is a constant depending on $\xi$ and
$\kappa$.

Substituting this into (3.8) gives that for $s\in(-\frac{1}{16(N+1)^{2}},0)$
and $t\in(s,0]$,

$\displaystyle\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\geq\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{1}^{\xi}v\|_{2}^{2}\,d\tau-
c_{2}(s,t,a,\xi,\kappa),$ (3.11)

where,

$c_{2}(s,t,a,\xi,\kappa)=\frac{C_{\xi,\kappa}\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|v\|_{L^{2}\left(B_{0}((1+\xi)\theta_{a}(\tau))\setminus
B_{0}(\theta_{a}(\tau))\right)}^{2}\,d\tau+c_{1}(s,t,a,\xi).$

By the same reasoning as for $c_{1}$

$-c_{2}(s,t,a,\xi,\kappa)\geq-C_{a,\gamma,\xi,\kappa}M^{2},$

where $C_{a,\gamma,\xi,\kappa}\in(0,\infty)$ is a constant depending on $a$,
$\gamma$, $\xi$ and $\kappa$. Now, we use the identity

$\|\Psi_{1}^{\xi}v\|_{2}^{2}=\|\Psi_{N}v\|_{2}^{2}-2\big{(}\Psi_{N}^{\xi}v,\Psi_{N,2}^{\xi}v\big{)}_{L^{2}}+\|\Psi_{N,2}^{\xi}v\|_{2}^{2}$

and (3.11) to get

$\displaystyle\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\geq\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v\|_{2}^{2}\,d\tau-
c_{3}(s,t,a,\xi,\kappa),$ (3.12)

where,

$\displaystyle
c_{3}(s,t,a,\xi,\kappa)=\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\left[2(\Psi_{N}^{\xi}v,\Psi_{N,2}^{\xi}v)_{2}-\|\Psi_{N,2}^{\xi}v\|_{2}^{2}\right]d\tau+c_{2}(s,t,a,\xi,\kappa).$

As $\Psi_{N,2}^{\xi}(\cdot,\tau)$ is supported on
$B((N+1)\theta_{a}(\tau))\setminus B(\theta_{a}(\tau))$, we have by Hunt’s
inequality

$-c_{3}(s,t,a,\xi,\kappa)\geq-C_{a,\gamma,\xi,\kappa}NM^{2},$

where the constant $C_{a,\gamma,\xi,\kappa}\in(0,\infty)$ depends on $a$,
$\gamma$, $\xi$ and $\kappa$. We finally find from (3.12), for
$s\in(-\frac{1}{16(N+1)^{2}},0)$ and $t\in(s,0]$,

$\int\limits_{s}^{t}\frac{\theta_{a}(\tau)^{\gamma}}{\theta_{a}(s)^{\gamma+1}}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau\geq\,\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}\,g_{\gamma}(s,t)-C_{a,\gamma,\xi,\kappa}NM^{2},$
(3.13)

where the constant $C_{a,\gamma,\xi,\kappa}\in(0,\infty)$ depends on $a$,
$\gamma$, $\xi$ and $\kappa$.

###  3.2. Gronwall estimate

With the Poincaré-type inequality obtained in (3.13), the estimate (3.6)
becomes for $s\in(-\frac{1}{16(N+1)^{2}},0)$, and $t\in(s,0]$,

$\displaystyle\frac{\theta_{a}(t)^{\gamma+1}}{\theta_{a}(s)^{\gamma+1}}f(t)\,+\,$
$\displaystyle\theta_{a}(s)^{2}\frac{\partial g_{\gamma}}{\partial
s}(s,t)+\left(\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}-\frac{a}{2}\big{(}\frac{\gamma}{2}+1\big{)}-\frac{C_{*}M}{N}\right)g_{\gamma}(s,t)$
$\displaystyle\leq$ $\displaystyle
C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2})+C_{a,\gamma,\xi,\kappa}NM^{2}.$

