Quantitative regularity for the Navier-Stokes equations via spatial concentration

This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound $\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$ is a first blow-up time and $(0,T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (2012), which says that if $u$ is a smooth finite-energy solution to the Navier-Stokes equations on $\mathbb{R}^3\times (0,1)$ with $$\sup_{n}\|u(\cdot,t_{(n)})\|_{L^{3}(\mathbb{R}^3)}<\infty\,\,\,\textrm{and}\,\,\,t_{(n)}\uparrow 1,$$ then $u$ does not blow-up at $t=1$. To prove our results we develop a new strategy for proving quantitative bounds for the Navier-Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and \v{S}ver\'{a}k (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.

necessary conditions for such a solution to lose smoothness or to 'blow-up' at time T * > 0 2 were given in the seminal paper of Leray [26]. In particular, in [26] it is shown that if T * is a first blow-up time of u then we necessarily have , for p ∈ (3, ∞]. The L 3 (R 3 ) norm is scale-invariant or 'critical' 3 with respect to the Navier-Stokes rescaling. Its role in the regularity theory of the Navier-Stokes equations is much more subtle than that of the subcritical L p (R 3 ) norms with 3 < p ≤ ∞. In particular, it is demonstrated by a elementary scaling argument in [4] 4 that there cannot exist a universal function f : (0, ∞) → (0, ∞) such that the following analogue of (3) holds true: and 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 for all t ∈ [0, T * ).
In the celebrated paper [13] of Escauriaza, Seregin and Šverák, it was shown that if a finiteenergy solution u first blows-u at T * > 0 then necessarily The proof in [13] is by contradiction. A rescaling procedure or 'zoom-in' is performed 5 using (2) and a compactness argument is applied. This gives a non-zero limit solution to the Navier-Stokes equations that vanishes at the final moment in time. The contradiction is achieved by showing that the limit function must be zero by applying a Liouville type theorem based on backward uniqueness for parabolic operators satisfying certain differential inequalities. By now there are many generalizations of (6) to cases of other critical norms. See, for example, [12], [16], [34] and [47]. Let us mention the arguments in [13] and the aforementioned works are by contradiction and hence are qualitative. It is worth noting that the result in [13], together with a proof by contradiction based on the 'persistence of singularities' lemma in [35] (specifically Lemma 2.2 in [35]), gives the following. Namely, that there exists an F : (0, ∞) → (0, ∞) such that if u is a finite-energy solution to the Navier-Stokes equations then (7) u L ∞ (0,1;L 3 (R 3 )) < ∞ ⇒ u L ∞ (R 3 ×( 1 2 ,1)) ≤ F ( u L ∞ (0,1;L 3 (R 3 )) ). Such an argument is obtained by a compactness method and gives no explicit 6 information about F . In a remarkable recent development [45], Tao used a new approach to provide the 2 We say that a solution u to the Navier-Stokes equations first blows-up at T * > 0 if u ∈ first explicit quantitative estimates for solutions of the Navier-Stokes equations belonging to the critical space L ∞ (0, T ; L 3 (R 3 )). As a consequence of these quantitative estimates, Tao showed in [45] that if a finite-energy solution u first blows-up at T * > 0 then for some absolute constant c > 0 log log log 1 Since there cannot exist f such that (4)- (5) holds true, at first sight (8) may seem somewhat surprising, though it is not conflicting with such a fact. Notice that log log log 1 is not invariant with respect to the Navier-Stokes scaling (2) but is slightly supercritical 7 due to the presence of the logarithmic denominator. Let us also mention that prior to Tao's paper [45], in the presence of axial symmetry, a different slightly supercritical regularity criteria was obtained in [33].
The contribution of our present paper is to develop a new strategy for proving quantitative estimates (see Propositions 2 and 3) for the Navier-Stokes equations, which then enables us to build upon Tao's work [45] to quantify critical norms. Our first Theorem involves applying the backward propagation of concentration stated in Proposition 2 below to give a new necessary condition for solutions to the Navier-Stokes equations to possess a Type I blow-up. Note that if u is a finite-energy solution that first blows-up at T * > 0 we say that T * is a Type I blow-up if (9) u L ∞ (0,T * ;L 3,∞ (R 3 )) ≤ M.
In the case of a Type I blow-up at T * the nonlinearity in (2) is heuristically balanced with the diffusion. Despite this, it remains a long standing open problem whether or not Type I blow-ups can be ruled out when M is large. Let us now state our first theorem.
Theorem 1 (rate of blow-up, Type I). There exists a universal constant M 0 ∈ [1, ∞) such that for all M ≥ M 0 and δ ∈ (0, 1) the following holds true. Assume that u is a mild solution to the Navier-Stokes equations on R 3 × [0, T * ) with u ∈ L ∞ loc ([0, T * ); L ∞ (R 3 )). 8 Assume that (1) u L ∞ t L 3,∞ x (R 3 ×(0,T * )) ≤ M (2) u has a singular point at (x, t) = (0, T * ). In particular u / ∈ L ∞ x,t (Q (0,T * ) (r)) for all sufficiently small r > 0. 7 We say a quantity F (u, p) is supercritical if, for the rescaling (2), we have F (u λ , p λ ) = λ −β F (u λ , p λ ) for some β > 0. 8 Under these assumptions, u is smooth on the epoch (0, T ) for any T < T * and belongs to L ∞ ((0, T ); L 4 (R 3 ))∩L ∞ ((0, T ); L 5 (R 3 )) by interpolation, which enables us to satisfy the hypothesis needed in Sections 6-7. Furthermore using Lemma 2.4 in [20], it gives that u coincides with all local energy solutions (we refer to footnote 41 for a definition), with initial data u(·, s), on R 3 × (s, T * ) for any 0 < s < T * . We call such a solution a 'smooth solution with sufficient decay' on the interval [0, T ], for T < T * . A mild solution with Schwartz class initial data and maximal time of existence T * will be such a solution, and so Theorem 1 applies to that setting. Notice that smoothness is needed here in order to get estimate (10) for all t in the ad hoc interval. The framework of 'smooth solutions with enough decay' is needed to apply Theorem 1 to the setting of Corollary 1, where the solution is not of finite energy.
Then the above assumptions imply that there exists c(δ, M, T * ) ∈ (0, ∞), which we will specify in the proof, such that for any t ∈ (max( T * 2 , T * − c(δ, M, T * )), T * ) we have This theorem is proved in Subsection 2.2 below. Notice that in Theorem 1, not only is the rate new but also the fact that the L 3 norm blows up on a ball of radius O((T * − t) 1 2 − ) around any Type I singularity. Previously in [27] (specifically Theorem 1.3 in [27]), it was shown that if a solution blows up (without Type I bound) then the L 3 norm blows up on certain non-explicit concentrating sets.
In our second main theorem, we apply Proposition 3 to fully quantify Seregin's result in [36], which generalizes Theorem 1.2 in [45]. Now let us state our second theorem.
This theorem is proved in Subsection 2.2 below. 10 Assume for contradiction that (16) does not hold. Then we have a sequence of solutions (u k ) k∈N such that u (k) (·, 1) L ∞ (R 3 ) ↑ ∞ and, for each k, a sequence of associated time slices t k (n) ↑ 1 such that The main block for the contradiction argument to go through is that the sequence (t k (n) ) n∈N may be different for distinct indices k. 11 In particular, M is chosen such that the following is true. If (u, p) is a suitable finite-energy solution (defined in Section 1.4 for t ∈ (0, T ) and M larger than a universal constant. See Lemma 6. 12 Notice that smoothness is needed here to have the energy inequality starting from every time t k , and not for almost every t ∈ (−1, 0), see (43) as would be the case if (u, p) was just a suitable finite-energy solution.
Further applications. Section 4 contains three further applications of the technology developed in the present paper: (i) Proposition 7, a regularity criteria based on an effective 13 relative smallness condition in the Type I setting, (ii) Corollary 9, an effective bound for the number of singular points in a Type I blow-up scenario, (iii) Proposition 10, a regularity criteria based on an effective relative smallness condition on the L 3 norm at initial and final time. Non-effective quantitative bounds of the above results were previously obtained by compactness methods: for (ii) see [12,Theorem 2], for (iii) see [2, Theorem 4.1 (i)]. 1. 1. Comparison to previous literature and novelty of our results. Theorems 1 and 2 in this paper follow from new quantitative estimates for the Navier-Stokes equations (Propositions 2 and 3), which build upon recent breakthrough work by Tao in [45]. In particular, Tao shows that for classical 14 solutions to the Navier-Stokes equations Before describing our contribution, we first find it instructive to outline Tao's approach in [45]. Fundamental to Tao's approach for showing (22) is the following fact 15 (see Section 6 in [45]). There exists a universal constant ε 0 such that if u is a classical solution to the Navier-Stokes equations with and then u L ∞ x,t (R 3 ×( 7 8 ,1)) can be estimated explicitly in terms of A and N * . Related observations were made previously in [11], but in a slightly different context and without the bounds explicitly stated.
In this perspective, Tao's aim is the following: Tao's goal: Under the scale-invariant assumption (23), if (24) fails for ε 0 = A −O (1) and N = N 0 , what is an upper bound for N 0 ?
In [45] (Theorem 5.1 in [45]), it is shown that N 0 exp exp exp(A O(1) ), which implies (22) by means of the quantitative regularity mechanism (24) with N * = 2N 0 . We emphasize that since the regularity mechanism (24) is global: all quantitative estimates obtained in this way are in terms of globally defined quantities.
The strategy in [45] for showing Tao's goal with N 0 exp(exp(exp(A O(1) ))) can be summarized in four steps. We refer the reader to the Introduction in [45] for more details.

