Partial regularity for Navier-Stokes and liquid crystals inequalities without maximum principle
aa r X i v : . [ m a t h . A P ] O c t Partial regularity for Navier-Stokes and liquid crystalsinequalities without maximum principle
Gabriel S. Koch
University of SussexBrighton BN1 9QH, UK [email protected]
October 21, 2020
Abstract
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 wellas the local and global energy inequalities and the pressure equation which may be derived formallyfrom the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokessystem itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in themid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokessystem when the “director field” d is taken to be zero. In addition to an extended Navier-Stokessystem, the Lin-Liu model includes a further parabolic system which implies a maximum principlefor d which they use to establish partial regularity of solutions. For the analogous “inequality” oneloses this maximum principle, but here we establish certain partial regularity results nonetheless.Our results recover in particular the partial regularity results of Caffarelli-Kohn-Nirenberg for“suitable weak solutions” of the Navier-Stokes system, and we verify Scheffer’s assertion that thesame hold for solutions of the weaker “inequality” as well. In [LL95] and [LL96], Fang-Hua Lin and Chun Liu consider the following system, which reduces tothe classical Navier-Stokes system in the case d ≡ u t − ∆ u + ∇ T · [ u ⊗ u + ∇ d ⊙ ∇ d ] + ∇ p = 0 ∇ · u = 0 d t − ∆ d + ( u · ∇ ) d + f ( d ) = 0 (1.1)with f = ∇ F for a scalar field F given by F ( x ) := ( | x | − , so that f ( x ) = 4( | x | − x (and in particular f (0) = 0). We take the spatial dimension to be three, so that for some Ω ⊆ R and T >
0, we are considering maps of the form u, d : Ω × (0 , T ) → R , p : Ω × (0 , T ) → R , and here F : R → R , f : R → R u represents the velocity vector field of a fluid, p is the scalar pressurein the fluid, and, as in nematic liquid crystals models, d corresponds roughly to the “director field”representing the local orientation of rod-like molecules, with u also giving the velocities of the centersof 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 theones 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 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 vectorand ∇ T as a row vector, so that each entry of (the column vector) ∇ T · J is the divergence of thecorresponding row of J . In what follows, for a vector field v we similarly denote by ∇ T v the matrixfield with i -th row given by ∇ T v i := ( ∇ v i ) T , i.e.,( ∇ T v ) ij = v i,j := ∂v i ∂x j , so that for smooth vector fields v and w we always have ∇ T · ( v ⊗ w ) = ( ∇ T v ) w + v ( ∇ · w ) = ( w · ∇ ) v + v ( ∇ · w ) . (1.2)For a scalar field φ we set ∇ 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 thematrices ( J : K = tr( J T K )). We set | J | := √ J : J and | v | := √ v · v , and to minimize cumbersomenotation will often abbreviate by writing ∇ v := ∇ T v for a vector field v where the precise structureof 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: ddt Z Ω (cid:20) | u | |∇ d | F ( d ) (cid:21) dx + Z Ω (cid:2) |∇ u | + | ∆ d − f ( d ) | (cid:3) dx ≤ t ∈ (0 , T ), as well as a localized version In principle, for d to only represent a “direction” one should have | d | ≡
1. As proposed in [LL95], F(d) is used tomodel a Ginzburg-Landau type of relaxation of the pointwise constraint | d | ≡
1. For further discussions on the modelingassumptions leading to systems such as the one above, see e.g. [LW14] or the appendix of [LL95] and the referencesmentioned therein. Many authors simply write ∇ · J , which is perhaps more standard. For sufficiently regular solutions one can show that equality holds. Note that in [LL96], the term “ −R f ( d, φ )” in (1.5) actually appears incorrectly as “+ R f ( d, φ )”. See Section 2 formore details. dt Z Ω (cid:20)(cid:18) | u | |∇ d | (cid:19) φ (cid:21) dx + Z Ω (cid:0) |∇ u | + |∇ d | (cid:1) φ dx ≤ Z Ω (cid:20) (cid:18) | u | |∇ d | (cid:19) ( φ t + ∆ φ ) + (cid:18) | u | |∇ d | p (cid:19) u · ∇ φ + u ⊗ ∇ φ : ∇ d ⊙ ∇ d − φ ∇ T [ f ( d )] : ∇ T d | {z } =: R f ( d,φ ) (cid:21) dx (1.5)for t ∈ (0 , T ) and each smooth, compactly supported in Ω and non-negative scalar field φ ≥
0. (ForNavier-Stokes, i.e. when d ≡
0, one may omit all terms involving d , even though 0 = F (0) / ∈ L ( R ).)In [LL95], for smooth and bounded Ω, the global energy inequality (1.4) is used to construct globalweak solutions to (1.1) for initial velocity in L (Ω), along with a similarly appropriate condition onthe initial value of d which allows (1.4) to be integrated over 0 < t < T . This is consistent withthe pioneering result of J. Leray [Ler34] for Navier-Stokes (treated later by many other authors usingvarious methods, but always relying on the natural energy as in [Ler34]).In [LL96], the authors establish a partial regularity result for weak solutions to (1.1) belonging tothe natural energy spaces which moreover satisfy the local energy inequality (1.5). The result is ofthe same type as known partial regularity results for “suitable weak solutions” to the Navier-Stokesequations. The program for such partial regularity results for Navier-Stokes was initiated in a seriesof papers by V. Scheffer in the 1970s and 1980s (see, e.g., [Sch77, Sch80] and other works mentionedin [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 ( S ) = 0, see Definition 1 below), implying that singularities (if they exist) cannot forexample form any smooth one-parameter curve in space-time. The method of proof in [LL96] largelyfollows 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]. Infact, it is surprising that one even obtains the same type of result ( P ( S ) = 0) as in [CKN82]. Theexplanation for this seems to be that although (1.1) is more complex than Navier-Stokes in view ofthe additional d components, one can derive a maximum principle for d because of the third equationin (1.1) which substantially offsets this complexity from the viewpoint of regularity. Therefore, undersuitable boundary and initial conditions on d , one may assume that d is in fact bounded, a fact whichis significantly exploited in [LL96]. More recently, the authors of the preprint [DHW19] establish thesame type of result for a related but more complex “Q-tensor” system; however there, as well, onemay obtain a maximum principle which is of crucial importance for proving partial regularity. One istherefore 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 whichlack any maximum principle?
In the Navier-Stokes setting, it was asserted by Scheffer in [Sch85] that in fact the proof of thepartial regularity result in [CKN82] does not require the full set of equations in (1.1). He mentionsthat 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 ≡ f which satisfies f · u ≤ d . 3n this paper, we explore what happens if one considers the analog of Scheffer’s “Navier-Stokes in-equality” for the system (1.1) when d = 0. That is, we consider triples ( u, d, p ) with global regularitiesimplied (at least when Ω is bounded and under suitable assumptions on the initial data) by (1.4) whichsatisfy (1.3) and ∇ · u = 0 weakly as well as (a formal consequence of) (1.5), but are not necessarilyweak 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 thirdequation in (1.1). We see that without further assumptions, the result is substantially weaker than the P ( S ) = 0 result for Navier-Stokes: following the methods of [LL96, CKN82] we obtain (see Theorem1 below) P ( S ) < ∞ , which does not even rule out a singular set S with positive space-time Lebesguemeasure (though it is shown to be small if the global energy norms are small). This reinforces ourintuition that the situation here is substantially more complex than that of Navier-Stokes. On theother hand, we show that under a suitable local decay condition on | d | (see (1.14) below, which inparticular holds when d ≡ P ( S ) = 0 as in [LL96] and [CKN82].In particular, we verify the above-mentioned assertion made by Scheffer in [Sch85] regarding partialregularity for Navier-Stokes inequalities.Our key observation which allows us to work without any maximum principle is that, in view ofthe global energy (1.4) and the particular forms of F and f , it is reasonable (see Section 2) to assume(1.9); this implies that d ∈ L ∞ (0 , T ; L (Ω)) which is sufficient for our purposes.As alluded to above, for our purposes we actually do not require all of the information which ap-pears in (1.5) above. In view of the fact that |R f ( d, φ ) | = | φ ∇ T [ f ( d )] : ∇ T d | ≤ | d | |∇ d | φ + 8 (cid:18) |∇ d | φ (cid:19) (1.6)(see (2.21) below), a consequence of (1.5) is that A ′ ( t ) + B ( t ) ≤ A ( t ) + C ( t ) for 0 < t < T , (1.7)with A , B , C ≥ R Ω ×{ t } g := R Ω g ( · , t ) dx ) as A ( t ) := Z Ω ×{ t } (cid:18) | u | |∇ d | (cid:19) φ , B ( t ) := Z Ω ×{ t } (cid:0) |∇ u | + |∇ d | (cid:1) φ and C ( t ) := Z Ω ×{ t } (cid:20) (cid:18) | u | |∇ d | (cid:19) | φ t + ∆ φ | + 12 | d | |∇ d | φ (cid:21) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ×{ t } (cid:20)(cid:18) | u | |∇ d | p (cid:19) u · ∇ φ + u ⊗ ∇ φ : ∇ d ⊙ ∇ d (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (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 aminus sign) actually being, for technical reasons, the only troublesome term. We therefore use aGr¨onwall-type argument to hide this term to the left-hand side of (1.7) so that (if φ | t =0 ≡ A ′ ( t ) + B ( t ) ≤ C ( t ) + 8 e T Z t C ( τ ) dτ for 0 < t < T . (1.8) In fact, one can also show that d ∈ L s loc (0 , T ; L ∞ (Ω)) for any s ∈ [2 , See Footnote 4. In fact, the appearance of | d | on the right-hand side of (1.6), and hence of (1.7) as well, is handled precisely by theassumption that d ∈ L ∞ (0 , T ; L (Ω)), and is the reason for the slightly weaker results compared to the Navier-Stokessetting (i.e., when d ≡ Note that if R f ( d, φ ) had appeared with a plus sign in (1.5), one could have simply dropped this troublesome termas a non-positive quantity. C T ∼ T e T + 1), which is sufficient for our purposes. (In fact, for all elements of the proof otherthan Proposition 3, a weaker form as in (3.5) is sufficient.)In order to state our main result, we first recall the definition of the outer parabolic Hausdorff measure P k (see [CKN82, pp.783-784]): Definition 1 (Parabolic Hausdorff measure) . For any
S ⊂ R × R and k ≥ , define P k ( S ) := lim δ ց P kδ ( S ) , where P kδ ( S ) := inf ∞ X j =1 r kj (cid:12)(cid:12)(cid:12)(cid:12) S ⊂ ∞ [ j =1 Q r j , r j < δ ∀ j ∈ N and Q r is any parabolic cylinder of radius r > , i.e. Q r = Q r ( x, t ) := B r ( x ) × ( t − r , t ) ⊂ R × R for some x ∈ R and t ∈ R . P k is an outer measure, and all Borel sets are P k -measurable. Our main result is the following:
Theorem 1.
