Partial regularity for Navier-Stokes and liquid crystals inequalities without maximum principle

In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the"Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the"director field"$d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$ which they use to establish partial regularity (specifically, $\mathcal{P}^{1}(\mathcal{S})=0$) of solutions. For the analogous"inequality"one loses this maximum principle, but here we nonetheless establish certain partial regularity results (namely $\mathcal{P}^{\frac 92 + \delta}(\mathcal{S})=0$, so that in particular the putative singular set $\mathcal{S}$ has space-time Lebesgue measure zero). Under an additional assumption on $d$ for any fixed value of a certain parameter $\sigma \in (5,6)$ (which for $\sigma =6$ reduces precisely to the boundedness of $d$ used by Lin and Liu), we obtain the same partial regularity ($\mathcal{P}^{1}(\mathcal{S})=0$) as do Lin and Liu. In particular, we recover the partial regularity result ($\mathcal{P}^{1}(\mathcal{S})=0$) of Caffarelli-Kohn-Nirenberg (1982) for"suitable weak solutions"of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for solutions of the weaker"inequality"as well.


Introduction
In [LL95] and [LL96], Fang-Hua Lin and Chun Liu consider the following system, which reduces to the classical Navier-Stokes system in the case d ≡ 0 (here we have set various parameters equal to one for simplicity): (1.1) with f = ∇F for a scalar field F given by F (x) := (|x| 2 − 1) 2 , so that f (x) = 4(|x| 2 − 1)x (and in particular f (0) = 0). We take the spatial dimension to be three, so that for some Ω ⊆ R 3 and T > 0, we are considering maps of the form u, d : Ω × (0, T ) → R 3 , p : Ω × (0, T ) → R , and here F : R 3 → R , f : R 3 → R 3 are fixed as above. As usual, u represents the velocity vector field of a fluid, p is the scalar pressure in the fluid, and, as in nematic liquid crystals models, d corresponds roughly 1 to the "director field" representing the local orientation of rod-like molecules, with u also giving the velocities of the centers of mass of those anisotropic molecules.
In (1.1), for vector fields v and w, the matrix fields v ⊗ w and ∇v ⊙ ∇w are defined to be the ones with entries (v ⊗ w) ij = v i w j and (∇v ⊙ ∇w) ij = v ,i · w ,j := ∂v k ∂x i ∂w k ∂x j (summing over the repeated index k as per the Einstein convention), and for a matrix field J = (J ij ), we define 2 the vector field ∇ T · J by (∇ T · J) i := J ij,j := ∂J ij ∂x j (summing again over j). We think formally of ∇ (as well as any vector field) as a column vector and ∇ T as a row vector, so that each entry of (the column vector) ∇ T · J is the divergence of the corresponding row of J. In what follows, for a vector field v we similarly denote by ∇ T v the matrix field with i-th row given by ∇ T v i := (∇v i ) T , i.e., so that for smooth vector fields v and w we always have For a scalar field φ we set ∇ 2 φ := ∇ T (∇φ), and for matrix fields J = (J ij ) and K = (K ij ), we let J : K := J ij K ij (summing over repeated indices) denote the (real) Frobenius inner product of the matrices (J : K = tr(J T K)). We set |J| := √ J : J and |v| := √ v · v, and to minimize cumbersome notation will often abbreviate by writing ∇v := ∇ T v for a vector field v where the precise structure of the matrix field ∇ T v is not crucial; for example, |∇v| := |∇ T v|.
We note that by formally taking the divergence ∇· of the first line in (1.1) we obtain the usual "pressure equation" − ∆p = ∇ · (∇ T · [u ⊗ u + ∇d ⊙ ∇d]) . (1.3) As in the Navier-Stokes (d ≡ 0) setting, one may formally deduce (see Section 2 for more details) from (1.1) the following global and local energy inequalities which one may expect "sufficiently nice" solutions of (1.1) to satisfy: 3 In principle, for d to only represent a "direction" one should have |d| ≡ 1. As proposed in [LL95], F(d) is used to model a Ginzburg-Landau type of relaxation of the pointwise constraint |d| ≡ 1. For further discussions on the modeling assumptions leading to systems such as the one above, see e.g. [LW14] or the appendix of [LL95] and the references mentioned therein.
