and on the other hand

$$ \begin{array}{l} |\xi(x, t) - \xi(y, t)| \le \eta(|x - y|) \quad \text{in } \Omega_d \cap C_{\delta, \bar{x}} \quad (\text{CA}) \\ \Downarrow \\ (\bar{x}, T) \text{ is a regular point of } U. \end{array} $$

The condition (CA) is dubbed the ‘continuous alignment condition on concentrating balls’. We can read this result as a ‘geometric concentration’ result. Indeed, the regularity criteria based on continuous alignment of the vorticity direction involves the direction of the vorticity field. It is hence of geometric nature. Here we demonstrate that vorticity alignment is sufficient on concentrating sets. The contrapositive means that the direction of the vorticity breaks any modulus of continuity on sets concentrating on the singularity following the parabolic scaling, see Figure 4.1. In Chapter 5 we investigate ‘norm concentration’ near potential singularities.

**Remark 4.4** (on localized criteria). In [36] we also obtain local variations of Theorem 4.3. To the best of our knowledge those are the first local results regarding regularity under the vorticity alignment condition, for a Navier-Stokes solution having no-slip on the flat part of the boundary. Previously all localisations were known for only interior cases [174, 171, 162]. For precise statements of our local results, we refer to [36, Theorem 4 in Section 6].

**Remark 4.5** (on the converse statement). When $\bar{x}$ is a regular point of $U$ belonging to the interior of the half-space, it can be seen that $U$ has Hölder continuous spatial derivatives near $\bar{x}$ and the converse statement to Theorem 4.3 is true, i.e. the vorticity alignment condition (CA) holds. It is not clear to us whether or not that remains to be the case when $\bar{x}$ lies on $\partial\mathbb{R}_+^3$. The difficulty is that there are examples of Navier-Stokes solutions, see Subsection 4.1.4, with no-slip boundary condition on the flat part of the boundary that demonstrate that boundedness of $U$ near the flat boundary does not imply boundedness of $\nabla U$. It is interesting to note that these examples do not provide a counterexample to the converse statement of Theorem 4.3. In fact in [301] the construction is based upon a monotone shear flow, whose vorticity direction is constant.

### Novelty of our result

To the best of our knowledge, the only vorticity alignment regularity criteria known for the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, T)$ with no-slip boundary condition, was proved by Giga, Hsu and Maekawa in [159] for mild solutions under the additional assumption that $U$ satisfies the Type I condition (4.20). In particular, in [159] it was shown that if $\eta: [0, \infty) \to \mathbb{R}$ is a nondecreasing continuous function with $\eta(0) = 0$ then the following holds true. Namely, the assumption

$$ |\xi(x, t) - \xi(y, t)| \le \eta(|x - y|) \quad \text{for all } (x, t), (y, t) \in \Omega_d \quad (4.24) $$

implies that $U$ is bounded up to $t = T$. Notice that (4.24) on the contrary to (CA) states vorticity alignment on the whole of $\Omega_d$ rather than on the concentrating sets $\Omega_d \cap C_{\delta, \bar{x}}$.

As far as we know, the global or local statements with vorticity alignment on concentrating sets are new even for the case of the whole-space $\mathbb{R}^3$.

Given the strong nonlocal effects of the pressure in the half-space, see subsections 4.1.6, 4.1.4 and 4.1.2, and Subsection 5.3.2 in the next chapter, it is a priori far from clear that such
a result is true for the half-space, although its equivalent for the whole-space is well known. Roughly speaking, thanks to the Type I assumption we manage to tame a bit the nonlocality of the pressure. Beyond that regime, no analog of Constantin and Fefferman's result [107] (see Subsection 4.1.5 above) is known for solutions of the Navier-Stokes equations in the half-space with no-slip boundary conditions. This was actually mentioned as an open problem in Subsection 4.1.6 above.

## 4.3 New ideas and strategy for the proofs

### 4.3.1 Resolvent pressure formulas in the half-space and pressure estimates

We outline the four main ingredients that we use to give an elementary proof of the fractional pressure estimates of Theorem 4.1. Our proof builds upon the formulas obtained in collaboration with Maekawa and Miura [249].

**First ingredient: formulas for the harmonic pressure** We decompose the pressure of the Stokes system (4.3) with the no-slip boundary condition (4.4) on the boundary of the half-space into a Helmholtz pressure $P_{Helm}^F$ and a harmonic pressure $P_{harm}^F$. The treatment of the Helmholtz pressure is standard. We hence focus on the harmonic pressure that formally solves the elliptic Neumann boundary value problem (4.5). We rely on a resolvent pressure formula derived in our work in collaboration with Maekawa and Miura [247, Section 2]; see also Subsection 5.3.2 below. We use resolvent kernel estimates established in the companion paper [249] joint with Maekawa and Miura, which is based on an earlier decomposition of the pressure for the Stokes resolvent problem carried out by Desch, Hieber and Prüss in [124]. We have the following formula for the harmonic pressure:

$$P_{harm}^{F}(x', x_3, t) = \frac{1}{2\pi i} \int_{0}^{t} \int_{\Gamma} e^{\lambda(t-s)} \int_{\mathbb{R}_{+}^{3}} \tilde{q}_{\lambda}(x' - z', x_3, z_3) (\mathbb{P}\nabla \cdot F(z', z_3, s))_3 dz' dz_3 d\lambda ds, \quad (4.25)$$

for all $(x', x_3) \in \mathbb{R}_+^3$. Here $\mathbb{P}$ denotes the Helmholtz-Leray projection and $\Gamma = \Gamma_\rho$ with $\rho \in (0, 1)$ is the curve

$$\{\lambda \in \mathbb{C} \mid |\arg \lambda| = \eta, |\lambda| \ge \rho\} \cup \{\lambda \in \mathbb{C} \mid |\arg \lambda| \le \eta, |\lambda| = \rho\} \quad (4.26)$$

for some $\eta \in (\frac{\pi}{2}, \pi)$. The kernel $\tilde{q}_\lambda$ is defined by for all $x' \in \mathbb{R}^2$ and $x_3, z_3 > 0$,

$$\tilde{q}_{\lambda}(x', x_3, z_3) := - \int_{\mathbb{R}^2} e^{ix' \cdot \xi} e^{-|\xi|x_3} e^{-\sqrt{\lambda+|\xi|^2}(\xi)z_3} \left( \frac{\sqrt{\lambda+|\xi|^2}(\xi)}{|\xi|} + 1 \right) d\xi. \quad (4.27)$$

We will see below that the use of formula (4.25), involving the vertical component of the source term rather than the tangential one (on that subject see also Remark 5.14) is more adapted to the study of $P_{harm}^F$. Indeed the vertical component of $\mathbb{P}\nabla \cdot F$ vanishes on the boundary of $\mathbb{R}_+^3$, which is not necessarily the case for the tangential component. Notice that the estimates for $\tilde{q}_\lambda$ are derived along the exact same lines as [249, Proposition 3.7].
**Second ingredient: a formula for the source term** We compute the Helmholtz-Leray projection. From the work of Koch and Solonnikov [220, Proposition 3.1], there exists $G \in L^q(0, T; W^{1,p}(\mathbb{R}_+^3))$ such that

$$ \mathbb{P}\nabla \cdot F = \nabla \cdot G. \qquad (4.28) $$

Moreover, we have the following formula for $G$: for all $1 \le \alpha, \beta \le 3$,

$$ G_{\alpha\beta} = F_{\alpha\beta} - \delta_{\alpha\beta}F_{33} + (1 - \delta_{3\alpha})\partial_{x\beta} \left( \int_{\mathbb{R}_+^3} \partial_{z\gamma} N(x, z) F_{\alpha\gamma}(z) dz \right. \\ \left. + \int_{\mathbb{R}_+^3} (\partial_{z3} N(x, z) F_{3\alpha}(z) - \partial_{z\alpha} N(x, z) F_{33}(z)) dz \right), \qquad (4.29) $$

where $N$ is the Neumann kernel for the half-space. This formula implies by the Calderón-Zygmund theory that for all $p \in (1, \infty)$ the $W_x^{1,p}$ norm of $G$ can be estimated in terms of the $W_x^{1,p}$ norm of $F$. Furthermore, notice that

$$ (\mathbb{P}\nabla \cdot F)_3 = \sum_{\alpha=1}^{2} \partial_{\alpha} G_{\alpha 3} \qquad (4.30) $$

because $G_{33} = 0$ by the formula (4.29). This cancellation property is essential since it enables to avoid the derivative of $\tilde{q}_\lambda$ in the vertical direction, which has less decay in $x'$; see [249, Proposition 3.7]. Finally, the fact that the trace of $F(\cdot, t)$ vanishes on $\partial\mathbb{R}_+^3$ implies that the trace of $G_{\alpha 3}(\cdot, t)$ also vanishes on $\partial\mathbb{R}_+^3$.

**Third ingredient: a Calderón-Zygmund estimate for the harmonic pressure** Our starting point to estimate the harmonic part of the pressure is formula (4.25), which we combine with the formula (4.30) for the source term. We have for almost all $x_3 \in (0, \infty)$, $x' \in \mathbb{R}^2$, $s \in (0, T)$,

$$ \begin{aligned} & \int_0^\infty \int_{\mathbb{R}^2} \tilde{q}_\lambda(x' - z', x_3, z_3) (\mathbb{P}\nabla \cdot F)_3(z', z_3, s) dz' dz_3 \\ &= \sum_{\alpha=1}^2 \int_0^\infty \int_{\mathbb{R}^2} \partial_{x_\alpha} \tilde{q}_\lambda(x' - z', x_3, z_3) G_{\alpha 3}(z', z_3, s) dz' dz_3 \\ &=: I(x', x_3, s). \end{aligned} $$

In [36] we show that for all $\alpha \in \{1, 2\}$, $x_3, z_3 > 0$, the kernel $k_{x_3, z_3}$ defined by for all $x' \in \mathbb{R}^2$

$$ k_{\alpha, x_3, z_3}(x') = \left(1 + \frac{1}{|\lambda|^{\frac{1}{2}} z_3}\right)^{-1} |\lambda|^{-\frac{1}{2}} e^{c^* |\lambda|^{\frac{1}{2}} z_3} \partial_{x_\alpha} \tilde{q}_\lambda(x', x_3, z_3) $$

is a Calderón-Zygmund kernel.
**Fourth ingredient: Hardy's inequality** We now turn to estimating the harmonic pressure. We have for almost all $s \in (0, T)$,

$$ \|I(\cdot, s)\|_{L_{\infty}(0, \infty; L_p(\mathbb{R}^2))} \le C \sum_{\alpha=1}^{2} \left\| \int_{0}^{\infty} e^{-c|\lambda|^{\frac{1}{2}} z_3} \frac{\|G_{\alpha 3}(\cdot, z_3, s)\|_{L_p(\mathbb{R}^2)}}{z_3} dz_3 \right\|_{L_{\infty}(0, \infty)}, $$

for some $c \in (0, c^*)$. Thanks to the fact that $G_{\alpha 3}$ has zero trace on the boundary, we can use Hardy's inequality in the vertical direction, see [36, Lemma 8], and get for almost every $s \in (0, T)$,

$$ \|I(\cdot, s)\|_{L_{\infty}(0, \infty; L_p(\mathbb{R}^2))} \le C|\lambda|^{-\frac{1}{2}(\kappa - \frac{1}{p})} \|G(\cdot, s)\|_{L_p(\mathbb{R}_+^3)}^{1-\kappa} \|\partial_{z_3} G(\cdot, s)\|_{L_p(\mathbb{R}_+^3)}^{\kappa}, $$

with a constant $C(\kappa, p) \in (0, \infty)$. It remains to compute the integral in $\lambda$ and estimate the convolution in time. This finishes the proof of the fractional pressure estimates of Theorem 4.1.

### 4.3.2 Boundedness of scale-invariant energies in Type I blow-up scenarios

In order to prove Theorem 4.2 a number of innovations are needed. Indeed, as we already mentioned at several places, the case near a flat boundary is considerably more intricate than the interior case. The proof relies on the local energy inequality; see below (5.20) in the context of local energy solutions, or [36, Definition 6] in the context of suitable solutions. The main difficulty lies in the estimate for the pressure term

$$ I(U, P, r) := \int_{-r^2}^{0} \int_{B_+(r)} P U \cdot \nabla \Phi \, dxds \quad (4.31) $$

where $\Phi \in C_0^\infty(B(r) \times (-r^2, \infty))$ is a positive test function.

To highlight the difficulties concerned with the half-space, let us first discuss the proof of the simpler interior case [310, 303]. For the Navier-Stokes equations in the whole-space (with sufficient decay), the Calderón-Zygmund theory gives that the pressure $P$ can be estimated directly in terms of $U \otimes U$, which we write in a formal way $P \simeq U \otimes U$. For the interior case of a suitable weak solution in $Q(1)$, this fact implies the decay estimate

$$ D_{\frac{3}{2}}(P, \tau r) \le C(\tau D_{\frac{3}{2}}(P, r) + \tau^{-2}C(U, r)), \quad (4.32) $$

for a given $\tau \in (0, 1)$ and all $r \in (0, 1]$. To prove the interior version of Theorem 4.2, it then suffices to bound the right hand side of (4.32). The game to play is to combine energy type quantities and the Type I bound (4.18) to obtain

$$ C(U, r) \lesssim_M A^\mu(U, r) \left( \frac{1}{r^3} \int_{Q(r)} |U|^2 dxds + \frac{1}{r} \int_{Q(r)} |\nabla U|^2 dxds \right)^{\theta} \quad (4.33) $$

↓

$$ C(U, r) + D_{\frac{3}{2}}(P, \tau r) \lesssim_M \varepsilon \left( \frac{1}{r^3} \int_{Q(r)} |U|^2 dxds + \frac{1}{r} \int_{Q(r)} |\nabla U|^2 dxds \right) + c(M, \varepsilon, \kappa, \tau) + C \tau D_{\frac{3}{2}}(P, r) \quad (4.34) $$
with

$$0 < \mu, \theta, \quad \kappa := \mu + \theta < 1, \quad \varepsilon \in (0, 1) \quad \text{and} \quad c(M, \varepsilon, \kappa, \tau) > 0. \quad (4.35)$$

This estimate is based on interpolation. Figure 4.2 shows how to obtain it in a clear way. Estimate (4.33) is critical to get the decay estimate for the energy for $\varepsilon$ and $\tau$ small enough; see [303, estimate (54)].

<figure><img src="image_11.png" /><figcaption>Figure 4.2 – Interpolation for $L_t^{\hat{l}} L_x^{\hat{s}}(Q_+(1))$</figcaption></figure>

In the case of the half-space, on the contrary, we cannot estimate $P$ in terms of $U \otimes U$; in short $P \neq U \otimes U$. This issue is brought up in particular in the work of Koch and Solonnikov [220, Theorem 1.3]; see also Subsection 4.1.2 above. Considering the unsteady Stokes system in the half-space with no-slip boundary conditions, they prove that there are divergence-form source terms $\nabla \cdot F$, with $F \in L_q^{t,x}$, for which the pressure is not integrable in time. A possible alternative is to use maximal regularity for the Stokes system [163], i.e. $\nabla P \simeq U \cdot \nabla U$. Such an estimate was used for the local boundary regularity theory of the Navier-Stokes theory by Seregin in [306]. It remains a cornerstone for estimating the pressure locally near the boundary in particular given the localized version proved in [308]. Formally applying the maximal regularity estimate, together with the Poincaré-Sobolev and Hölder inequalities, the estimate of (4.31) (with $r = 1$) turns into

$$\|U\|_{L_t^l L_x^s(Q_+(1))} \|P - (P)_{B_+(1)}(t)\|_{L_t^{l'} L_x^{s'}(Q_+(1))} \\ \le \|U\|_{L_t^l L_x^s(Q_+(1))} \|U\|_{L_t^{l'} L_x^{s'}(Q_+(1))} \|\nabla U\|_{L^2(Q_+(1))} \quad (4.36)$$

with

$$\frac{1}{\hat{l}} + \frac{1}{\hat{l}} = \frac{1}{2} \quad \text{and} \quad \frac{1}{s} + \frac{1}{\hat{s}} = \frac{5}{6}.$$

The presence of the gradient of $U$ makes this quantity ‘too supercritical’. Using sharp interpolation arguments with the Type I bound (4.16) gives

$$\|U\|_{L_t^l L_x^s(Q_+(1))} \|U\|_{L_t^{l'} L_x^{s'}(Q_+(1))} \lesssim_M (A(U, 1))^\lambda \quad (4.37)$$
with any $\lambda$ slightly greater than $1 - (\frac{1}{l} + \frac{1}{\bar{l}}) = \frac{1}{2}$. Such a strategy will always produce a
total power of energy quantities $\kappa = \lambda + \frac{1}{2} \ge 1$, as can be seen by combining (4.36) with
(4.37). The analysis of Figure 4.2 enables to understand the limitation on the exponent $\kappa$.
Interpolating the $L_{t,x}^4$ norm of $U$ between the energy, i.e. the point $L_t^\infty L_x^2$, and the Type
I condition, i.e. the region $L_t^q L_x^p$, $q < 2$, yields a power of the energy too large. Thus,
any attempt to get an estimate similar to (4.34), with a small $\varepsilon$, in the half-space by using
maximal regularity estimates for the pressure and the Type I condition (4.16) will not work.

We overcome this difficulty in the half-space thanks to the new fractional estimate on the pressure discovered by Chang and Kang [89, Theorem 1.2] and reproved by us in [36], see Theorem 4.1 above. The idea is to have an intermediate estimate between $P \simeq U \otimes U$ which is known to be false (see above the comment about the result of Koch and Solonnikov [220]), and the maximal regularity estimate $\nabla P \simeq U \cdot \nabla U$ which is too supercritical. The fractional pressure estimates can be formally written as $\nabla^\beta P \simeq \nabla^\beta (U \otimes U)$, for $1 \ge \beta$ sufficiently close to 1. With the help of the fractional pressure estimate, we are able to implement the strategy of [310, 303]. We believe that this is the first instance where the estimate of Chang and Kang [89] is used and is really pivotal.

4.3.3 A new strategy for regularity under vorticity alignment in a scale-
invariant regime

The purpose of our work is to pave the way for a new method to prove regularity under
the vorticity alignment and Type I condition. This new approach allows more flexibility
in the rescaling procedure, and hence enables us to get geometric criteria on concentrating
balls, as stated in Theorem 4.3 above. To reach a contradiction, we do not need to show
that the blow-up profile $U_\infty$ is zero as in [162, 159]. Instead it suffices to prove that it
is bounded at a specific point, which is much easier. The vorticity alignment condition
serves the purpose of showing that $U_\infty$ is close to a two-dimensional flow in certain weak
topologies. This is sufficient to infer that $U_\infty$ is bounded at the desired point. We first recall
the method of Giga, Hsu and Maekawa [159] for the half-space and then outline our new
strategy.

A strategy based on a Liouville theorem

The proof by Giga, Hsu and Maekawa [159] for regularity under vorticity alignment in
the half-space is inspired by the whole-space strategy of Giga and Miura [162]. The proof
is by contradiction. One assumes vorticity alignment and the existence of a blow-up point.
The proof relies on two steps:

(1) Use the assumption on vorticity alignment and a blow-up procedure to reduce to a non-zero two-dimensional nontrivial bounded limit solution with positive vorticity, which is defined on either the whole-space or the half-space. This limiting solution to the Navier-Stokes equations belongs to one of two special classes called ‘whole-space mild bounded ancient solutions’ or ‘half-space mild bounded ancient solutions’, see [36, Section 1.3].

(2) Obtain a contradiction by showing the limit function must be zero. This is done in [162] and [159] by proving a Liouville theorem for two-dimensional whole-space and half-space mild bounded ancient solutions having positive vorticity and satisfying an ODE blow-up Type I bound.
The rescaling used in the first step is very specific. Let $t_k \uparrow T$, $y_k$ and $R_k \downarrow 0$ be selected such that

$$ \frac{1}{R_{(k)}} - 1 \le |U(y_k, t_k)| \le \sup_{0 \le s \le t_k} \|U(\cdot, s)\|_{L_\infty(\mathbb{R}^3)} := \frac{1}{R_k}. $$

Then $U_k(y, t) = R_k U(R_k(y + y_k), (R_k)^2(t + t_k))$ produces either a whole-space or half-space mild bounded ancient solution $U_\infty$ in the limit. By [39, Theorem 1.3], $U_\infty$ is smooth in space-time. To reach a contradiction, it is hence necessary to prove a strong fact about the limit, namely that the mild bounded ancient solution is zero. This is the purpose of the second step. For the case of the half-space, this step is nontrivial and involves a delicate analysis of the vorticity equation and its boundary condition.