Fix an open set Ω ⊂ R and T ∈ (0 , ∞ ) , set Ω T := Ω × (0 , T ) and suppose u, d : Ω T → R and p : Ω T → R satisfy the following four assumptions:1. u , d and p belong to the following spaces: u, d, ∇ d ∈ L ∞ (0 , T ; L (Ω)) , ∇ u, ∇ d, ∇ d ∈ L (Ω T ) (1.9) and p ∈ L (Ω T ) ; (1.10) u is weakly divergence-free: ∇ · u = 0 in D ′ (Ω T ) ; (1.11)
3. the following pressure equation holds weakly: − ∆ p = ∇ · [ ∇ T · ( u ⊗ u + ∇ d ⊙ ∇ d )] in D ′ (Ω T ) ; (1.12)
4. for some C T > (depending only on T ), the following local energy inequality holds: R Ω ×{ t } (cid:0) | u | + |∇ d | (cid:1) φ dx + R t R Ω (cid:0) |∇ u | + |∇ d | (cid:1) φ dx dτ ≤ C T R t (cid:8) R Ω ×{ τ } (cid:2)(cid:0) | u | + |∇ d | (cid:1) | φ t + ∆ φ | + | d | |∇ d | φ (cid:3) dx + (cid:12)(cid:12) R Ω ×{ τ } (cid:2)(cid:0) | u | + |∇ d | + p (cid:1) u · ∇ φ + u ⊗ ∇ φ : ∇ d ⊙ ∇ d (cid:3) dx (cid:12)(cid:12) (cid:9) dτ for a.e. t ∈ (0 , T ) and ∀ φ ∈ C ∞ (Ω × (0 , ∞ )) s.t. φ ≥ . (1.13) For a vector field f or matrix field J and scalar function space X , by f ∈ X or J ∈ X we mean that all componentsor entries of f or J belong to X ; by ∇ f ∈ X we mean all second partial derivatives of all components of f belong to X ; etc. Locally integrable functions will always be associated to the standard distribution whose action is integration againsta suitable test function so that, e.g., [ ∇ · u ]( ψ ) = − [ u ]( ∇ ψ ) := − R u · ∇ ψ for ψ ∈ D (Ω T ). Note that u ⊗ u + ∇ d ⊙ ∇ d ∈ L (Ω T ) ⊂ L (Ω T ), see (2.18) - (2.19). For brevity, for ω ⊂ R , we set Z ω ×{ t } g dx := Z ω g ( x, t ) dx . et S ⊂ Ω T be the (potentially empty) set of singular points where | u | and |∇ d | are not essentiallybounded in any neighborhood of each z ∈ S , and let P k be the k -dimensional parabolic Hausdorff outermeasure (see Definition 1). The following are then true:1. P ( S ) < ∞ , with µ ( S ) . P ( S ) . Z Z Ω T (cid:16) | u | + |∇ d | + | p | + | d | (cid:17) dz < ∞ where µ is the Lebesgue outer measure on R .2. There exists a universal constant ǫ > such that if, moreover, sup z ∈ Ω T lim sup r ց r Z Z Q r ( z ) | d | dz ! < ǫ , (1.14) then µ ( S ) = 0 and, moreover, P ( S ) = 0 . Note that in the case d ≡
0, we regain the classical result of P ( S ) = 0 for Navier-Stokes as obtainedin, for example, [CKN82], and more specifically for the (weaker) Navier-Stokes inequalities mentionedin [Sch85]. Remark 1.
In the case
Ω = R , 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) impliesthat u, ∇ d ∈ L (Ω T ) by interpolation (see (2.18)) and Sobolev embeddings, and then (1.12) gives p ∈ L (Ω T ) ⊂ L loc (Ω T ) by Calderon-Zygmund estimates. For a more general Ω , the existence of sucha 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). We note that Theorem 1 does not immediately recover the result of [LL96] as a special example:although the Morrey-type norm of d in our condition (1.14) will be finite under the assumption that d ∈ L ∞ , it will not necessarily be small. Although such finiteness is sufficient for part of the proof(Proposition 3 below), if one replaces (1.14) by the weaker assumption d ∈ L ∞ and still hopes for thesame conclusion, then in Proposition 2 below one would need to adjust the argument in the proof toensure that one need not include the | d | term in E in the “ ǫ -regularity” Lemma 1 below. Heuristi-cally, however, one can argue as follows:If d were bounded, then taking for example D := 24 k d k L ∞ (Ω T ) + 8 < ∞ one would deduce from(1.6) that |R f ( d, φ ) | ≤ D (cid:18) |∇ d | (cid:19) φ . Adjusting the Gr¨onwall-type argument leading to (1.8), one could then deduce from (1.5) that (if A (0) = 0) A ′ ( t ) + B ( t ) ≤ e C ( t ) + De DT Z T e C ( τ ) dτ for 0 < t < T , A . B means that A ≤ CB for some suitably universal constant C >
0, and in general we set z = ( x, t ) ∈ Ω T , dz := dx dt . Recall from Definition 1 that Q r ( x , t ) := B r ( x ) × ( t − r , t ). We assume this is roughly the argument in [LL96], although the details are not explicitly given; see, in particular,[LL96, (2.45)] which appears without the “remainder” term denoted in [LL96] by R ( f, φ ), and here by R f ( d, φ ). e C ( t ) := Z Ω ×{ t } (cid:18) | u | |∇ d | (cid:19) | φ t +∆ φ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ×{ t } (cid:20)(cid:18) | u | |∇ d | p (cid:19) u · ∇ φ + u ⊗ ∇ φ : ∇ d ⊙ ∇ d (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Using such an energy inequality, one would not need to include the | d | term in E (see (3.6)) as onewould not need to consider the term coming from R f ( d, φ ) at all in Proposition 2, and (noting thatthe L ∞ norm is invariant under the re-scaling on d in (3.22)) one could then adjust Lemmas 1 and 2appropriately 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, withmany essentially appearing in [LL96] or [CKN82]. However we feel that our presentation is particu-larly transparent and may be a helpful addition to the literature, and we include all details so thatour results are easily verifiable. Acknowledgment:
The author would like to offer his sincere thanks to Prof. Arghir Zarnescufor many insightful discussions, for introducing him to the field of liquid crystals models, and forsuggesting a problem which led to this publication.
We will show in this section that the assumptions in Theorem 1 are at least formally satisfied bysmooth solutions to the system (1.1).