Now, we drop the first term in the left hand side and we consider $t=0$. Let
us write

$A:=\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}-\frac{a}{2}\Big{(}\frac{\gamma}{2}+1\Big{)}-\frac{C_{*}M}{N}$

and

$B:=C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2})+C_{a,\gamma,\xi,\kappa}NM^{2}.$

Thus, we study

$\frac{d}{ds}(g_{\gamma}(\cdot,0))(s)+\frac{A}{\theta_{a}(s)^{2}}g_{\gamma}(s,0)\leq\frac{B}{\theta_{a}(s)^{2}}.$

We multiply this equation by the function

$k(s)=\left(\frac{\theta_{a}(s)}{\theta_{a}(s_{0})}\right)^{-\frac{2A}{a}},$

where $s_{0}:=-\frac{1}{32(N+1)^{2}}$. The function $k(s)$ satisfies

$\frac{dk}{ds}(s)=\frac{A}{\theta_{a}(s)^{2}}\left(\frac{\theta_{a}(s)}{\theta_{a}(s_{0})}\right)^{-\frac{2A}{a}}=\frac{A}{\theta_{a}(s)^{2}}k(s).$

Then, we find

$\frac{d}{ds}\left(kg_{\gamma}(\cdot,0)\right)(s)\leq\frac{B}{A}\left(\frac{A}{\theta_{a}(s)^{2}}k(s)\right).$

Let $s\in[s_{0},0)$. Integrating over $[s_{0},s]$ and multiplying by
$k^{-1}(s)$ we obtain

$g_{\gamma}(s,0)\leq
g_{\gamma}(s_{0},0)\frac{1}{k(s)}+\frac{B}{A}\left(1-\frac{1}{k(s)}\right).$
(3.14)

Estimate (3.14) will be useful in the next section to demonstrate Proposition
3.1. We want $A>0$ in order to have a damping effect, which will be achieved
by taking $\xi,\,\kappa$ small and $N$ large.

###  3.3. Proof of Proposition 3.1

Our objective is to prove the boundedness of $f$. Here we take $\gamma=1$. We
rely on the estimate (3.14). As $a=\lambda_{S}(B_{0}(1))$ according to the
choice made in Theorem A,111111Here we see that any
$0<a<\frac{4}{3}\lambda_{S}(B_{0}(1))$ works; see Footnote 3. We remark here
that there is a possibility to get the full range $0<a<4\lambda_{S}(B_{0}(1))$
as in [24]. Indeed, it suffices to choose different parameters in Young’s
inequality leading to (2.15) and (2.16). This results in a small parameter
$\varepsilon$ in front of the third term in the right hand side of (2.15) and
of the second term in the right hand side of (2.16). We do not carry out this
technical modification here so as to keep the number of parameters to a
minimum. we first take $0<\kappa,\xi\ll 1$ and then the parameter
$N(M,\xi,\kappa)$ (see the definition (2.3) of $\Psi_{N}$) large enough to get

$A=\frac{\lambda_{S}(B_{0}(1))}{(1+\kappa)(1+\xi)^{2}}-\frac{3a}{4}-\frac{C_{*}M}{N}>0.$
(3.15)

Hence, from (3.14), we obtain $1/k(s)$ goes to $0$ when $s\uparrow 0$. Thus,
we get $g_{1}(\cdot,0)$ is bounded on $[s_{0},0)$ with as defined above
$s_{0}=-\frac{1}{32(N+1)^{2}}$.