## Our new strategy based on persistence of singularities

Our strategy in [36] in collaboration with Barker allows to prove regularity under the Type I condition, assuming vorticity alignment on concentrating sets, as in Theorem 4.3. It also makes it possible to achieve local versions of the result of Giga, Hsu and Maekawa [159, Theorem 1.3], and of our main result, Theorem 4.3.

The keystone in our scheme is the stability of singularities for the Navier-Stokes equations. Stability of singularities and compactness arguments were also used: (i) to prove the existence of potential blow-up solutions with minimal $\dot{H}^{\frac{1}{2}}(\mathbb{R}^3)$ data [287], or $L^3$ data [196]; (ii) to abstractly quantify Escauriaza-Seregin-Šverák type results as in the papers [7, Theorem 4.1 and Remark 4.2] and [9, Theorem 3.1 and Lemma 3.3]; see also Chapter 6; (iii) to give an abstract proof of a mild criticality breaking result in [37]; see also Lemma 6.7 in Subsection 6.3.5 below. The following lemma is taken from Barker's Ph.D. thesis [32, Proposition 5.5]; see also [9, Proposition A.5] and [280, Proposition 5.21] for subsequent generalisations. The stability of singular points was first proved in the interior case by Rusin and Šverák [287]. It is a consequence of $\varepsilon$-regularity type results.

**Lemma 4.6** (persistence of singularities; [36, Lemma 3]). Suppose $(U_k, P_k)$ are suitable solutions to the Navier-Stokes equations in $Q_+(1)$. Suppose that there exists a finite $M \in (0, \infty)$ such that

$$ \sup_k \left( \|U_k\|_{L_t^2 L_x^\infty(Q_+(1))} + \|\nabla U_k\|_{L^2(Q_+(1))} + \|P_k\|_{L_{\frac{3}{2}}(Q_+(1))} \right) = M < \infty. \quad (4.38) $$

Furthermore assume that

$$ \lim_{k \to \infty} \|U_k - U_\infty\|_{L^3(Q_+(1))} = 0, \quad (4.39) $$

$$ P_k \to P_\infty \text{ in } L_{\frac{3}{2}}(Q_+(1)), \quad (4.40) $$

($0, 0$) is a singular point of $U_k$ for all $k$

Then the above assumptions imply that $(U_\infty, P_\infty)$ is a suitable solution to the Navier-Stokes equations in $Q_+(1)$ with $(0, 0)$ being a singular point of $U_\infty$.

To show more precisely how our proof of Theorem 4.3 works, let us assume for contradiction that $U$ is singular at the space-time point $(0, T)$. Let $R_k \downarrow 0$ be any sequence. Schematically our method is in three steps.
**Step 1: rescaling and a priori bounds** We rescale in the following way

$$ U_k(y, s) := R_k U(R_k y, T + R_k^2 s) \quad \text{and} \quad P_k(y, s) := R_k^2 P(R_k y, T + R_k^2 s), \quad (4.41) $$

for all $y \in \mathbb{R}_+^3$ and $s \in (-T/R_k^2, 0)$. Theorem 4.2 enables us to show that the ODE blow-up Type I condition (4.20) implies the boundedness of the following scale-invariant quantities

$$ \sup_{0<r<1} \left\{ \frac{1}{r} \sup_{T-r^2<t<T} \int_{B_+(r)} |U(x,t)|^2 dx + \frac{1}{r} \int_{T-r^2}^{T} \int_{B_+(r)} |\nabla U|^2 dxdt + \frac{1}{r^2} \int_{T-r^2}^{T} \int_{B_+(r)} |P - (P(\cdot,t))_{B_+(r)}|^{\frac{3}{2}} dxdt \right\} < \infty. \quad (4.42) $$

Hence (4.38) holds for $(U_k, P_k)$ and we can apply Lemma 4.6.

**Step 2: passing to the limit and persistence of singularities** Applying Lemma 4.6 about the persistence of singularities gives us the following. Namely the blow-up profile $U_\infty$ is an ancient mild solution in $\mathbb{R}_+^3 \times (-\infty, 0)$, which has a singularity at $(0, 0)$, satisfies the Type I assumption (4.20) with $T = 0$ and has bounded scaled energy.

**Step 3: continuous alignment and contradiction via regularity of 2D flows** The continuous alignment condition for the vorticity implies that the vorticity direction of $U_\infty$ is constant in a large ball. We can then reach a contradiction by the following lemma for $\bar{U} = U_\infty$, proved in [36, Section 4].

**Lemma 4.7** (regularity via reduction to 2D; [36, Proposition 4]). Suppose that $(\bar{U}, \bar{P})$ is an ancient mild solution to the Navier-Stokes equations on $\mathbb{R}_+^3 \times (-\infty, 0)$ with $\bar{U}|_{\partial \mathbb{R}_+^3} = 0$. Let $\bar{\omega} := \nabla \times \bar{U}$ and suppose $(\bar{U}, \bar{P})$ is a suitable solution to the Navier-Stokes equations on $Q_+(r)$ for any $r > 0$. Furthermore suppose that $\bar{U}$ satisfies the ODE blow-up Type I assumption (4.20) with $T = 0$ and

$$ \sup_{0<r<\infty} \left\{ \frac{1}{r} \sup_{-r^2<t<0} \int_{B_+(r)} |\bar{U}(x,t)|^2 dx + \frac{1}{r} \int_{Q_+(r)} |\nabla \bar{U}|^2 dxdt + \frac{1}{r^2} \int_{Q_+(r)} |\bar{P} - (\bar{P}(\cdot,t))_{B_+(r)}|^{\frac{3}{2}} dxdt \right\} \le M'. \quad (4.43) $$

Under the above assumptions the following holds true. For all $M, M' \in (0, \infty)$, there exists $\gamma(M, M') \in (0, \infty)$ such that if

$$ \bar{\omega}(x, -t_0) \cdot \vec{e}_i = 0 \quad \text{in } B_+(\gamma(M, M')\sqrt{t_0}) \text{ for } i = 2, 3 \quad \text{and some } t_0 \in (0, \infty) \quad (4.44) $$
then $(x, t) = (0, 0)$ is a regular point for $\bar{U}$.

The proof of Lemma 4.7 goes again by contradiction. Notice that suitability of $(\bar{U}, \bar{P})$ is needed to use Lemma 4.6 on the persistence of singularities. Notice that arguments from [39] demonstrate that $\bar{U}$ is $C^\infty$ in space-time on $\mathbb{R}_+^3 \times (-\infty, 0)$, hence the pointwise condition on the vorticity (4.44) is well defined.
*Remark 4.8 (main flexibility in our method).* The main flexibility of our method lies in the fact that we can use any sequence $R_k \downarrow 0$ in the rescaling procedure. Therefore, we can tune the sequence to our needs. In the case of the whole-space, we can take advantage of this versatility on time slices of the cone $C_{t_n, \delta, \bar{x}}$ defined in (4.22), see [36, Theorem 3 in Section 5], and not the whole-cone $C_{\delta, \bar{x}}$ defined in (4.21) as in Theorem 4.3.

*Remark 4.9 (on the ODE blow-up rate Type I).* Under the ODE blow-up rate Type I assumption (4.20) with $T = 0$, we gain additional information about the blow-up profile. Namely, $U_\infty \in L_{\text{loc}}^\infty(-\infty, 0; L^\infty(\mathbb{R}_+^3))$. This plays an important role in our proof. Specifically, it ensures that for $t_0 > 0$, $U_\infty$ is the unique strong solution to the Navier-Stokes equations on $\mathbb{R}_+^3 \times (-t_0, 0)$ with initial data $U_\infty(\cdot, -t_0)$. It is not known if such considerations apply to the case when $U_\infty$ satisfies (4.43) only.

*Remark 4.10 (a difficulty for localized results and a new idea).* We discuss some new idea needed to prove a localized version of Theorem 4.3, see [36, Theorem 4]. For the whole proof, we assume that $(U, P)$ is a suitable solution in $Q_+(1)$. We assume that $(0, 0)$ is a space-time blow-up point. Our aim is to reach a contradiction under a localized continuous alignment condition, see [36, equation (134)]. In the local setting close to boundaries, we can have boundedness of the velocity but unboundedness of the vorticity, as can be seen from the construction of bounded flows with unbounded derivative in the half-space; see Subsection 4.1.4 above. Hence, contrary to mild solutions in the half-space there is no a priori bound for the space-time regularity of the solution that ensures that the vorticity direction converges in the space of continuous functions. But this is a crucial ingredient to use the vorticity alignment condition and to thus show that the limiting flow resulting from the rescaling procedure is two-dimensional. We go around the use of strong convergence for the vorticity by using Egoroff’s theorem.
# Chapter 5

# Norm concentration

This chapter relies mainly on the papers:

— [10], with Dallas Albritton and Tobias Barker, *Localized smoothing and concentration for the Navier-Stokes equations in the half space, submitted* (2021).

— [35], with Tobias Barker, *Localized Smoothing for the Navier-Stokes Equations and Concentration of Critical Norms Near Singularities, Arch. Ration. Mech. Anal.* (2020).

## Contents

<table><tr><td>5.1</td><td>Context: state of the art and obstacles</td><td>88</td></tr><tr><td>5.1.1</td><td>Why study infinite energy solutions?</td><td>88</td></tr><tr><td>5.1.2</td><td>Local energy solutions in the whole-space</td><td>89</td></tr><tr><td>5.1.3</td><td>Regularity results for local energy solutions</td><td>91</td></tr><tr><td>5.2</td><td>Main results</td><td>92</td></tr><tr><td>5.2.1</td><td>Local smoothing in the whole-space</td><td>92</td></tr><tr><td></td><td>Novelty of our results</td><td>93</td></tr><tr><td></td><td>Further development</td><td>94</td></tr><tr><td>5.2.2</td><td>Concentration in the whole-space</td><td>94</td></tr><tr><td></td><td>Novelty of our result</td><td>96</td></tr><tr><td></td><td>Further developments</td><td>96</td></tr><tr><td>5.2.3</td><td>Local energy solutions in the half-space</td><td>97</td></tr><tr><td></td><td>Novelty of our result</td><td>98</td></tr><tr><td></td><td>Further developments</td><td>99</td></tr><tr><td>5.2.4</td><td>Local smoothing and concentration in the half-space</td><td>99</td></tr><tr><td></td><td>Novelty of our results</td><td>101</td></tr><tr><td>5.3</td><td>New ideas and strategy for the proofs</td><td>101</td></tr><tr><td>5.3.1</td><td>Local-in-space short-time smoothing in the whole-space via a Caffarelli-Kohn-Nirenberg scheme</td><td>101</td></tr><tr><td>5.3.2</td><td>Existence of global-in-time local energy solutions in the half-space</td><td>103</td></tr><tr><td>5.3.3</td><td>Local-in-space short-time smoothing in the half-space via a compactness method</td><td>108</td></tr></table>

This Chapter is devoted to the study of concentration effects for certain critical norms near potential singularities of the three-dimensional Navier-Stokes equations in the whole-space and in the half-space. The central thread is to identify scales involved in singularity
formation, describe possible asymptotic self-similarity or understand the shape of blow-up solutions, if they exist. This study complements the analysis of Chapter 4, where we studied geometric concentration phenomena.

Norm concentration is a by-product of local smoothing properties for ‘local energy solutions’ with locally critical or subcritical initial data. Local energy solutions, that have uniformly bounded local energy may arise from blow-up limits around Type I singularities. In order to investigate norm concentration in the half-space, we therefore need to extend the existence results for local energy solutions, that were previously known in the whole-space, to the half-space. This poses considerable difficulties related to the strong nonlocal effects of the pressure in the half-space. The problem was mentioned as an open question in several papers [40, 41]. In collaboration with Maekawa and Miura [247], we solve this problem by introducing new estimates for the pressure with data having uniformly bounded local energy.

## 5.1 Context: state of the art and obstacles

### 5.1.1 Why study infinite energy solutions?

Let us list a number of reasons that motivate the study of non-decaying solutions:

(1) The global energy is supercritical for the scaling of the Navier-Stokes equations in 3D. Hence the energy blows-up when zooming in on potential singularities. In certain settings, for example if one considers a Type I singularity, the blow-up limit is of uniformly locally finite energy which is for instance used in the work of Seregin [294] concerned with the blow-up of the critical $L^3$ norm and in my work with Barker [35] concerned with concentration near Type I singularities.

(2) Such solutions are adapted to the study of forward self-similar solutions and discretely self-similar solutions, which are good candidates for producing non-unique Leray-Hopf solutions [197, 198, 179].

(3) Such solutions are interesting from the point of view of dynamics because they can generate their own dynamics. The large-scale effects of the pressure are stronger and may even drive the flow, see the examples of parasitic solutions (5.1) in the whole-space and (4.2) in the half-space. They can include a strain representing the effect of other vortices on a vortex filament modeled for instance by the Burgers vortex, see [74, 145, 162].

(4) In the same spirit as the last item, one can easily find examples of blow-up solutions with infinite energy. There are for instance singular Burgers vortices [263, 275] that blow-up in a scale-invariant way and grow linearly at space infinity. In collaboration with Maekawa and Miura we constructed a family of blow-up solutions around these singular Burgers vortices [248]. It is however unclear if such solutions have any relevance for potential singularities of Leray-Hopf solutions since they have constant vorticity directions, see Chapter 4.

(5) The framework of spatially homogeneous statistical solutions is a good setting to study probabilistic solutions with a law invariant under space translations. Basson [43] studies the existence of homogeneous statistical solutions without source term and proves their suitability (see also Višik and Fursikov [345, 346]). In [44] Basson proves the existence of spatially homogeneous statistical solutions with additive white noise source term.
5.1.2 Local energy solutions in the whole-space

The study of ‘local energy solutions’ was pioneered by Lemarié-Rieusset [235] in the early 2000s. Roughly speaking, these are weak solutions with locally finite energy, that satisfy a local energy inequality. Hence they form a larger class than the Leray-Hopf solutions [238, 189] that have finite global energy. The notion of local energy solutions became very popular for a number of reasons that we detail above in Subsection 5.1.1. In a nutshell, they provide a framework for the study of solutions with special symmetries, such as forward (discretely) self-similar solutions, while keeping enough properties to have standard tools such as ε-regularity.

To work with the local energy inequality (see (5.20) for the half-space), it is important to be able to compute the pressure to estimate the term

$$\int_0^t \int_{\mathbb{R}_+^3} (|U|^2 + 2P)U \cdot \nabla \Phi \,dxds.$$

Since the energy is assumed to be barely locally finite, the availability of pressure formulas becomes a non trivial question. Indeed, there are many examples of solutions of locally finite energy that have zero initial data

$$U(x, t) = f(t) \quad \text{and} \quad P(x, t) = -f'(t) \cdot x \qquad (5.1)$$

for $f \in C_0^\infty((0, \infty); \mathbb{R}^3)$; see (4.2) for an example in the half-space. These are the so-called ‘parasitic solutions’ or Serrin’s examples [311] driven by the pressure. Decomposing the pressure $P = P_{Helm} + P_{harm}$ into a Helmholtz pressure that is formally equal to

$$P_{Helm} = (-\Delta)^{-1} \nabla \cdot \nabla \cdot (U \otimes U)$$

(this formula requires some adaptation for the case of non-decaying data see (5.2)) and a harmonic pressure $P_{harm}$ that is harmonic on $\mathbb{R}^3$, the parasitic flows correspond to $P_{Helm} = 0$ and to a non trivial harmonic pressure that grows linearly. In order to have a pressure that is equal to the Helmholtz pressure up to a constant harmonic pressure, one needs to rule out the parasitic solutions. There are two approaches for this, which lead to two different definitions of local energy solutions:

(i) Either some mild decay of the solution is assumed at space infinity, such as

$$\lim_{|\bar{x}| \to +\infty} \int_0^{R^2} \int_{B_{\bar{x}}(R)} |U(x, t)|^2 dxdt = 0 \quad \text{for any } R \in (0, \infty).$$

A Liouville theorem [194] enables then to rule out the parasitic solutions and ensures the validity of the pressure formula. This approach is adopted for instance by Jia and Šverák [196, 197] and Tsai [339, page 160].

(ii) Or one includes the pressure formula directly in the definition of the solutions. For all
$\bar{x} \in \mathbb{R}^3$, there exists a function $c_{\bar{x}}(t) \in L^{\frac{3}{2}}(0, T)$ such that for all $(x, t) \in \mathbb{R}^3 \times (0, T)$,

$$ P(x,t) - c_{\bar{x}}(t) = -\frac{1}{3}|U(x,t)|^2 + \frac{1}{4\pi} \int_{B_{\bar{x}}(2)} K(x-y) \cdot U(y,t) \otimes U(y,t) dy \\ + \frac{1}{4\pi} \int_{\mathbb{R}^3 \setminus B_{\bar{x}}(2)} (K(x-y) - K(\bar{x}-y)) \cdot U(y,t) \otimes U(y,t) dy, \qquad (5.2) $$

where the kernel is defined by $K := \nabla^2(\frac{1}{|x|})$ and the splitting between a 'small-scale' or a 'local' part and a 'large-scale' or a 'nonlocal' part is needed to have enough decay on the kernel to handle the non-decaying solution $U$. This thread is followed by for instance by Kikuchi and Seregin [217] and Kwon and Tsai [226].

The equivalence between the class of solutions satisfying a pressure formula and the class of mild solutions is thoroughly investigated in several papers: by Furioli, Lemarié-Rieusset and Terraneo [139], Lemarié-Rieusset [235] and Bradshaw and Tsai [69] for data in $L_{uloc}^2$, by Fernández-Dalgo and Lemarié-Rieusset [135] for data in the weighted space $L^2(\mathbb{R}^3, (1+|x|)^{-4} dx)$ that allows for growth at space infinity.

As for the global-in-time existence of local energy solutions, it was proved for data decaying mildly in different senses. Lemarié-Rieusset [235] proved the global-in-time existence for data $U_0 \in L_{uloc}^2(\mathbb{R}^3)$ satisfying

$$ \lim_{|\bar{x}| \to +\infty} \int_{\bar{x} \in \mathbb{R}^3} |U_0|^2 dx = 0. \qquad (5.3) $$

This is equivalent to taking the data in the closure of $C_0^\infty(\mathbb{R}^3)$ for the $L_{uloc}^2$ norm. That class contains $L^{3,\infty}(\mathbb{R}^3)$ [67, Lemma 6.3]. Bradshaw and Tsai [67, Theorem 1.5] prove the global existence under the assumption that $U_0 \in L_{uloc}^2(\mathbb{R}^3)$ satisfies

$$ \lim_{R \to \infty} \sup_{\bar{x} \in \mathbb{R}^3} \frac{1}{R^2} \int_{B_{\bar{x}}(1)} |U_0|^2 dx = 0. \qquad (5.4) $$

That class includes critical Morrey data $\dot{M}^{2,1}$, hence data (even discretely self-similar) that does not satisfy the mild decay (5.3). In the work [226], Kwon and Tsai show the existence of global local energy solutions for data that satisfies the following mild decay of oscillation

$$ \lim_{|\bar{x}| \to \infty} \int_{B_{\bar{x}}(1)} |U_0 - (U_0)_{B_{\bar{x}}(1)}|^2 dx = 0. $$

Beyond the class of uniformly locally finite energy, global-in-time local energy solutions were studied for locally finite data $U_0 \in L_{uloc}^2(\mathbb{R}^3)$ that in addition is discretely self-similar. In that vein, let us cite the work of Chae and Wolf [85], Bradshaw and Tsai [66] and the book of Lemarié-Rieusset [237]. Fernández-Dalgo and Lemarié-Rieusset [134] prove global-in-time existence of weak solutions for data in the space $U_0 \in L^2(\mathbb{R}^3, (1+|x|)^{-\gamma} dx)$, $\gamma \in (0, 2]$. Notice that constants are excluded from that space. Bradshaw and Kukavica [62] (local-in-time) and Bradshaw, Kukavica and Tsai [64] (global-in-time) introduced a class of initial data that they call 'intermittent data' that contains all $L_{uloc}^2$ discretely self-similar data. Such data is allowed to grow at space infinity in an intermittent way. This
class includes the class considered in [67, Theorem 1.5], see (5.4), and the data considered in [134].