As in [LL96], let us assume that we have smooth solutions to (1.1) which vanish or decay sufficientlyat ∂ Ω (assumed smooth, if non-empty) and at spatial infinity as appropriate so that all boundaryterms vanish in the following integrations by parts, and proceed to establish smooth versions of (1.4)and (1.5). First, noting the simple identities ∇ T · ( ∇ d ⊙ ∇ d ) = ∇ (cid:18) |∇ d | (cid:19) + ( ∇ T d ) T ∆ d (2.1)and [( ∇ T d ) T ∆ d ] · u = [( ∇ T d ) u ] · ∆ d = [( u · ∇ ) d ] · ∆ d , (2.2)at a fixed t one may perform various integrations by parts (keeping in mind that ∇ · u = 0) to see that0 = Z Ω [ u t − ∆ u + ∇ T · ( u ⊗ u ) + ∇ p + ∇ T · ( ∇ d ⊙ ∇ d )] · u dx = Z Ω (cid:20) ∂∂t (cid:18) | u | (cid:19) + |∇ u | + [( u · ∇ ) d ] · ∆ d | {z }(cid:21) dx (2.3)and, recalling that f = ∇ F so that [ d t + ( u · ∇ ) d ] · f ( d ) = (cid:0) ∂∂t + u · ∇ (cid:1) [ F ( d )], that0 = − Z Ω [ d t + ( u · ∇ ) d − (∆ d − f ( d ))] · (∆ d − f ( d )) dx = − Z Ω (cid:20) − ∂∂t (cid:18) |∇ d | F ( d ) (cid:19) + [( u · ∇ ) d ] · ∆ d | {z } −| ∆ d − f ( d ) | (cid:21) dx . (2.4)Adding the two gives the 7lobal energy identity for (1.1): ddt Z Ω (cid:20) | u | |∇ d | F ( d ) (cid:21) dx + Z Ω (cid:2) |∇ u | + | ∆ d − f ( d ) | (cid:3) dx = 0 (2.5)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 )) byonly a part of its localized version in divergence-form, namely by ∇ T · ( φ ∇ T d ), at the expense of theappearance of | ∆ d − f ( d ) | anywhere in the local energy.Recalling (2.1) and (2.2) and noting further that[( u · ∇ ) d ] · [ ∇ T · ( φ ∇ T d )] = [( u · ∇ ) d ] · [ φ ∆ d ] + [( u · ∇ ) d ] · [( ∇ φ · ∇ ) d ]= [( u · ∇ ) d ] · [ φ ∆ d ] + u ⊗ ∇ φ : ∇ d ⊙ ∇ d and that [∆( ∇ T d )] : ∇ T d = ∆ (cid:18) |∇ d | (cid:19) − |∇ d | , one may perform various integrations by parts to deduce (as ∇ · u = 0) that0 = Z Ω [ u t − ∆ u + ∇ T · ( u ⊗ u ) + ∇ p + ∇ T · ( ∇ d ⊙ ∇ d )] · uφ dx = Z Ω (cid:20) ∂∂t (cid:18) | u | φ (cid:19) + |∇ u | φ − | u | φ t + ∆ φ ) − (cid:18) | u | |∇ d | p (cid:19) u · ∇ φ + [( u · ∇ ) d ] · (∆ d ) φ | {z }(cid:21) dx and 0 = − Z Ω [ d t + ( u · ∇ ) d − (∆ d − f ( d ))] · [ ∇ T · ( φ ∇ T d )] dx = − Z Ω (cid:20) − ∂∂t (cid:18) |∇ d | φ (cid:19) − |∇ d | φ + |∇ d | φ t + ∆ φ ) −∇ T [ f ( d )] : φ ∇ T d + [( u · ∇ ) d ] · (∆ d ) φ | {z } + u ⊗ ∇ φ : ∇ d ⊙ ∇ d (cid:21) dx for smooth and compactly-supported φ , upon adding which and noting again the cancelation of theindicated terms we obtain theLocal energy identity for (1.1): ddt Z Ω (cid:20)(cid:18) | u | |∇ d | (cid:19) φ (cid:21) dx + Z Ω (cid:0) |∇ u | + |∇ d | (cid:1) φ dx = (2.6)= Z Ω (cid:20) (cid:18) | u | |∇ d | (cid:19) ( φ t + ∆ φ ) + (cid:18) | u | |∇ d | p (cid:19) u · ∇ φ + u ⊗ ∇ φ : ∇ d ⊙ ∇ d − φ ∇ T [ f ( d )] : ∇ T d | {z } =: R f ( d,φ ) (cid:21) dx . − ” preceding R f ( d, φ ), and the term“(( u · ∇ ) d ⊙ ∇ d ) · ∇ φ ” which appears in [LL96] has been more accurately written here as u ⊗ ∇ φ : ∇ d ⊙ ∇ d , and that u ⊗ ∇ φ : ∇ d ⊙ ∇ d = [( ∇ d ⊙ ∇ d ) ∇ φ ] · u = [( u · ∇ ) d ] · [( ∇ φ · ∇ ) d ]. 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 , and for T ∈ (0 , ∞ ] let us set Ω T := Ω × (0 , T ) and L rt L qx ( T ) := L r (0 , T ; L q (Ω) . According to (2.5), we expect, so long as M := k u ( · , k L (Ω) + k∇ d ( · , k L (Ω) + k F ( d ( · , k L (Ω) < ∞ , (which we would assume as a requirement on the initial data), to construct solutions with u in theusual Navier-Stokes spaces: u ∈ L ∞ t L x ( ∞ ) and ∇ u ∈ L t L x ( ∞ ) . (2.7)As for d we expect as well in view of (2.5) that ∇ d ∈ L ∞ t L x ( ∞ ) , F ( d ) ∈ L ∞ t L x ( ∞ ) and [∆ d − f ( d )] ∈ L t L x ( ∞ ) . (2.8)The norms of all quantities in the spaces given in (2.7) and (2.8) are controlled by either M (the F ( d ) term) or ( M ) (all other terms), by integrating (2.5) over t ∈ (0 , ∞ ). Recalling that F ( d ) := ( | d | − and f ( d ) := 4( | d | − d , (2.9)one sees that | f ( d ) | = 16 F ( d ) | d | , and one can easily confirm the following simple estimates: k d k L ∞ t L x ( ∞ ) ≤ k F ( d ) k / L ∞ t L x ( ∞ ) + k k L ∞ t L x ( ∞ ) , (2.10) k F ( d ) k / L ∞ t L / x ( ∞ ) ≤ k d k L ∞ t L x ( ∞ ) + k k L ∞ t L x ( ∞ ) , (2.11) k f ( d ) k L ∞ t L x ( ∞ ) ≤ k F ( d ) k L ∞ t L / x ( ∞ ) k d k L ∞ t L x ( ∞ ) (2.12)and k ∆ d k L (Ω T ) ≤ k ∆ d − f ( d ) k L (Ω T ) + T / k f ( d ) k L ∞ t L x ( ∞ ) . (2.13)Therefore, if we assume that | Ω | < ∞ , (2.14)and hence 1 ∈ L ∞ (0 , ∞ ; L (Ω)) ∩ L ∞ (0 , ∞ ; L (Ω)) , (2.8) along with (2.10) implies that d ∈ L ∞ (0 , ∞ ; L (Ω)) ( . ) ⊂ L ∞ (0 , ∞ ; L (Ω)) . (2.15)so that (2.8) and (2.15) imply d ∈ L ∞ (0 , ∞ ; H (Ω)) ֒ → L ∞ (0 , ∞ ; L (Ω)) (2.16)by the Sobolev embedding, from which (2.11) implies that F ( d ) ∈ L ∞ t L / x ( ∞ )9hich, along with (2.12) and (2.16), implies that f ( d ) ∈ L ∞ t L x ( ∞ ) , from which, finally, (2.13) and the last inclusion in (2.8) implies that∆ d ∈ L (Ω T ) for any T < ∞ , (2.17)with the explicit estimate (2.13) which can then further be controlled by M 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 thatweak 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 L ∞ ( I ; L (Ω)) ∩ L ( I ; L (Ω)) ⊂ L α ( I ; L − α (Ω)) for any α ∈ [0 ,
1] (2.18)(for example, one may take α = so that α = − α = ). Using this along with the Sobolevembedding we expect (as mentioned in Remark 1) that( u and ) ∇ d ∈ L α (0 , T ; L − α (Ω)) for any α ∈ [0 , , T < ∞ (2.19)with the explicit estimate k∇ d k α L αt L − αx ( T ) . T k∇ d k α L ∞ t L x ( ∞ ) + k∇ d k α − L ∞ t L x ( ∞ ) k∇ d k L (Ω T ) . Here, we will justify the well-posedness of the terms appearing in the local energy equality (2.6), basedon the expected global regularity discussed in the previous section. In fact, all but the final termin (2.6) (where one can furthermore take the essential supremum over t ∈ (0 , T )) can be seen to bewell-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 ∇ T [ f ( d )] = ∇ T [( | d | − d ] = 2 d ⊗ [ d · ( ∇ T d )] + ( | d | − ∇ T d , (2.20)Recalling that R f ( d, φ ) := φ ∇ T [ f ( d )] : ∇ T d , we therefore have R f ( d, φ ) = φ (cid:18) d ⊗ [ d · ( ∇ T d )] : ∇ T d + | d | |∇ d | (cid:19) − φ |∇ d | (2.21)where we have to be careful how we handle the appearance of, essentially, | d | in the first term (thesecond term is integrable in view of (2.8)). We have, for example, that k φ | d | |∇ d | k L (Ω T ) ≤ k φ k L ∞ (Ω T ) k d k L (Ω T ) k∇ d k L (Ω T ) and that k d k L (Ω T ) < ∞ for any T ∈ (0 , ∞ ) (2.22)by (2.16), and either k φ |∇ d | k L (Ω T ) ≤ k φ k L ∞ (Ω T ) k∇ d k L (Ω T ) or k φ |∇ d | k L (Ω T ) ≤ k φ k L (Ω T ) k∇ d k L (Ω T ) , (recall that φ is assumed to have compact support) and, for example, that k∇ d k L / (Ω T ) < ∞ for any T ∈ (0 , ∞ ) (2.23)by (2.19). 10 Proof of Theorem 1
The first part of Theorem 1 is a consequence of the following “ L ǫ -regularity” Lemma 1, while thesecond part is a consequence of the “ ˙ H ǫ -regularity” Lemma 2 below which is itself a consequenceof Lemma 1. In the following, for a given z = ( x , t ) ∈ R × R and r >