Now, we want to prove that $f(s)$ is bounded. Take any
$t_{1}\in[\frac{s_{0}}{2},0)$ and define $s_{1}:=2t_{1}$. Thus,
$s_{1}<t_{1}<0$ and

$2(t_{1}-s_{1})=-s_{1}=\frac{\theta_{a}(s_{1})^{2}}{a}.$

Moreover, for $\tau\in(s_{1},t_{1})$ we have

$\frac{1}{\sqrt{2}}\theta_{a}(s_{1})=\theta_{a}(t_{1})<\theta_{a}(\tau)<\theta_{a}(s_{1})=\sqrt{2}\theta_{a}(t_{1}).$

Then, we get

$\displaystyle\frac{1}{(t_{1}-s_{1})}\int\limits_{s_{1}}^{t_{1}}f(\tau)d\tau$
$\displaystyle\leq\frac{2a}{\theta_{a}(s_{1})^{2}}\int\limits_{s_{1}}^{t_{1}}f(\tau)d\tau\leq
2a\int\limits_{s_{1}}^{0}\frac{1}{\theta_{a}(s_{1})^{2}}f(\tau)d\tau$
$\displaystyle=2ag_{1}(s_{1},0).$

We can find $s_{1}^{\prime}\in(s_{1},t_{1})$ such that

$f(s_{1}^{\prime})\leq\frac{1}{(t_{1}-s_{1})}\int\limits_{s_{1}}^{t_{1}}f(\tau)d\tau\leq
2a\sup_{s\in[s_{0},0)}g_{1}(s,0).$ (3.16)

Inequality (3.5), with $t=t_{1}$ and $s=s_{1}^{\prime}$, gives

$\displaystyle\frac{1}{2}\,f(t_{1})\leq$
$\displaystyle\frac{\theta_{a}(t_{1})^{2}}{\theta_{a}(s_{1}^{\prime})^{2}}f(t_{1})$
$\displaystyle\leq$ $\displaystyle
f(s_{1}^{\prime})+\frac{C_{*}M}{N}g_{1}(s_{1}^{\prime},0)+C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2})$
$\displaystyle\leq$ $\displaystyle
2a\sup_{s\in[s_{0},0)}g_{1}(s,0)+\frac{C_{*}M}{N}g_{1}(s_{1}^{\prime},0)+C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2}),$

where we used (3.16) in the last line. Therefore $f$ is bounded on a small non
empty interval $[\frac{s_{0}}{2},0)$, and eventually on $(-1,0)$ by using the
boundedness of the energy for $s\in(-1,\frac{s_{0}}{2})$, i.e.

$\operatorname*{ess\,sup}_{s\in(-1,0)}\frac{1}{\theta_{a}(s)}\|\Psi_{N}v(\cdot,s)\|_{2}^{2}<\infty.$
(3.17)

This directly implies (3.1) which concludes the proof of Proposition 3.1.

##  4\. Proof of the main results

###  4.1. Case with axisymmetry

We prove (1.3) in Theorem A by contraposition. Hence, we assume by
contraposition that

$\operatorname*{ess\,sup}_{s\in(-1,0)}\|v(\cdot,s)\|_{L^{3,\infty}(B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{a(-s)}))}\leq M.$ (4.1)

There are two steps in this proof. Note that we only assume axisymmetry at the
end of the second step.

#### Step 1: control of a scale-invariant Morrey-type quantity

This step works without the assumption of axisymmetry. We start form the
control (3.17). Observe that for $\gamma>-1$,

$\operatorname*{ess\,sup}_{s\in(-1,0)}\int\limits_{s}^{0}\frac{\theta_{a}(\tau)^{\gamma-2}}{\theta_{a}(s)^{\gamma+1}}\|\Psi_{N}v(\tau)\|_{L^{2}}^{2}\,d\tau<+\infty.$
(4.2)

By (4.2) we can now repeat the computations that lead to the estimate (2.2)
with $\gamma=0$. Hence, we have (2.2) with $\gamma=0$ and $t=0$, i.e.