1) Frequency bubbles of concentration (Proposition 3.2 in
13 See footnote 6 for the definition of 'effective' bounds. 14 In [45], these are solutions that are smooth in R 3 × (0, 1) and such that all derivatives of u and p lie in x (R 2 × (0, 1)). 15 Let ϕ ∈ C ∞ 0 (B0(1)) with ϕ ≡ 1 on B0( 1 2 ). The Littlewood-Paley projection PN is defined for any 2) Localized lower bounds on vorticity (p.37 in [45]). For certain scales S > 0 and an 'epoch of regularity' where the solution enjoys 'good' quantitative estimates on R 3 × I S (in terms of A and S), Tao shows the following: the previous step and u L ∞ Here, α, β and γ are positive universal constants.
3) Lower bound on the L 3 norm at the final moment in time t 0 (p. 37-40 in [45]

4) Conclusion
: summing scales to bound T N 2 0 . Letting S vary for certain permissible S, the annuli in (29) become disjoint. The sum of (29) over such disjoint permissible annuli is bounded above by u(·, t 0 ) L 3 (R 3 ) and the lower bound due to the summing of scales is exp(− exp(A O(1) )) log(T N 2 0 ). This gives the desired bound on N 0 , namely Let us emphasize once more that the approach in [45] produces quantitative estimates involving globally defined quantities, since the quantitative regularity mechanism (24) is inherently global. We would also like to emphasize that the fact that u L ∞ t L 3 x < A is crucial for showing steps 1-2 in the above strategy.
The goal of the present paper is to develop a new robust strategy for obtaining new quantitative estimates of the Navier-Stokes equations, which are then applied to obtain Theorems 1 and 2. The main novelty (which we explain in more detail below) is that our strategy allows us to obtain local quantitative estimates and even applies to certain situations where we are outside the regime of scale-invariant controls. For simplicity, we will outline the strategy for the case when u L ∞ t L 3,∞ x (R 3 ×(t 0 −T,t 0 )) ≤ M , before remarking on this strategy for cases without such a scale-invariant control (Theorem 2).
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 [21] (see also [6] for extensions to critical cases). In particular, the result of [21], together with rescaling arguments from [6], implies the following. If u : R 3 × [t 0 − T, t 0 ] → R 3 is a smooth solution with sufficient decay 16 of the Navier-Stokes equations and u L ∞ can be estimated explicitly in terms of M and t 0 − t * 0 . Here, S (M ) = O(1)M −100 is as in Theorem 3. In this perspective, the aim of our strategy is the following what is a lower bound for t 0 − t 0 ? This is the main aim of Proposition 2. Taking s 0 such that t 0 − t 0 ≥ 2T s 0 , we can then apply (30)-(31) with t * 0 = t 0 − T s 0 . One might think of the main goal of our strategy as a physical space analogy to Tao's goal with In contrast to (24), the regularity mechanism (30)-(31) produces quantitative estimates that are in terms of locally defined quantities, which is crucial for obtaining the localized results as in Theorem 1. Our strategy for obtaining a lower bound of t 0 − t 0 (see Proposition 2) can be summarized in three steps; see also