Let us point to two approaches used in these works to extend the solutions globally in time:

(1) An approach that is inspired from Calderón's decomposition [77]. For example in case of $L_{uloc}^2(\mathbb{R}^3)$ data such that (5.3), one proves the existence of a local-in-time solution that has improved integrability at time $t_1 > 0$ and then splits $U(\cdot, t_1)$ into a large part in $C_0^\infty(\mathbb{R}^3)$ for which one has a global-in-time Leray-Hopf solution [238] and a small part in the uniformly local Lebesgue space $L_{uloc}^3(\mathbb{R}^3)$ for which one has a global mild solution [250]. Fundamental to this strategy is to be able to transfer the mild decay (5.3) of the data to the solution itself via local energy estimates. This scheme, or variants, is used by Lemarié-Rieusset [235] and Kwon and Tsai [226]; see also our Theorem 5.10 below with Maekawa and Miura for local energy solutions in the half-space and its proof in Subsection 5.3.2.

(2) The other approach is based on scaling. One combines the local existence with local energy inequalities on expanding balls $B(n)$ with $n \to \infty$ which push the existence time like $n^2$. The assumption on mild decay on the data is used to get a representation formula for the pressure. This scheme, or variants, is used by Lemarié-Rieusset [236] for critical Morrey-Campanato spaces, Bradshaw and Tsai [67]. Scaling also plays a key role in the argument of Fernández-Dalgo and Lemarié-Rieusset [134].

### 5.1.3 Regularity results for local energy solutions

The regularity of Leray-Hopf solutions starting from smooth Schwartz or $C_0^\infty(\mathbb{R}^3)$ data being a famous open problem, it is obvious that the regularity problem is even more difficult for local energy solutions. However, local energy solutions form a large enough class in which it is interesting to have even partial results, for at least two reasons:

(i) this class contains forward self-similar and discretely self-similar data,

(ii) it is a good framework for local-in-space short-time smoothing.

**Regularity of self-similar solutions** Forward self-similar solutions were shown to be smooth by Grujić [170] using the partial regularity of Caffarelli, Kohn and Nirenberg [75]. This argument does not apply to discretely self-similar solutions though. Regularity of discretely self-similar solutions for data in $L^{3,\infty}(\mathbb{R}^3)$ and a similarity parameter $\bar{\lambda}$ close to 1 was proved by Kang, Miura and Tsai [212, Theorem 1.8].

For Leray-Hopf solutions, Leray proved eventual regularity in [238, paragraph 34]. In the context of local energy solutions, such results are not known and it is actually conjectured that there are counter-examples to such properties, for instance for a discretely self-similar solution having a local singularity with data in $L^{3,\infty}$ [65] (provided that such an object exists). Bradshaw and Tsai [67] investigated several initial regularity, eventual regularity, far field regularity properties for local energy solutions under smallness of critical Morrey-type quantities. In [68], Bradshaw and Tsai studied a class, intermediate between the Leray-Hopf class and the local energy class for which eventual regularity holds.

**Local-in-space short-time smoothing** This topic pioneered by Jia and Šverák [197] is a part of the general questioning about what data produce smooth solutions. Here we consider
global supercritical data such as $U_0 \in L_{uloc}^2(\mathbb{R}^3)$. In addition, we assume that locally, the data belongs to a subcritical space $U_0 \in L^m(B_0(1))$ for $m > 3$. Then Jia and Šverák [197, Theorem 3.2] prove that local energy solutions with such data are Hölder continuous in space and time away from the initial time (up to initial time if instead of $U_0 \in L^m(B_0(1))$ one assumes that the data is Hölder continuous) locally in space, i.e. in a cylinder $B_0(\frac{1}{2}) \times (\beta, S)$ for some $S \in (0, \infty)$ and any $0 < \beta < S$. This result follows from a decomposition of the solution reminiscent of Calderón [77] into a mild solution $a$ originating roughly from $U_0|_{B_0(1)} \in L^m(\mathbb{R}^3)$ and $V$ a local energy solution to a perturbed Navier-Stokes system with subcritical drift originating from $U_0|_{\mathbb{R}^3 \setminus B_0(1)} \in L_{uloc}^2(\mathbb{R}^3)$. The mild solution $a$ is smooth, while the perturbation $V$ is smooth locally in space and time, up to initial time, thanks to an $\varepsilon$-regularity result for the Navier-Stokes system with subcritical drift. A key point is to show that the local energy of the perturbation $V$ is small in $B_0(\frac{1}{2}) \times (0, S)$ by controlling the non-local effects of the pressure. In [197], the $\varepsilon$-regularity result is proved via a compactness scheme inspired from Lin's paper [243].

Extension of this result to the critical case, see below Theorem 5.1 and Subsection 5.3.1 for the whole-space and Theorem 5.11 and Subsection 5.3.3 for the half-space, requires new tools due to the lack of improvement of flatness at the limit.

In the past decade local-in-space smoothing results became an important tool for several applications:

**Forward self-similar solutions** In [197], the local-in-space regularity is used to prove certain a priori estimates for forward self-similar solutions [197, Theorem 4.1]. Using the self-similarity one can transfer local regularity away from the origin into decay at space infinity. This enabled the authors to prove the breakthrough result about the existence of large-data forward self-similar solutions.

**Concentration** In collaboration with Barker [35], see Theorem 5.5 below for the whole-space, and with Albritton and Barker [10], see Theorem 5.13 below for the half-space, we show concentration of certain scale-critical norms near potential Type I singularities. Such results are direct corollaries of local-in-space smoothing results.

**Quantitative regularity** Local-in-space smoothing is the pivotal tool in our scheme for quantitative regularity developed in [38] in collaboration with Barker, see Chapter 6.

## 5.2 Main results

### 5.2.1 Local smoothing in the whole-space

The following result shows that certain regularization properties that are known globally for mild solutions also hold locally in space and time. Our theorem below is stated in the locally finite-energy framework in view of the applications to concentration, see Theorem 5.5 below. For a definition of local energy solutions in the whole-space, we refer to [197, Definition 3.1]; see also [235, Chapter 32 and 33] and [217] and Definition 5.8 below for the half-space. Notice that contrary to [217], the definition that we use does not incorporate the pressure formula. This is the reason for the mild decay assumption (5.5) in the theorem below.

**Theorem 5.1** (local-in-space short-time smoothing in the whole-space; [35, Theorem 1], in collaboration with Barker). For all $M \in (0, \infty)$, there exists $S(M) \in (0, \frac{1}{4}]$ and an independent universal constant $\gamma_{univ} \in (0, \infty)$ such that the following holds true.
Consider any local energy solution $U$ to the Navier-Stokes equations with initial data $U_0 \in L_{uloc,\sigma}^2(\mathbb{R}^3)$ satisfying

$$ \begin{gathered} \|U_0\|_{L_{uloc}^2(\mathbb{R}^3)} \le M, \\ \lim_{|\bar{x}| \to +\infty} \sup_{\bar{x} \in \mathbb{R}^3} \int_{B_{\bar{x}}(1)} |U(\cdot, t)|^2 dx = 0, \qquad (5.5) \\ U_0 \in L^3(B_0(1)) \quad \text{and} \quad \|U_0\|_{L^3(B_0(1))} \le \gamma_{\text{univ}}. \end{gathered} $$

Then the above assumptions imply that

$$ U \in L^{\infty}(B_0(\frac{1}{2}) \times (\beta, S(M))), \qquad (5.6) $$

for all $\beta \in (0, S(M))$.

This result asserts that the regularity of local energy solutions is a somewhat local property, near initial time. Indeed the solution $U$ is bounded hence smooth in space. In some sense this means that the nonlocal effects of the pressure are not strong enough to perturb the regularizing properties of the parabolic part of the equation in short time; see also the comment (S) in [197, p. 234].

**Remark 5.2** (a stronger statement). The result proved in [35] is actually stronger than the statement above. Indeed, considering the mild solution $a$ associated to an $L^3$ continuous divergence-free extension of the critical data $U_0|_{B_0(1)}$, we prove that

$$ U - a \in C_{par}^{0,\nu}(\overline{B_0(\frac{1}{2})} \times [0, S(M)]), \qquad (5.7) $$

for some $\nu \in (0, \frac{1}{2})$.

**Remark 5.3** (quantitative version). Theorem 5.1 is qualitative. It is possible to state a quantitative version of this theorem as is done below for the analogous result in the half-space, see Theorem 5.11. Such quantitative versions of local-in-space smoothing, especially for locally subcritical data, are key to the strategy developed in Chapter 6 to prove quantitative regularity estimates.

**Remark 5.4** (local smoothing in borderline endpoint Lorentz and Besov spaces). In addition we prove local-in-space short-time smoothing for data locally small in borderline endpoint spaces. Namely, we prove a version of Theorem 5.1: (i) for $U_0 \in L^{3,\infty}(B_1(0))$, see [35, Appendix B], and (ii) for $U_0 \in \dot{B}_{p,\infty}^{-1+\frac{3}{p}}(B_1(0))$ for $p \in (3, \infty)$, see [35, Appendix B]. We also state a localized version of Theorem 5.1 in the framework of suitable solutions covering the case of critical Lebesgue, Lorentz and Besov data, see [35, Theorem 3].

### Novelty of our results

Our theorem can be seen as an extension to the scale-critical case of the pioneering result of Jia and Šverák [197, Theorem 3.1] for subcritical $U_0 \in L^m(B_0(1))$, $m > 3$. Such an extension poses certain challenges. Compactness arguments as in [197, Theorem 3.1] cannot be used directly to prove the Höder regularity (5.7) of the perturbation or even the boundedness (5.6) of $U$. Indeed, in the critical case there is a lack of improvement of flatness for the limiting perturbed linear equation with scale-critical drift terms. Difficulties with using compactness arguments are also found when proving $\varepsilon$-regularity statements
for the Navier-Stokes equations in higher dimensions, see [127]-[126]. This is the main difficulty we have to overcome to prove Theorem 5.1. We handle this difficulty by: (i) proving a subcritical Morrey bound thanks to a Caffarelli-Kohn-Nirenberg type scheme, (ii) bootstrapping the regularity via linear estimates. These points are explained in more details in Subsection 5.3.1 below.

## Further development

Shortly after our paper [35] appeared on arXiv, Kang, Miura and Tsai released a paper where they prove a localized version of Theorem 5.1 above, see [211, Theorem 1.1]. Their theorem is stated for suitable solutions to the Navier-Stokes equations and for data locally in $L^3$. They manage to obtain the boundedness property via a compactness scheme, but not the Hölder continuity of the perturbation as in (5.7). This is consistent with the remark above and the lack of improvement of flatness for the linear limit equation (5.27). Their method actually relies on improvement of integrability for the linear limit system (5.27) in the Lebesgue scale [211, proposition 4.1], rather than improvement of flatness. This then turns into a subcritical Morrey type bound for the solution of the perturbed Navier-Stokes system (5.26). The boundedness of the perturbation follows from standard $\epsilon$-regularity for the Navier-Stokes equation without perturbation term. We use this method in our work with Albritton and Barker [10] to establish a version of Theorem 5.1 for the half-space, see Theorem 5.11 below.

### 5.2.2 Concentration in the whole-space

We investigate accumulation behavior of norms near blow-up times on balls whose radius shrinks to zero as time approaches the singularity. In the next theorem we state ‘norm concentration’ near scale-critical potential singularities of the Navier-Stokes equations. We assume that the solution satisfies the following generalized Type I bound: for fixed $M, T^* \in (0, \infty)$ and a fixed radius $r_0 \in (0, \infty]$,

$$ \sup_{\bar{x} \in \mathbb{R}^3} \sup_{r \in (0, r_0)} \sup_{T^* - r^2 < t < T^*} r^{-\frac{1}{2}} \left( \int_{B_{\bar{x}}(r)} |U(x, t)|^2 dx \right)^{\frac{1}{2}} \leq M. \quad (5.8) $$

In order to make sense of the condition (5.8) in the case when $r_0 > \sqrt{T^*}$, we may extend $U$ by zero in negative times. In the Type I blow-up regime the diffusion is heuristically balanced with the nonlinearity. Despite this it remains a long-standing open problem whether or not Type I blow-ups can be ruled out when $M$ is large.

**Theorem 5.5** (concentration near scale-critical singularities in the whole-space; [35, Theorem 2], in collaboration with Barker). Let $\gamma_{univ}$ and $S(M)$ be given by Theorem 5.1 above. There exists $t_*(T^*, M, r_0) \in [0, \infty)$ such that the following holds true. Let $U$ be a Leray-Hopf solution to the Navier-Stokes equations in $\mathbb{R}^3 \times (0, \infty)$ satisfying the generalized Type I bound (5.8). Furthermore, suppose $U$ first blows-up at $T^*$ and has a singular point at the space-time point $(0, T^*)$.

Then the above assumptions imply that

$$ \|U(\cdot, t)\|_{L^3(B_0(\sqrt{T^*-t/S(M)/2}))} > \gamma_{univ}, \quad (5.9) $$

for all $t \in (t_*(T^*, M, r_0), T^*)$.
<figure><img src="image_12.png" /><figcaption>Figure 5.1 – Quantitative local-in-space short-time smoothing and concentration near potential Type I singularities</figcaption></figure>

Figure 5.1 illustrates how the concentration result of Theorem 5.5 follows from the local-in-space short-time smoothing stated in Theorem 5.1.

Remark 5.6 (on the generalized Type I condition). Let $M' \in (0, \infty)$. It is clear that the generalized Type I condition (5.8) is satisfied by Leray-Hopf solutions blowing-up at time $T^* > 0$ and such that

$$|U(x,t)| \le \frac{M'}{|x|}, \quad \text{for all } (x,t) \in \mathbb{R}^3 \times (0, T^*).$$

More generally, it is also satisfied for Leray-Hopf solutions $U$ blowing-up at time $T^* > 0$ and satisfying a scale-critical Morrey-type bound, i.e.

$$\|U(\cdot, t)\|_{\dot{M}^{2,3}} := \sup_{\bar{x} \in \mathbb{R}^3} \sup_{r \in (0, \infty)} r^{-\frac{1}{2}} \left( \int_{B_0(r)} |U(x, t)|^2 dx \right)^{\frac{1}{2}} \le M'$$

for all $t \in (0, T^*)$. This condition corresponds to (5.8) with $r_0 = \infty$ and $M = M'$. In this case, the concentration in Theorem 5.5 holds for any $t \in (0, T^*)$. It is less obvious to see that type I blow-ups satisfying the bound

$$\sqrt{T^* - t} |U(x, t)| \le M' \qquad (5.10)$$
or

$$ \sqrt{T^* - t}^\theta |x|^{1-\theta} |U(x,t)| \le M' \quad (5.11) $$

for some $\theta \in (0, 1)$ also enter the framework of Theorem 5.5. Yet, (5.10) and (5.11) imply that there exists $r_0 \in (0, \infty)$ and $M(M', U_0, r_0) \in (0, \infty)$ such that (5.5) holds. This is proved in [310]; see also the review article [303, pages 844-849].

Remark 5.7 (concentration in borderline endpoint in Lorentz and Besov spaces). This remark is the pendant of Remark 5.4. Local-in-space short-time smoothing for data locally small in the borderline endpoint spaces $L^{3,\infty}$ and $\dot{B}_{p,\infty}^{-1+\frac{3}{p}}$ for $p \in (3, \infty)$ implies concentration in the same spaces on a paraboloïd of concentration near potential singularities satisfying the generalized Type I bound (5.8). The proof goes along the same lines as for the $L^3$ case, following the rescaling used showed in Figure 5.1.

## Novelty of our result

Norm concentration phenomena were investigated for dispersive equations in the wake of the pioneering work by Merle and Tsutsumi [255], see [165, 256, 330, 186, 188].

In [240] an interesting concentration result is proved for a weak Leray-Hopf solution $U$ which first blows-up at $T^* > 0$. In particular, this result imply that there exists $t_n \uparrow T^*$ and $x_n \in \mathbb{R}^3$ such that

$$ \|U(\cdot, t_n)\|_{L^m(B_{x_n}(\sqrt{C(m)(T^*-t_n)}))} \ge \frac{C(m)}{(T^*-t_n)^{\frac{1}{2}(1-\frac{3}{m})}}, \quad 3 \le m \le \infty. \quad (5.12) $$

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

By using a rescaling argument and an estimate of the existence time of mild solutions in terms of the size of the initial data in $L_{\text{uloc}}^m(\mathbb{R}^3)$, $m > 3$, in collaboration with Maekawa and Miura we improved (5.12). In particular, see [249, Corollary 1.1], we showed that for every $t \in (0, T^*)$ (not just a sequence $t_n \uparrow T$) there exists $x(t) \in \mathbb{R}^3$ such that

$$ \|U(\cdot, t)\|_{L^m(B_{\sqrt{x(t)(C(m)(T^*-t))}})} \ge \frac{C(m)}{(T^*-t)^{\frac{1}{2}(1-\frac{3}{m})}}, \quad 3 \le m \le \infty. \quad (5.13) $$

This result holds for the whole-space relying on the mild solutions constructed by Maekawa and Terasawa [250] and in the half-space relying on the solutions we constructed with Maekawa and Miura in [249]. Notice that both (5.12) and (5.13) hold without any additional Type I assumption.

In (5.12) or (5.13) no information is provided on $x_n$ and $x(t)$. It is natural to ask whether the concentration phenomenon occurs on balls $B_x(R)$ with $R = O(\sqrt{T^* - t})$ and with $(x, T^*)$ being a singular point. Theorem 5.5 answers this in the affirmative for the $L^3$ for Leray-Hopf solutions which first blow-up at time $T^*$ and which satisfy the Type I bound.

## Further developments

In [212, Theorem 1.6], Kang, Miura and Tsai state two results that can be read as concentration/accumulation results for the supercritical $L^2$ norm near potential singularities of the Navier-Stokes equations. The first result is in the spirit of the result (5.13) from [249].
There exists $\gamma_{univ} \in (0, \infty)$, $S \in (0, \infty)$ and a function $x = x(t) \in \mathbb{R}^3$ such that for all $t \in (0, T^*)$,

$$ \frac{1}{\sqrt{T^* - t}} \int_{B_{x(t)}(\sqrt{\frac{T^*-t}{S}})} |U(x,t)|^2 dx > \gamma_{univ}. \quad (5.14) $$

This result is in the vein of the one of Grujić and Xu [173, Theorem 4.1] and Bradshaw and Tsai [69, Theorem 8.2]. The second result of [212, Theorem 1.6] is in the spirit of (5.5) from [35]. It holds under the generalized Type I condition (5.8). There exists $\gamma_{univ} \in (0, \infty)$ and $S(M) \in (0, \infty)$ such that for all $t \in (0, T^*)$,

$$ \frac{1}{\sqrt{T^* - t}} \int_{B_0(\sqrt{\frac{T^*-t}{S(M)}})} |U(x,t)|^2 dx > \gamma_{univ}. \quad (5.15) $$

Notice that both results involve scale-invariant Morrey type quantities. Considering their contrapositive, they can also be read as dynamically restricted regularity criteria since the scales on which smallness is needed are smaller and smaller as one approaches the final time.

In a different direction, let us also mention the result of Miller [260, Theorem 1.10] that states accumulation behavior of $\|U(x,t)\}_{\{|U(x,t)|>h(t)\}}^L(\mathbb{R}^3)$, where $h = h(t)$ is any function $L^2(0, T^*; \mathbb{R}_+)$.

### 5.2.3 Local energy solutions in the half-space

In collaboration with Maekawa and Miura we extended the notion of local energy solution to the half-space $\mathbb{R}_+^3$. We prove local-in-time as well as global-in-time existence results. For simplicity, let us only define global-in-time local energy solutions. The definition of local energy solutions on $\mathbb{R}_+^3 \times (0, T)$ can be adapted mutatis mutandis; see [247, Definition 1.1].