$\displaystyle\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)}\int\limits_{s}^{0}\|\nabla(\Psi_{N}v)\|_{2}^{2}$
$\displaystyle\leq\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\Bigg{(}\frac{1}{\theta_{a}(s)}\|\Psi_{N}v(s)\|_{2}^{2}+\frac{C_{*}M}{N}\int\limits_{s}^{0}\frac{\theta_{a}(\tau)^{-2}}{\theta_{a}(s)}\|\Psi_{N}v\|_{2}^{2}\,d\tau\Bigg{)}$
$\displaystyle\qquad+C_{a,\gamma}(M^{4}+SMN^{2}+M^{2}N^{2}).$

Therefore,

$\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)}\|{\Psi_{N}v}(s)\|^{2}_{L^{2}}+\frac{1}{\theta_{a}(s)}\int\limits_{s}^{0}\|\nabla(\Psi_{N}v)\|_{2}^{2}\,d\tau<+\infty.$

By interpolation between $L^{2}$ and $L^{6}$, this implies that

$\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)^{2}}\int\limits_{s}^{0}\int\limits_{B_{0}({\sqrt{a}})}|\Psi_{N}{v}(x,\tau)|^{3}\,dx\,d\tau<+\infty.$
(4.3)

Therefore,

$\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)^{2}}\int\limits_{s}^{0}\int\limits_{B_{0}(\theta_{a}(\tau))}|v(x,\tau)|^{3}\,dx\,d\tau<+\infty.$
(4.4)

Moreover, the assumption (4.1) implies that for $\delta\in(0,\frac{1}{5})$

$\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)^{5(1-\frac{1+\delta}{2})}}\int\limits_{s}^{0}\int\limits_{B_{0}(\theta_{a}(s))\setminus
B_{0}(\theta_{a}(\tau))}|v(x,\tau)|^{\frac{5}{2}(1+\delta)}\,dx\,d\tau<+\infty.$
(4.5)

Combining (4.4) and (4.5) yields

$\operatorname*{ess\,sup}_{r\in(0,\frac{1}{4(N+1)})}\frac{1}{r^{5(1-\frac{1+\delta}{2})}}\int\limits_{-r^{2}}^{0}\int\limits_{B_{0}(r)}|v|^{\frac{5}{2}(1+\delta)}\,dx\,d\tau<+\infty.$
(4.6)

#### Step 2: conclusion

By [30, Theorem 6] (see also [31]), we have that (4.6) for $0<\delta\ll 1$
implies the boundedness of

$\operatorname*{ess\,sup}_{r\in(0,\frac{1}{4(N+1)})}\Bigg{(}\operatorname*{ess\,sup}_{s\in(-r^{2},0)}\frac{1}{r}\int\limits_{B_{0}(r)}|v(x,\tau)|^{2}\,dx\,d\tau+\frac{1}{r^{2}}\int\limits_{-r^{2}}^{0}\int\limits_{B_{0}(r)}|v|^{3}\,dx\,d\tau\\\
+\frac{1}{r}\int\limits_{-r^{2}}^{0}\int\limits_{B_{0}(r)}|\nabla
v|^{2}\,dx\,d\tau\Bigg{)}<+\infty.$ (4.7)

Let us now assume that $v$ is axisymmetric. Then (4.7) and [28, Theorem 2.1]
imply that $(0,0)$ is a regular point.

###  4.2. General case without axisymmetry

We prove (1.2) in Theorem A by contraposition. Hence, we assume by
contraposition that

$\operatorname*{ess\,sup}_{s\in(-1,0)}\|v(\cdot,s)\|_{L^{3}(B_{0}(\sqrt{a})\setminus
B_{0}(\sqrt{a(-s)}))}\leq M.$ (4.8)

There are two steps in this proof.

#### Step 1: control of a scale-invariant Morrey-type quantity

Note that (4.8) implies (4.1). Hence estimate (4.3) holds. Combining (4.3)
with (4.8) gives

$\operatorname*{ess\,sup}_{s\in(-\frac{1}{16(N+1)^{2}},0)}\frac{1}{\theta_{a}(s)^{2}}\int\limits_{s}^{0}\int\limits_{B_{0}(\sqrt{a})}|{v}(x,\tau)|^{3}\,dx\,d\tau<+\infty.$
(4.9)

#### Step 2: conclusion

From (4.9) and a pigeonholing argument, we find a sequence of times
$t_{k}\in(-\frac{1}{16(N+1)^{2}},0)$ such that $t_{k}\uparrow 0$ and

$\sup_{k}\int\limits_{B_{0}(\sqrt{a})}|v(x,t_{k})|^{3}\,dx<\infty.$

Applying [1, Theorem 1.1] immediately yields that $(0,0)$ is a regular point.