1) Backward propagation of vorticity concentration (Lemma 4).
Let is not too close to t 0 − T and is such that We show that for all t 0 ∈ (t 0 −T, t 0 ), such that t 0 −t 0 is sufficiently large compared to t 0 − t 0 (in other words t 0 is well-separated from t 0 ), we have We refer the reader to Lemma 4 for precise statements for the rescaled/translated situation 2) Lower bound on localized L 3 norm at the final moment in time t 0 . Using the previous step, together with the same arguments as [45] involving quantitative Carleman inequalities, we show that for certain permissible annuli that We wish to mention that the role of the Type I bound is to show the solution u obeys good quantitative estimates in certain space-time regions, which is needed to apply the Carleman inequalities to the vorticity equation. 16 In this paper, 'smooth solution with sufficient decay' always denotes the notion described in footnote 8. 3) Conclusion: summing scales to bound t 0 − t 0 from below. Summing (34) over all permissible disjoint annuli finally gives us the desired lower bound for t 0 − t 0 in Proposition 2. We note that the localized L 3 norm of u at time t 0 plays a distinct role to that of Type I condition described in the previous step. Its sole purpose is to bound the number of permissible disjoint annuli that can be summed, which in turn gives the lower bound of t 0 − t 0 . Together with the assumed global Type I assumption, this is essentially why the lower bound in Theorem 1 on the localized L 3 norm near a Type I singularity is a single logarithm and holds at pointwise times.
Although the above relates to the case of Proposition 2 and Theorem 1 where we stress that the above strategy (with certain adjustments) is robust enough to apply to certain settings without a Type I control (Theorem 2).
Recall that Theorem 2 is concerned with quantitative estimates on u : First we remark that the local quantitative regularity statement (30)-(31) remains true (with t * 0 replaced by t k ) if u is a C ∞ (R 3 × (−1, 0]) finite-energy solution and the Type I condition is replaced by the weaker assumption that u(·, t (k) ) L 3 (R 3 ) ≤ M . Our goal then becomes the following Our second goal: If (30) fails for t * 0 = t j (with t 0 = 0 and T = 1), what is an upper bound for j?
In this setting '1) Backward propagation of vorticity concentration' still remains valid if a sufficiently well-separated subsequence of t (k) is taken (see Lemma 6 and Proposition 3). To show this we use energy estimates in [38] for solutions to the Navier-Stokes equations with L 3 (R 3 ) initial data. 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. The price one pays in this setting (when compared to the estimates in [45]), is a gain of an additional exponential in the estimates. The reason is the control on the energy of u(·, t) − e t∆ u 0 ( with u 0 ∈ L 3 (R 3 )) requires the use of Gronwall's lemma.
In the strategy in [45] the lower bound on vorticity (28), which is needed for getting a lower bound on the localized L 3 norm at t 0 via quantitative Carleman inequalities, is obtained from the frequency bubbles of concentration. In order for this transfer of scaleinvariant information to take place, it appears essential that the solution has a scale-invariant control such as u L ∞ t L 3 x ≤ A. In our strategy, we instead work directly with quantities involving vorticity (similar to (28)), 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.

Final Remarks and Comments.
We give some heuristics about the quantitative estimates of the form (36) u L ∞ (R 3 ×( 1 2 ,1)) ≤ G( u X ) that one can expect for the Navier-Stokes equations, when a finite-energy solution u solution belongs to certain normed spaces X ⊂ S (R 3 × (0, 1)).

Critical case.
In the subcritical norm case, we saw that seeking estimates of the form (36) that are invariant with respect to the scaling (2), gives a suitable candidate for G that can be realised. The case when the norm · X is critical is more subtle, since a scaling argument does not provide a suitable candidate for G. We first mention that the case of sufficiently small critical norms, for example (38) u L 5 (R 3 ×(0,1)) < ε 0 , 17 A quantity F (u) ∈ [0, ∞) is said to be subcritical if, for the rescaling (2), there exists α > 0 such that is essentially of a similar category to the subcritical case (though a scaling argument is not applicable). Indeed, a similar argument outlined as before (based on [9], see also Proposition 15) gives that in this case we have (36) with G(x) ∼ x. Is this consistent with the fact that solutions with small scale-invariant norms exhibit similar behaviour to the linear system and hence are typically expected to satisfy linear estimates.
For obtaining quantitative estimates of the form (36) when the scale-invariant norm is large, it is less clear what the candidate for G might be. This seems to be the case even for large global scale-invariant norms that exhibit smallness at small local scales 18 (for example L 5 (R 3 ×(0, 1))). Such local smallness properties have been utilized to prove qualitative regularity by essentially linear methods. See [43], for example. For the case of a smooth finite-energy solution u having finite scale-invariant L 5 (R 3 ×(0, 1)) norm, one way to obtain quantitative estimates 19 is to consider the vorticity equation (37) with initial vorticity ω 0 ∈ L 2 (R 3 ). Performing an energy estimate yields for t ∈ [0, 1] where the second term in right-hand side is due to the vortex stretching term ω · ∇u in (37). For the case that u ∈ L 5 (R 3 × (0, 1)), application of Hölder's inequality, Lebesgue interpolation, Sobolev embedding theorems and Young's inequality lead to (40) ω(·, t) 2 Gronwall's lemma, followed by arguments similar to the subcritical case, yields Though this is not exactly of the form (36), a slightly different argument gives that for any finite-energy solution u in L 5 (R 3 ×(0, 1)) we get that (36) holds with G(x) ∼ exp(O(1)x 5 ).
In particular, this can be achieved using L q energy estimates in [30], the pidgeonhole principle and reasoning in the previous subsection. The above argument (39)- (41) shows that being able to substantially improve upon G(x) ∼ exp(O(1)x 5 ) would most likely require the utilization of a nonlinear mechanism that reduces the influence of the vortex stretching term ω · ∇u in (37). It seems plausible that the discovery of such a mechanism would have implications for the regularity theory of the Navier-Stokes equations (such as Type I blow-ups). 1. 3. Outline of the paper. In each of the Sections 2-7, we distinguish between cases where: (i) one assumes a Type I control on the solution and (ii) one assumes a control on the velocity field on time slices only.
In Section 2, we state our main quantitative estimates (Propositions 2 and 3) and we demonstrate how these statements imply the main results of this paper: Theorem 1, Corollary 1 and Theorem 2. Section 3 is devoted to the proof of Propositions 2 and 3. Section 4 contains three further applications of the technology developed in the present paper, in particular Corollary 9 concerning a quantitative bound for the number of singularities in a Type I blow-up scenario. In Section 5, we quantify Jia and Šverák's results regarding local-in-space short-time smoothing, which is a main tool for proving the quantitative estimates in Section 3. The main result in Section 5 is Theorem 3. In Section 6, we give a new proof of Tao's result that solutions possess 'quantitative annuli of regularity', which is required for proving our main propositions in Section 3. The central results in Section 6 are Lemma 20 and Lemma 22. Section 7 is concerned with the utilization of arguments from the papers of Leray and Tao to show existence of quantitative epochs of regularity (Lemma 27 and Lemma 29). In Appendix A we recall known results about mild solutions and local energy solutions, and we give pressure formulas. In Appendix B, we recall the quantitative Carleman inequalities proven by Tao Universal constants. For universal constants in the statements of propositions and lemmas associated to the Type I case (specifically Proposition 2 and Lemma 4), we adopt the convention of a superscript . For universal constants in the statements of propositions and lemmas associated to the Type I case (specifically Proposition 3 and Lemma 6), we adopt the convention of a superscript .
In Lemma 4, Lemma 6 and Section 5, we track the numerical constants arising. Elsewhere in this paper, we adopt the convention that C denotes a positive universal constant which may vary from line to line.
We use the notation X Y , which means that there exists a positive universal constant C such that X ≤ CY.
In several places in this paper (notably Section 3 and Appendix B) the notation O(1) is used to denote a positive universal constant and −O(1) denotes a negative universal constant.
Whenever we refer to a quantity (M for example) being 'sufficiently large', we understand this as M being larger than some universal constant that can (in principle) be specified. 1.4.2. Vectors and Domains. For a vector a, a i denotes the i th component of a.
Let us stress that in Section 5 only we use cubes instead of balls: B x (r) = x+(−r, r) 3 . This is for computational convenience, since we track numerical constants in Section 5. We emphasize that the results in Section 5 hold for spherical balls too, with certain universal constants adjusted. 1.4.3. Mild, suitable and finite-energy solutions to the Navier-Stokes equations. Throughout this paper, we refer to u : R 3 × [0, T ] as a mild solution of the Navier-Stokes equations (1) if it satisfies the Duhamel formula: Here, e t∆ is the heat semigroup, P is the projection onto divergence-free vector fields. A mild solution on [0, T * ) is a function that is a mild solution on [0, T ] for any T ∈ (0, T * ).
Let Ω ⊂ R 3 . We say that (u, p) is a suitable weak solution to the Navier-Stokes equations (1) in Ω × (T 1 , T ) if it fulfills the properties described in [18] (Definition 6.1 p.133 in [18]) . We say that u is a suitable finite-energy solution to the Navier-Stokes equations on R 3 × (T 1 , T ) if it is a solution to (1) in the sense of distributions and • it satisfies the global energy inequality (42) u(·, t) 2 It is known that the above defining properties of suitable weak solutions imply that there for all t ∈ [t , T ] and t ∈ Σ.