**Definition 5.8** (local energy solutions in the half-space; [247, Definition 1.1] in collaboration with Maekawa and Miura). A pair $(U, P)$ is called a ‘local energy solution’ to the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, \infty)$ with no-slip boundary condition on $\partial \mathbb{R}_+^3 \times (0, \infty)$ and the initial data $U_0 \in L_{uloc,\sigma}^2(\mathbb{R}_+^3)$ satisfying the mild decay assumption

$$ \lim_{|\bar{x}| \to +\infty} \int_{\bar{x} \in \mathbb{R}_+^3} |U_0|^2 dx = 0, \quad (5.16) $$

if $(U, P)$ satisfies the following conditions:

$$ (1) \text{ We have } U \in L_{\text{loc}}^{\infty}([0, \infty); L_{\text{uloc},\sigma}^2(\mathbb{R}_+^3)), P \in L_{\text{loc}}^{\frac{3}{2}}(\overline{\mathbb{R}_+^3} \times (0, \infty)) \text{ and} $$

$$ \sup_{\bar{x} \in \mathbb{R}_+^3} \int_0^{T'} \|\nabla U\|_{L^2(B_{\bar{x}}(1) \cap \mathbb{R}_+^3)}^2 dt + \sup_{\bar{x} \in \mathbb{R}_+^3} \left( \int_{\delta}^{T'} \|\nabla p\|_{L^{\frac{9}{8}}(B_{\bar{x}}(1) \cap \mathbb{R}_+^3)}^{\frac{3}{2}} dt \right)^{\frac{2}{3}} < \infty, \quad (5.17) $$

$$ \lim_{|\bar{x}| \to +\infty} \sup_{\bar{x} \in \mathbb{R}_+^3} \int_{B_{\bar{x}}(1) \cap \mathbb{R}_+^3} |U(\cdot, t)|^2 dx = 0, \quad (5.18) $$

for all $T' \in (0, \infty)$ and $\delta \in (0, T')$.
(2) For any $\varphi \in C_0^\infty(\overline{\mathbb{R}_+^3})$, $\varphi U \in L^2(0, T; H_0^1(\mathbb{R}_+^3))$.

(3) The pair $(U, P)$ is a solution in the sense of distributions.

(4) The function $t \mapsto \int_{\mathbb{R}_+^3} U(\cdot, t) \cdot w \, dx$ belongs to $C([0, T))$ for any compactly supported $w \in L^2(\mathbb{R}_+^3)^3$. Moreover, for any compact set $K \subset \overline{\mathbb{R}_+^3}$,

$$ \lim_{t \to 0} \|U(\cdot, t) - U_0\|_{L^2(K)} = 0. \qquad (5.19) $$

(5) The pair $(U, P)$ satisfies the local energy inequality: for all non-negative $\Phi \in C_0^\infty((0, T) \times \overline{\mathbb{R}_+^3})$ and for a.e. $t \in (0, T)$,

$$ \begin{aligned} & \int_{\mathbb{R}_+^3} |U(x, t)|^2 \Phi(x, t) \, dx + 2 \int_0^t \int_{\mathbb{R}_+^3} |\nabla U|^2 \Phi(x, t) \, dxds \\ & \le \int_0^t \int_{\mathbb{R}_+^3} |U|^2 (\partial_t + \Delta) \Phi \, dxds + \int_0^t \int_{\mathbb{R}_+^3} (|U|^2 + 2P) U \cdot \nabla \Phi \, dxds. \end{aligned} \qquad (5.20) $$

*Remark 5.9 (on the mild decay assumption).* Our definition is in the vein of the one for the whole-space used in [196, 197], where the authors defined local energy weak solutions in $\mathbb{R}^3 \times (0, \infty)$ which decay at spatial infinity. Contrary to [217] they do not include the representation formula for the pressure. Similarly, in the class considered here, the solutions decay mildly at spatial infinity thanks to (5.18), and thus the parasitic solutions are automatically excluded (via Liouville theorems, see for instance [194, 195, 249]), which guarantees the validity of a representation formula for the pressure. For a definition, in the spirit of [217], of local energy solutions in the half-space that avoids any mild decay assumption and hence includes the representation formula for the pressure, we refer to [63, Definition 1.1].

**Theorem 5.10 (global-in-time local energy solutions in the half-space; [247, Theorem 1] in collaboration with Maekawa and Miura).** For any initial data $U_0 \in L_{uloc,\sigma}^2(\mathbb{R}_+^3)$ satisfying the mild decay assumption (5.16), there exists a local energy weak solution $(U, P)$ to the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, \infty)$ in the sense of Definition 5.8.

## Novelty of our result

This result states the global-in-time existence of local energy weak solutions in the sense of Definition 5.8. It is the analog for the half-space of the theorem of Lemarié-Rieusset [235, Theorem 33.1] and of Kikuchi and Seregin [217, Theorem 1.5] for the whole-space $\mathbb{R}^3$; see also Subsection 5.1.2. Our Theorem 5.10 answers an open problem mentioned by Barker and Seregin in [40, Section 1]:

Unfortunately, and analogue of Lemarié-Rieusset type solutions for the half-space is not known yet. In fact it is an interesting open problem.

The main innovation making this progress possible is a new pressure estimate based on resolvent pressure formulas for non-decaying data, see Subsection 5.3.2. This pressure estimate gives some Lebesgue integrability in time of the harmonic pressure near initial time, hence the term involving the pressure can be estimated in the local energy inequality.
Interestingly, such pressure estimates do not seem to be obtained via previously known pressure representation formulas which give a non-integrable singularity near initial time. This was the main block to the open problem mentioned above. We refer to Subsection 5.3.2 below for further insights on this point.

In addition, let us remark that our tools enable us to reprove the result of Barker and Seregin [40] on the extension of Seregin's result [294] to solutions in the half-space bounded in $L^3(\mathbb{R}_+^3)$ along a sequence of times.

### Further developments

In the work [63], Bradshaw, Kukavika and Ożański prove the global existence of local energy solutions in the half-space for non-uniformly locally square integrable data that may grow at large-scales in an intermittent sense. In this setting, one has a Morrey-type local energy. Such data was also considered for instance in [62, 64, 135] in the whole-space. This framework is adapted to the study of forward discretely self-similar solutions, see also Subsection 5.1.2 above. In [63, Theorem 1.6] existence of forward discretely self-similar solutions is stated for the Navier-Stokes equations in the half-space with a no-slip boundary condition. The analysis of the paper [63] builds upon our work with Maekawa and Miura, namely the pressure formulas discovered in [247] and the kernel estimates from [249]. The decomposition of the pressure, see [249, Section 2] and Subsection 5.3.2 below, is actually included in the definition of local energy solutions. These formulas are very useful to Bradshaw, Kukavika and Ożański, as they were to us, in order to avoid the singularity in short-time of the harmonic pressure estimates [163, 324]. The pressure estimates though differ from ours, because contrary to [247] (see Definition 5.8 and Theorem 5.10 above) the local kinetic plus dissipation energy functional in [63] is not uniformly locally finite.

Finally, let us emphasize that the pressure formulas from our work [247] with Maekawa and Miura are central to the proof of local-in-space short-time smoothing in the half-space, see Theorem 5.11 in the next subsection.

### 5.2.4 Local smoothing and concentration in the half-space

In the paper [10] in collaboration with Albritton and Barker, we prove the analog of Theorem 5.1 (local-in-space short-time smoothing) and Theorem 5.5 (concentration near a scale-critical singularity) for solutions of the Navier-Stokes equations in the half-space with no-slip boundary conditions.

The following result states local smoothing for locally critical and subcritical data at the level of local energy solutions defined in Definition 5.8 and shown to exist in Theorem 5.10 above.

**Theorem 5.11** (local-in-space short-time smoothing in the half-space; [10, Theorem 1.1], in collaboration with Albritton and Barker). There exists $\gamma_{univ} \in (0, \infty)$ such that the following holds. Let $T \in (0, \infty)$ and $m \in [3, \infty)$ be fixed. Let $U$ be a local energy solution on $\mathbb{R}_+^3 \times (0, T)$ with initial data $U_0 \in L_{uloc}^2(\mathbb{R}_+^3)$ satisfying

$$ \|U_0\|_{L_{uloc}^2(\mathbb{R}_+^3)} \leq M, $$
and the mild decay condition (5.16). Assume that

$$ \textit{either} \quad \|U_0\|_{L^m(B_+(1))} \le N, \quad \text{if } m \in (3, \infty), \qquad (5.21) $$

$$ \textit{or} \quad \|U_0\|_{L^3(B_+(1))} \le N \le \gamma_{\text{univ}}, \quad \text{if } m=3. \qquad (5.22) $$

Then there exists $S = S(M, N, m) \in (0, T]$ satisfying the following property:

$$ \sup_{t \in (0, S)} t^{\frac{3}{2m}} \|U(\cdot, t)\|_{L^{\infty}(B_{+}(\frac{1}{2}))} \le C(m)(N + N^{\kappa(m)}), \qquad (5.23) $$

for constants $C(m) \in (0, \infty)$ and $\kappa(m) \ge 1$.

The result above is a global result. We also obtain localized results for suitable solutions, see [10, Theorem 1.4]. Such a result is in the spirit of the result of Kang, Miura and Tsai [211, Theorem 1.1]. Moreover, contrary to the whole-space result Theorem 5.1 that we stated above in a qualitative form, the half-space result (5.11) is stated in a quantitative way.

We now turn to the application of the above result to concentration of critical norms near potential scale-critical singularities in the half-space. As in the case of the whole-space, we state our result for solutions satisfying a generalized Type I condition

$$ \sup_{\bar{x} \in \mathbb{R}_+^3} \sup_{r \in (0, r_0)} \sup_{T^* - r^2 < t < T^*} r^{-\frac{1}{2}} \left( \int_{B_{\bar{x},+}(r)} |U(x, t)|^2 dx \right)^{\frac{1}{2}} \le M. \qquad (5.24) $$

In order to make sense of the condition (5.24) in the case when $r_0 > \sqrt{T^*}$, we may extend $U$ by zero in negative times.

**Remark 5.12** (on the generalized Type I condition in the half-space). As consequence of the unification of Type I blow-ups in the half-space achieved in [36, Theorem 2] in collaboration with Barker, see Theorem 4.2, we see that the ‘ODE blow-up Type I condition’

$$ \sqrt{T^* - t} |U(x, t)| \le M' $$

implies that (5.24) is satisfied for some $r_0 \in (0, \infty)$ and $M(M', U_0, r_0) \in (0, \infty)$.

**Theorem 5.13** (concentration near scale-critical singularities in the half-space; [10, Theorem 1.3], in collaboration with Albritton and Barker). Let $T^* > 0$. Let $\gamma_{\text{univ}}$ and $S(M, \gamma_{\text{univ}}, 3)$ be given by Theorem 5.11 above.

There exists $t_*(T^*, M, r_0) \in (0, \infty)$ such that the following holds true. Let $U$ be a Leray-Hopf solution to the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, \infty)$ with the no-slip boundary condition on $\partial \mathbb{R}_+^3 \times (0, T^*)$ and satisfying the generalized Type I bound (5.24). Furthermore, suppose $U$ first blows-up at $T^*$ and has a singular point at the space-time point $(0, T^*)$.

Then the above assumptions imply that

$$ \|U(\cdot, t)\|_{L^3(B_+(\sqrt{T^*-t}S(M)/2))} > \gamma_{\text{univ}}, \qquad (5.25) $$

for all $t \in (t_*(T^*, M, r_0), T^*)$.

In addition to this global concentration result, let us note that we obtained also a localized version of this concentration result for suitable solutions, see [10, Theorem 1.5].
## Novelty of our results

As we mentioned previously, one heuristic interpretation of local-in-space short-time smoothing is the following. The nonlocal effects of the pressure do not substantially hinder the parabolic smoothing properties of the Navier-Stokes equations, at least locally in space and time. It is not clear at first sight that such a property holds even in $\mathbb{R}^3$. It is all the more difficult to prove such local smoothing properties in the half-space, for a number of fundamental reasons related to the fact that the nonlocal effects of the pressure are much stronger in $\mathbb{R}^3_+$ than in $\mathbb{R}^3$. We refer to the difficulties outlined in Section 4.1. Our results Theorem 5.11 and Theorem 5.13 are the first results of this type for the half-space. These results build upon the step forward done in my paper [247] with Maekawa and Miura about the existence of local energy solutions in the half-space, see Theorem 5.10 above.

# 5.3 New ideas and strategy for the proofs

## 5.3.1 Local-in-space short-time smoothing in the whole-space via a Caffarelli-Kohn-Nirenberg scheme

In the subcritical case, the main part of the proof of Jia and Šverák [197] for proving local-in-space short-time smoothing relies upon establishing an $\varepsilon$-regularity criteria for suitable solutions of the perturbed Navier-Stokes equations with subcritical $a$, i.e. $a \in L_t^\infty L_x^m(B_0(1) \times (-1, 0))$, $m > 3$ and $\nabla \cdot a = 0$,

$$ \begin{gathered} \partial_t V - \Delta V + V \cdot \nabla V + a \cdot \nabla V + V \cdot \nabla a + \nabla Q = 0, \tag{5.26} \\ \nabla \cdot V = 0 \quad \text{in } B_0(1) \times (-1, 0). \end{gathered} $$

Notice that $a$ is the mild solution associated to $U_0|_{B_0(1)}$ properly extended to the whole-space in a divergence-free manner. In particular, they show that if certain scale-critical quantities involving $V$ and $Q$ on the unit cube $B_0(1) \times (-1, 0)$ are small then one has decay of the oscillation:

$$ \frac{1}{r^5} \int_{t_0-r^2}^{t_0} \int_{B_{\bar{x}}(r)} \left| V - \int_{t-r^2}^{t} \int_{B_{\bar{x}}(r)} V dyds \right|^3 dxds' \le Cr^\alpha $$

for all $(\bar{x}, t) \in B_0(\frac{1}{2}) \times (-\frac{1}{4}, 0)$ and for some $\alpha > 0$. This implies parabolic Hölder continuity of $V$ by Campanato's characterisation. The proof of the decay of the oscillation in Jia and Šverák's paper [197] is achieved by contradiction and by compactness arguments. Related arguments were previously used in the context of the Navier-Stokes equations by Lin [243] and by Ladyženskaja and Seregin in [230]. Such arguments applied to the system (5.26) crucially use that for $a$ in subcritical spaces, we have parabolic Hölder continuity in $B_0(\frac{1}{2}) \times (-\frac{1}{4}, 0)$ for the following limit linear system:

$$ \begin{gathered} \partial_t W - \Delta W + a \cdot \nabla W + W \cdot \nabla a + \nabla Q = \nabla \cdot G, \tag{5.27} \\ \nabla \cdot W = 0 \quad \text{in } B_0(1) \times (-1, 0), \end{gathered} $$

for a subcritical forcing term $G \in L^{\frac{5m}{3}}(B_0(1) \times (-1, 0))$.

Unfortunately, when $U_0$ is critical and hence $a$ and $G$ belong to scale-invariant spaces with respect to the Navier-Stokes scaling such as $L^5(B_0(1) \times (-1, 0))$, we do not expect
solutions of (5.27) to be Hölder continuous. This lack of improvement for the perturbed linear system (5.27) to prevents us from relying on compactness arguments to directly prove the boundedness of the perturbation V.

As is the case in [197] paper, the key point is to take advantage of the smallness of the local energy of the perturbation V locally in space near initial time, i.e. in $B_0(\frac{7}{8}) \times (0, S(M))$ for some $S(M) \in (0, \infty)$. There are then two main bricks in the proof:

(Step-1) We prove a subcritical Morrey bound on the perturbation. Smallness of the local energy together with the smallness of $\|a\|_{L^5_{t,x}}$ enables us to prove a subcritical Morrey bound on V: for $\delta \in (0, 3)$ fixed, for $(\bar{x}, t) \in B_0(\frac{3}{4}) \times (0, S(M))$,

$$ \sup_{r \in (0, \frac{3}{4})} \frac{1}{r^{5-\delta}} \int_{t-r^2}^{t} \int_{B_{\bar{x}}(r)} |V|^3 dx ds < \infty, \quad (5.28) $$

with V extended by 0 in negative times. Estimate (5.28) is based on a Caffarelli-Kohn-Nirenberg type iteration. The proof requires some technical innovations, in particular concerning the treatment of the pressure and of the perturbation terms $a \cdot \nabla V$ and $V \cdot \nabla a$. A major difficulty is that the decay of $\|a\|_{L^5(B_{\bar{x}}(r) \times (t-r^2,t))}$ does not improve when $r \to 0$. The $L^5_{t,x}$ norm for $a$ is the critical threshold for the iteration to work. Exploiting that $a$ is a solution to the Navier-Stokes equations and the bound (5.28), one could directly apply standard $\varepsilon$-regularity away from the initial time to get smoothness of the solution $U$. This is what we do for the half-space, see Subsection 5.3.3 below. Instead, in the whole-space, we aim at obtaining the boundedness of V up to the initial time, see (5.7).

(Step-2) We obtain the boundedness of the perturbation $V = U - a$ and eventually Hölder continuity in the parabolic metric up to initial time (5.7). It goes through the use of the Morrey bound (5.28) to control the nonlinear term in (5.26) and a bootstrap to get the boundedness. Related arguments were used by Seregin in [307]. We now comment on the final bootstrap to get the Hölder regularity. Let us point out that for the subcritical case $U_0 \in L^m(B_0(1))$ (which corresponds to $a$ belonging to subcritical spaces), Jia and Šverák prove in [197] that the perturbation $V = U - a$ is Hölder continuous in the parabolic metric up to the initial time. Moreover, in [197] the Hölder exponent arising from the proof degenerates as $m$ approaches the critical case $m = 3$. Perhaps at first sight it appears somewhat unexpected that one still obtains Hölder continuity of $V$ up to the initial time, for the critical initial data case. Our proof for showing this relies upon the structure of estimates for the mild solution $a$, in particular $\sup_{s \in (0, S(M))} s^{\frac{1}{5}} \|a(\cdot, s)\|_{L^5(\mathbb{R}^3)} \ll 1$ and the fact that $V$ has zero initial data. Such points allow us to obtain a decay of the $L^\infty$ norm of $V$ near the initial time, which is key for going from $V$ being bounded to Hölder continuous.

The extension of our results to the Besov case $\dot{B}_{p,\infty}^{-1+\frac{3}{p}}$ for $p \in (3, \infty)$, see Remark 5.4, relies on some ideas which are new as far as we know. In particular, in the Caffarelli-Kohn-Nirenberg-type iteration, we need to exploit the local decay of the kinetic energy near the initial time, because the critical drift is more singular in the Besov case than in the $L^3$ case. It satisfies namely for $S \in (0, \infty)$ and sufficiently small data

$$ \sup_{s \in (0, S)} \left( s^{\frac{1}{2}(1-\frac{3}{p})} \|a(\cdot, s)\|_{L^p} + s^{\frac{1}{2}} \|a(\cdot, t)\|_{L^\infty} \right) \le K(p) \sup_{s \in (0, S)} s^{\frac{1}{2}(1-\frac{3}{p})} \|e^{t\Delta} U_0\|_{L^p}. $$
In particular, we don't have a global $L_{t,x}^5$ bound on $a$. Rather than carrying out the Caffarelli-Kohn-Nirenberg iteration at the level of the subcritical Morrey-type bound (5.28), we propagate subcritical Morrey bounds (and local energy bound) that have decay near initial time: for some $\eta \in (0, 1)$, we show that

$$ \sup_{r \in (0, \frac{3}{4})} \frac{1}{r^{5-\delta}} \int_{t-r^2}^{s} \int_{B_{\bar{x}}(r)} |V|^3 dxd\hat{s} \le C_{\text{univ}} s_{+}^{\frac{3}{2}\eta}, \quad t-r^2 < s < t. $$

Such an insight was used before for global estimates by Barker [33] to prove weak-strong uniqueness, in Barker, Seregin and Šverák's paper [41] for global $L^{3,\infty}$ solutions and by Albritton and Barker [7] for global Besov solutions. However, to the best of our knowledge, it is the first time that the decay of the kinetic energy near initial time is used in local estimates, such as a Caffarelli-Kohn-Nirenberg type iteration. We believe this point is of independent interest.

### 5.3.2 Existence of global-in-time local energy solutions in the half-space

The key point of our work [247] and of the proof of the existence of local energy solutions in the half-space, see Theorem 5.10, is a new estimate for the pressure of Navier-Stokes in the half-space with non-decaying data. We rely on a decomposition of the pressure derived from the explicit representation obtained in collaboration with Maekawa and Miura [249]. This representation formula for the pressure of the Stokes resolvent problem builds upon an idea of Desch, Hieber and Prüss [124] that enables to circumvent the use of the Helmholtz-Leray projector known to be ill-behaved on spaces of non-decaying functions. The resolvent kernel is decomposed into a Dirichlet-Laplace part and a part corresponding to the nonlocal pressure, see [249, Subsection 2.3].