2 Many authors simply write ∇ · J, which is perhaps more standard. 3 For sufficiently regular solutions one can show that equality holds.
In [LL96], the authors establish a partial regularity result for weak solutions to (1.1) belonging to the natural energy spaces which moreover satisfy the local energy inequality (1.5). The result is of the same type as known partial regularity results for "suitable weak solutions" to the Navier-Stokes equations. The program for such partial regularity results for Navier-Stokes was initiated in a series of papers by V. Scheffer in the 1970s and 1980s (see, e.g., [Sch77,Sch80] and other works mentioned in [CKN82]), and subsequently improved by various authors (e.g. [CKN82,Lin98,LS99,Vas07]), perhaps most notably by L. Caffarelli, R. Kohn and L. Nirenberg in [CKN82]. They show (as do [LL96]) that the one-dimensional parabolic Hausdorff measure of the (potentially empty) singular set S is zero (P 1 (S) = 0, see Definition 1 below), implying that singularities (if they exist) cannot for example form any smooth one-parameter curve in space-time. The method of proof in [LL96] largely follows the method of [CKN82].
Of course the general system (1.1) is (when d = 0) substantially more complex than the Navier-Stokes system, and one therefore could not expect a stronger result than the type in [CKN82]. In fact, it is surprising that one even obtains the same type of result (P 1 (S) = 0) as in [CKN82]. The explanation for this seems to be that although (1.1) is more complex than Navier-Stokes in view of the additional d components, one can derive an a priori maximum principle for d because of the third equation in (1.1) which substantially offsets this complexity from the viewpoint of regularity. Therefore, under suitable boundary and initial conditions on d, one may assume that d is in fact bounded, a fact which is significantly exploited in [LL96]. More recently, the authors of the preprint [DHW19] establish the same type of result for a related but more complex "Q-tensor" system; however there, as well, one may obtain a maximum principle which is of crucial importance for proving partial regularity. One is therefore led to the following natural question, which we will address below: Can one deduce any partial regularity for systems similar in structure to (1.1) but which lack any maximum principle?
In the Navier-Stokes setting, it was asserted by Scheffer in [Sch85] that in fact the proof of the partial regularity result in [CKN82] does not require the full set of equations in (1.1). He mentions that the key ingredients are membership of the global energy spaces, the local energy inequality (1.5), the divergence-free condition ∇ · u = 0 and the pressure equation (1.3) (with d ≡ 0 throughout). Scheffer called vector fields satisfying these four requirements solutions to the "Navier-Stokes inequal-ity", equivalent to solutions to the Navier-Stokes equations with a forcing f which satisfies f · u ≤ 0 everywhere. In contrast, the results in [LL96] do very strongly use the third equation in (1.1) in that it implies a maximum principle for d.
In this paper, we explore what happens if one considers the analog of Scheffer's "Navier-Stokes inequality" for the system (1.1) when d = 0. That is, we consider triples (u, d, p) with global regularities implied (at least when Ω is bounded and under suitable assumptions on the initial data) by (1.4) which satisfy (1.3) and ∇ · u = 0 weakly as well as (a formal consequence of) (1.5), but are not necessarily weak solutions of the first and third equations (i.e., the two vector equations) in (1.1). In particular, we will not assume that d ∈ L ∞ (Ω × (0, T )), which would have been reasonable in view of the third equation in (1.1). We see that without further assumptions, the result is substantially weaker than the P 1 (S) = 0 result for Navier-Stokes: following the methods of [LL96,CKN82] we obtain (see Theorem 1 below) P 9 2 +δ (S) = 0 for any δ > 0. This reinforces our intuition that the situation here is substantially more complex than that of Navier-Stokes. On the other hand, we show that under a suitable uniform local decay condition on |d| σ (|u| 3 + |∇d| 3 ) (1− σ 6 ) with σ ∈ (5, 6) (see (1.14) below, which in particular holds when d ≡ 0 as in [CKN82]), one in fact obtains P 1 (S) = 0 as in [LL96] and [CKN82]. In particular, we verify the above-mentioned assertion made by Scheffer in [Sch85] regarding partial regularity for Navier-Stokes inequalities.