Lorentz spaces.
For a measurable subset Ω ⊆ R d and a measurable function f : , is the set of all measurable functions g on Ω such that the quasinorm g L p,q (Ω) is finite. The quasinorm is defined by Notice that there exists a norm, which is equivalent to the quasinorm defined above, for which L p,q (Ω) is a Banach space. For p ∈ [1, ∞) and 1 ≤ q 1 < q 2 ≤ ∞, we have the following continuous embeddings and the inclusion is strict.

Quantitative estimates in the Type I and time slices case.
Proposition 2 (main quantitative estimate, Type I). There exists a universal constant 1 4 ], such that the following holds. Let (u, p) be a smooth solution with sufficient decay 20 to the Navier- 20 See footnote 8. Notice that by this definition u is bounded up to t0.
and such that the vorticity concentrates at time t 0 in the following sensê Then, we have the following lower bound Furthermore, for we also have the bound for universal constants C 1 , C 2 ∈ (0, ∞). Here C and S (M ) are the constants given by Lemma 4. Furthermore, if for fixed λ ∈ (0, exp(M 1023 )) we additionally assume that then we instead have the lower bound Furthermore, for we also have the bound

Figure 1.1 illustrates Proposition 2.
Proposition 3 (main quantitative estimate, time slices). There exists a universal constant We define M by (17). There exists S (M ) ∈ (0, 1 4 ], such that the following holds. Let (u, p) be a C ∞ (R 3 × (−1, 0)) finite-energy solution to the Navier-Stokes equations (1) Select any "well-separated" subsequence (still denoted t (k) ) such that For this well-separated subsequence, assume that there exists j + 1 such that the vorticity concentrates at time t (j+1) in the following sense Then, we have the following upper bound on j Here S (M ) is the constant given by Lemma 6.

Proofs of the main results.
In this section we prove the main results stated in the Introduction.
Proof of Theorem 1. Take M ≥ M 0 . Here, M 0 := max(M 2 , M 6 ) ≥ 1 with M 2 being from Proposition 2 and M 6 being from Corollary 12. We prove here a slightly stronger statement than (10), namely First we note that [6] (specifically Theorem 2 in [6]), together with assumptions (1)-(2) in the statement of Theorem 1 imply that there exist S BP (M ) and γ univ > 0 such that univ for all t ∈ (0, T * ).
Suppose for contradiction that (60) does not hold for some for an appropriate c(δ) ∈ (0, ∞). This implies that for M sufficiently large Considering u on R 3 × [0, t] and observing Proposition 2, we see that (61) implies that the assumption (52) is satisfied with λ := (T * − t) 1−δ 2 and T := t. Furthermore, by (62)-(63) we have that λ ∈ (0, exp(M 1023 )). Hence, we can apply Proposition 2. Namely by (55) for we have the bound Using that (0, T * ) is a singular point of u, the Type I bound on u and Corollary 12, we see that there exists a universal constant C univ such that From our contradiction assumption (which implies (64)) we see that Using this and (62) for an appropriate c(δ) ∈ (0, ∞), we get that With these two facts, we see that (65) contradicts (66).
Integration of this then immediately gives the upper bound of (12), which in fact holds true for t ∈ (−1, 0). Next, note that (68) implies We also remark that since u is non-zero and λ-DSS we must have for all sufficiently small r.
Proof of Theorem 2. Applying Proposition 3 we see that for j = exp(exp((M ) 1224 )) +1 we have the contrapositive of (58). In particular, Almost identical arguments to those utilized in the proof of Lemma 4, except using the bound (174) instead of (175), give Since all estimates are independent of the spatial point where (58) occurs, we conclude that This concludes the proof of the theorem. Lemma 4 (backward propagation of concentration, Type I). There exists two universal 1 4 ], such that the following holds. Let (u, p) be a 'smooth solution with sufficient decay' 21 of the Navier-Stokes equations (1) in I = [−1, 0] satisfying the Type I bound (48). Assume that there exists t 0 ∈ [−1, 0) such that t 0 is not too close to −1 in the sense

PROOFS OF THE MAIN
and such that the vorticity concentrates at time t 0 in the following sense Then, the vorticity concentrates in the following sense Here Proof of Lemma 4. The proof is by contradiction. It relies on Theorem 3 below about localin-space short-time smoothing. We define S ∈ (0, 1 4 ] in the following way: where S * is the constant defined in Theorem 3 (see also the formula (206)), C Sob ∈ (0, ∞) is the best constant in the Sobolev embedding H 1 (B 0 (2)) ⊂ L 6 (B 0 (2)) and C ellip ∈ (0, ∞) is the best constant in the estimate for weak solutions to See, for example, Lemma 6.2 in Bradshaw and Tsai's paper [8].
For all sufficiently large M ∈ [1, ∞), let S (M ) ∈ (0, 1 4 ] be the constant defined by (74). Let (u, p) be a suitable finite-energy solution 22 of the Navier-Stokes equations (1) in I = [−1, 0] satisfying the Type I bound (48). Assume that the space-time point (0, 0) is a singularity for u. Then, for all t ∈ Σ, where Σ is a full measure subset of [−1, 0) defined in Subsection 1.4.3, the vorticity concentrates in the following sensê Lemma 6 (backward propagation of concentration, time slices). There exists a universal constant There exists S (M ) ∈ (0, 1 4 ], such that the following holds.
Suppose further that the vorticity concentrates at time t 0 in the following sense With the additional separation condition that the above assumptions imply that for any s 0 ∈ [t 0 , t 0 8α 201 ] the vorticity concentrates in the following sense We then have Furthermore, arguments from [17] imply that Using (81)-(83) and Gronwall's lemma, we infer (for M larger than some universal constant) that Here M is defined by (17) for an appropriate universal constant L * ∈ (0, ∞) coming from the Gronwall estimate. In particular, using that s ∈ [t 0 , Here, we used the fact that α ≥ M . Next assume for contradiction the under the assumptions of Lemma 6, we have the converse of (79). Namely, there exists and rescale to get U λ : U λ (y, t) := λu(λy, λ 2 t + s 0 ) and P λ (y, t) := λ 2 p(λy, λ 2 t + s 0 ).
Using (81) and (84), we see that Similarly to Lemma 4, we define where S * is the constant defined in Theorem 3, C Sob ∈ (0, ∞) is the best constant in the Sobolev embedding H 1 (B 0 (2)) ⊂ L 6 (B 0 (2)) and C ellip ∈ (0, ∞) is the best constant in the estimate 23 Based on the energy method, Lebesgue interpolation, Sobolev embedding, Hölder's inequality and Young's inequality.