##  Appendix A Known results about Lorentz spaces

Given a measurable subset $\Omega\subseteq\mathbb{R}^{d}$, let us define the
Lorentz spaces. For a measurable function $f:\Omega\rightarrow\mathbb{R}$
define:

$d_{f,\Omega}(\alpha):=\mu(\\{x\in\Omega:|f(x)|>\alpha\\}),$ (A.1)

where $\mu$ denotes the Lebesgue measure. The Lorentz space $L^{p,q}(\Omega)$,
with $p\in[1,\infty)$, $q\in[1,\infty]$, is the set of all measurable
functions $g$ on $\Omega$ such that the quasinorm $\|g\|_{L^{p,q}(\Omega)}$ is
finite. Here:

$\|g\|_{L^{p,q}(\Omega)}:=\Big{(}p\int\limits_{0}^{\infty}\alpha^{q}d_{g,\Omega}(\alpha)^{\frac{q}{p}}\frac{d\alpha}{\alpha}\Big{)}^{\frac{1}{q}},$
(A.2) $\|g\|_{L^{p,\infty}(\Omega)}:=\sup_{\alpha>0}\alpha
d_{g,\Omega}(\alpha)^{\frac{1}{p}}.$ (A.3)

It is known there exists a norm, which is equivalent to the quasinorm defined
above, for which $L^{p,q}(\Omega)$ is a Banach space. For $p\in[1,\infty)$ and
$1\leq q_{1}<q_{2}\leq\infty$, we have the following continuous embeddings

$L^{p,q_{1}}(\Omega)\hookrightarrow L^{p,q_{2}}(\Omega)$ (A.4)

and the inclusion is known to be strict.

Our main tool in this paper is the following Hölder’s inequality for Lorentz
spaces. The statement below and proof can be found in Hunt’s paper [17,
Theorem 4.5, p.271]; see also [26, Theorems 3.4-3.5, page 141].

######  Proposition A.1.

Suppose that $1\leq p,q,r\leq\infty$ and $1\leq s_{1},s_{2}\leq\infty$.
Furthermore, suppose that $p$, $q$, $r$, $s_{1}$ and $s_{2}$ satisfy the
following relations:

$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$

and

$\frac{1}{s_{1}}+\frac{1}{s_{2}}=\frac{1}{s}.$

Then the assumption that $f\in L^{p,s_{1}}(\Omega)$ and $g\in
L^{q,s_{2}}(\Omega)$ imply that $fg\in L^{r,s}(\Omega)$, with the estimate

$\|fg\|_{L^{r,s}(\Omega)}\leq
C(p,q,s_{1},s_{2})\|f\|_{L^{p,s_{1}}(\Omega)}\|g\|_{L^{q,s_{2}}(\Omega)}.$
(A.5)

## Conflict of interest

The authors declare that they have no conflict of interest.

## Data availability statement

Data sharing is not applicable to this article as no datasets were generated
or analyzed during the current study.

## Acknowledgment

CP and PFD are partially supported by the Agence Nationale de la Recherche,
project BORDS, grant ANR-16-CE40-0027-01. CP is also partially supported by
the Agence Nationale de la Recherche, project SINGFLOWS, grant ANR-
18-CE40-0027-01, project CRISIS, grant ANR-20-CE40-0020-01, by the CY
Initiative of Excellence, project CYNA (CY Nonlinear Analysis) and project
CYFI (CYngular Fluids and Interfaces). PFD is also supported by the Labex MME-
DII. TB and CP thank the Institute of Advanced Studies of Cergy Paris
University for their hospitality.

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