Our key observation which allows us to work without any maximum principle is that, in view of the global energy (1.4) and the particular forms of F and f , it is reasonable (see Section 2) to assume (1.9); this implies 5 that d ∈ L ∞ (0, T ; L 6 (Ω)) which is sufficient for our purposes.
As alluded to above, for our purposes we actually do not require all of the information which appears in (1.5) above. In view of the fact that (see (2.21) below), a consequence of (1.5) is that with A, B, C ≥ 0 defined (denoting Ω×{t} g := Ω g(·, t) dx) as (1.7) is nearly sufficient, with the appearance of A(t) on the right-hand side (in fact, even with u omitted, which cannot be avoided as "R f (d, φ)" appears on the right-hand side of (1.5) with a minus 6 sign) actually being, for technical reasons, the only 7 troublesome term. 8 We therefore use a Grönwall-type argument to hide this term to the left-hand side of (1.7) so that (if φ| t=0 ≡ 0) (1.8) 5 In fact, one can also show that d ∈ L s loc (0, T ; L ∞ (Ω)) for any s ∈ [2, 4). 6 See Footnote 4. 7 In fact, the appearance of |d| 2 on the right-hand side of (1.6), and hence of (1.7) as well, is handled precisely by the assumption that d ∈ L ∞ (0, T ; L 6 (Ω)), and is the reason for the slightly weaker results compared to the Navier-Stokes setting (i.e., when d ≡ 0). 8 Note that if R f (d, φ) had appeared with a plus sign in (1.5), one could have simply dropped this troublesome term as a non-positive quantity.
9 For a vector field f or matrix field J and scalar function space X, by f ∈ X or J ∈ X we mean that all components or entries of f or J belong to X; by ∇ 2 f ∈ X we mean all second partial derivatives of all components of f belong to X; etc.
Note that in the case d ≡ 0, we regain the classical result of P 1 (S) = 0 for Navier-Stokes as obtained in, for example, [CKN82], and more specifically for the (weaker) Navier-Stokes inequalities mentioned in [Sch85].
We recall that the definition of the outer parabolic Hausdorff measure P k is given as follows (see [CKN82,): Definition 1 (Parabolic Hausdorff measure). For any S ⊂ R 3 × R and k ≥ 0, define and Q r is any parabolic cylinder of radius r > 0, i.e.
for some x ∈ R 3 and t ∈ R. P k is an outer measure, and all Borel sets are P k -measurable.
Remark 1. In the case Ω = R 3 , the condition (1.10) on the pressure follows (locally, at least) from (1.9) and (1.12) if p is taken to be the potential-theoretic solution to (1.12), since (1.9) implies that u, ∇d ∈ L 10 3 (Ω T ) by interpolation (see (2.18)) and Sobolev embeddings, and then (1.12) gives p ∈ L 5 3 (Ω T ) ⊂ L 3 2 loc (Ω T ) by Calderon-Zygmund estimates. For a more general Ω, the existence of such a p can be derived from the motivating equation (1.1) (e.g. by estimates for the Stokes operator), see [LL96] and the references therein. Here, however, we will not refer to (1.1) at all and simply assume p satisfies (1.10) and address the partial regularity of such a hypothetical set of functions satisfying (1.9) -(1.13).
Using such an energy inequality, one would not need to include the |d| 6 term in E 3,6 (see (3.6)) as one would not need to consider the term coming from R f (d, φ) at all in Proposition 2, and (noting that the L ∞ norm is invariant under the re-scaling on d in (3.25)) one could then adjust Lemmas 1 and 2 appropriately to recover the result in [LL96] using the proof of Theorem 1 below.