0, as in [CKN82] we willadopt the following the notation for the standard parabolic cylinder Q r ( z ) as well as the followingtime intervals and their “centered” versions (indicated with a star): I r ( t ) := ( t − r , t ) , I ∗ r ( t ) := ( t − r , t + r ) ,Q r ( z ) := B r ( x ) × I r ( t ) and Q ∗ r ( z ) := B r ( x ) × I ∗ r ( t ) . (3.1) Lemma 1 ( L ǫ -regularity, cf. Theorem 2.6 of [LL96] and Proposition 1 of [CKN82]) . There exists ¯ ǫ ∈ (0 , sufficiently small such that for any ¯ z = (¯ x, ¯ t ) ∈ R × R , the following holds:Suppose (see (3.1)) u, d : Q (¯ z ) → R and p : Q (¯ z ) → R with u, d, ∇ d ∈ L ∞ ( I (¯ t ); L ( B (¯ x ))) , ∇ u, ∇ d, ∇ d ∈ L ( Q (¯ z )) and p ∈ L ( Q (¯ z )) (3.2) satisfy ∇ · u = 0 in D ′ ( Q (¯ z )) , (3.3) − ∆ p = ∇ · ( ∇ T · [ u ⊗ u + ∇ d ⊙ ∇ d ]) in D ′ ( Q (¯ z )) (3.4) and, for some constants ¯ C > and ¯ ρ ∈ (0 , , the following local energy inequality holds: R B (¯ x ) ×{ t } (cid:0) | u | + |∇ d | (cid:1) φ dx + R t ¯ t − R B (¯ x ) (cid:0) |∇ u | + |∇ d | (cid:1) φ dx dτ ≤ ¯ C R t ¯ t − (cid:8) R B (¯ x ) ×{ τ } (cid:2)(cid:0) | u | + |∇ d | (cid:1) | φ t + ∆ φ | + ( | u | + |∇ d | ) |∇ φ | + ¯ ρ | d | |∇ d | φ (cid:3) dx + (cid:12)(cid:12) R B (¯ x ) ×{ τ } pu · ∇ φ dx (cid:12)(cid:12) (cid:9) dτ for a.e. t ∈ I (¯ t ) and ∀ φ ∈ C ∞ ( B (¯ x ) × (¯ t − , ∞ )) s.t. φ ≥ . (3.5) Set E := Z Z Q (¯ z ) ( | u | + |∇ d | + | p | + | d | ) dz . (3.6) If E < ¯ ǫ , then u, ∇ d ∈ L ∞ ( Q (¯ z )) with k u k L ∞ ( Q / (¯ z )) , k∇ d k L ∞ ( Q / (¯ z )) ≤ ¯ ǫ . In order to prove Lemma 1, we will require the following two technical propositions. In order to statethem, let us fix (recalling (3.1)), for a given z = ( x , t ) (to be clear by the context), the abbreviatednotations r k := 2 − k , B k := B r k ( x ) , I k := I r k ( t ) and Q k := B k × I k (3.7)(so that Q k = Q − k ( z )) and, for each k ∈ N , we define the quantities L k = L k ( z ) and R k = R k ( z ) These are defined in such a way that Q ∗ r ( x , t ) = Q r ( x , t + r ), and subsequently Q r ( x , t + r ) = B r ( x ) × ( t − r , t + r )is a “centered” cylinder with center ( x , t ). See Footnote 12, and note that (1.13) implies (3.5) with ¯ C ∼ C T and ¯ ρ = 1 if Q (¯ z ) ⊆ Ω T . Note that E < ∞ by (3.2) and standard embeddings, see Section 2. z = ( x , t ) will be clear by context) by L k := ess sup t ∈ I k Z B k − (cid:0) | u ( t ) | + |∇ d ( t ) | (cid:1) dx + Z I k Z B k − (cid:0) |∇ u | + |∇ d | (cid:1) dx dt (3.8)and R k := Z − Z Q k − (cid:0) | u | + |∇ d | (cid:1) dz + r / k Z − Z Q k − | u || p − ¯ p k | dz (3.9)where ¯ p k ( t ) := Z B k − p ( x, t ) dx .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 > such thatthe following holds:Fix any ¯ z = (¯ x, ¯ t ) ∈ R × R , suppose u , d and p satisfy (3.2) and (3.4), and set E as in (3.6).Then for any z ∈ Q (¯ z ) we have (see (3.7), (3.8), (3.9)) R n +1 ( z ) ≤ C A (cid:18) max ≤ k ≤ n L / k ( z ) + k p k / L / ( Q / ( z )) | {z } ≤ E (cid:19) ∀ n ≥ . (3.10)The proof of Proposition 1 uses only the H¨older and Poincar´e inequalities, Sobolev embedding andCalderon-Zygmund estimates along with a local decomposition of the pressure (see (4.20)) using thepressure equation (3.4). Proposition 2 (Cf. Lemma 2.8 of [LL96]) . There exists a large universal constant C B > such thatthe following holds:Fix any ¯ z = (¯ x, ¯ t ) ∈ R × R , suppose u , d and p satisfy (3.2), (3.3) and (3.5), and set E as in(3.6).Then for any z ∈ Q (¯ z ) we have (see (3.7), (3.8), (3.9)) L n ( z ) ≤ C B (cid:18) max k ≤ k ≤ n R k ( z ) + E / + (1 + k k ) E (cid:19) ∀ n ≥ for any k ∈ { , . . . , n − } . 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 senseof (3.22)) the same way (as do those in (3.10)), which is why the energy inequality is necessary. We use the standard notation for averages, e.g. Z B − f ( x ) dx := 1 | B | Z B f ( x ) dx . Proof of Lemma 1:
We first note that for any φ ≥ ρ ≤ ρ Z Z Q | d | |∇ d | φ ≤ Z Z Q | d | + Z Z Q |∇ d | φ . Taking φ in particular such that φ ≡ Q = Q / ( z ), we see easily from this that L . ) . E + E / ∀ z ∈ Q / (¯ z ) . (3.12)It is also easy to see that L n +1 ≤ L n for any n ∈ N . (3.13)Hence we may pick C >> z ∈ Q (¯ z ) (and suppressing the dependence on z inwhat follows) we have L , L , L . ) , ( . ) ≤ ( C ) / (cid:16) E + E / (cid:17) , (3.14) C A ≤ C ) C B ≤ ( C ) / for C A and C B as in Propositions 1 and 2. Having fixed C (uniformly over z ∈ Q / (¯ z )), we thenchoose ¯ ǫ ∈ (0 ,
1) so small that ¯ ǫ < C ) ⇐⇒ C ¯ ǫ < ¯ ǫ / . Noting first that ¯ ǫ ≤ (¯ ǫ ) / , under the assumption E < ¯ ǫ we in particular see from (3.14) that L , L , L ≤ ( C ¯ ǫ ) / . Then, by Proposition 1 with n ∈ { , } we have R , R . ) ≤ C { L / , L / , L / } + ¯ ǫ ) ≤ C ( C + 1)2 ¯ ǫ ≤ C ¯ ǫ < ¯ ǫ / which implies due to Proposition 2 with n = 4 and k = 3 that L . ) ≤ C B (max { R , R } + E / + (1 + 3 · ) E ) ≤ ( C ¯ ǫ ) / . Then in turn, Proposition 1 with n = 4 gives L , L , L , L ≤ ( C ¯ ǫ ) / . ) = ⇒ R < ¯ ǫ / , from which Proposition 2 with n = 5 and, again, k = 3 gives R , R , R < ¯ ǫ / . ) = ⇒ L ≤ ( C ¯ ǫ ) / , and continuing we see by induction that Proposition 1 and Proposition 2 (with k = 3 fixed through-out) imply that R n ( z ) < ¯ ǫ / , L n ( z ) ≤ ( C ¯ ǫ ) / ∀ n ≥ . This, in turn, implies (for example) that (see, e.g., [WZ77, Theorem 7.16]) | u ( x , t ) | + |∇ d ( x , t ) | ≤ ¯ ǫ / for all Lebesgue points z ∈ Q (¯ z ) of | u | + |∇ d | which implies the L ∞ statement, and Lemma 1 is proved. (cid:3) Lemma 1 will be used to prove the first assertion in Theorem 1 as well as the next lemma, which inturn will be used to prove the second assertion in Theorem 1.13 emma 2 ( ˙ H ǫ -regularity, cf. Theorem 3.1 of [LL96] and Proposition 2 of [CKN82]) . There existsmall universal constants ǫ > and ǫ > such that the following holds. Fix Ω T := Ω × (0 , T ) as inTheorem 1, and suppose u , d and p satisfy assumptions (1.9) - (1.13). If (recall (3.1)) lim sup r ց r Z Z Q ∗ r ( z ) | d | dz < ǫ (3.15) and lim sup r ց r Z Z Q ∗ r ( z ) (cid:0) |∇ u | + |∇ d | (cid:1) dz < ǫ , (3.16) for some z ∈ Ω T , then z is a regular point, i.e. | u | and |∇ d | are essentially bounded in someneighborhood of z . For the proof of Lemma 2, for z = ( x , t ) ∈ Ω T and for r > A z , B z , C z , D z , E z , F z (cf. [LL96, (3.3)]) and G z using the cylinders Q ∗ r ( z ) (whose “centers” z are inthe interior, see (3.1)) by A z ( r ) := 1 r ess sup t ∈ I ∗ r ( t ) Z B r ( x ) (cid:0) | u ( t ) | + |∇ d ( t ) | (cid:1) dx ,B z ( r ) := 1 r Z Z Q ∗ r ( z ) (cid:0) |∇ u | + |∇ d | (cid:1) dz ,C z ( r ) := 1 r Z Z Q ∗ r ( z ) (cid:0) | u | + |∇ d | (cid:1) dz , D z ( r ) := 1 r Z Z Q ∗ r ( z ) | p | / dz ,E z ( r ) := 1 r Z Z Q ∗ r ( z ) | u | n(cid:12)(cid:12)(cid:12) | u | − | u | r (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) |∇ d | − |∇ d | r (cid:12)(cid:12)(cid:12)o