Next notice that from
Using (88)-(89), together with a similar reasoning to Lemma 4, we can apply Theorem 3 with M ≥ M 4 being sufficiently large. Specifically, we apply Remark 11 with β = α −212 . This gives Using that s 0 < t 0 < 0 and that S (M ) = O(1)M −100 , we have for M sufficiently large that So for s 0 256 < t 0 , we see that (91) implies that Here, we used s 0 ∈ [t 0 , Therefore, if we contradict (77). Thus, if (92) holds we must have that 8α 201 ] as desired. Note that (78) implies (92).

3.2.
Proof of the main quantitative estimate in the Type I case. This part is devoted to the proof of Proposition 2. Following Tao [45], the idea of the proof is to transfer the concentration of the enstrophy at times t 0 far away in the past to large-scale lower bounds for the enstrophy at time t 0 . This is done in Step 1-3 below. The last step, Step 4 below, consists in transferring the lower bound on the enstrophy at time t 0 to a lower bound for the L 3 norm at time t 0 and summing appropriate scales. In Step 5 we sum scales under the additional assumption (52). Without loss of generality, we now take t 0 = 0. We also assume that T = 1. The general statement is obtained by scaling In the course of the proof we will need to take M larger, always larger than universal constants. Let u : R 3 × [−1, 0] → R 3 be a 'smooth solution with sufficient decay' 24 of the Navier-Stokes equations (1) in I = [−1, 0] satisfying the Type I bound (48). Assume that there exists t 0 ∈ [−1, 0) such that t 0 is not too close to −1 in the sense and such that the vorticity concentrates at time t 0 in the following sense The rest of the proof relies on the Carleman inequalities of Proposition 34 and Proposition 35. These are the tools used to transfer the concentration information (94) from the time t to time 0 and from the small scales B 0 (4 2 ) to large scales.
Step 1: quantitative unique continuation. The purpose of this step is to prove the following estimate: for all T 1 and R such that Thus, we can apply Lemma 27 and Remark 28 with 24 See footnote 8. 25 Notice that the whole argument of Section 3.2 goes through assuming that (94) holds for almost any The bound (261) in Remark 28 implies that there exists an epoch of regularity and for j = 0, 1, 2, We apply the second Carleman inequality, Proposition 35 (quantitative unique continuation), on the cylinder Notice that the quantitative regularity (99) and the vorticity equation (37) implies that on C 1 so that (312) is satisfied with S = S 1 := T 1 and C Carl = 16 3 . Let For M sufficiently large we have 0 <š 1 ≤ŝ 1 ≤ T 1 10000 . Hence by (314) we have We first use the concentration (94) for times s ∈ [s − 20000 ] to bound Z 1 from below. By (100), we have 20000 ] and for M sufficiently large. We have Second, we bound from above X 1 . We rely on the quantitative regularity (99) to obtain Hence, Third, for Y 1 we decompose and estimate as follows where we used the quantitative regularity (99). Hence, Gathering these bounds and combining with (101) yields Using (98) and |x 1 | ≥ M 100 ( T 1 2 ) 1 2 , we see that for M sufficiently large Hence, for all which yields the claim (96) of Step 1.
Step 2: quantitative backward uniqueness. The goal of this step and Step 3 below is to prove the following claim: for all 8 C M 749 (−t 0 ) < T 2 ≤ 1 and M sufficiently large. Here, R 2 , R 2 and C(100) are as in (104)-(106). This is the key estimate for Step 4 below and the proof of Proposition 2.
We apply here the results of Section 6 for the quantitative existence of an annulus of regularity. Although the parameter µ in Section 6 is any positive real number, here we need to take µ sufficiently large in order to have a large enough annulus of quantitative regularity, and hence a large r + below in the application of the first Carleman inequality Proposition 34. To fix the ideas, we take µ = 100. 26 Let T 1 and T 2 such that for a universal constant K ≥ 1 to be chosen sufficiently large below. In particular it is chosen in Step 3 such that (123) holds, which makes it possible to absorb the upper bound (122) of X 3 in the left hand side of (120 and a good cylindrical annulus (106) such that for j = 0, 1, We apply now the quantitative backward uniqueness, Proposition 34 to the function w : 26 More specifically, we see that µ is chosen so that 10µ > 350 in order to obtain (111) from (109) and (110). 27 The reason for this is to ensure we can apply Step 1 to get a lower bound (110) for Z2.
An important remark is that although we have a large cylindrical annulus of quantitative regularity A 2 , we apply the Carleman estimate on a much smaller annulus, namely (108) Choosing M sufficiently large such that 2C j C(100)M −300 ≤ 1 and 2 3 2C 2 C(100)M −300 ≤ 1, we see that the bounds (107) imply that the differential inequality (309) is satisfied with S = S 2 := T 2 M 201 and C Carl = M 201 . Take on condition that M is sufficiently large: one needs c(100)M 1000 > 1280. Note also that Thanks to the separation condition (103) and to the fact that for M large enough (104) implies we can apply the concentration result of Step 1, taking there T 1 = T 2 4M 201 = S 2 4 and R = 20r − . By (96) we have that Therefore, one of the following two lower bounds holds where we used the upper bound (105) for (112). The bound (112) can be used directly in Step 4 below. On the contrary, if (111) holds more work needs to be done to transfer the lower bound on the enstrophy at time 0. This is the objective of Step 3 below.
Step 3: a final application of quantitative unique continuation. Assume that the bound (111) holds. We will apply the pigeonhole principle three times successively in order to end up in a situation where we can rely on the quantitative unique continuation to get a lower bound at time 0. We first remark that this with the definition (108) of the annulus A 2 implies the following lower bound By the pigeonhole principle, there exists Using the bounds (107), we have that By the pigeonhole principle, there exists We finally cover the annulus balls of radius (−t 3 ) 1 2 , and apply the pigeonhole principle a third time to find that there exists We apply now the second Carleman inequality, Proposition 35, to the function w : Let S 3 := −20000t 3 . We take 28 (116) Notice that due to (104)- (105) and (113), we have that so that (313) is satisfied. Furthermore, from (114) we have (114), we see that for M large enough S 3 ≤ T 2 32 , hence the bounds (107) imply that the differential inequality (309) is satisfied on B 0 (r) × [0, S] with S = S 3 , r = r 3 and C Carl = 1. Therefore, by (314) we have dxdt.
We choose K sufficiently large so that where C univ ∈ (0, ∞) is the universal constant appearing in the last inequality of (122). Therefore, the term in the right hand side of (122) is negligible with respect to the lower bound (121) of Z 3 . Combining now (120) with the lower bound (121), we obtain 29 Hence, |ω(x, 0)| 2 dx.
Using (104), (119) and the upper bound it follows that Step 4, conclusion: summing the scales and lower bound for the global L 3 norm. The key estimate is (102). From (104)-(105), we see that the volume of B 0 3 4 C(100)M 1000 R 2 \ B 0 2R 2 is less than or equal to T  29 Here one notices a key advantage of taking r3 to depend linearly on −t3 as in (116). Otherwise the trivial bound (124) where we used the lower bound (114) on −t3, would lead one more exponential in the final estimate. Taking r3 as in (116) is Tao's idea; see footnote 28. and that ω i (x, 0) has constant sign in B r 4 (x 4 ). This along with Hölder's inequality yields that for a fixed positive ϕ ∈ C ∞ c (B 0 (1)). Hence, using (104)-(105) we get Next we divide into two cases.
In this case, we use (126) with T 2 = 1 to immediately get (for M greater than a sufficiently large universal constant) In this case we sum (126) on the |u(x, 0)| 3 dx , we see we have the contrapositive of (93). In particular, almost identical arguments to those utilized in the proof of Lemma 4, except using the bound (174) instead of (175), give Step 5, conclusion: summing of scales under additional assumption (52).
In this case, we use the additional assumption (52) to immediately get First notice that in this case In this case we sum (126) on the k + 1 ≥ 2 scales T 2 , Using (128)-(129) we obtain This gives which was also obtained in Case 1 and hence applies in all cases. Defining we see we have the contrapositive of (93). In particular, almost identical arguments to those utilized in the proof of Lemma 4, except using the bound (174) instead of (175), give This concludes the proof of Proposition 2. In the course of the proof we will need to take M larger, always larger than universal constants. Let u : 0)) finite-energy solution to the Navier-Stokes equations (1) Selecting any "well-separated" subsequence (still denoted t (k) ) such that 30 . 30 This separation condition is stronger than that of Lemma 6. This stronger condition is needed to sum disjoint annuli in Step 4 below.
For this well-separated subsequence, assume that there exists j + 1 such that the vorticity concentrates at time t (j+1) in the following sense Lemma 6 then implies imply that the vorticity concentrates in the following sense for any Step 1: quantitative unique continuation. The purpose of this step is to prove the following estimate: for all T 1 , s 0 and R such that and for j = 0, 1, 2, and r 2 1 ≥ 4000T 1 . We apply the second Carleman inequality, Proposition 35 (quantitative unique continuation), on the cylinder Notice that the quantitative regularity (138) and the vorticity equation (37) implies that on C 1 so that (312) is satisfied with S = S 1 := T 1 and C Carl = 16 3 . Let For M sufficiently large we have 0 <š 1 ≤ŝ 1 ≤ T 1 10000 . Hence by (314) we have We first use the concentration (133) for times