Finally, we remark that the majority of the arguments in the proofs given below are not new, with many essentially appearing in [LL96] or [CKN82]. However we feel that our presentation is particularly transparent and may be a helpful addition to the literature, and we include all details so that our results are easily verifiable. many insightful discussions, for introducing him to the field of liquid crystals models, and for suggesting a problem which led to this publication. The author would also like to thank Prof. Camillo De Lellis for introducing him to Scheffer's notion of Navier-Stokes inequalities. Finally, the author would like to thank the anonymous referee for insightful comments about a previous draft of this article.

Motivation
We will show in this section that the assumptions in Theorem 1 are at least formally satisfied by smooth solutions to the system (1.1).

Energy identities
As in [LL96], let us assume that we have smooth solutions to (1.1) which vanish or decay sufficiently at ∂Ω (assumed smooth, if non-empty) and at spatial infinity as appropriate so that all boundary terms vanish in the following integrations by parts, and proceed to establish smooth versions of (1.4) and (1.5). First, noting the simple identities at a fixed t one may perform various integrations by parts (keeping in mind that ∇ · u = 0) to see that and, recalling that (2.4) Adding the two gives the Global energy identity for (1.1): in view of the cancelation of the indicated terms in (2.3) and (2.4).
It is not quite straightforward to localize the calculations in (2.3) and (2.4), for example replacing the (global) multiplicative factor (∆d − f (d)) by (∆d − f (d))φ for a smooth and compactly supported φ. Arguing as in [LL96], one can deduce a local energy identity by instead replacing (∆d − f (d)) by only a part of its localized version in divergence-form, namely by ∇ T · (φ∇ T d), at the expense of the appearance of |∆d − f (d)| 2 anywhere in the local energy.
Recalling (2.1) and (2.2) and noting further that one may perform various integrations by parts to deduce (as ∇ · u = 0) that for smooth and compactly-supported φ, upon adding which and noting again the cancelation of the indicated terms we obtain the Local energy identity for (1.1): dx .

Global energy regularity heuristics
Let us first see where the global energy identity (2.5) leads us to expect weak solutions to (1.1) to live (and hence why we assume (1.9) in Theorem 1).
To ease notation, in what follows let's fix Ω ⊂ R 3 , and for T ∈ (0, ∞] let us set Ω T := Ω × (0, T ) and According to (2.5), we expect, so long as (which we would assume as a requirement on the initial data), to construct solutions with u in the usual Navier-Stokes spaces: As for d we expect as well in view of (2.5) that The norms of all quantities in the spaces given in (2.7) and (2.8) are controlled by either M 0 (the one sees that |f (d)| 2 = 16F (d)|d| 2 , and one can easily confirm the following simple estimates: (2.10) Therefore, if we assume that |Ω| < ∞ , (2.14) and hence (2.8) along with (2.10) implies that (2.15) so that (2.8) and (2.15) imply by the Sobolev embedding, from which (2.11) implies that which, along with (2.12) and (2.16), implies that from which, finally, (2.13) and the last inclusion in (2.8) implies that ∆d ∈ L 2 (Ω T ) for any T < ∞ , (2.17) with the explicit estimate (2.13) which can then further be controlled by M 0 via (2.8), (2.10), (2.11) and (2.12).
We therefore see that it is reasonable (in view of the usual elliptic regularity theory) to expect that weak solutions to (1.1) should have the regularities in (1.9) of Theorem 1.
Note further that various interpolations of Lebesgue spaces imply, for example, that for any interval I ⊂ R one has (for example, one may take α = 3 5 so that 2 α = 6 3−2α = 10 3 ). Using this along with the Sobolev embedding we expect (as mentioned in Remark 1) that with the explicit estimate 15

Local energy regularity heuristics
Here, we will justify the well-posedness of the terms appearing in the local energy equality (2.6), based on the expected global regularity discussed in the previous section. In fact, all but the final term in (2.6) (where one can furthermore take the essential supremum over t ∈ (0, T )) can be seen to be well-defined by (2.19) under the assumptions in (1.9) and (1.10).