Indeed, we generalize the whole-space representation formula (5.2) to the case of the half-space. Due to the boundary $\partial\mathbb{R}_+^3$, in addition to the Helmholtz pressure $P_{Helm}$, a harmonic pressure $P_{harm}$ has to be taken into account. Indeed, the nonlinear term in the Navier-Stokes equations can be decomposed into

$$ \nabla \cdot (U(\cdot, t) \otimes U(\cdot, t)) = \nabla P_{Helm} + \mathbb{P} \nabla \cdot (U(\cdot, t) \otimes U(\cdot, t)). $$

Hence the harmonic pressure solves the Neumann boundary value problem

$$ \begin{cases} -\Delta P_{\text{harm}}(\cdot, t) = 0 & \text{in } \mathbb{R}_+^3, \\ \partial_3 P_{\text{harm}}(\cdot, t) = \gamma|_{x_3=0} \Delta U_3(\cdot, t) & \text{on } \partial \mathbb{R}_+^3. \end{cases} \quad (5.29) $$

where $\mathbb{P}$ is the Helmholtz-Leray projection on divergence-free vector fields. We are able to provide an explicit representation for the Helmholtz part of the pressure, as well as for the harmonic part. Since we handle non-decaying data, each pressure part has to be split, as in the case of the whole-space, into a local part and a nonlocal part. Notice that the formulas for the nonlocal pressure terms, see $P_{\text{nonloc}}^{U_0}$, $P_{\text{Helm,nonloc}}^{U \otimes U}$ and $P_{\text{harm,nonloc}}^{U \otimes U}$ below, are obtained by subtracting a constant, as for the whole-space. This is done in order to get a kernel with enough decay to handle non-decaying data.

Let $T > 0$ and $\bar{x} \in \mathbb{R}_+^3$. We decompose the solution $(U, P)$ to the Navier-Stokes
equations into $(U^{U_0}, P^{U_0})$ the solution of the homogeneous Stokes system

$$ egin{cases} 
\partial_t U^{U_0} - \Delta U^{U_0} + \nabla P^{U_0} = 0 & \text{in } \mathbb{R}_+^3 \times (0, T), \\
\nabla \cdot U^{U_0} = 0, & \\
U^{U_0} = 0 & \text{on } \mathbb{R}_+^3 \times (0, T), \\
U^{U_0}(\cdot, 0) = U_0, & 
\end{cases} 
\quad (5.30) $$

and $(U^{U \otimes U}, P^{U \otimes U})$ the solution of the Stokes system with source term $-\nabla \cdot (U \otimes U)$,

$$ egin{cases} 
\partial_t U^{U \otimes U} - \Delta U^{U \otimes U} + \nabla P^{U \otimes U} = -\nabla \cdot (U \otimes U) & \text{in } \mathbb{R}_+^3 \times (0, T), \\
\nabla \cdot U^{U \otimes U} = 0, & \\
U^{U \otimes U} = 0 & \text{on } \partial \mathbb{R}_+^3 \times (0, T), \\
U^{U \otimes U}(\cdot, 0) = 0. & 
\end{cases} 
\quad (5.31) $$

The pressure $P^{U_0}$ is a harmonic pressure since there is no source term in (5.30), hence no Helmholtz part. We split it into a local part around $\bar{x}$ and a nonlocal part away from $\bar{x}$: for all $(x, t) \in \mathbb{R}_+^3 \times (0, T)$,

$$ 
\begin{aligned} 
P^{U_0}(x, t) &= \frac{1}{2\pi i} \int_{\Gamma} e^{\lambda t} \int_{\mathbb{R}_+^3} q_{\lambda}(x' - z', x_3, z_3) \cdot U'_0|_{B_{\bar{x},+}(2)}(z', z_3) dz' dz_3 d\lambda \\ 
&\quad + \frac{1}{2\pi i} \int_{\Gamma} e^{\lambda t} \int_{\mathbb{R}_+^3} (q_{\lambda}(x' - z', x_3, z_3) - q_{\lambda}(\bar{x}' - z', \bar{x}_3, z_3)) \cdot U'_0|_{B_{\bar{x},+}(2)}(z', z_3) dz' dz_3 d\lambda \\ 
&=: P_{\text{loc}}^{U_0}(x, t) + P_{\text{nonloc}}^{U_0}(x, t). 
\end{aligned} 
\quad (5.32) $$

This formula is obtained by Dunford's formula which gives an inverse of the Laplace transform. Here for all $x' \in \mathbb{R}^2$ and $x_3, z_3 > 0$,

$$ q_{\lambda}(x', x_3, z_3) := i \int_{\mathbb{R}^2} e^{ix' \cdot \xi} e^{-|\xi|x_3} e^{-\sqrt{\lambda+|\xi|^2}z_3} \left( \frac{\xi}{|\xi|} + \frac{\xi}{\sqrt{\lambda+|\xi|^2}} \right) d\xi \quad (5.33) $$

is the harmonic pressure kernel for the resolvent problem defined in [249], (2.8e) and Proposition 3.7], and $\Gamma = \Gamma_{\kappa}$ with $\kappa \in (0, 1)$ is the curve of $\mathbb{C}$ defined previously in (4.26).

Remark 5.14 (on the formulas for the harmonic pressure terms). Notice that the formula (5.32) involves the tangential component of $U_0$ rather than a vertical component as in the formula (4.25). Hence the resolvent pressure kernels $q_{\lambda}$ and $\tilde{q}_{\lambda}$ that appear in the formulas are different. The two formulas are derived from [249]. The same comment applies to $P_{\text{harm}}^{U \otimes U}$ defined below in (5.35).

As for the pressure term $P^{U \otimes U}$, we decompose it into a Helmholtz pressure $P_{\text{Helm}}^{U \otimes U}$ and a harmonic pressure $P_{\text{harm}}^{U \otimes U}$ and split again into local and nonlocal terms. We have

$$ P^{U \otimes U} = P_{\text{Helm}}^{U \otimes U} + P_{\text{harm}}^{U \otimes U}, $$
with for the Helmholtz pressure

$$
\begin{align}
P_{Helm}^{U \otimes U}(x, t) & \\
&= \int_{\mathbb{R}_+^3} \nabla_z^2 N(x' - z', x_3, z_3) U \otimes U|_{\mathbb{R}_+^3 \setminus B_{\bar{x},+}(2)}(z', z_3, t) dz' dz_3 \nonumber \\
&\quad + \int_{\mathbb{R}_+^3} \nabla_z^2 (N(x' - z', x_3, z_3) - N(\bar{x}' - z', \bar{x}_3, z_3)) U \otimes U|_{\mathbb{R}_+^3 \setminus B_{\bar{x},+}(2)}(z', z_3, t) dz' dz_3 \nonumber \\
&= p_{Helm,loc}^{U \otimes U}(x, t) + p_{Helm,nonloc}^{U \otimes U}(x, t) \tag{5.34}
\end{align}
$$

where $N$ is the Neumann kernel in the half-space and for the harmonic pressure

$$
\begin{align}
P_{harm}^{U \otimes U}(x, t) & \\
&= \frac{1}{2\pi i} \int_0^t \int_{\Gamma} e^{\lambda(t-s)} \int_{\mathbb{R}_+^3} q_{\lambda}(x' - z', x_3, z_3) \cdot (\mathbb{P}\nabla \cdot (U \otimes U|_{B_{\bar{x},+}(2)}))'(z', z_3, s) dz' dz_3 d\lambda ds \nonumber \\
&\quad + \frac{1}{2\pi i} \int_0^t \int_{\Gamma} e^{\lambda(t-s)} \int_{\mathbb{R}_+^3} (q_{\lambda}(x' - z', x_3, z_3) - q_{\lambda}(\bar{x}' - z', \bar{x}_3, z_3)) \nonumber \\
&\qquad \cdot (\mathbb{P}\nabla \cdot (U \otimes U|_{\mathbb{R}_+^3 \setminus B_{\bar{x},+}(2)}))'(z', z_3, s) dz' dz_3 d\lambda ds \nonumber \\
&:= p_{harm,loc}^{U \otimes U}(x, t) + p_{harm,nonloc}^{U \otimes U}(x, t), \tag{5.35}
\end{align}
$$

where $q_\lambda$ and $\Gamma$ are defined as above in (5.33) and (4.26). It becomes clear from the formula (5.35) that the pressure in the half-space is more nonlocal than in the whole-space. Indeed, in addition to being nonlocal in space, there is nonlocality in time, because the harmonic pressure depends on the history of the flow.

It is essential to keep in mind that every term in decomposition above depends on $\bar{x}$. However, for two points $\bar{x}$ and $\hat{x}$, the difference of the associated pressures is a constant that depends only on time. Finally, let us emphasize that this level of precision in the description of the pressure can be achieved due to the special structure of $\mathbb{R}_+^3$, which allows to use the Fourier transform in the horizontal direction and hence to obtain explicit formulas. Notice that the (5.32) and (5.35) for the harmonic pressure terms $P_{\text{harm}}^{U0}$ and $P_{\text{harm}}^{U \otimes U}$ and (5.34) for the Helmholtz pressure have a convolution structure in the direction tangential to the boundary, but not in the vertical direction.

For estimating the pressure terms, we rely on the results for the linear theory in the half-space obtained in or companion paper [249]. The local pressure $P_{\text{loc}}^{U0}$ is certainly the most subtle term to analyze. Interestingly, if one uses the formulas obtained by Solonnikov [324, formula (1.9)], see also the papers [321, 323], one gets a non-integrable singularity near initial time for points $\bar{x} \in \partial \mathbb{R}_+^3$,

$$
\| P_{\text{loc}}^{U_0} \|_{L^2(B_{\bar{x},+}(1))} \le C \frac{(1+t)^{\frac{1}{2}}}{t} \| U_0 \|_{L^2(B_{\bar{x},+}(2))}.
$$
We are able to improve this bound near initial time by computing the integral in space variables with the resolvent pressure kernel $q_λ$ first, and then computing the Dunford integral over the curve $\Gamma$. We obtain the following proposition.

**Proposition 5.15** (estimates for the linear pressure terms; [247, Proposition 2.1]). Let $T > 0$. There exists a constant $C(T) \in (0, \infty)$ such that for $t \in (0, T)$,

$$ \frac{t}{\log(e+t)} \|\nabla P^{U_0}(\cdot, t)\|_{L^2_{\text{uloc}}(\mathbb{R}_+^3)} \le C \|U_0\|_{L^2_{\text{uloc}}(\mathbb{R}_+^3)}, \quad (5.36) $$

$$ t^{\frac{3}{4}} \| P_{\text{loc}}^{U_0}(\cdot, t) \|_{L^2(B_{\bar{x},+}(1))} \le C(T) \|U_0\|_{L^2(B_{\bar{x},+}(2))}, \quad (5.37) $$

$$ t^{\frac{3}{4}} \| P_{\text{nonloc}}^{U_0}(\cdot, t) \|_{L^{\infty}(B_{\bar{x},+}(1))} + t^{\frac{3}{4}} \| \nabla P_{\text{nonloc}}^{U_0}(\cdot, t) \|_{L^{\infty}(B_{\bar{x},+}(1))} \le C(T) \|U_0\|_{L^2_{\text{uloc}}(\mathbb{R}_+^3)}. \quad (5.38) $$

Moreover, (5.36) holds with $C$ independent of $T$.

*Remark 5.16* (erratum). In the published version of this result [247, Proposition 2.1], there is a mistake in the estimate (5.38). We claimed a singularity like $t^{\frac{1}{2}}$ instead of $t^{\frac{3}{4}}$. The slightly worse singularity in (5.38) does not change however substantially the other estimates in the paper since what matters is that one has some integrability of $\|P_{\text{nonloc}}^{U_0}(\cdot, t)\|_{L^{\infty}}$ in short-time. The latest version on arXiv https://arxiv.org/abs/1711.04486 is correct.

Estimate 5.36 above is directly obtained from the estimates for the Stokes semigroup in $L_{\text{uloc}}^2$ studied in our companion paper [249, Proposition 5.3]. The estimate (5.37) of $P_{\text{loc}}^{U_0}$ leads to the bound

$$ \| P_{\text{loc}}^{U_0} \|_{L^{\frac{4}{3}}(0,T; L^2(B_{\bar{x},+}(1)))} \le C(T) \| U_0 \|_{L^2(B_{\bar{x},+}(2))}. $$

This integrability in time, although with a small Lebesgue exponent, enables to use the local energy inequality (5.20) to control the solution $U$ of the Navier-Stokes equations.

Let us notice that the singularity $O(t^{-\frac{3}{4}})$ in (5.37) is consistent with the one obtained in [273, 336] for the Stokes resolvent problem in a bounded domain $\Omega$ with no-slip boundary condition and source term $f$:

$$ \|p\|_{L^2(\Omega)} \lesssim_{\alpha} |\lambda|^{-\alpha} \|f\|_{L^2(\Omega)} \quad \text{for } \alpha \in [0, 1/4). \quad (5.39) $$

In [335] the optimality of the threshold $1/4$ is established. The bound (5.39) turns, via Dunford's formula, into a short-time estimate for the pressure associated to the unsteady Stokes problem with a singularity $O(t^{-3/4-\delta})$ for $\delta > 0$. We emphasize that in the case of bounded domains on the one hand and of localized estimates on the other hand, the estimates that are obtained break the natural scaling of the equations in the whole-space. This is not a contradiction because in both situations there is no scale-invariance. For more discussion on this topic, we refer to Subsection 4.1.2 above.

We do not rewrite here the estimates for $P_{\text{Helm,loc}}^{U \otimes U}$, $P_{\text{Helm,nonloc}}^{U \otimes U}$, $P_{\text{harm,loc}}^{U \otimes U}$ and $P_{\text{harm,nonloc}}^{U \otimes U}$ that can be found in the paper, see [247, Proposition 2.2 and Proposition 2.3].

With these estimates for the pressure, the proof of Theorem 5.10 goes roughly as follows. The evolution starts with a rough data barely locally integrable, $U_0 \in L_{\text{uloc},\sigma}^2(\mathbb{R}_+^3)$, with the mild decay condition (5.16). The local-in-time local energy solution instantly becomes slightly more regular: $U(\cdot, t_0) \in L_{\text{uloc},\sigma}^4(\mathbb{R}_+^3)$ for almost all $t_0$ in the existence time
interval with the mild decay

$$ \lim_{|\bar{x}| \to +\infty} \int_{\bar{x} \in \mathbb{R}_+^3 \cap B_{\bar{x},+}(1) \cap \mathbb{R}_+^3} |U(\cdot, t_0)|^4 dx = 0. \quad (5.40) $$

This allows to decompose the data $U(\cdot, t_0)$ into a large $C_c^\infty(\mathbb{R}_+^3)$ part for which we have global-in-time Leray-Hopf solutions, and a small part in $L_{uloc,\sigma}^4(\mathbb{R}_+^3)$ for which we have local-in-time existence of mild solutions thanks to our work in collaboration with Maekawa and Miura [249, Proposition 7.1]. The difficult part of this reasoning is to transfer the decay of the initial data $U_0$ satisfying (5.16) to the solution $U$, i.e. to prove (5.40). This issue is addressed in [235, Proposition 32.2] and [217, Theorem 1.4] in the case of the whole-space. We handle this question for the half-space, see (5.41) below, which is more involved due to the strong nonlocal nature of the pressure. Our main results in this direction are summarized in the following proposition.

**Proposition 5.17** (main proposition for the existence of global-in-time local energy solutions; [247, Corollary 5.9, Lemma 4.1 and Theorem 2]). This proposition is divided into two parts:

(1) For all $M > 0$, there exist $T(M) \in (0, \infty)$ and $A_M \in [1, \infty)$ such that for all $U_0 \in L_{uloc,\sigma}^2(\mathbb{R}_+^3)$ satisfying the mild decay (5.16), for all local energy solution $U$ to the Navier-Stokes equations on $\mathbb{R}_+^3 \times (0, T(M))$, if $\|U_0\|_{L_{uloc}^2(\mathbb{R}_+^3)} \le M$, then

$$ \sup_{\bar{x} \in \mathbb{R}_+^3} \sup_{t \in (0, T(M))} \int_{B_{\bar{x},+}(1)} |U(\cdot, t)|^2 + \int_0^{T(M)} \int_{B_{\bar{x},+}(1)} |\nabla U|^2 + \left( \int_0^{T(M)} \int_{B_{\bar{x},+}(1)} |U|^3 \right)^{\frac{2}{3}} \le A_M. $$

(2) For $M$, $T(M)$, $U_0$ and $U$ as above we have: for all $R \ge 1$,

$$ \sup_{\bar{x} \in \mathbb{R}_+^3} \sup_{t \in (0, T(M))} \int_{B_{\bar{x},+}(1)} |(1 - \theta(\dot{\bar{R}}))U(\cdot, t)|^2 + \int_0^T \int_{B_{\bar{x},+}(1)} |(1 - \theta(\dot{\bar{R}}))\nabla U|^2 \\ + \left( \int_0^{T(M)} \int_{B_{\bar{x},+}(1)} |(1 - \theta(\dot{\bar{R}}))U|^3 \right)^{\frac{2}{3}} + \left( \int_{\delta}^{T(M)} \int_{B_{\bar{x},+}(1)} |P|^{\frac{3}{2}} \right)^{\frac{2}{3}} \xrightarrow{R \to \infty} 0, \quad (5.41) $$

for all $\delta \in (0, T(M))$. Here $\theta \in C_c^\infty(\mathbb{R}^3)$ is a non-negative cut-off such that $\theta \equiv 1$ on $B_0(1)$ and $\text{supp}(\theta) \subseteq B_0(2)$.

The proof of the decay estimate at large-scales (5.41) goes through the proof of the following Gronwall-type inequality: for all $R \ge 1$, for all $t \in [0, T(M)]$,

$$ \alpha_R(t) + \beta_R(t) \le C(M) \left( \left( \int_0^t \alpha_R^{21}(s) ds \right)^{\frac{1}{21}} + R^{-1}(\log R) + \|(1 - \theta(\dot{\bar{R}}))U_0\|_{L_{uloc}^2(\mathbb{R}_+^3)} \right). $$

Here

$$ \alpha_R(t) := \sup_{\bar{x} \in \mathbb{R}_+^3 \cup B_{\bar{x},+}(1)} \int_{B_{\bar{x},+}(1)} |(1 - \theta(\dot{\bar{R}}))U(\cdot, t)|^2, \quad \beta_R(t) := \sup_{\bar{x} \in \mathbb{R}_+^3 \cup B_{\bar{x},+}(1)} \int_0^t \int_{B_{\bar{x},+}(1)} |(1 - \theta(\dot{\bar{R}}))\nabla U|^2. $$
### 5.3.3 Local-in-space short-time smoothing in the half-space via a compactness method

For the proof of local-in-space smoothing in the half-space, Theorem 5.11, the general idea is the same as for the whole-space, see Subsection 5.3.1. We decompose the solution $U$ to the Navier-Stokes equation into $U = a + V$, where $a$ is the mild solution roughly corresponding to the initial data $U_0|_{B_+(1)}$ and $V$ is a solution to a perturbed equation. The key is again to prove smallness of the perturbation $V$ locally near $B_+(1) \times \{0\}$ via local energy estimates and then smoothness of $V$ by an $\varepsilon$-regularity result. Instead of relying on a Caffarelli-Kohn-Nirenberg type scheme as in Subsection 5.3.1 for the whole-space, we use here a compactness proof, inspired by Lin [243], Seregin and Ladyženskaja [230] for the $\varepsilon$-regularity of solutions to the Navier-Stokes equations on the one hand, and Jia and Šverák [197] and Kang, Miura and Tsai [211] for the $\varepsilon$-regularity of solutions to the Navier-Stokes equations with subcritical or scale-critical drift terms on the other hand.

A novelty of our approach compared to [211] is that we build smallness of $a$ into our compactness arguments, which completely bypasses the estimates in [211, Section 4] for the linear perturbed equation (5.27). Therefore our limit system in the compactness proof is simply a linear Stokes system.