8(M ) 201
to bound Z 1 from below. By (139), we have Second, we bound from above X 1 . We rely on the quantitative regularity (138) to obtain Hence, Third, for Y 1 we decompose and estimate as follows where we used the quantitative regularity (138). Hence, |ω(x, s )| 2 dx Gathering these bounds and combining with (140) yields Hence, for all s ∈ I 1 = [t 1 − which yields the claim (135) of Step 1.
Step 2: quantitative backward uniqueness. The goal of this step and Step 3 below is to prove the following claim: Here, R 2 and R 2 are as in (143)-(144). This is the key estimate for Step 4 below and the proof of Proposition 3.
We apply here the results of Section 6 for the quantitative existence of an annulus of regularity. Although the parameter µ in Section 6 is any positive real number, here we need to take µ sufficiently large in order to have a large enough annulus of quantitative regularity, and hence a large r + below in the application of the first Carleman inequality, Proposition 34. To fix the ideas, we take µ = 120. Let  and a good cylindrical annulus (145) such that for j = 0, 1, We apply now the quantitative backward uniqueness, Proposition 34 to the function w : An important remark is that although we have a large cylindrical annulus of quantitative regularity A 2 , we apply the Carleman estimate on a much smaller annulus, namely (147) The reason for this is to ensure we can apply Step 1 to get a lower bound (149) for Z 2 .
Choosing M sufficiently large such that 2C j (M ) −360 ≤ 1 and 2 3 2C 2 (M ) −360 ≤ 1, we see that the bounds (146) imply that the differential inequality (309) is satisfied with S = S 2 := T 2 (M ) 201 and C Carl = (M ) 201 . Take on condition that M is sufficiently large: one needs (M ) 1200 > 1280. By (311), we get Therefore, one of the following two lower bounds holds where we used the upper bound (144) for (151). The bound (151) can be used directly in Step 4 below. On the contrary, if (150) holds more work needs to be done to transfer the lower bound on the enstrophy at time 0. This is the objective of Step 3 below.
Step 3: a final application of quantitative unique continuation. Assume that the bound (150) holds. We will apply the pigeonhole principle three times successively in order to end up in a situation where we can rely on the quantitative unique continuation to get a lower bound at time 0. We first remark that this with the definition (147) of the annulus A 2 implies the following lower bound By the pigeonhole principle, there exists Using the bounds (146), we have that By the pigeonhole principle, there exists We finally cover the annulus balls of radius (−t 3 ) 1 2 , and apply the pigeonhole principle a third time to find that there exists We apply now the second Carleman inequality, Proposition 35, to the function w : w(x, t) = ω(x + x 3 , −t).
Step 4, conclusion: summing the scales and lower bound for the global L 3 norm.
The key estimate is (141). From (143)-(144), we see that the volume of the annulus is less than or equal to T and that ω i (x, 0) has constant sign in B r 4 (x 4 ). This along with Hölder's inequality yields that for a fixed non-negative ϕ ∈ C ∞ c (B 0 (1)). Recalling (142)-(144) we conclude that, for all k ∈ {1, . . . j}. Note that (131) implies that for distinct k the spatial annuli in (164) are disjoint. Summing (164) over such k we obtain that This gives This concludes the proof of Proposition 3.