The R f (d, φ) term of (2.6) requires some further consideration: in view of (2.9) we see that Recalling that where we have to be careful how we handle the appearance of, essentially, |d| 2 in the first term (the second term is integrable in view of (2.8)). We have, for example, that and that d L 6 (ΩT ) < ∞ for any T ∈ (0, ∞) (2.22) by (2.16), and either φ|∇d| 2 , (recall that φ is assumed to have compact support) and, for example, that ∇d L 10/3 (ΩT ) < ∞ for any T ∈ (0, ∞) (2.23) by (2.19). 15 A B means that A ≤ CB for some suitably universal constant C > 0.
In order to prove Lemma 1, we will require the following two technical propositions. In order to state them, let us fix (recalling (3.1)), for a given z 0 = (x 0 , t 0 ) (to be clear by the context), the abbreviated notations ) and, for each k ∈ N, we define the quantities (again, the dependence on z 0 = (x 0 , t 0 ) will be clear by context) by 20 and L k and R k correspond roughly to the left-and right-hand sides of the local energy inequality (3.5).
We now state the technical propositions, whose proofs we will give in Section 4: Proposition 1 (Cf. Lemma 2.7 of [LL96]). There exists a large universal constant C A > 0 such that the following holds: Fix anyz = (x,t) ∈ R 3 × R, suppose u, d and p satisfy (3.2) and (3.4).
The proof of Proposition 2 uses only the local energy inequality (3.5), the divergence-free condition (3.3) on u and elementary estimates. The quantities on either side of (3.11) do not scale (in the sense of (3.25)) the same way (as do those in (3.10)), which is why the energy inequality is necessary.
Let us now prove Lemma 1 using Propositions 1 and 2.
Proof of Lemma 1: Let us fix some q ∈ (5, 6] andC ∈ (0, ∞). We first note that for any φ ≥ 0 as in (3.5) we have 21 (recalling thatρ ≤ 1) It is also easy to see that L n+1 ≤ 8L n for any n ∈ N . (3.13) Hence we may pick C 0 = C 0 (q,C) >> 1 such that for any z 0 ∈ Q 1 2 (z) (and suppressing the dependence on z 0 in what follows) we have (3.14) for C A and C B as in Propositions 1 and 2. Having fixed C 0 (uniformly over z 0 ∈ Q 1/2 (z)), we then chooseǭ q ∈ (0, 1) so small thatǭ Noting first thatǭ q ≤ (ǭ q ) 2/3 , under the assumption E 3,q ≤ǭ q we in particular see from (3.14) that Then, by Proposition 1 with n ∈ {2, 3} we have q which implies due to Proposition 2 with n = 4 and k 0 = 3 that Then in turn, Proposition 1 with n = 4 gives from which Proposition 2 with n = 5 and, again, k 0 = 3 gives The inequality in fact holds for any q ∈ (2, 6]. and continuing we see by induction that Proposition 1 and Proposition 2 (with k 0 = 3 fixed throughout) imply that This, in turn, implies (for example) that (see, e.g., [WZ77,Theorem 7.16]) q for all Lebesgue points z 0 ∈ Q 1 2 (z) of |u| 3 + |∇d| 3 which implies the L ∞ statement, and Lemma 1 is proved.
Lemma 1 will be used to prove the first assertion in Theorem 1 as well as the next lemma, which in turn will be used to prove the second assertion in Theorem 1.
For the proof of Lemma 2, for z 0 = (x 0 , t 0 ) ∈ Ω T and for r > 0 sufficiently small, we define A z0 , B z0 , 3)]) and G z0 using the cylinders Q * r (z 0 ) (whose "centers" z 0 are in the interior, see (3.1)) by (note that G 0,z0 ≡ C z0 ) and, for q ∈ [0, 6), define M q,z0 (r) := 1 2 C z0 (r) + G The statement in Lemma 2 will follow from Lemma 1 along with the following technical "decay estimate" which will be proved in Section 4.
Let's now use Proposition 3 and Lemma 1 to prove Lemma 2.
In particular, by definition, z 0 = (x 0 , t 0 ) is a regular point, i.e. |u| and |∇d| are essentially bounded in a neighborhood of z 0 , so long as (3.24) holds for some sufficiently small r > 0.