We prove a one-scale $\varepsilon$-regularity criteria for suitable solutions of the perturbed Navier-Stokes equation. In the half-space, it is standard that $\varepsilon$-regularity also depends on the pressure $P$, and a typical choice of quantity is $\|P\|_{L^3/2}$. This choice is convenient for treating the term involving $UP$ in the local energy inequality. This comment brings us to our first difficulty, namely, that for solutions of the linear Stokes equations in the half-space with initial data in $L^2$, the pressure estimates are only known in $L_{t,loc}^{4/3-} L_x^2$, see Proposition 5.15 above. We already commented on the low time integrability in connection with the proof of existence local energy solutions in the half-space in Subsection 5.3.2. Therefore, we must prove $\varepsilon$-regularity for the perturbed Navier-Stokes system under a new assumption on the pressure. Essentially, we require smallness of the pressure in $L_t^{1+\delta_t} L_x^{2-\delta_x}$ with $0 < \delta_x \le \delta_t \ll 1$. The integrability of the velocity with the Hölder conjugate exponents compensates for the low time integrability of the pressure but remains controlled, with room to spare, by the energy space $L_t^\infty L_x^2 \cap L_t^2 H_x^1$.

There is a second difficulty, which concerns only the critical case $m = 3$ and was also encountered in [35, 211]. In this case, due to the lack of improvement of flatness, see Subsection 5.3.1 about the proof of local-in-space short-time smoothing in the whole-space, the compactness argument yields a subcritical Morrey bound just below $L^\infty$, but does not give boundedness let alone Hölder continuity. In [35], we overcome this difficulty by using parabolic regularity theory to bootstrap the regularity of the perturbation $V$, from subcritical Morrey to Hölder. In principle this is also possible here, with some more technicalities because we have to bootstrap the regularity in the half-space. However, it is not necessary for our application. Rather, following [211], we combine (i) the subcritical Morrey estimates for the perturbation $V$, (ii) the critical estimates for $a$, and (iii) the standard $\varepsilon$-regularity criterion (without lower order terms) for the solution $U$ of the Navier-Stokes equations so as to conclude the $L^\infty$-smoothing (5.23).
# Chapter 6

# Quantitative regularity

This chapter relies mainly on the papers:

- [37], with Tobias Barker, *Mild criticality breaking for the Navier-Stokes equations*, J. Math. Fluid Mech. (2021).

- [38], with Tobias Barker, *Quantitative regularity for the Navier-Stokes equations via spatial concentration*, Comm. Math. Phys. (2021).

## Contents

<table><tbody><tr><td><strong>6.1 Context: state of the art and obstacles</strong></td><td><strong>110</strong></td></tr><tr><td>6.1.1 Blow-up of borderline critical norms near potential singularities</td><td>110</td></tr><tr><td>6.1.2 Effective quantitative regularity in the subcritical and non borderline critical case</td><td>111</td></tr><tr><td>6.1.3 Abstract quantitative regularity in the borderline critical case</td><td>112</td></tr><tr><td>6.1.4 Tao's quantitative result</td><td>113</td></tr><tr><td><strong>6.2 Main results</strong></td><td><strong>113</strong></td></tr><tr><td>6.2.1 Localized blow-up rate for a scale-critical singularity</td><td>113</td></tr><tr><td>Novelty of our results</td><td>115</td></tr><tr><td>Further developments</td><td>115</td></tr><tr><td>6.2.2 Quantification of Seregin's 2012 result</td><td>116</td></tr><tr><td>Novelty of our result</td><td>117</td></tr><tr><td>6.2.3 Slight breaking of criticality in the borderline case: a conjecture of Tao</td><td>117</td></tr><tr><td>Novelty of our result</td><td>118</td></tr><tr><td><strong>6.3 New ideas and strategy for the proofs</strong></td><td><strong>118</strong></td></tr><tr><td>6.3.1 A new strategy for quantitative estimates: a toy model</td><td>119</td></tr><tr><td>6.3.2 A new strategy for quantitative estimates under a Type I bound</td><td>120</td></tr><tr><td>6.3.3 A comparison to Tao's strategy</td><td>123</td></tr><tr><td>6.3.4 Robustness of our new strategy: quantification without a Type I bound</td><td>125</td></tr><tr><td>6.3.5 Mild criticality breaking</td><td>127</td></tr></tbody></table>

Our interest in quantitative estimates under a priori control of critical norms goes back to the summer of 2019 and the publication by Tao [334] of his breakthrough paper on the quantification of the Escauriaza, Seregin and Šverák [129] result, see Subsection 6.1.4 below. Our focus is on regularity estimates of the generic form

$$ \|U\|_{L^{\infty}(\mathbb{R}^3 \times (-\frac{1}{2}, 0))} \leq \mathcal{G}(\|U\|_A, \|U(\cdot, 0)\|_B) \qquad (6.1) $$
for solutions $U$ of the Navier-Stokes equations, with $A \subseteq L^1(-1, 0; C)$ and $B, C$ are
certain Banach spaces contained in $S'(\mathbb{R}^3)$. The point is to derive an explicit formula for
$\mathcal{G}$. Together with Barker we develop a generic approach based on the concentration of local
scale-critical quantities in the physical space. We study quantitative regularity:

(i) **Under a Type I (scale-critical) control**, see Theorem 6.1, in the vein of Tao's result; in our result $A = L^\infty(-1, 0; L^{3,\infty}(\mathbb{R}^3))$ and $B \subseteq L^3(\mathbb{R}^3)$, while in Tao's result $A = L^\infty(-1, 0; L^3(\mathbb{R}^3))$ and there is no dependence in the second variable of $\mathcal{G}$;

(ii) **Outside a scale-critical regime** under control of the $L^3$ norm at discrete times, see Theorem 6.3; in this case the form of the quantitative estimate (6.1) has to be adapted a bit, see (6.22).

Explicit quantitative estimates in the critical case also enable us to slightly break the criti-
cality barrier using ideas from nonlinear dispersive equations, see Theorem 6.5. This whole
chapter is concerned with the three-dimensional Navier-Stokes equations in the whole-
space.

**6.1 Context: state of the art and obstacles**

**6.1.1 Blow-up of borderline critical norms near potential singularities**

This paragraph is concerned with qualitative results. We focus on borderline critical
norms that cannot be made small by shrinking the time interval. We make a difference
between two families:

(i) **Non endpoint spaces** such as $L_t^\infty L_x^3$ or $L_t^\infty L_x^{3,r}$, $r \in (3, \infty)$, for which regularity is known for arbitrarily large norm; this is the case for norms where a certain smallness is hidden that enables to prove that the limit ‘blow-up profile’, obtained by zooming in on a potential singularity, is zero; for instance in the case of the $L^3$ norm, we have the property that $\|g\|_{L^3(B(r))} \to 0$ when $r \downarrow 0$;

(ii) **Endpoint spaces**, sometimes called ‘ultra-critical spaces’ (see [47]) such as $L_t^\infty L_x^{3,\infty}$, $L_t^\infty \dot{B}_{p,\infty,x}^{-1+\frac{3}{p}}$, $p \in (3, \infty)$, or $L_t^\infty BMO_x^{-1}$ that do require some extra assumption to ensure the vanishing of the limiting blow-up profile.

Notice that these two categories of spaces also appear in the mild solution theory. Indeed the type of results that we can prove are of a different nature: (i) local-in-time existence for any data, however large, and global-in-time existence for small data in the first category, (ii) only global-in-time existence results for small data are known in the second category. Note though that it is possible to prove the existence of global weak solutions for arbitrary data in all cases [304, 41, 6, 7].

Let us first consider the case of non endpoint norms. In the breakthrough paper [129],
Escauriaza, Seregin and Šverák showed that if $(\bar{x}, T^*)$ is a singular point then

$$
\limsup_{t \uparrow T^*} \|U(\cdot, t)\|_{L^3(B_{\bar{x}}(r))} = \infty \quad \text{for any fixed } r > 0. \quad (6.2)
$$

This is equivalent to showing regularity under $L_t^\infty L_x^3$ control. The result was reproved by Gallagher, Koch and Planchon [141] via a profile decomposition technique inspired from Kenig and Koch [215]. The result was also extended to non endpoint Lorentz spaces $L_t^\infty L_x^{3,r}$, $r \in (3, \infty)$, by Phuc [281] and to non endpoint Besov spaces $L_t^\infty \dot{B}_{p,q,x}^{-1+\frac{3}{p}}$, $p, q \in$
(3, $\infty$), by Gallagher, Koch and Planchon [142] using a profile decomposition method and by Albritton [6] using the stability of global weak Besov solutions. About teen years after (6.2) was proved, Seregin [294] improved (6.2):

$$ \lim_{t \uparrow T^*} \|U(\cdot, t)\|_{L^3(\mathbb{R}^3)} = \infty. \qquad (6.3) $$

This result goes beyond the critical case because it tells that boundedness of the $L^3(\mathbb{R}^3)$ norm along a sequence of times, rather than for all times, is enough to get the regularity.

Second we consider the case of endpoint spaces. Let us point that it is unknown in general whether the Type I blow-up condition $U \in L_t^\infty L_x^{3,\infty}$ implies regularity. Exceptions are when this quasinorm is small [221], when the flow is axisymmetric [94, 93, 219], or when the blow-up profile vanishes thanks to some smallness condition, see [41, 297]. Finally, Albritton and Barker [7] show the analog of Seregin's result [294] for endpoint Besov norms. Namely they prove that if the endpoint Besov norm $\dot{B}_{p,\infty}^{-1+\frac{3}{p}}$, $p \in (3, \infty)$, is bounded along a sequence of times $t_n \uparrow T$ and if at final time $T$ the blow-up profile satisfies

$$ \sqrt{T - t_n} U(\sqrt{T - t_n}(\cdot - \bar{x}), T) \xrightarrow{*} 0 \quad \text{in} \quad \mathcal{D}'(\mathbb{R}^3) $$

then $(\bar{x}, T)$ is a regular point.

### 6.1.2 Effective quantitative regularity in the subcritical and non borderline critical case

It is known since the major work of Leray [238] that subcritical $L^p$ norms, $p \in (3, \infty]$ blow-up with the following rate

$$ (T^* - t)^{\frac{3}{2}(\frac{1}{3} - \frac{1}{p})} \|U(\cdot, t)\|_{L^p} \le c_p \quad \text{for all} \quad t \in (0, T^*), \qquad (6.4) $$

for $c_p \in (0, \infty)$ and $T^*$ a blow-up time. These blow-up estimates are the other side of the coin of regularity estimates. We present here two methods to obtain quantitative regularity estimates under subcritical and critical controls. A further strategy is presented in Subsection 6.3.1.

**Subcritical case** If $U$ is a finite-energy solution on $\mathbb{R}^3 \times (0, 1)$ with a finite subcritical norm, then it is known that $U$ must belong to $C^\infty(\mathbb{R}^3 \times (0, 1])$. See, for example, [227]. Moreover, one typically has a quantitative estimate of the form (6.1) with

$$ \mathcal{G}(x, y) = cx^\beta \quad \text{with} \quad \beta > 0. $$

To demonstrate this, consider $U$ belonging to $L_{x,t}^{5+\delta}(\mathbb{R}^3 \times (0, 1))$ for $\delta > 0$. An application of Caffarelli, Kohn and Nirenberg's result [75] gives that (6.1) holds true with $\mathcal{G}(x, y) \simeq x^{\frac{\delta+5}{\delta}}$. Such a quantitative estimate is invariant with respect to the Navier-Stokes scaling.

**Critical case** In the subcritical norm case, we saw that seeking estimates of the form (6.1) that are invariant with respect to the scaling gives a suitable candidate for $\mathcal{G}$. The case when the norm is critical is more subtle, since a scaling argument does not provide a suitable candidate for $\mathcal{G}$. We first mention that the case of sufficiently small critical norms, for example

$$ \|U\|_{L^5(\mathbb{R}^3 \times (0,1))} < \varepsilon_*, \qquad (6.5) $$
is essentially of a similar category to the subcritical case (though a scaling argument is not applicable). Indeed, a similar argument as before based on [75] gives that in this case we have (6.1) with $\mathcal{G}(x, y) \simeq x$. This is consistent with the fact that solutions with small scale-invariant norms exhibit similar behavior to the linear system and hence are typically expected to satisfy linear estimates. For obtaining quantitative estimates of the form (6.1) when the scale-invariant norm is large, it is less clear what the candidate for $\mathcal{G}$ might be. For the case of a smooth finite-energy solution $U$ having finite but large scale-invariant $L^5(\mathbb{R}^3 \times (0, 1))$ norm, one way to obtain quantitative estimates is to consider the vorticity equation (4.12) with initial vorticity $\omega_0 \in L^2(\mathbb{R}^3)$; see also [237, Chapter 11]. Performing an energy estimate yields for $t \in [0, 1]$

$$ \|\omega(\cdot, t)\|_{L^2(\mathbb{R}^3)}^2 + 2 \int_0^t \int_{\mathbb{R}^3} |\nabla \omega|^2 dx dt' \le \|\omega_0\|_{L^2(\mathbb{R}^3)}^2 + 2 \int_0^t \int_{\mathbb{R}^3} (\omega \cdot \nabla U) \cdot \omega dx dt', \quad (6.6) $$

where the second term in the right-hand side is due to the vortex stretching term $\omega \cdot \nabla U$ in (4.12). For the case that $U \in L^5(\mathbb{R}^3 \times (0, 1))$, application of Hölder's inequality, Sobolev embedding theorems and Young's inequality lead to

$$ \|\omega(\cdot, t)\|_{L^2(\mathbb{R}^3)}^2 + \int_0^t \int_{\mathbb{R}^3} |\nabla \omega|^2 dx dt' \le \|\omega_0\|_{L^2(\mathbb{R}^3)}^2 + C \int_0^t \|U(\cdot, t')\|_{L^5(\mathbb{R}^3)}^5 \|\omega(\cdot, t')\|_{L^2(\mathbb{R}^3)}^2 dt'. \quad (6.7) $$

Gronwall's lemma, followed by arguments similar to the subcritical case, yields

$$ \|U\|_{L^{\infty}(\mathbb{R}^3 \times (\frac{1}{2}, 1))} \lesssim \|\omega\|_{L^{\infty}(0, 1; L^2(\mathbb{R}^3))}^2 \le \|\omega_0\|_{L^2(\mathbb{R}^3)}^2 \exp \left( \|U\|_{L^5(\mathbb{R}^3 \times (0, 1))}^5 \right). \quad (6.8) $$

Though this is not exactly of the form (6.1), a slightly different argument gives that for any finite-energy solution $U$ in $L^5(\mathbb{R}^3 \times (0, 1))$ we get that (6.1) holds with $\mathcal{G}(x, y) \simeq \exp(Cx^5)$. In particular, this can be achieved using the strategy of Subsection 6.3.1.

The above argument (6.6)-(6.8) shows that being able to substantially improve upon $\mathcal{G}(x, y) \simeq \exp(Cx^5)$ would most likely require the utilization of a nonlinear mechanism that reduces the influence of the vortex stretching term $\omega \cdot \nabla U$ in (4.12) (such a mechanism is used for instance in Constantin and Fefferman's result [107], see Subsection 4.1.5). It seems plausible that the discovery of such a mechanism would have implications for the regularity theory of the Navier-Stokes equations.

### 6.1.3 Abstract quantitative regularity in the borderline critical case

The argument of Escauriaza, Seregin and Šverák [129] is by contradiction and hence qualitative. It can, though, be quantified abstractly using the ‘persistence of singularities’ lemma in [287, Lemma 2.2] (note that the persistence of singularities was also a key tool in our strategy for regularity under vorticity alignment, see Subsection 4.3.3). Namely, there exists a function $\mathcal{G}$ such that if $U$ is a finite-energy solution to the Navier-Stokes equations then

$$ \|U\|_{L^{\infty}(0,1; L^3(\mathbb{R}^3))} < \infty \Rightarrow \|U\|_{L^{\infty}(\mathbb{R}^3 \times (\frac{1}{2}, 1))} \le \mathcal{G}(\|U\|_{L^{\infty}(0,1; L^3(\mathbb{R}^3))}). \quad (6.9) $$

Such an argument is by contradiction and hence gives no explicit information about $\mathcal{G}$.

On a tangential note, it is demonstrated [32] by an elementary scaling argument inspired from a talk of Seregin at the University of Sussex on 03 March 2014, that if the set of
finite-energy solutions to the Navier-Stokes equations (with Schwartz class initial data) that blows-up is non empty, there cannot exist a positive universal function $\mathcal{F}$ such that the following analogue of (6.4) holds true:

$$ \lim_{s \to 0^+} \mathcal{F}(s) = \infty, \qquad (6.10) $$

and for all $T^* > 0$, if $U$ is a finite-energy solution to the Navier-Stokes equations (with Schwartz class initial data) that first blows-up at $T^* > 0$ then $U$ necessarily satisfies

$$ \|U(\cdot, t)\|_{L^3(\mathbb{R}^3)} \ge \mathcal{F}(T^* - t) \qquad (6.11) $$

for all $t \in [0, T^*)$.

### 6.1.4 Tao's quantitative result

In a remarkable recent development [334], Tao used a new approach to provide the first explicit quantification of the seminal result of Escauriaza, Seregin and Šverák [129]. In particular, Tao shows that for classical solutions to the Navier-Stokes equations on $\mathbb{R}^3 \times (-1, 0)$ belonging to the critical space $L^\infty(-1, 0; L^3(\mathbb{R}^3))$,

$$ \|U(\cdot, t)\|_{L^\infty(\mathbb{R}^3)} \le \exp\left(\exp\left(\exp\left(\|U\|_{L^\infty_t L^3_x(\mathbb{R}^3 \times (0,t))}^C\right)\right)\right)(-t)^{-\frac{1}{2}} \quad \text{for all } -1 \le t < 0, \qquad (6.12) $$

where $C \in (0, \infty)$ is a universal constant. Here $\mathcal{G}(x, y) \simeq \exp(\exp(\exp(x^C)))$ in the notation of the general form estimate (6.1).

Combining these quantitative estimates and the Leray blow-up rate (6.4) for the subcritical $L^\infty$ norm, Tao showed that if a finite-energy solution $U$ first blows-up at $T^* > 0$ then for some universal constant $c \in (0, \infty)$,

$$ \limsup_{t \uparrow T^*} \frac{\|U(\cdot, t)\|_{L^3(\mathbb{R}^3)}}{(\log \log \log \frac{1}{T^*-t})^c} = \infty. \qquad (6.13) $$

Since there cannot exist $\mathcal{F}$ such that (6.10)-(6.11) holds true, at first sight (6.13) may seem somewhat surprising, though it is not conflicting with such a fact. Notice that

$$ \frac{\|U(\cdot, t)\|_{L^3(\mathbb{R}^3)}}{(\log \log \log \frac{1}{T^*-t})^c} $$

is not invariant with respect to the Navier-Stokes scaling but is slightly supercritical due to the presence of the logarithmic denominator. The $\limsup_{t \uparrow T^*}$ in (6.13) is due to the fact that the whole $L^\infty_t L^3_x(\mathbb{R}^3 \times (-1, 0))$ norm appears in the quantitative regularity (6.12).

We refer to Subsection 6.3.3 for a description of Tao's scheme and a comparison to ours.

## 6.2 Main results

### 6.2.1 Localized blow-up rate for a scale-critical singularity

**Theorem 6.1** (rate of blow-up, Type I; [38, Theorem A], in collaboration with Barker). For all $M \in [1, \infty)$ sufficiently large, the following holds true.