4.1.
Effective regularity criteria based on the local smallness of the L 3,∞ at blow-up time.
Proposition 7. For all M ∈ [1, ∞) sufficiently large the following result holds true. Consider a suitable finite-energy solutions (u, p) to the Navier-Stokes equations on R 3 × [−1, 0] that satisfies the following Type I bound  [29] for example) we have that any suitable finite-energy solution with Type I bound is a mild solution on x,t (R 3 × (−1, 0)).

4.2.
Estimate for the number of singular points in a Type I scenario. The technology developed in the present paper also enables us to give an effective bound for the number singularities in a Type I scenario. The following proposition and its corollary are effective versions of the results by Choe, Wolf and Yang [12] and Seregin [37]. For all suitable finite-energy solutions 32 (u, p) to the Navier-Stokes equations on R 3 × [−1, 0] that satisfy the following Type I bound Let x 0 ∈ R 3 . Assume that there exists r ∈ (0, exp(M 1021 )), Then This result is a variant of Theorem 1 in [12] and Proposition 1.3 in [37]. Our contribution is to provide the explicit formula (168) for ε(M ) in terms of M .
Corollary 9. Let T * ∈ (0, ∞) and M ∈ [1, ∞) be sufficiently large. Assume that (u, p) is a suitable finite-energy solution to the Navier-Stokes equations on R 3 × [0, T * ] that satisfies the following Type I bound Then u has at most exp(exp(M 1024 )) blow-up points at time T * .
Proof of Corollary 9. We follow here the argument of [37]. Without loss of generality we can assume that u is defined on [−1, 0] rather than [0, T * ]. Let σ denote the set of all singular points at time 0. We take a finite collection of p points There exists r ∈ (0, exp(M 1021 )) such that B x i (r) ∩ B x j (r) = ∅ for all i = j. Then, Proposition 8 implies that This yields the result. 32 For a definition of suitable finite-energy solutions we refer to Section 1.4 'Notations'.
Proof of Proposition 8. Without loss of generality we assume that x 0 = 0. As in the proof of Proposition 7, we assume for contradiction that (0, 0) is a singular point. Using verbatim reasoning as in the proof of Proposition 7, we see that the outcome of Step 1-3 in Section 3.2 holds, in particular estimate (102), which holds for all 0 < T 2 ≤ 1.
Arguing as in Step 4, and using the same notation, we get that there exists such that for r 4 := T |u(x, 0)| 3 .
Hence, there exists x 5 ∈ B x 4 (r 4 ) such that This holds for r = T for K chosen such that (162) holds. Reasoning as in Step 4 of Section 3.3, we then obtain This concludes the proof.

MAIN TOOL 1: LOCAL-IN-SPACE SHORT-TIME SMOOTHING
The role of the next result is central in our paper.  1 4 ] such that the following holds. Consider an initial data u 0 satisfying the global control and, in addition, u 0 ∈ L 6 (B 0 (2)) with u 0 L 6 (B 0 (2)) ≤ N.
Then, for any global-in-time added 'global-in-time' local energy solution 33 (u, p) to (1) with initial data u 0 we have the estimate Remark 11. As a conclusion to the hypothesis in the above Theorem, one can also obtain general version of (175). Specifically, for a local energy solution with β ∈ (0, S * ), we get 33 We recall that the definition of a 'local energy solution' is given in footnote 8. We will require this more general estimate. The computations producing it as identical to those used to show Theorem 3 and hence are omitted.  (180)) such that the following holds. Suppose (u, p) is a 'smooth solution with sufficient decay' 34 on R 3 × [0, T ] for any T ∈ (0, T ) and satisfies Then we conclude that Proof. We define S * * ∈ (0, 1 4 ] in the following way: . We then apply Theorem 3 to U and then rescale according to (181). This gives (179) as desired.
Theorem 3 is proven in Section 5.2 below. It relies on an ε-regularity result for suitable weak solutions 35 to the perturbed Navier-Stokes equations around a subcritical drift a ∈ L m (Q (0,0) (1)), m > 5. We recover the result of Jia and Šverák [21, Theorem 2.2] by a Caffarelli, Kohn and Nirenberg scheme [9] already used in [6] for critical drifts. We also point out here that local-in-space short-time regularity estimates near locally critical initial data were recently proved in [22] using compactness arguments. Contrary to the critical case, here we can prove boundedness directly.
One takes advantage of the subcriticality of the drift a in the following way: This plays a key role in the estimate of I 4 and I 5 in Step 3, J 2 and J 4 in Step 4, using the same notations as in the proof of [6, Theorem 3]. The restriction κ < 2 comes from handling J 5 and J 6 , while the restriction κ < 3 − 15 m comes from bounding J 2 and J 4 .
We assume in addition that u 0 ∈ L 6 (B 0 (2)). Moreover, Let u be any local energy solution to (1) with such a data u 0 . The goal is to prove the local-in-space short-time smoothing for u stated in Theorem 3.
Step 1: decomposition of the initial data.
Step 2: control of the local energy of the perturbation. We use the decomposition given by Lemma 14 for u 0 as above. Let a be the mild solution given by Proposition 31 associated to the data u 0,a ∈ L 6 (R 3 ). The mild solution a exists at least on the time interval (0, S a mild ), where S a mild : where q a is the pressure associated to a. Moreover, since u is a local energy solution with the initial data u 0 , Proposition 33 implies the following control of the local energy where S u locen (N ) := k 1 min(M −4 , 1). As a consequence, the perturbation v = u − a is a local energy solution to (184) where K 1 ∈ [1, ∞) is a universal constant and Moreover, we have the following pressure estimate with a universal constant K 1 ∈ [1, ∞). This bound follows from (195), (305) and (308).
Step 3: smallness of the local energy in short time. Let φ ∈ C ∞ c (R 3 ) be a cut-off function such that where K 3 ∈ [1, ∞). We estimate the local energy Let t ∈ (0, S v ). Let us estimate each term in the right hand side. For that purpose, we rely on the bounds (197) for the local energy and (199) for the pressure. For the terms involving only v, we have using that |B 0 ( 3 2 )| = 3 3 , and 10 .
For the terms involving a and v we use (301) in Proposition 31, more precisely the bound a L 10 (R 3 ×(0,S a mild )) ≤ K 0 N . This in turn implies the controls 10 .
Notice that K * can be taken as where K 0 is defined in Proposition 31, K 1 in (197) and (199), and K 3 is the constant in (200).
Step 4: boundedness of the perturbation. Let ε ∈ (0, min(ε * , 2 −9 )). Our objective is now to apply the ε-regularity result Corollary 13 in order to get the boundedness of the perturbation. As in [21] and [6], we extend v by 0 in the past. The extension v is a suitable weak solution on B 0 (1) × (−∞, S v ) to the Navier-Stokes equations (184) with a drift a defined to be the zero extension of a to R 3 × (−∞, 0). The bound on the local energy (201) is crucial here, as is emphasized in [21]. Notice that the extended a is not a mild solution to the Navier-Stokes system (1) on R 3 × (−∞, S a mild ) but in Corollary 13 this fact is not required. We have the bound a L 10 (R 3 ×(−∞,S a mild )) ≤ K 0 N. By the control (197) on the local energy and (199) on the pressure, we have t t−1B 0 (1) For the rest of the proof we take this definition of S * . It follows from (188) that Combining this estimate with (301) enables to obtain for all β ∈ (0, S * ) which implies estimate (174) as a particular case.
Step 5: estimates of the gradient of the perturbation. In this step, we take β = 3S * 4 . Our goal is to prove the following claim Moreover, the local energy estimate (196) implies that ∇U L 2 (Q (0,0) (1)) = r − 1 2 ∇u L 2 (Q (0,S * ) (r)) ≤ 2 For the pressure, we decompose p − C 0 (t) according to (305). We have according to the estimates in Proposition 33 on the one hand, and using (306) and Calderón-Zygmund on the other hand. Hence, Notice that these are rough bounds, but they are enough for our purposes. Therefore, Using the local maximal regularity [39,Proposition 2.4] leads to where we used the bounds (210) and (211). A simple local energy estimate for ∇U then leads to the bound 22 . Scaling back to the original variables leads to (209) and concludes the proof. 6. MAIN TOOL 2: QUANTITATIVE ANNULI OF REGULARITY In this section we prove that a solution u, satisfying the hypothesis of Propositions 2-3, enjoys good quantitative bounds in certain spatial annuli. This was crucially used in the aforementioned propositions in two places. Namely for applying the Carleman inequalities (Propositions 34-35), as well as in 'Step 4' for transferring the lower bound of the vorticity to a lower bound on the localized L 3 norm.
In the context of classical solutions to the Navier-Stokes equations in L ∞ t L 3 x (R 3 × (t 0 − T, t 0 )), a related version was proven by Tao in [45] using a delicate analysis of local enstrophies from [44]. Our proof is somewhat different and elementary (though we use the 'pidgeonhole principle' as in [45]), instead we utilize known ε-regularity criteria.
Bearing in mind (268), the energy estimates (266), (272) and Calderón-Zygmund estimates for the pressure, the following Lemma is obtained as an immediate corollary to the above Proposition. We also use the known fact that mild solutions to the Navier-Stokes equations in L 4 x,t (R 3 × (0, T )) are suitable weak solutions on R 3 × (0, T ), which can be seen by using a mollification argument along with Calderón-Zygmund estimates for the pressure. and universal constantsC j ∈ (0, ∞) for j = 0, 1, 2 such that for j = 0, 1 A simple rescaling gives the following corollary, which is directly used in the proof of Proposition 2.
there exists R (u, p, M, µ, R) with (235) and universal constantsC j for j = 0, 1, 2 such that for j = 0, 1 Bearing in mind (295), the energy estimate in footnote 11 (see also Lemma 6) and Calderón-Zygmund estimates for the pressure, the following Lemma is obtained as an immediate corollary to Proposition 19. and universal constantsC j for j = 0, 1, 2 such that for j = 0, 1  and universal constantsC j for j = 0, 1, 2 such that for j = 0, 1 Remark 24. According to Remark 17, in Proposition 19-Corollary 23 we have that w, ∂ t w, ∇w and ∇ 2 w are continuous in space and time on the annuli considered. This remark is needed to apply the first Carleman inequality, Proposition 34, in Section 3 and 4.