In order to prove Theorem 1, we now prove the following general lemma, from which Lemma 1 and Lemma 2 will have various consequences (including Theorem 1 as well as various other historical results, which we point out for the reader's interest). As a motivation, note first that, for r > 0 and z 1 := (x 1 , t 1 ) ∈ R 3 × R, according to the notation in (3.25) a change of variables gives for any q ∈ [1, ∞).

Lemma 3.
Fix any open and bounded Ω ⊂⊂ R 3 , T ∈ (0, ∞), k ≥ 0 and C k > 0, and suppose S ⊆ Ω T := Ω × (0, T ) and that U : Ω T → [0, ∞] is a non-negative Lebesgue-measurable function such that the following property holds in general: then (recall Definition 1) P k (S) < ∞ (and hence the parabolic Hausdorff dimension of S is at most k) with the explicit estimate where µ is the Lebesgue outer measure, and if k < 5, then in fact P k (S) = µ(S) = 0.
Before proving Lemma 3, let's first use it along with Lemma 1 and Lemma 2 to give the
On the other hand, we know slightly more than (3.42). The assumptions on u and d in (1.9) imply (for example, by (2.18) with α = 3 5 , along with Sobolev embedding) that u, ∇d ∈ L 10 3 (Ω T ). Suppose we also knew (as in the case when Ω = R 3 ) that p ∈ L .
All of the above follows from Lemma 1 alone. We will now show that Lemma 2 allows one (under assumption (1.14) for some σ ∈ (5, 6), and even if p / ∈ L 5 3 (Ω T )) to further decrease the dimension of the parabolic Hausdorff measure, with respect to which the singular set has measure zero, from 5 3 to 1. This was essentially the most significant contribution of [CKN82] in the Navier-Stokes setting d ≡ 0.
Let us now give the Proof of Lemma 3. Fix any δ > 0, and any open set V such that (3.43) For each z := (x, t) ∈ S, according to (3.33) we can choose r z ∈ (0, δ) sufficiently small so that Q * rz (z) ⊂ V and 1 r k By a Vitalli covering argument (see [CKN82,Lemma 6.1]), there exists a sequence (z j ) ∞ j=1 ⊆ S such that and such that the set of cylinders {Q * rz j (z j )} j are pair-wise disjoint. We therefore see from (3.44) that which is finite (and uniformly bounded in δ) by (3.34). Note that according to Definition 1 of the parabolic Hausdorff measure P k , (3.46) implies due to (3.46), which establishes (3.35).
Let us now assume that k ≤ 5. Letting µ be the Lebesgue (outer) measure, note that since we have chosen r z < δ for all z ∈ S. If k = 5, (3.48) along with Definition 1 gives the explicit estimate (3.36) on µ(S). If k < 5, since δ > 0 was arbitrary, sending δ → 0 we conclude (by (3.34)) that µ(S) = 0 and hence S is Lebesgue measurable with Lebesgue measure zero. We may therefore take V to be an open set such that µ(V ) is arbitrarily small but so that (3.43) still holds, and deduce that P k (S) = 0 by (3.34) and (3.47).

Proofs of technical propositions
In order to prove Proposition 1 as well as Proposition 3, we will require certain local decompositions of the pressure (cf. [CKN82,(2.15)]) as follows: 4.1 Localization of the pressure be the fundamental solution of −∆ in R 3 so that, in particular, for any fixed x ∈ R 3 , and set G x ψ,1 : Suppose Π ∈ C 2 (Ω; R), v ∈ C 1 (Ω; R 3 ) and K ∈ C 2 (Ω; R 3×3 ).
Regularizing the linear equation (1.12) using a standard spatial mollifier at any t ∈ (0, T ) where (1.12) holds in D ′ (Ω) and where the inclusions in (4.6), (4.7) and (4.8) hold, applying Claim 1 and passing to limits gives the almost-everywhere convergence (after passing to a suitable subsequence) due, in particular, to the boundedness of the linear operator S on L 5 3 (Ω 2 ).