Assume that $U$ is a mild solution to the Navier-Stokes equations on $\mathbb{R}^3 \times [0, T^*)$ with $U \in$
<figure><img src="image_13.png" /><figcaption>Figure 6.1 – Blow-up of the $L^3$ norm on concentrating sets</figcaption></figure>

$$L_{\text{loc}}^{\infty}([0, T^*); L^{\infty}(\mathbb{R}^3)).$$

Assume in addition that $(0, T^*)$ is a Type I blow-up, i.e. $U$ has a singular point at $(x, t) = (0, T^*)$, in particular $U \notin L_{x,t}^{\infty}(Q_{(0,T^*)}(r))$ for all sufficiently small $r > 0$, and

$$\|U\|_{L_t^\infty L_x^{3,\infty}(\mathbb{R}^3 \times (0,T^*))} \leq M. \qquad (6.14)$$

Then the above assumptions imply that there exists $S(M) \simeq M^{-30} \in (0, \frac{1}{4}]$ such that for any $t \in (\frac{T^*}{2}, T^*)$ and

$$R \in \left( \sqrt{\frac{T^* - t}{S(M)}}, e^{M^{1022} \sqrt{T^*}} \right) \qquad (6.15)$$

we have

$$\int_{|x|<R} |U(x,t)|^3 dx \geq \frac{\log\left(\frac{R^2}{M^{802}|T^*-t|}\right)}{\exp(\exp(M^{1025}))}. \qquad (6.16)$$

The proof of Theorem 6.1 relies on the combination, as is shown in Figure 6.1, of the quantitative regularity in the Type I case on the one hand, see Subsection 6.3.2 below and [38, Proposition 2.1], with concentration estimates near a potential singularity for the local $L^3$ norm and for the $L^\infty$ norm on the other hand, see the results of Chapter 5.

*Remark 6.2* (further quantitative results). In addition to Theorem 6.1, our techniques enable us to obtain two further results:
(1) a regularity criteria based on the relative smallness of the $L^{3,\infty}$ quasinorm at final time, see [38, Proposition 4.1]. Namely, we prove that for a global-in-time suitable finite-energy solution to the Navier-Stokes equations on $\mathbb{R}^3 \times [-1, \infty)$ satisfying the Type I bound (6.14), if for some $T^* \in (-1, 0]$,

$$ \lim_{r \to 0} \|U(\cdot, T^*)\|_{L^{3,\infty}(B_0(r))} \le \exp(-\exp(M^{1023})), \quad (6.17) $$

then, $(0, T^*)$ is a regular point.

(2) a quantification of the number of singularities in a Type I blow-up scenario, see [38, Corollary 4.3]. We prove that a global-in-time suitable finite-energy solution to the Navier-Stokes equations on $\mathbb{R}^3 \times [0, \infty)$ satisfying the Type I bound (6.14) has at most $\exp(\exp(M^{1024}))$ blow-up points at time $T^*$.

## Novelty of our results

The innovations in the results above lie in the following aspects, that all rely on the fact that we assume the scale-invariant assumption (6.14).

First, in Theorem 6.1 not only is the rate new but also the fact that the $L^3$ norm blows up on a ball concentrating on the potential singularity. Indeed, given the range (6.15), it is possible to take $R = O((T^* - t)^{\frac{1-\delta}{2}})$ for $\delta \in (0, 1)$ and times $t$ sufficiently close to the blow-up time $T^*$. Previously, see [240, Theorem 1.3], it was shown that if a solution blows up without a Type I bound then the $L^3$ norm blows up on certain non-explicit concentrating sets. This localization is possible because we work with quantities defined locally in physical space, rather than global quantities involving the Fourier transform as in [334].

Second, the lower bound (6.16) holds for all times $t \in (\frac{T^*}{2}, T^*)$ and not only for a sequence tending to $T^*$ as is the case in (6.13) obtained in [334]. Indeed, under the Type I condition (6.14), we are able to get a quantitative regularity result that only involves the $L^3$ norm at final time.

Third, we manage to remove two logarithms from the lower bound (6.13). Using a result by Chae and Wolf [84], we notice that our rate is optimal for backward discretely self-similar solutions with sufficient decay, see [38, Corollary 1.1]. On this topic notice that our estimates do not succeed in ruling our this blow-up scenario, because there is a huge gap between the lower bound in 6.16 in which the constant behaves like $\exp(-\exp(M^{1025}))$ and the upper bound one gets using the Type I a priori bound in which the constant behaves like $M^3$, see [38, equation (12)]. Moreover, notice that estimate (6.16) is written in a scale-invariant form. Hence this estimate does not contradict Seregin's remark, see Subsection 6.1.3, about the non-existence of a universal blow-up rate in the borderline case.

Fourth, we are able to explicitly quantify the number of blow-up points in the Type I scenario, see remark 6.2, which improves upon the non effective bounds in [100, 297].

## Further developments

In the recent paper [276], Palasek was also able to improve upon the triple log rate (6.13) obtained by Tao in [334]. In the case of axisymmetric solutions for instance, the triple log is replaced by a double log. Without any symmetry assumption on the solution, a similar improvement can be obtained by replacing the $L^3$ norm by the norm $\|r^{1-\frac{3}{q}}U\|_{L^\infty_t L^q_x}$ for $q \in (3, \infty)$ and $r := |x_h|$.
The quantitative regularity for solutions $U \in L_t^\infty L_x^d$ to the Navier-Stokes equations in higher dimensions $d \ge 4$ was handled by Palasek in [277]. This work gives an effective quantification of the qualitative result by Dong and Du [125]. For the blow-up rate, one pays the price of an additional logarithm compared to the result in dimension three (6.13).

Let us also mention the result of Feng, He and Wang [133] which quantifies the blow-up of borderline non-endpoint Lorentz norms $L^{3,q}(\mathbb{R}^3)$, for $q \in (3, \infty)$, hence quantifying the result by Phuc [281].

Concerning the quantification of the number of blow-up points in a Type I blow-up scenario, Barker [34, Theorem 2] was able to drastically improve the double exponential bound that we obtained in [38], see Remark 6.2. Indeed, the bound for the number of blow-up points is reduced to $O(M^{20})$ under the weaker assumption that the $L^{3,\infty}$ quasi-norm of the solution is uniformly bounded by $M$ on a sequence tending to the blow-up time.

### 6.2.2 Quantification of Seregin's 2012 result

**Theorem 6.3** (quantification of Seregin's result; [38, Theorem B] in collaboration with Barker). For all $M \in [1, \infty)$ sufficiently large, the following holds true.
We define

$$M^b := \exp\left(\frac{LM^5}{2}\right), \qquad (6.18)$$

for an appropriate universal constant $L \in (0, \infty)$. Let $U$ be a finite-energy $C^\infty(\mathbb{R}^3 \times (-1, 0))$ solution to the Navier-Stokes equations on $\mathbb{R}^3 \times [-1, 0]$.
Assume that there exists $t_k \in [-1, 0)$ such that

$$t_k \uparrow 0 \text{ with } \sup_k \|U(\cdot, t_k)\|_{L^3(\mathbb{R}^3)} \le M. \qquad (6.19)$$

Select any 'well-separated' subsequence (still denoted $t_k$) such that

$$\sup_k \frac{-t_{k+1}}{-t_k} < \exp(-2(M^b)^{1223}). \qquad (6.20)$$

Then for

$$j := \lceil \exp(\exp((M^b)^{1224})) \rceil + 1, \qquad (6.21)$$

we have the bound

$$\|U\|_{L^\infty(\mathbb{R}^3 \times (\frac{t_{j+1}}{4}, 0))} \le \frac{CM^{-23}}{(-t_{j+1})^{\frac{1}{2}}}, \qquad (6.22)$$

for a universal constant $C \in (0, \infty)$.

Figure 6.2 is an illustration of this theorem. Of course the zone of quantitative regularity depends on the sequence $t_k$ on which the assumption (6.19) holds via the time $t_{j+1}$, $j$ being the index defined by (6.21). The quadruple exponential bound

$$0 > t_{j+1} \gtrsim -\exp(-\exp \exp \exp(M^6))$$

is a consequence of the definition (6.21) of $j$, the well separation property (6.20) and the definition (6.18) for $M^b$.
<figure><img src="image_14.png" /><figcaption>Figure 6.2 – Quantification of Seregin’s 2012 result</figcaption></figure>

*Remark 6.4* (a further quantitative result). We also obtain a quantitative regularity criteria in terms of the relative smallness of the solution at final time and initial time. Namely, if

$$ \|U(\cdot, -1)\|_{L^3(\mathbb{R}^3)} \leq M, $$

then

$$ \|U(\cdot, 0)\|_{L^3(B_0(\exp((M^b)^{1221})), B_0(1))} \leq \exp(-\exp((M^b)^{1223})), $$

implies that $(0, 0)$ is a regular point. A ‘non-effective’ version of this result is in [7, Theorem 4.1 (i)].

### Novelty of our result

**Theorem 6.3** is a quantitative version of the result of Seregin [294, Theorem 1.1]. Let us emphasize that our result, as well as Seregin’s previously, holds without a scale-critical condition because the control (6.19) of the $L^3$ norm is only on time slices. This is in stark contrast with Theorem 6.16, Tao’s [334] quantitative regularity see (6.12) above and the quantitative regularity results of [276, 277, 133].

### 6.2.3 Slight breaking of criticality in the borderline case: a conjecture of Tao

We know discuss the possibility of transferring the slight supercriticality in time of Tao’s result (6.13) to slight supercriticality in space.

**Theorem 6.5** (blow-up of slightly supercritical Orlicz norms; [37, Theorem 2]). There exists a universal constant $\theta \in (0, 1)$ such that the following holds.

Let $U$ be a weak Leray-Hopf solution to the Navier-Stokes equations on $\mathbb{R}^3 \times (0, \infty)$ with initial data $U_0 \in L^2(\mathbb{R}^3) \cap L^4(\mathbb{R}^3)$. Assume that $U$ first blows-up at $T^* > 0$, namely

$$ U \in L_{\text{loc}}^{\infty}(0, T^*; L^{\infty}(\mathbb{R}^3)) \text{ and } U \notin L^{\infty}(\frac{1}{2}T^*, T^*; L^{\infty}(\mathbb{R}^3)). $$
Then the above assumptions imply that

$$ \limsup_{t \uparrow T^*} \int_{\mathbb{R}^3} \frac{|U(x, t)|^3}{\left( \log \log \log \left( \left( \log(e^{3e^e} + |U(x, t)|) \right)^{\frac{1}{3}} \right) \right)^{\theta}} dx = \infty. \quad (6.23) $$

## Novelty of our result

Recently, in [334, Remark 1.6], Tao conjectured that if a solution first loses smoothness at time $T^* > 0$, then the Orlicz norm $||U(\cdot, t)||_{L^3(\log \log \log L)^{-c}(\mathbb{R}^3)}$ must blow-up as $t$ tends to $T^*$. Theorem 6.5 provides a positive answer to Tao's conjecture, albeit with an extra logarithm in the denominator.

As far as we know, Theorem 6.5 is the first result of this type concerned with slight criticality breaking in borderline spaces. Previously, it was shown in [87] that if $U$ is a weak Leray-Hopf solution satisfying

$$ \int_0^\infty \int_{\mathbb{R}^3} \frac{|U|^5}{\log(1 + |U|)} dxdt < \infty $$

then $U$ is smooth on $\mathbb{R}^3 \times (0, \infty)$. Subsequent improvements were obtained in [234] and [52]; see also [265]. Let us mention that the techniques used in these papers cannot be used to treat the borderline case considered in Theorem 6.5. We also mention the paper of Chan and Yoneda [88] that mixes the boundedness of a strongly supercritical norm $L_t^\infty L_x^{\alpha, \infty}$ for $\alpha \in (2.343, 3)$ with a geometrical information involving $\nabla \cdot (U/|U|)$.

For other partial differential equations, it is often the case that a refined understanding of critical regimes can be used to prove ‘slightly supercritical’ results. Such slightly supercritical results occur for the nonlinear wave equation [331, 104], the hyperdissipative Navier-Stokes equations [332, 31, 102, 103], the supercritical SQG equation [113, 109] and the fractional Burgers equation [112] to name a few. In these works the slight supercriticality is obtained by varying the power of the nonlinearity or the strength of the fractional dissipation.

Our method is indeed inspired by the recent result of Bulut [73] for the nonlinear supercritical Schrödinger equation. In particular our proof of Theorem 6.5 relies on the proof of another statement, see Lemma 6.7 below, that we name ‘mild criticality breaking’, which is the counterpart for Navier-Stokes of Bulut’s result for Schrödinger.

## 6.3 New ideas and strategy for the proofs

The goal of our paper [38] is to develop a new robust strategy for obtaining new quantitative estimates of the Navier-Stokes equations. The main novelty is that our strategy allows us to obtain local quantitative estimates, Theorem 6.1, and applies to certain situations where we are outside the regime of scale-invariant controls, Theorem 6.3. Before showing how our strategy can be applied to prove our two main results, let us outline the main idea on a toy model.
### 6.3.1 A new strategy for quantitative estimates: a toy model

In this section, we outline the strategy that is used for quantitative regularity in the borderline case, see Subsection 6.3.2, in the simpler non borderline critical space $L_{t,x}^5$.

Schematically, there are two main parts in the reasoning:

(1) Assume certain critical a priori bounds, such as

$$ \|U\|_{L^5(\mathbb{R}^3 \times (-1,0))} < \infty \quad \text{or} \quad \|U\|_{L^\infty(-1,0; L^3(\mathbb{R}^3))} < \infty $$

as in Tao's result (see Subsection 6.1.4), or the Type I assumption

$$ \|U\|_{L^\infty(-1,0; L^{3,\infty}(\mathbb{R}^3))} \le M < \infty \quad \text{and} \quad \|U(\cdot, 0)\|_{B_0(R(M))} < \infty $$

where $R(M) \gg 1$ as in Subsection 6.3.2 below. Then, certain scale-invariant quantities (Kato-type norms, Fourier-based quantities, scale-invariant entrophies) cannot concentrate for times $0 > t \ge t_*$ or for frequencies $N \ge N_*$, where $t_*$ (resp. $N_*$) is a given quantitative time (resp. frequency). That time $t_*$ or frequency $N_*$ can be interpreted as Kolmogorov dissipative scales. Tao's point and ours is to quantify them explicitly in terms of $M$. Such results are achieved using various tools (bilinear estimates, local smoothing results, Carleman inequalities). The main insight is to propagate the concentration from a given time slice $t'$ to backward time slices $t''$ and to have enough such slices that one can then sum to gain coercivity.

(2) A regularity result: non concentration implies smallness implies quantitative regularity via $\varepsilon$-regularity type results.

Let us now sketch the proof in the toy model case, see Figure 6.3. Assume that

$$ \|U\|_{L^5(\mathbb{R}^3 \times (-1,0))} < \infty. $$

For the first part of the reasoning, we argue in the following two-step way:

#### (Step-1) Backward propagation of Kato-norm concentration

Using bilinear estimates for the heat semigroup [78] we get

$$ \|U(\cdot, t')\|_{L^5(\mathbb{R}^3)} > \frac{\varepsilon}{(-t')^{\frac{1}{5}}} \Rightarrow \|U(\cdot, t'')\|_{L^5(\mathbb{R}^3)} > \frac{\varepsilon/2}{(-t'')^{\frac{1}{5}}}, \quad \text{for all } t'' \in (-1, 2t'). $$

#### (Step-2) Summation of scales

Summing the concentration for $t'' \in (-1, 2t')$,

$$ \|U\|_{L^5(\mathbb{R}^3 \times (-1,0))}^5 \ge \int_{-1}^{2t'} \|U(\cdot, t'')\|_{L^5(\mathbb{R}^3)}^5 dt'' \ge -\frac{\varepsilon^5}{32} \log(-2t'). $$

Hence,

$$ \frac{1}{2} > -t' > \frac{1}{2} \exp \left( - \frac{32 \|U\|_{L^5(\mathbb{R}^3 \times (-1,0))}^5}{\varepsilon^5} \right) =: -t_* $$

For the second part of the reasoning we use $\varepsilon$-regularity on $Q_{(\bar{x},0)}(\sqrt{-t_*/2}) = B_{\bar{x}}(\sqrt{-t_*/2}) \times (-\frac{t_*}{2}, 0)$ for all $\bar{x} \in \mathbb{R}^3$ to obtain

$$ \|U\|_{L^\infty(\mathbb{R}^3 \times (-\frac{t_*}{2}, 0))} \lesssim \frac{\varepsilon^{\frac{1}{3}}}{\sqrt{-t_*}} \lesssim \varepsilon^{\frac{1}{3}} \exp \left( \frac{16 \|U\|_{L^5(\mathbb{R}^3 \times (-1,0))}^5}{\varepsilon^5} \right). \quad (6.24) $$

Notice that in the quantitative estimate (6.24) the subcritical $L_{t,x}^\infty$ norm is estimated in terms of a single exponential of the $L_{t,x}^5$ norm; compare to (6.12) which involves a triple exponential.
<figure><img src="image_15.png" /><figcaption>Figure 6.3 – Quantitative regularity via concentration of scale-critical quantities: a toy model</figcaption></figure>

### 6.3.2 A new strategy for quantitative estimates under a Type I bound

We outline here the strategy for the case when one has a scale-invariant control in the form

$$ \|U\|_{L_t^\infty L_x^{3,\infty}(\mathbb{R}^3 \times (-1,0))} \leq M. \qquad (6.25) $$

**Local-in-space local-in-time smoothing in our approach** Fundamental to our strategy is the use of local-in-space smoothing near the initial time for the Navier-Stokes equations pioneered by Jia and Šverák in [197] (see Subsection 5.1.3). In particular, the result of [197], together with rescaling arguments from [35], implies the following. Assume $U$ is a smooth solution with sufficient decay of the Navier-Stokes equations on $\mathbb{R}^3 \times [-1, 0]$ and satisfies the Type I bound (6.25). If the scale-invariant entrophy is small for a certain time $t' \in (-1, 0)$ in the following way

$$ (-t')^{\frac{1}{2}} \int_{B_0(4\sqrt{S(M)}^{-1}(-t')^{\frac{1}{2}})} |\omega(x, t')|^2 dx \leq M^2 \sqrt{S(M)}, \qquad (6.26) $$

then

$$ \|U\|_{L_{x,t}^{\infty} \left(B_0\left(\frac{1}{2}\sqrt{S(M)}^{-1}(-t')^{\frac{1}{2}}\right) \times \left(\frac{3}{4}(-t') + t', 0\right)\right)} \qquad (6.27) $$
can be estimated explicitly in terms of $M$ and $-t'$ via an $\varepsilon$-regularity result (local-in-space smoothing for subcritical data). Here, $S(M) = CM^{-100}$. We refer to Figure 6.4 where $t' = t_*/2$.

<figure><img src="image_16.png" /><figcaption>Figure 6.4 – Quantitative regularity via concentration of a scale-invariant enstrophy: maximal time of enstrophy concentration and quantitative boundedness</figcaption></figure>

**Our main goal** In this perspective, the aim of our strategy is the following

**Our goal:** If (6.26) fails for $t'$, what is a lower bound $-t_*$ for $-t'$ in terms of $M$ and $\|U(\cdot, 0)\|_{L^3(B_0(R(M)))}$ for a certain $R(M) \gg 1$?

This upper bound $-t_*$ for the enstrophy concentration can be interpreted as the time at which dissipation effects take over the nonlinearity. Notice that for that to happen, we need that the solution belongs to $L^3$ at final time, because a mere Type I assumption is at this point not enough to beat the scaling. Our strategy for obtaining a lower bound of $-t_*$, stated in [38, Proposition 2.1], can be summarized in three steps.

**Three steps strategy to show our main goal**

**(Step-1) Backward propagation of vorticity concentration**

For this step and the next one, we refer to Figure 6.6. Suppose $t' \in (-1, 0)$ is not too close to $-1$ and is such that

$$ \int_{B_0(4\sqrt{S(M)})^{-1}(-t')^{\frac{1}{2}}} |\omega(x, t')|^2 dx > \frac{M^2 \sqrt{S(M)}}{(-t')^{\frac{1}{2}}} \quad (6.28) $$

We show that for all $t'' \in (-1, t')$ such that $-t''$ is well-separated from $-t'$, we have

$$ \int_{B_0(4\sqrt{S(M)})^{-1}(-t'')^{\frac{1}{2}}} |\omega(x, t'')|^2 dx > \frac{M^2 \sqrt{S(M)}}{(-t'')^{\frac{1}{2}}} \quad (6.29) $$
<figure><img src="image_17.png" /><figcaption>Figure 6.5 – Quantitative regularity via concentration of a scale-invariant enstrophy: backward, large-scale and forward propagation of enstrophy concentration</figcaption></figure>

(Step-2) **Lower bound on localized $L^3$ norm at the final moment in time**

Using the previous step, together with the same arguments as in Tao's paper [334] involving quantitative Carleman inequalities, we show that for certain permissible annuli that

$$ \int_{R \le |x| \le R'} |U(x, 0)|^3 dx \ge \exp(-\exp(M^C)). \quad (6.30) $$

The role of the Type I bound is to show that the solution $U$ obeys good quantitative estimates in certain space-time regions, ‘epochs of regularity’ and ‘annuli of regularity’, which is needed to apply the Carleman inequalities to the vorticity equation.