dz ,F z ( r ) := 1 r Z Z Q ∗ r ( z ) | u || p | dz and G z ( r ) := 1 r Z Z Q ∗ r ( z ) | d | dz where g r ( t ) := Z B r ( x ) − g ( y, t ) dy , (3.17)and define M z ( r ) := C z ( r ) + D z ( r ) + E z ( r ) + F z ( r ) . (3.18)In particular, (3.15) says that lim sup r → G z ( r ) < ǫ and (3.16) says that lim sup r → B z ( r ) < ǫ . Thestatement in Lemma 2 will follow from Lemma 1 along with the following technical “decay estimate”which will be proved in Section 4. Proposition 3 (Decay estimate, cf. Lemma 3.1, [LL96]) . There exists a non-decreasing function ¯ c : [0 , ∞ ) → (0 , ∞ ) such that the following holds: if u , d and p satisfy (1.9) - (1.13) for Ω T as inTheorem 1, and z ∈ Ω T and ρ ∈ (0 , are such that Q ∗ ρ ( z ) ⊆ Ω T and furthermore sup ρ ∈ (0 ,ρ ] B z ( ρ ) ≤ and sup ρ ∈ (0 ,ρ ] G z ( ρ ) ≤ g (3.19) for some finite g ≥ , then M z ( γρ ) ≤ ¯ c ( g ) h γ M z + γ − B z ( M z + M z + M z ) i ( ρ ) (3.20) for any ρ ∈ (0 , ρ ] and γ ∈ (0 , ] . Let’s now use Proposition 3 and Lemma 1 to prove Lemma 2.14 roof of Lemma 2:
Let us first note the following important consequence of Lemma 1. Fix z := ( x , t ) ∈ Ω T and ¯ ǫ ∈ (0 ,
1) as in Lemma 1, and suppose that M z ( r ) < (cid:16) ¯ ǫ (cid:17) and G z ( r ) < ¯ ǫ r ∈ (0 ,
1] such that Q ∗ r ( z ) ⊆ Ω T . Setting u z ,r ( x, t ) := ru ( x + rx, t + r t ) , p z ,r ( x, t ) := r p ( x + rx, t + r t )and d z ,r ( x, t ) := d ( x + rx, t + r t ) , (3.22)a change of variables from z = ( x, t ) to( y, s ) := ( x + rx, t + r t ) (3.23)implies that Z Q ∗ (0 , (cid:16) | u z ,r | + |∇ d z ,r | + | p z ,r | + | d z ,r | (cid:17) dz = C z ( r ) + D z ( r ) + G z ( r ) < ¯ ǫ . Since Q ∗ (0 ,
0) = Q (0 , ), it follows from assumptions (1.9) - (1.13) that u z ,r , d z ,r and p z ,r satisfythe assumptions of Lemma 1 with ¯ z = (¯ x, ¯ t ) := (0 , ) and ¯ ρ := r ∈ (0 , E = E ( u z ,r , d z ,r , p z ,r , ¯ z ) < ¯ ǫ , we therefore conclude by Lemma 1 that | u z ,r ( z ) | , |∇ d z ,r ( z ) | ≤ ¯ ǫ for a.e. z ∈ Q (0 , ) = B (0) × ( − , )and hence | u ( y, s ) | , |∇ d ( y, s ) | ≤ ¯ ǫ r for a.e. ( y, s ) ∈ B r ( x ) × ( t − r , t + r ) . In particular, by definition, z = ( x , t ) is a regular point, i.e. | u | and |∇ d | are essentially boundedin a neighborhood of z , so long as (3.21) holds for some sufficiently small r > ǫ := ¯ ǫ c := (cid:16) ¯ ǫ (cid:17) , we choose γ ∈ (0 , ] so small that furthermore4¯ c ( ǫ ) γ ≤ , (3.24)where ¯ c ( ǫ ) is the constant from Proposition 3 with g := ǫ ; finally, we choose ǫ > ǫ ≤ γ c . (3.25) For example, if one fixes an arbitrary φ ∈ C ∞ ( Q ∗ (0 , φ z ,r ( x, τ ) := φ (cid:18) x − x r , τ − t r (cid:19) , then φ z ,r ∈ C ∞ ( Q ∗ r ( z )) ⊂ C ∞ (Ω T ). One can therefore use the test function φ z ,r in (1.13), make the change ofvariables ( ξ, s ) := (cid:16) x − x r , τ − t r (cid:17) (so ( x, τ ) = ( x + rξ, t + r s )) and divide both sides of the result by r to obtainthe local energy inequality (3.5) for the re-scaled functions with ¯ ρ = r (as all terms scale the same way except for | d | |∇ d | φ z ,r ) and ¯ z = (0 , ). The other assumptions are straightforward. z ∈ Ω T is such that (3.15) and (3.16) hold, it implies in particular that there exists some ρ ∈ (0 , Q ∗ ρ ( z ) ⊆ Ω T and, furthermore,sup ρ ∈ (0 ,ρ ] G z ( ρ ) < ǫ and sup ρ ∈ (0 ,ρ ] B z ( ρ ) < ǫ . (3.26)Suppose now that z is not a regular point. Then we must have M z ( ρ ) ≥ c > ρ ∈ (0 , ρ ] , (3.27)or else (3.21) would hold for some r ∈ (0 , ρ ] in view of (3.26) which would imply that z is a regularpoint as we established above using Lemma 1.It then follows from (3.25), (3.26) and (3.27) (and the facts that γ, c ≤
1) that γ − B z ( ρ ) ( . ) ≤ γ − ǫ . ) ≤ ( γc ) ≤ γ · min { , c , c } ( . ) ≤ γ min { , M z ( ρ ) , M z ( ρ ) } for all ρ ≤ ρ . Using this along with (3.20), we conclude by Proposition 3 and (3.24) that M z ( γρ ) ≤ cγ M z ( ρ ) ≤ M z ( ρ ) for all ρ ∈ (0 , ρ ] . In particular, since γ k ρ ∈ (0 , ρ ] for any k ∈ N , by iterating the estimate above we see that M z ( γ n ρ ) ≤ M z ( γ n − ρ ) ≤ M z ( γ n − ρ ) ≤ · · · ≤ n M z ( ρ ) < c for a sufficiently large n ∈ N which contradicts (3.27) (with ρ = γ n ρ ), and hence contradictsour assumption that z is not a regular point. Therefore z must indeed be regular, which provesLemma 2. (cid:3) In order to prove Theorem 1, we now prove the following general lemma, from which Lemma 1and Lemma 2 will have various consequences (including Theorem 1 as well as various other historicalresults, which we point out for the reader’s interest). As a motivation, note first that, for r > z := ( x , t ) ∈ R × R , according to the notation in (3.22) a change of variables gives Z Q ∗ (0 , | u z ,r | q + | p z ,r | q = 1 r − q Z Q ∗ r ( x ,t ) | u | q + | p | q , Z Q ∗ (0 , |∇ u z ,r | q = 1 r − q Z Q ∗ r ( x ,t ) |∇ u | q and Z Q ∗ (0 , | d z ,r | q = 1 r Z Q ∗ r ( x ,t ) | d | q for any q ∈ [1 , ∞ ). 16 emma 3. Fix any open and bounded Ω ⊂⊂ R , T ∈ (0 , ∞ ) , k ≥ and C k > , and suppose S ⊆ Ω T := Ω × (0 , T ) and that U : Ω T → [0 , ∞ ] is a non-negative Lebesgue-measurable function suchthat the following property holds in general: ( x , t ) ∈ S = ⇒ lim sup r ց r k Z Q ∗ r ( x ,t ) U dz ≥ C k . (3.28) If, furthermore, U ∈ L (Ω T ) , (3.29) then (recall Definition 1) P k ( S ) < ∞ (and hence the parabolic Hausdorff dimension of S is at most k ) with the explicit estimate P k ( S ) ≤ C k Z Ω T U dz ; (3.30) moreover, if k = 5 , then µ ( S ) ≤ π P ( S ) ≤ · π C Z Ω T U dz (3.31) where µ is the Lebesgue outer measure, and if k < , 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
Proof of Theorem 1:
First note that for any r > z := ( x , t ) ∈ R × R such that Q r ( z ) ⊆ Ω T , it follows (asin the proof of Lemma 2) that the re-scaled triple ( u z ,r , d z ,r , p z ,r ) (see (3.22)) satisfies the condi-tions of Lemma 1 with ¯ z := (0 , r Z Q r ( x ,t ) | u | + |∇ d | + | p | + 1 r Z Q r ( x ,t ) | d | == Z Q (0 , | u z ,r | + |∇ d z ,r | + | p z ,r | + | d z ,r | < ¯ ǫ (3.32)(with ¯ ǫ as in Lemma 1), it follows that | u z ,r | , |∇ d z ,r | ≤ C on Q (0 ,
0) for some
C >