On the other hand, if (4.4) holds, then by (4.10) we have and one can write Since G x ∇ψ ∈ C ∞ 0 for x ∈ Ω 1 , one can again integrate by parts in the final term to obtain for x ∈ Ω 1 in view of (4.12). Moreover, since ψK ∈ C 2 0 and G x ∈ L 1 loc , as usual for convolutions one can change variables to obtain which gives us (4.5) for any x ∈ Ω 1 , where (see, e.g., [GT01, Theorem 9.9]) S is a singular integral operator as claimed. (Note that ∇ 2 G x / ∈ L 1 loc so that one cannot simply integrate by parts twice in this term putting all derivatives on G x , but G x ψK is the Newtonian potential of ψK which can be twice differentiated in various senses depending on the regularity of K.)

Proof of Proposition 1
In what follows, for O ⊆ R 3 and I ⊆ R, we will use the notation · q;O := · L q (O) , · s;I := · L s (I) , · q,s;O×I := · L s (I;L q (O)) = · L q (O) L s (I) and we will abbreviate by writing · q;O×I := · q,q;O×I = · L q (O×I) .
We first note some simple inequalities. Letting B r ⊂ R 3 be a ball of radius r > 0, from the embedding W 1,2 (B 1 ) ֒→ L 6 (B 1 ) applied to functions of the form g r (x) = g(rx) (or suitably shifted, if the ball is not centered as zero), we obtain g r 6;B1 g r 2;B1 + ∇g r 2;B1 = g r 2;B1 + r (∇g) r 2;B1 whereupon, noting by a simple change of variables that g r q;B1 = r − 3 q g q;Br for any q ∈ [1, ∞), we obtain for any ball B r of radius r > 0 and any g that g 6;Br 1 r g 2;Br + ∇g 2;Br (4.13) where the constant is independent of r as well as the center of B r . Next, for any v(x, t), using Hölder to interpolate between L 2 and L 6 we have (4.14) Then for I r ⊂ R with |I r | = r 2 and Q r := B r × I r , Hölder in the t variable gives (the first of which is sometimes called the "multiplicative inequality") with a constant independent of r. From these, noting that |B r | ∼ r 3 , |Q r | ∼ r 5 , it follows easily that, for example, (4.15) Note also that a similar scaling argument applied to Poincaré's inequality gives the estimate g − g Br q;Br r ∇g q;Br ∼ |B r | 1 3 ∇g q;Br (4.16) for any r > 0 and q ∈ [1, ∞], where g O is the average of g in O for any O ⊂ R 3 with |O| < ∞. Note finally that a simple application of Hölder's inequality gives (4.17) Proceeding now with the proof, fix someφ ∈ C ∞ 0 (R 3 ) such that φ ≡ 1 in B r2 (0) = B 1 4 (0) and supp(φ) ⊆ B r1 (0) = B 1 2 (0) . Now fixz = (x,t) ∈ R 3 × R and z 0 = (x 0 , t 0 ) ∈ Q 1 2 (z), define B k , I k and Q k by (3.7) for this z 0 and define φ by φ(x) :=φ(x − x 0 ). So 2 (x). The following estimates will clearly depend only onφ, i.e. constants will be uniform for all z 0 ∈ Q 1 2 (z)).
First, applying (4.15) to v ∈ {u, ∇d} and recalling (3.8) we see that for any n.