(Step-3) **Conclusion: summing scales to bound $-t'$ from below**

We refer to Figure 6.6 for this step. Summing (6.30) over all permissible disjoint annuli finally gives us the desired lower bound for $-t'$. We note that the localized $L^3$ norm of $U$ at time 0 plays a distinct role to that of the Type I condition described in the previous step. Its sole purpose is to bound the number of permissible disjoint annuli that can be summed. This turns into a lower bound of $-t'$, which reads:

$$ -t' \ge CM^{-749} \exp \left( -4M^{1023} \exp(\exp(M^{1024})) \int_{B_0(M^{1023})} |U(x, 0)|^3 dx \right). \quad (6.31) $$

The single exponential of $\int_{B_0(M^{1023})} |U(x, 0)|^3 dx$ in the lower bound (6.31) is why the lower bound in Theorem 6.1 on the localized $L^3$ norm near a Type I singularity is a single
<figure><img src="image_18.png" /><figcaption>Figure 6.6 – Quantitative regularity via concentration of a scale-invariant enstrophy: summation of scales</figcaption></figure>

logarithm. The fact that (6.31) just depends on $U$ at the final time explains why (6.16) holds at pointwise times.

**Remark 6.6** (on the exponentials). We emphasize that under the Type I assumption (6.25), the single exponential dependence in $\int_{B_0(M^{1023})} |U(x, 0)|^3 dx$ in the estimate (6.31) is due to the summation of scales in the third step ‘Conclusion: summing scales to bound $-t'$ from below’ above. This is hard to improve because the fact that the concentration zones at final time need to be non-overlapping, see Figure 6.6, causes that we can only sum a geometric sequence of times $t''$. The double exponential dependence in the Type I bound $M$ in (6.31) comes from the fact that we use linear tools, namely quantitative Carleman inequalities for backward uniqueness and unique continuation, and pigeonhole to locate good scales, such as for instance to find a good annulus of quantitative regularity. This double exponential in $M$ may be improved in principle but this does not seem to be possible with the techniques of our paper.

### 6.3.3 A comparison to Tao's strategy

The high-level strategies of Tao and ours are almost parallel. The differences between the two approaches lie in the quantities (concentration in Fourier space vs. local concentration in physical space) that are considered and in the tools that we use.

**Global-in-space local-in-time smoothing in Tao's approach** Fundamental to Tao's approach for showing (6.12) is the following fact, see [334, Section 6]. If $U$ is a classical
solution to the Navier-Stokes equations on $\mathbb{R}^3 \times [-1, 0]$ with

$$ \|U\|_{L_t^\infty L_x^3 (\mathbb{R}^3 \times (-1,0))} \le A \qquad (6.32) $$

$$ \text{and} \quad N^{-1} \|P_N U\|_{L_{x,t}^{\infty}(\mathbb{R}^3 \times (-\frac{1}{2},0))} < \varepsilon(A) \text{ for all } N \ge N_*, \qquad (6.33) $$

for a certain $0 < \varepsilon(A) \ll 1$, then $\|U\|_{L_{x,t}^\infty(\mathbb{R}^3 \times (-\frac{1}{8},0))}$ can be estimated explicitly in terms of A and $N_*$.

Here $P_N$ is the Littlewood-Paley projection on the frequency $N > 0$. Related observations were made previously by Cheskidov and Shvydkoy [98, 99] and Cheskidov and Dai [97], but without the bounds explicitly stated. There the frequency $N_*$ is called the Kolmogorov scale and denoted $\Lambda$. Heuristically, see [333], if (6.33) holds, that is if $N^{-1}\|P_N U\|_{L_{x,t}^\infty} \ll 1$ is small, then $N\|P_N U\|_{L_{t,x}^2} \ll N^2\|P_N U\|_{L_{t,x}^\infty}$ so that the diffusion dominates the nonlinearity.

*Notice that assumption (6.32) replaces the Type I assumption (6.25) and the smallness of the scale-invariant quantity (6.33) at high frequencies replaces the smallness of the scale-invariant entrophy (6.26).*

**Tao's main goal** In this perspective, Tao's aim is the following:

**Tao's goal:** Under the scale-invariant assumption (6.32), if (6.33) fails for $\varepsilon(A) = A^{-C}$ and $N = N_0$, what is an upper bound $N_*$ for $N_0$?

In Tao's paper [334, Theorem 5.1], it is shown that $N_0 \lesssim \exp \exp \exp(A^C)$, which implies (6.12) by means of the quantitative regularity mechanism (6.33). We emphasize that since the regularity mechanism (6.33) is global, all quantitative estimates obtained in this way are in terms of globally defined quantities.

*One might think of the main goal of our strategy as a physical space analogy to Tao's goal with*

$$ N_0 \approx (-t')^{-\frac{1}{2}}. $$

*In contrast to (6.33), the regularity mechanism (6.26)-(6.27) produces quantitative estimates that are in terms of locally defined quantities, which is crucial for obtaining the localized results as in Theorem 6.1.*

**Four steps strategy to show Tao's main goal** The strategy in [334] for showing Tao's goal with $N_0 \lesssim \exp(\exp(\exp(A^C)))$ can be summarized in four steps. We refer the reader to the Introduction in [334] for more details.

**(Step-1) Frequency bubbles of concentration [334, Proposition 3.2]**

Suppose $\|U\|_{L_t^\infty L_x^3 (\mathbb{R}^3 \times (-1,0))} \le A$ is such that

$$ N_0^{-1} |P_{N_0} u(x_0, 0)| > A^{-C}. \qquad (6.34) $$

Then for all $n \in \mathbb{N}$ there exists $N_n > 0$, $(x_n, t_n) \in \mathbb{R}^3 \times (-1, t_{n-1})$ such that

$$ N_n^{-1} |P_{N_n} u(x_n, t_n)| > A^{-C} \qquad (6.35) $$

with

$$ x_n = x_0 + O((-t_n)^{\frac{1}{2}}), \quad N_n \approx |-t_n|^{-\frac{1}{2}}. \qquad (6.36) $$

*This is the analogue of (Step-1) above in our proof in collaboration with Barker.*
(Step-2) Localized lower bounds on vorticity [334, p.37]

For certain scales $S > 0$ and an 'epoch of regularity', $I_S \subseteq [-S, -A^{-\alpha}S]$, where the solution enjoys 'good' quantitative estimates on $\mathbb{R}^3 \times I_S$ (in terms of A and S), Tao shows the following. The previous step and $\|U\|_{L_t^\infty L_x^3 (\mathbb{R}^3 \times [-1,0])} \le A$ imply

$$ \int_{B_{x_0}(A^\beta S^{\frac{1}{2}})} |\omega(x, t)|^2 dx \ge A^{-\gamma} S^{-\frac{1}{2}} \quad \text{for all } t \in I_S. \quad (6.37) $$

Here, $\alpha, \beta$ and $\gamma$ are positive universal constants.

This is an additional step where frequency information is transferred to the scale-invariant entrophy. This is not needed in our proof because we work directly with the scale-invariant entrophy.

(Step-3) Lower bound on the $L^3$ norm at the final moment in time $t_0$ [334, p.37-40]

Using quantitative versions of the Carleman inequalities in [129], see [334, Proposition 4.2 and Proposition 4.3], Tao shows that the lower bounds in (Step-2) can be transferred to a lower bound on the $L^3$ norm of $U$ at the final moment of time 0. The applicability of the Carleman inequalities to the vorticity equation requires the 'epochs of regularity' in the previous step and the existence of 'good spatial annuli' where the solution enjoys good quantitative estimates. Specifically, Tao shows that (Step-2) on $I_S$ implies

$$ \int_{R_S \le |x-x_0| \le R'_S} |U(x, 0)|^3 dx \ge \exp(-\exp(A^C)). \quad (6.38) $$

This is the analogue of (Step-2) above in our proof.

(Step-4) Conclusion: summing scales to bound $TN_0^2$

Letting $S$ vary for certain permissible $S$, the annuli in (6.38) become disjoint. The sum of (6.38) over such disjoint annuli is bounded from above by $\|U(\cdot, 0)\|_{L^3(\mathbb{R}^3)}$ and the lower bound due to the summing of scales is $\exp(-\exp(A^C)) \log(TN_0^2)$. This gives the desired bound on $N_0$, namely

$$ TN_0^2 \lesssim \exp(\exp(\exp(A^C))). $$

This is the analogue of (Step-3) above in our proof.

Let us emphasize once more that the approach in [334] produces quantitative estimates involving globally defined quantities, since the quantitative regularity mechanism (6.33) is inherently global. We also emphasize that the fact that $\|U\|_{L_t^\infty L_x^3} \le A$ is crucial for showing (Step-1) and (Step-2) in the above strategy.

### 6.3.4 Robustness of our new strategy: quantification without a Type I bound

Our strategy above, see Subsection 6.3.2 is robust enough (with certain adjustments) to apply to certain settings without a Type I control as for the quantification of Seregin's result [294] in Theorem 6.3.

In the strategy of Tao [334] the lower bound on vorticity (6.37), which is needed for getting a lower bound on the localized $L^3$ norm at final time 0 via quantitative Carleman inequalities, is obtained from the frequency bubbles of concentration. In order for this
transfer of scale-invariant information to take place, it appears essential that the solution has a scale-invariant control such as $||U||_{L^\infty_t L^3_x} \le A$, see [334, Proposition 3.1]. In our strategy, we instead work directly with quantities similar to (6.37) involving vorticity, which are tailored for the immediate use of quantitative Carleman inequalities. In this way, we crucially avoid any need to transfer scale-invariant information, giving our strategy a certain degree of robustness.

Recall that Theorem 6.3 is concerned with quantitative estimates, where we assume

$$t_k \uparrow 0 \text{ with } \sup_k ||U(\cdot, t_k)||_{L^3(\mathbb{R}^3)} \le M. \qquad (6.39)$$

First we remark that the local quantitative regularity statement (6.26)-(6.27) remains true (with $t'$ replaced by $t_k$) if $U$ is a $C^\infty(\mathbb{R}^3 \times [-1, 0])$ finite-energy solution and the Type I condition is replaced by the weaker assumption that $||U(\cdot, t_k)||_{L^3(\mathbb{R}^3)} \le M$. Our goal then becomes the following

**Our second goal:** If (6.26) fails for $t' = t_j$, what is an upper bound for $j$?

A key point is that we are able to backward propagate concentration as in (Step-1) in Subsection 6.3.2 despite the fact that the $L^3$ norm is bounded only at a countable number of times. As above, this requires a separation between the time $t'$ where initial concentration happens and the times $s$ of backward propagation of concentration, see (Step-1 time slices). This is why in Theorem 6.3 we need to take a sufficiently well-separated subsequence of $t_k$, see (6.20).

(Step-1 time slices) **Backward propagation of vorticity concentration in the time slices case**

Fix any $\alpha \ge M^b$ and let $t', t'' \in [-1, 0)$ be such that

$$\frac{t''}{\alpha^{1051}} < t' < 0.$$

Assume that

$$||U(\cdot, t')||_{L^3} \le M \text{ and } ||U(\cdot, t'')||_{L^3} \le M.$$

If the vorticity concentrates at time $t'$ in the following way

$$\int_{B_0(4\sqrt{S(M)}^{-1}(-t')^{\frac{1}{2}})} |\omega(x, t')|^2 dx > M^2(-t')^{-\frac{1}{2}}\sqrt{S(M)},$$

then for any $s \in [t'', \frac{t''}{8\alpha^{201}}]$ the vorticity concentrates in the following sense

$$\int_{B_0(4(-s)^{\frac{1}{2}}\alpha^{106})} |\omega(x, s)|^2 dx > \frac{(M+1)^2}{(-s)^{\frac{1}{2}}\alpha^{106}}.$$

To show this we use energy estimates in [304] for solutions to the Navier-Stokes equations with $L^3(\mathbb{R}^3)$ initial data. We decompose

$$U(\cdot, s) = e^{(s-t_0)\Delta} U(\cdot, t'') + V(\cdot, s) \quad \text{for any } 0 \ge s \ge t''$$

and get

$$||V(\cdot, s)||_{L^2(\mathbb{R}^3)}^2 + \int_{t''}^{s} \int_{\mathbb{R}^3} |\nabla V(x, t)|^2 dx dt \le C(M^b)^4 (s - t'')^{\frac{1}{2}}.$$
The price one pays in this setting when compared to the estimates in Tao's paper [334], is a gain of an additional exponential in the estimates. The reason is the control on the energy of $U(\cdot, s) - e^{(s-t_0)\Delta}U(\cdot, t'')$ with $U(\cdot, t'') \in L^3(\mathbb{R}^3)$ requires the use of Gronwall's lemma. Such estimates are also central to gain good quantitative control of the solution in certain space-time regions, which are required for applying the quantitative Carleman inequalities.

### 6.3.5 Mild criticality breaking

Our method for proving Theorem 6.5 relies on the following lemma and on a careful tuning of the parameters (estimating the $L^{3-\mu}$ norm for a well-chosen parameter $\mu$). Lemma 6.7 is directly inspired by the recent result of Bulut [73] for a nonlinear supercritical defo-cusing Schrödinger equation.

**Lemma 6.7.** For all $M \in [1, \infty)$ and $E \in [1, \infty)$ sufficiently large, there exists $\delta(M, E) \in (0, \frac{1}{2}]$ such that the following holds. Let $U$ be a suitable weak Leray-Hopf solution to the Navier-Stokes equations on $\mathbb{R}^3 \times (0, \infty)$ with initial data $U_0 \in L^2(\mathbb{R}^3) \cap L^4(\mathbb{R}^3)$. Assume that

$$ \|U_0\|_{L^2}, \|U_0\|_{L^4} \le M, $$

and that

$$ \|U\|_{L^{\infty}(0,\infty;L^{3-\delta}(M,E)(\mathbb{R}^3))} \le E. \qquad (6.40) $$

Then, the above assumptions imply that $U$ is smooth on $\mathbb{R}^3 \times (0, \infty)$. Moreover, there is an explicit formula for $\delta(M, E)$, see [37, equation (26)], and $\delta(M, E) \to 0$ when $E \to \infty$ or $M \to \infty$.

We call the result of Lemma 6.7 a 'mild breaking of the criticality', or a 'mild supercritical regularity criteria' as opposed to strong criticality breaking results obtained for instance in the axisymmetric case [279, 298, 299, 95]. Indeed, the supercritical space $L_t^\infty L^3 - \delta(M,E)$ in which we break the scaling depends on the size $E$ of the solution in this supercritical space via $\delta(M, E)$. In other words this can be considered as a non effective regularity criteria, hence the term 'mild'. Moreover, given a solution $U$, assume that you knew all the $L_t^\infty L_x^{3-\delta}$ norms for $\delta \to 0$. Then the question whether Lemma 6.7 applies to $U$ or not becomes a question about how fast

$$ \|U\|_{L^{\infty}(0,\infty;L^{3-\delta}(\mathbb{R}^3))} $$

grows when $\delta \to 0$. We also emphasize that the larger the $M$, the closer the exponent $3 - \delta(M, E)$ is from the critical exponent 3. Of course we would have regularity if the solution was a priori bounded in the critical space $L_t^\infty L_x^3$. Our result shows that with $L^4$ initial data we can relax the exponent 3 to a slightly supercritical $3 - \delta(M, E)$. Let us also remark that the condition $U_0 \in L^4(\mathbb{R}^3)$ can be replaced by any subcritical condition $U_0 \in L^{3+}(\mathbb{R}^3)$.

The main idea of the proof of Lemma 6.7 is to transfer subcritical information from the initial time forward in time. In that perspective our main goal is

**Our main goal:** Prove that there exists $\delta(M, E) \in (0, \frac{1}{2}]$ and $K(M, E) \in [1, \infty)$ such that for all $U_0 \in L^2(\mathbb{R}^3) \cap L^4(\mathbb{R}^3)$ and any suitable weak Leray-Hopf solution associated to the initial data $U_0$, if

$$ \|U_0\|_{L^2}, \|U_0\|_{L^4} \le M, $$
and

$$ \begin{gathered} \|U\|_{L^{\infty}(0,\infty;L^{3-\delta(M,E)}(\mathbb{R}^3))} \le E, \\ \|U\|_{L^{\infty}(0,T;L^4(\mathbb{R}^3))} \le K(M,E). \end{gathered} \tag{6.41} $$

then

This then obviously implies the result stated in Lemma 6.7. The crucial point is that $K(M, E)$ is uniform in time.

The only a priori globally controlled quantity is a supercritical $L_t^\infty L_x^{3-\delta}$ norm. We are not aware of any regularity mechanism enabling to brake the critically barrier based on the sole knowledge of such a supercritical bound. Therefore, the idea, following Bulut [73] is to transfer the subcritical information coming from the initial data $U_0 \in L^4(\mathbb{R}^3)$ to arbitrarily large times by using three ingredients:

(1) the control of the critical $L_t^\infty L_x^3$ norm via interpolation between the supercritical norm $L_t^\infty L_x^{3-\delta(M,E)}$ and the subcritical $L_t^\infty L_x^4$ norm

$$ \begin{aligned} \|U\|_{L^{\infty}(0,T;L^3(\mathbb{R}^3))} &\le \|U\|_{L^{\infty}(0,T;L^{3-\delta}(\mathbb{R}^3))}^{\frac{3-\delta}{3+3\delta}} \|U\|_{L^{\infty}(0,T;L^4(\mathbb{R}^3))}^{\frac{4\delta}{3+3\delta}} \\ &\le E^{\frac{3-\delta}{3+3\delta}} K^{\frac{4\delta}{3+3\delta}}; \end{aligned} $$

(2) the quantitative control of the critical non borderline $L_{t,x}^5$ norm (see [37, Proposition 3]) in terms of the critical norm $\|U\|_{L^\infty(0,\infty;L^3(\mathbb{R}^3))}$, and the supercritical $L^2$ and subcritical $L^4$ norms of the initial data $U_0$

$$ \|U\|_{L^5(0,T;L^5(\mathbb{R}^3))} \le C(M) \exp \exp \exp \left( C_{\text{univ}} \left( E^{\frac{3-\delta}{3+3\delta}} K^{\frac{4\delta}{3+3\delta}} \right)^c \right); $$

this hinges on the quantitative bounds on solutions belonging to the critical space $L_t^\infty L_x^3$, which were established by Tao in [334], see Subsection 6.1.4 above; this step enables the slicing of the interval $(0, T)$ into a $T$-independent number $m$ of disjoint epochs $I_j = (t_j, t_{j+1})$,

$$ \begin{aligned} \varepsilon^5 m = \sum_{j=1}^{m} \|U\|_{L^5(I_j; L^5(\mathbb{R}^3))}^5 &\le \|U\|_{L^5(0, T; L^5(\mathbb{R}^3))}^5 \\ &\le C(M) \exp \exp \exp \left( \left( E^{\frac{3-\delta}{3+3\delta}} K^{\frac{4\delta}{3+3\delta}} \right)^c \right); \end{aligned} $$

(3) an $L^4$ energy estimate [37, Proposition 4] under the $L_{t,x}^5$ control of $U$, which enables the transfer the subcritical information from time $t_j$ to $t_{j+1}$

$$ \begin{aligned} \mathcal{E}_{4,t_{j+1}} &\le \|U(\cdot, t_j)\|_{L^4(\mathbb{R}^3)}^4 + C \|U\|_{L^5(\mathbb{R}^3 \times I_j)} \mathcal{E}_{4,t_{j+1}} \\ &\le \|U(\cdot, t_{j+1})\|_{L^4(\mathbb{R}^3)}^4 + C \varepsilon \mathcal{E}_{4,t_{j+1}}, \end{aligned} $$

where $\mathcal{E}_{4,t_{j+1}}$ is the $L^4$ energy, see [37, equation (13)] and eventually to $T$

$$ \|U\|_{L^{\infty}(0,T;L^4(\mathbb{R}^3))}^4 = \max_{1 \le j \le m+1} \{\|U\|_{L^{\infty}(I_j;L^4(\mathbb{R}^3))}^4\} \le 64M^4 2^m. $$

One then designs the number $K(M, E)$ to bound the right hand side above.
Chapter 7

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