MAIN TOOL 3: QUANTITATIVE EPOCHS OF REGULARITY
In this section, we prove that a solution satisfying the hypothesis of Propositions 2-3 enjoys good quantitative estimates in certain time intervals. In the literature, these are commonly referred to as 'epochs of regularity'. Such a property is crucially used in 'Step 1' of the above propositions, when applying a quantitative Carleman inequality based on unique continuation (Proposition 35).
To show the existence of such epochs of regularity, we follow Leray's approach in [26]. In particular, we utilize arguments involving existence of mild solutions for subcritical data and weak-strong uniqueness. We provide full details for the reader's convenience.

Suppose that
(252) f ∈ L p 1 ,q 1 (R d ) and g ∈ L p 2 ,q 2 (R d ). Then We will also use an inequality that we will refer to as 'Hunt's inequality'. The statement below and its proof can be found in Hunt's paper [19] (Theorem 4.5, p.271 of [19]).

|I|.
Proof. The first part of the proof closely follows arguments in Tao's paper [45]. The only difference in the first part of the proof is that we exploit the above facts regarding Lorentz spaces. For completeness, we give full details.
As observed by Tao in [45], (256)-(259) are invariant with respect to the Navier-Stokes scaling and time translation. So we can assume without loss of generality Step 1: a priori energy estimates. Clearly we have from the standing assumptions that It is known that e t∆ P∇· has an associated convolution kernel K. Furthermore, from Solonnikov's paper [42], this satisfies the estimate ds ≤ M 2 (t + 1)  .
Here, C univ is a universal constant.
Using that the pressure is given by a Riesz transform acting on u⊗u, we can apply Calderón-Zygmund to get that the pressure p associated to u satisfies (280) p L ∞ t L 3 x (R 3 ×(τ (0),τ (1))) ≤ CM 6 .
Moreover, similar arguments as those used in Proposition 2.2 of [38] yield that for t ∈ (0,T ), w(·, t) 2 |e t∆ u(·, 0)| 4 dxdt + C t 0 w(·, t ) 2 L 2 x e t ∆ u(·, 0) 5 L 5 x dt . 40 The timeT is the image of 0 by the scalings and translations. Its precise value does not matter at all, since the proof is carried out on the time interval [0,t + 1 2 ].
Note that the energy inequality for w, which is used to produce this estimate, can be justified rigorously using (298) and similar arguments as those used in Proposition 14.3 in [24]. Using (294)-(298) and Gronwall's lemma, we infer that (299) sup Here we used (289). Now let Σ ⊂ [0,T ] be such that (43) is satisfied for all t ∈ [t ,T ] and t ∈ Σ. Since u is a suitable finite-energy solution we have that |Σ| =T . Furthermore, Σ can be chosen without loss of generality such that R 3 |∇w(x, t )| 2 dx < ∞ for all t ∈ Σ.
Using (299), the Sobolev embedding theorem, the pidgeonhole principle and (297), we see that there exists t 1 ∈ [t,t + 1 4 ] ∩ Σ such that (300) u(·, t 1 ) 2 L 6 ≤ Cα 105 . Making use of the fact that u satisfies the energy inequality starting from t 1 and (300), we can utilize similar arguments to those used in Lemma 27 replacing M by α 18 .

APPENDIX A. AUXILIARY RESULTS
We first state the existence result of mild solutions with subcritical data.