Note first that, by the classical Calderon-Zygmund estimates, there is a universal constant C cz > 0 such that, for all n ∈ N, we have p 1,n (t) 3 2 ;B n+1 ≤ C cz χ n φJ(t) 3 2 ;R 3 ≤ C cz φ ∞;R 3 J(t) 3 2 ;B n . (4.22) Next, since the appearance of ∇φ in p 3 exactly cuts off a neighborhood of the singularity of G x (see (4.1)) uniformly for all x ∈ B 1 8 (x 0 ) (as we integrate over |x 0 − y| ≥ 1 4 , hence |x − y| ≥ 1 8 ), we see that p 3,n (·, t) ∈ C ∞ (B 1 8 (x 0 )) for t ∈ I 1 8 (t 0 ) with, in particular, In the term p 2,n , the singularity coming from G x is also isolated due to the appearance of χ n , but it is no longer uniform in n so we must be more careful. As we are integrating over a region which avoids a neighborhood of the singularity at y = x of G x , we can pass the derivatives in S under the integral sign to write and note, in view of (4.1) that we see that for all t ∈ I 1 8 (t 0 ). Now, recalling the notationf for a function f (x, t) and k ∈ N, for any t ∈ I 2 = (t 0 − ( 1 4 ) 2 , t 0 ) and n ≥ 2, we estimate Note further that, setting , (4.26) we have since |I n+1 | = r 2 n+1 and n−1 Integrating over t ∈ I n+1 in (4.25), applying Hölder in the variable t and recalling by (4.19) that It follows now from (4.21) that (where the constant is universal). This along with (3.13) easily implies (3.10) and proves Proposition 1.

Proof of Proposition 3
In this section we prove the technical decay estimate (Proposition 3) used to prove Lemma 2. In all of what follows, recall the definitions in (3.17) and (3.18) of A z0 , B z0 , C z0 , D z0 , E z0 , F z0 , G q,z0 and M q,z0 . We will require the following three claims which essentially appear in [LL96] and which generalize certain lemmas in [CKN82]; however we include full proofs in order to clarify certain details, and to highlight the role of G q,z0 (not utilized in [LL96]) in Claim 4 which is therefore 25 a slightly refined version of what appears in [LL96].
To complete the proof of Proposition 3, we require the following: Claim 4 (Estimate requiring the local energy inequality (cf. Lemma 5.5 in [CKN82])). There exists a constant c 5 > 0 such that for any u, d and p which have the regularities in (1.9) and (1.10) for Ω T := Ω × (0, T ) as in Theorem 1 and such that u satisfies the weak divergence-free property (1.11) and the local energy inequality (1.13) holds for some constantC ∈ (0, ∞), the estimate holds for any q ∈ [2, 6) and any z 0 ∈ R 3+1 and ρ > 0 such that Q * ρ (z 0 ) ⊆ Ω T .
Postponing the proof of the claims, let us use them to prove the proposition.
In all of what follows, we note the simple facts that, for any ρ > 0 and α ∈ (0, 1], and (4.51) Proof of Proposition 3.
To first prove (4.48), we note that (4.58) implies (since r ≤ ρ 2 ) that Multiplying and dividing by (r/ρ) α 2 for any α ∈ R, Cauchy's inequality gives Since we want a positive power of γ = r/ρ in the first term and a negative one on the second (because it contains B which will be small), we want to take α > 0. Choosing α = 1 purely to make the following expression simpler, since p = p 3 + p 2 + p 1 , we see from (4.61), (4.62) and (4.63) that D(r) r ρ · [D + (AB) To prove (4.49), we note that F z0 (r) ≤ F 1 (r) + F 2 (r) + F 3 (r), where we set F j (r) := 1 r 2 Qr |p j ||u| dz .
(4.68) Clearly we have III := Q * ρ |pu · ∇φ| dz ρ −1 (ρ 2 F z0 (ρ)) = ρF z0 (ρ) . (4.69) Using the weak divergence-free condition ∇ · u = 0 in (1.11) to write (see (1.2)) (u · ∇)d = ∇ T · (d ⊗ u) (ρ) 26 Note that it is only the appearance of ∇ 2 d in the estimate of term IV which forces us to include u in the definition of Gq,z 0 . Indeed, switching the roles of u (which appears in Cz 0 along with ∇d) and ∇d (which appears in Gq,z 0 even with u omitted), one could otherwise control term IV in precisely the same way. If u is omitted in Gq,z 0 , one could still obtain the same estimate of IV if one takes q = 6, but this would dramatically weaken the statement of Theorem 1. The remainder of the proof of Theorem 1 does not require (but is not harmed by) the inclusion of u in Gq,z 0 . as long as 2 ≤ q < 6, as in that case we have