0, and hence | u | , |∇ d | ≤ Cr on Q r ( x , t ); in particular, every interior point of Q r ( x , t ) is a regular point, assuming(3.32) holds. Therefore, taking z := ( x , t ) such that Q r ( x , t ) = Q ∗ r ( x , t ) , (so x = x and t is slightly lower than t so that ( x , t ) is in the interior of the cylinder Q r ( x , t ))and letting S ⊂ Ω T be the singular set of the solution ( u, d, p ), we see (in particular) that, since r < r for r < x , t ) ∈ S = ⇒ lim sup r ց r Z Q ∗ r ( x ,t ) | u | + |∇ d | + | p | + | d | ≥ ¯ ǫ (3.33)(in fact, (3.33) must hold with lim inf instead of lim sup). Therefore, as long as( u, ∇ d, p, d ) ∈ L (Ω T ) × L (Ω T ) × L (Ω T ) × L (Ω T ) , (3.34)we may apply Lemma 3 (using a suitable covering argument, it is not hard to see that without lossof generality we can assume Ω is bounded) with U := | u | + |∇ d | + | p | + | d | , k = 5 and C k := ¯ ǫ tosee that P ( S ) < ∞ r Z Q r ( x ,t ) | u | + |∇ d | + | p | = Z Q (0 , | u z ,r | + |∇ d z ,r | + | p z ,r | < ¯ ǫ r Z Q r ( x ,t ) | d | = Z Q (0 , | d z ,r | < ¯ ǫ , (3.36)then it would follow that ( x , t ) / ∈ S for ( x , t ) as above. Therefore under the general assumption(1.14) with ǫ = ¯ ǫ , we would have (3.36) for sufficiently small r , and hence( x , t ) ∈ S = ⇒ lim sup r ց r Z Q ∗ r ( x ,t ) | u | + |∇ d | + | p | ≥ ¯ ǫ . (3.37)Therefore, as long as ( u, ∇ d, p ) ∈ L (Ω T ) × L (Ω T ) × L (Ω T ) , (3.38)we may apply Lemma 3 with U := | u | + |∇ d | + | p | , k = 2 and C k := ¯ ǫ/ P ( S ) = 0 . On the other hand, we know slightly more than (3.38). The assumptions on u and d in (1.9) implythat u, ∇ d ∈ L (Ω T )(for example, by (2.18) with α = , along with Sobolev embedding). Suppose we also knew (as in the case when Ω = R ) that p ∈ L (Ω T )(which essentially follows from (1.9) and (1.12), see [LL96, Theorem 2.5]). Then (3.29) holds with U := | u | + |∇ d | + | p | , and moreover H¨older’s inequality implies that ( r Z Q ∗ r ( z ) | u | + |∇ d | + | p | ) ≤ | Q | " r Z Q ∗ r ( z ) | u | + |∇ d | + | p | ( | Q | is the Lebesgue measure of the unit parabolic cylinder). In view of (3.37), one could thereforeapply Lemma 3 with U := | u | + |∇ d | + | p | , k = 53 and C k = ¯ ǫ | Q | . to deduce (similar to Scheffer’s result in [Sch80]) that P ( S ) = 0 . All of the above follows from Lemma 1 alone. We will now show that Lemma 2 allows one (under as-sumption (1.14), and even if p / ∈ L (Ω T )) to further decrease the dimension of the parabolic Hausdorffmeasure, with respect to which the singular set has measure zero, from to 1. This was essentiallythe most significant contribution of [CKN82] in the Navier-Stokes setting d ≡ d satisfies (1.14) with ǫ (= ¯ ǫ ) as in Lemma 2. Taking ǫ > x , t ) ∈ S = ⇒ lim sup r ց r Z Q ∗ r ( x ,t ) (cid:0) |∇ u | + |∇ d | (cid:1) ≥ ǫ , so that (3.28) holds with U := |∇ u | + |∇ d | and k = 1. The second assumption in (1.9) implies that(3.29) holds as well with U := |∇ u | + |∇ d | . Therefore P ( S ) = 0by Lemma 3 with U := |∇ u | + |∇ d | , k = 1 and C k = ǫ . This completes the proof of Theorem 1(assuming Lemma 3). (cid:3) Let us now give the
Proof of Lemma 3.
Fix any δ >
0, and any open set V such that S ⊆ V ⊆ Ω × (0 , T ) . (3.39)For each z := ( x, t ) ∈ S , according to (3.28) we can choose r z ∈ (0 , δ ) sufficiently small so that Q ∗ r z ( z ) ⊂ V and 1 r kz Z Q ∗ rz ( z ) U ≥ C k . (3.40)By a Vitalli covering argument (see [CKN82, Lemma 6.1]), there exists a sequence ( z j ) ∞ j =1 ⊆ S suchthat S ⊆ ∞ [ j =1 Q ∗ r zj ( z j ) (3.41)and such that the set of cylinders { Q ∗ r zj ( z j ) } j are pair-wise disjoint. We therefore see from (3.40) that ∞ X j =1 r kz j ≤ C k ∞ X j =1 Z Q ∗ rzj ( z j ) U ≤ C k Z V U ≤ C k Z Ω T U (3.42)which is finite (and uniformly bounded in δ ) by (3.29). Note that according to Definition 1 of theparabolic Hausdorff measure P k , (3.42) implies P k ( S ) ≤ k C k Z V U ≤ k C k Z Ω T U (3.43)due to (3.42), which establishes (3.30).Let us now assume that k ≤
5. Letting µ be the Lebesgue (outer) measure, note that µ ( Q ∗ r zj ) ≤ | B | (5 r z j ) so that µ ( S ) ( . ) ≤ | B | ∞ X j =1 (5 r z j ) ≤ | B | δ − k ∞ X j =1 r kz j ( . ) ≤ δ − k | B | C k Z Ω T U , (3.44)since we have chosen r z < δ for all z ∈ S . If k = 5, (3.44) along with Definition 1 gives the explicitestimate (3.31) on µ ( S ). If k <
5, since δ > δ → µ ( S ) = 0 and hence S is Lebesgue measurable with Lebesgue measure zero. We may thereforetake V to be an open set such that µ ( V ) is arbitrarily small but so that (3.39) still holds, and deducethat P k ( S ) = 0 by (3.29) and (3.43). (cid:3) Proofs of technical propositions
In order to prove Proposition 1 as well as Proposition 3, we will require certain local decompositionsof the pressure (cf. [CKN82, (2.15)]) as follows:
Claim 1.
Fix open sets Ω ⊂⊂ Ω ⊂⊂ Ω ⊂ R and ψ ∈ C ∞ (Ω ; R ) with ψ ≡ on Ω . Let G x ( y ) := 14 π | x − y | (4.1) be the fundamental solution of − ∆ in R so that, in particular, ∇ G x ∈ L q (Ω ) for any q ∈ [1 , ) for any fixed x ∈ R , and set G xψ, := − G x ∇ ψG xψ, := 2 ∇ G x · ∇ ψ + G x ∆ ψG xψ, := ∇ G x ⊗ ∇ ψ + ∇ ψ ⊗ ∇ G x + G x ∇ ψ , so that G xψ, , G xψ, , G xψ, ∈ C ∞ (Ω ) for any fixed x ∈ Ω . Suppose Π ∈ C (Ω; R ) , v ∈ C (Ω; R ) and K ∈ C (Ω; R × ) .If − ∆Π = ∇ · v in Ω , (4.2) then for any x ∈ Ω , Π( x ) = − Z ∇ G x · vψ + Z G xψ, · v + Z G xψ, Π . (4.3) Similarly, if − ∆Π = ∇ · ( ∇ T · K ) in Ω , (4.4) then for any x ∈ Ω , Π( x ) = S [ ψK ]( x ) + Z G xψ, : K + Z G xψ, Π (4.5) where S [ e K ]( x ) := ∇ x · (cid:18) ∇ Tx · Z G x e K (cid:19) = Z G x ∇ · (cid:16) ∇ T · e K (cid:17) ∀ e K ∈ C (Ω ; R × ) ; in particular (noting ∇ G x / ∈ L ), S : [ L q (Ω )] × → L q (Ω ) for any q ∈ (1 , ∞ ) is a bounded, linearCalderon-Zygmund operator. Remark 2.
We note, therefore, that under the assumptions (1.9), (1.10) and (1.12), by suitableregularizations one can see that for almost every fixed t ∈ (0 , T ) , (4.3) and (4.5) hold for a.e. x ∈ Ω with Π := p ( · , t ) , K := J ( · , t ) and v := ∇ T · J ( · , t ) where J := u ⊗ u + ∇ d ⊙ ∇ d . Indeed, under the assumptions (1.9), we have u, ∇ d ∈ L (Ω T ) so that (omitting the x -dependence) J ( t ) ∈ L (Ω) for a.e. t ∈ (0 , T ) . (4.6)20 oreover, since u, ∇ d ∈ L ∞ (0 , T ; L (Ω)) ∩ L (Ω T ) and ∇ u, ∇ d ∈ L (Ω T ) , we have ∇ T · J ∈ L (0 , T ; L (Ω)) ∩ L (Ω T ) so that ∇ T · J ( t ) ∈ L (Ω) ∩ L (Ω) for a.e. t ∈ (0 , T ) . (4.7) Finally, (1.10) implies that p ( t ) ∈ L (Ω) for a.e. t ∈ (0 , T ) . (4.8) Fix now any t ∈ (0 , T ) such that the inclusions in (4.6), (4.7) and (4.8) hold. Since G xψ,j ∈ C ∞ for x ∈ Ω , the terms in (4.3) and (4.5) containing G xψ,j are all well-defined for every x ∈ Ω since J ( t ) , ∇ T · J ( t ) , p ( t ) ∈ L loc (Ω) . The term in (4.3) containing ∇ G x is in L rx (Ω ) for any r ∈ [1 , ) by Young’s convolution inequality (since Ω is bounded), so that term is well-defined for a.e. x ∈ Ω .Indeed, for R > such that Ω ⊆ B R ( x ) for some x ∈ R , we have x − y ∈ B R := B R (0) for all x, y ∈ Ω . Letting G ( y ) := G ( y ) and χ B R the indicator function of B R , since ψ is supported in Ω we therefore have − Z ∇ G x · vψ = [([ ∇ G ] χ B R ) ∗ ( vψ )]( x ) for all x ∈ Ω . Therefore (cid:13)(cid:13)(cid:13)(cid:13)Z ∇ G x · vψ (cid:13)(cid:13)(cid:13)(cid:13) L rx (Ω ) ≤ k ([ ∇ G ] χ B R ) ∗ vψ k L r ( R ) ≤ k [ ∇ G ] χ B R k L q ( R ) k vψ k L s ( R ) = k∇ G k L q ( B R ) k vψ k L s (Ω ) < ∞ by Young’s inequality for any q ∈ [1 , ) , s ∈ [1 , ) and r such that r = q + s (note that + − ). Finally, S [ ψJ ( t )] ∈ L (Ω ) by the Calderon-Zygmund estimates (as < < ∞ ), soagain that term is defined for a.e. x ∈ Ω .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 andpassing 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 (Ω ) . Proof of Claim 1.
Since (extending Π by zero outside of Ω) ψ Π ∈ C ∞ ( R ), by the classicalrepresentation formula (see, e.g., [GT01, (2.17)]), for any x ∈ R we have ψ ( x )Π( x ) = − Z G x ∆( ψ Π) = − Z G x ( ψ ∆Π + 2 ∇ ψ · ∇ Π + Π∆ ψ ) . (4.9)In particular, for a fixed x ∈ Ω where ψ ≡
1, we have G x ∇ ψ ∈ C ∞ ( R ) so that integrating by partsin (4.9) we see that Π( x ) = Z G x ψ ( − ∆Π) + Z G xψ, Π . (4.10)If (4.2) holds, then by (4.10) we haveΠ( x ) = Z G x ψ ∇ · v + Z G xψ, Π (4.11)21or any x ∈ Ω . One can then carefully integrate by parts once in the first term of (4.11) as follows:for a small ǫ > Z | y − x | >ǫ G x ψ ∇ · v dy = − Z | y − x | >ǫ [ ∇ ( G x ψ )] · v dy + 14 πǫ Z | y − x | = ǫ ψv · ν y dS y | {z } = O ( ǫ ) and since the second term vanishes as ǫ → | ∂B ǫ ( x ) | . ǫ , we conclude (since ∇ G x ∈ L loc ) that Z G x ψ ∇ · v = − Z [ ∇ ( G x ψ )] · v = − Z ∇ G x · vψ + Z G xψ, · v which, along with (4.11), implies (4.3) for any x ∈ Ω .On the other hand, if (4.4) holds, then by (4.10) we haveΠ( x ) = Z G x ψ ∇ · ( ∇ T · K ) + Z G xψ, Π (4.12)and one can write ∇ · ( ∇ T · ( ψK )) = [ ∇ ψ ] T : K + ∇ T ψ · [ ∇ · K ] + ∇ ψ · [ ∇ T · K ] + ψ ∇ · ( ∇ T · K )so that (as ∇ ψ = ∇ T ( ∇ ψ ) = ∇ ( ∇ T ψ ) = [ ∇ ψ ] T since ψ ∈ C ) Z G x [ ψ ∇ · ( ∇ T · K )] = Z G x [ ∇ · ( ∇ T · ( ψK ))] − Z G x [ ∇ ψ : K ] − Z (cid:18) [ G x ∇ T ψ ] · [ ∇ · K ] + [ G x ∇ ψ ] · [ ∇ T · K ] (cid:19) . Since G x ∇ ψ ∈ C ∞ for x ∈ Ω , one can again integrate by parts in the final term to obtainΠ( x ) = Z G x [ ∇ · ( ∇ T · ( ψK ))] + Z G xψ, : K + Z G xψ, Πfor x ∈ Ω in view of (4.12). Moreover, since ψK ∈ C and G x ∈ L loc , as usual for convolutions onecan change variables to obtain Z G x ∇ · (cid:0) ∇ T · ( ψK ) (cid:1) = (cid:20) ∇ x · (cid:18) ∇ Tx · Z G x ψK (cid:19)(cid:21) ( x ) =: S [ ψK ]( x )which gives us (4.5) for any x ∈ Ω , where (see, e.g., [GT01, Theorem 9.9]) S is a singular integraloperator as claimed. (Note that ∇ G x / ∈ L loc so that one cannot simply integrate by parts twice inthis term putting all derivatives on G x , but R G x ψK is the Newtonian potential of ψK which can betwice differentiated in various senses depending on the regularity of K .) (cid:3) In what follows, for
O ⊆ R and I ⊆ R , we will use the notation k · k q ; O := k · k L q ( O ) , k · k s ; I := k · k L s ( I ) , k · k q,s ; O× I := k · k L s ( I ; L q ( O )) = (cid:13)(cid:13) k · k L q ( O ) (cid:13)(cid:13) L s ( I ) and we will abbreviate by writing k · k q ; O× I := k · k q,q ; O× I = k · k L q ( O× I ) .
22e first note some simple inequalities. Letting B r ⊂ R be a ball of radius r >
0, from the embedding W , ( B ) ֒ → L ( B ) applied to functions of the form g r ( x ) = g ( rx ) (or suitably shifted, if the ball isnot centered as zero), we obtain k g r k B . k g r k B + k∇ g r k B = k g r k B + r k ( ∇ g ) r k B whereupon, noting by a simple change of variables that k g r k q ; B = r − q k g k q ; B r for any q ∈ [1 , ∞ ), we obtain for any ball B r of radius r > g that k g k B r . r k g k B r + k∇ g k B r (4.13)where the constant is independent of r as well as the center of B r . Next, for any v ( x, t ), using H¨olderto interpolate between L and L we have k v ( t ) k B r ≤ k v ( t ) k B r k v ( t ) k B r ( . ) . r − k v ( t ) k B r + k v ( t ) k B r k∇ v ( t ) k B r . (4.14)Then for I r ⊂ R with | I r | = r and Q r := B r × I r , H¨older in the t variable gives k v k Q r . r − | I r | k v k , ∞ ; Q r + k v k , ∞ ; Q r h | I r | k∇ v k Q r i so that r − k v k Q r . k v k , ∞ ; Q r + k v k , ∞ ; Q r k∇ v k Q r . k v k , ∞ ; Q r + k∇ v k Q r (the first of which is sometimes called the “multiplicative inequality”) with a constant independent of r . From these, noting that | B r | ∼ r , | Q r | ∼ r , it follows easily that, for example, Z − Z Q n − | v | dz . (cid:18) ess sup t ∈ I n Z B n − | v ( t ) | dx (cid:19) + (cid:18)Z I k Z B k − |∇ v | dx dt (cid:19) . (4.15)Note also that a similar scaling argument applied to Poincar´e’s inequality gives the estimate k g − g B r k q ; B r . r k∇ g k q ; B r ∼ | B r | k∇ g k q ; B r (4.16)for any r > q ∈ [1 , ∞ ], where g O is the average of g in O for any O ⊂ R with |O| < ∞ . Notefinally that a simple application of H¨older’s inequality gives k g O k q ; O ≤ k g k q ; O . (4.17)Proceeding now with the proof, fix some ˜ φ ∈ C ∞ ( R ) such that˜ φ ≡ B r (0) = B (0)and supp( ˜ φ ) ⊆ B r (0) = B (0) . Now fix ¯ z = (¯ x, ¯ t ) ∈ R × R and z = ( x , t ) ∈ Q (¯ z ), define B k , I k and Q k by (3.7) for this z anddefine φ by φ ( x ) := ˜ φ ( x − x ). So φ ≡ B = B ( x )and supp( φ ) ⊆ B = B ( x ) ⊂ B (¯ x ) , since x ∈ B (¯ x ). The following estimates will clearly depend only on ˜ φ , i.e. constants will be uniformfor all z ∈ Q (¯ z )). 23irst, applying (4.15) to v ∈ { u, ∇ d } and recalling (3.8) we see that1 r n (cid:0) k u k Q n + k∇ d k Q n (cid:1) . Z − Z Q n − ( | u | + |∇ d | ) dz ( . ) . L / n (4.18)for any n , with a constant independent of n . In particular, k u k Q n + k∇ d k Q n . r n L / n (4.19)for any n .Next, by Claim 1 and Remark 2 with ψ := φ , Ω := B and Ω := B , at almost every( x, t ) ∈ Q = Q ( z ) = B ( x ) × ( t − ( ) , t ) (where p = φp ), as in (4.5) we have p ( x, t ) = S [ φJ ( t )]( x ) + Z B \ B (2 ∇ G x ⊗ σ ∇ φ + G x ∇ φ ) : J ( t ) dy + Z B \ B (2 ∇ G x · ∇ φ + G x ∆ φ ) p ( t ) dy , (4.20)where J := u ⊗ u + ∇ d ⊙ ∇ d , (4.21)2 a ⊗ σ b := a ⊗ b + b ⊗ a and the operator S consisting of second derivatives of the Newtonian potentialgiven by S [ ˜ K ]( x ) := ∇ x · (cid:18) ∇ Tx · Z B G x ˜ K (cid:19) for ˜ K ∈ L q ( B ) is a bounded linear Calderon-Zygmund operator on L q ( B ) for 1 < q < ∞ . Hencefor any n ∈ N , denoting by χ n the indicator function for the set B n = B − n ( x ) and splitting φ = χ n φ + (1 − χ n ) φ in the first term of (4.20), we can write p = p ,n + p ,n + p ,n ≡ p ,n + p ,n + p , where, for almost every ( x, t ) ∈ Q , p ( x, t ) = S [ χ n φJ ( t )]( x ) | {z } =: p ,n ( x,t ) + S [(1 − χ n ) φJ ( t )]( x ) | {z } =: p ,n ( x,t ) ++ Z B \ B (2 ∇ G x ⊗ σ ∇ φ + G x ∇ φ ) : J ( t ) dy + Z B \ B (2 ∇ G x · ∇ φ + G x ∆ φ ) p ( t ) dy | {z } =: p ,n ( x,t ) ≡ p ( x,t ) (where the last term is clearly independent of n , but we keep the notation p ,n for convenience).Note first that, by the classical Calderon-Zygmund estimates, there is a universal constant C cz > n ∈ N , we have k p ,n ( t ) k ; B n +1 ≤ C cz k χ n φJ ( t ) k ; R ≤ C cz k ˜ φ k ∞ ; R k J ( t ) k ; B n . (4.22)Next, since the appearance of ∇ φ in p exactly cuts off a neighborhood of the singularity of G x (see(4.1)) uniformly for all x ∈ B ( x ) (as we integrate over | x − y | ≥ , hence | x − y | ≥ ), we see that p ,n ( · , t ) ∈ C ∞ ( B ( x )) for t ∈ I ( t ) with, in particular, k∇ x p ,n ( t ) k ∞ ; B n +1 ( n ≥ ≤ k∇ x p ,n ( t ) k ∞ ; B ( x ) ≤ c ( ˜ φ ) (cid:0) k J ( t ) k B + k p ( t ) k B (cid:1) . (4.23)24n the term p ,n , the singularity coming from G x is also isolated due to the appearance of χ n , but it isno longer uniform in n so we must be more careful. As we are integrating over a region which avoidsa neighborhood of the singularity at y = x of G x , we can pass the derivatives in S under the integralsign to write ∇ x p ,n ( x, t ) = Z B \ B n ∇ x [( ∇ x G x ) T : φJ ( t )] dy = n − X k =1 Z B k \ B k +1 ∇ x [( ∇ x G x ) T : φJ ( t )] dy and note, in view of (4.1) that (cid:12)(cid:12) ∇ x G x ( y ) (cid:12)(cid:12) . | x − y | ≤ (2 k +2 ) . k | B k | ∀ x ∈ B k +2 , y ∈ (cid:0) B k +1 (cid:1) c . Therefore, since B n +1 = B ( n − ⊆ B k +2 for 1 ≤ k ≤ n − , we see that k∇ x p ,n ( · , t ) k ∞ ,B n +1 . c ( ˜ φ ) n − X k =1 k Z B k − | J ( y, t ) | dy (4.24)for all t ∈ I ( t ).Now, recalling the notation ¯ f k ( t ) := Z B k − f ( x, t ) dx for a function f ( x, t ) and k ∈ N , for any t ∈ I = ( t − ( ) , t ) and n ≥
2, we estimate Z B n +1 | u ( x, t ) || p ( x, t ) − ¯ p n +1 ( t ) | dx ≤ (4.25) ≤ X j =1 Z B n +1 | u ( x, t ) || p j,n ( x, t ) − ¯ p j,nn +1 ( t ) | dx ≤ k u ( · , t ) k B n +1 X j =1 k p j,n ( · , t ) − ¯ p j,nn +1 ( t ) k ; B n +1(4 . , (4 . , H¨older . k u ( t ) k B n +1 k p ,n ( t ) k ; B n +1 + | B n +1 | X j =2 k∇ p j,n ( t ) k ∞ ; B n +1 (4 . , (4 . , (4 . , H¨older . k u ( t ) k B n +1 k J ( t ) k ; B n + r n +1 ( n − X k =1 k Z B k − | J ( t ) | dy ! + k J ( t ) k ; B + k p ( t ) k ; B )! . Note further that, setting L J,k := (cid:13)(cid:13)(cid:13)(cid:13)Z B k − | J ( t ) | dy (cid:13)(cid:13)(cid:13)(cid:13) L ∞ t ( I k ) , (4.26)we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X k =1 k Z B k − | J ( t ) | dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t ( I n +1 ) ≤ | I n +1 | (cid:18) max ≤ k ≤ n − L J,k (cid:19) n − X k =1 k ≤ r n +1 max ≤ k ≤ n − L J,k , | I n +1 | = r n +1 and n − X k =1 k = 2 n − − < n = r − n . Integrating over t ∈ I n +1 in (4.25), applying H¨older in the variable t and recalling by (4.19) that k u k Q n +1 . r n +1 L / n +1 , we obtain Z Z Q n +1 | u || p − p n +1 | dz . (4.27) . r n +1 L / n +1 (cid:26) k J k ; Q n + r n +1 max ≤ k ≤ n − L J,k + r n +1 (cid:16) k J k ; Q + k p k ; Q (cid:17)(cid:27) . It follows now from (4.21) that k J k ; Q k ≤ k u k Q k + k∇ d k Q k ( . ) . (cid:16) r k L / k (cid:17) = r k L k (4.28)and L J,k ( . ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z B k − (cid:0) | u ( · ) | + |∇ d ( · ) | (cid:1) dy (cid:13)(cid:13)(cid:13)(cid:13) ∞ ; I k ≤ L k . (4.29)Now from (4.21), (4.27), (4.28), (4.29) and the simple fact that r n = r n +1 ≤ r n +1 Z − Z Q n +1 − | u || p − ¯ p n +1 | dz . L / n +1 (cid:26) r n +1 L n + r n +1 max ≤ k ≤ n − L k + r |{z} ≤ L + k p k ; Q (cid:27) . L / n +1 (cid:26) max ≤ k ≤ n L k + k p k ; Q (cid:27) . Since Z − Z Q n +1 − (cid:0) | u | + |∇ d | (cid:1) dz ( . ) . L n +1 , adding the previous estimates and recalling (3.8) and (3.9) we have R n +1 . L n +1 + L / n +1 (cid:18) max ≤ k ≤ n L k + k p k ; Q (cid:19) (where the constant is universal). This along with (3.13) easily implies (3.10) and provesProposition 1. (cid:3) For simplicity, take ¯ z = z = (0 , Q k = Q k (0 , φ in (3.5) such that φ = φ n := χψ n , where (recall that here Q = Q (0 ,
0) = B (0) × ( − ,
0) so χ will be zero in a neighborhood of the “parabolic boundary” of Q ) χ ∈ C ∞ (cid:16) B (0) × (cid:0) − , ∞ (cid:1)(cid:17) , χ ≡ Q , ≤ χ ≤ ψ n ( x, t ) := 1( r n − t ) / e − | x | r n − t ) for t ≤ . (4.31)26ote that the singularity of ψ n would naturally be at ( x, t ) = (0 , r n ) / ∈ Q , so ψ n ∈ C ∞ ( Q ) and wemay extend ψ n smoothly to t > n so that, inparticular, φ n ∈ C ∞ ( B (0) × ( − , ∞ )) as required in (3.5) (with (¯ x, ¯ t ) = (0 , ∇ ψ n ( x, t ) = − x r n − t ) ψ n ( x, t ) and ψ nt + ∆ ψ n ≡ Q . (4.32)Note first that for ( x, t ) ∈ Q n ( n ≥ ≤ | x | ≤ r n and r n ≤ [ r n − t ] ≤ r n so that r n = (cid:0) r n (cid:1) e r n ≤ ( r n − t ) / e | x | r n − t ) ≤ (cid:0) r n (cid:1) e r n r n = 2 e r n . Hence 12 e · r n ≤ ψ n ( x, t ) ≤ r n ∀ ( x, t ) ∈ Q n (4.33)and therefore (as r n − t > |∇ x ψ n ( x, t ) | = | x | r n − t ) | ψ n ( x, t ) | . r n r n · r n = 1 r n ∀ ( x, t ) ∈ Q n . (4.34)Next, note similarly that for 2 ≤ k ≤ n and ( x, t ) ∈ Q k − \ Q k , we have r k ≤ | x | ≤ r k − = 2 r k and r k ≤ r n + r k ≤ [ r n − t ] ≤ r n + r k − ≤ r k − = 8 r k , so that e r k = (cid:0) r k (cid:1) e r k r k ≤ ( r n − t ) / e | x | r n − t ) ≤ (cid:0) r k (cid:1) e (2 rk )24 r k = 2 er k . Therefore 12 e · r k ≤ ψ n ( x, t ) ≤ e · r k ∀ ( x, t ) ∈ Q k − \ Q k (2 ≤ k ≤ n ) (4.35)and hence, as in (4.34), |∇ x ψ n ( x, t ) | . r k r k · r k = 1 r k ∀ ( x, t ) ∈ Q k − \ Q k (2 ≤ k ≤ n ) . (4.36)We can therefore estimate (for n ≥ φ n = ψ n in Q n ):12 e · r n (cid:20) ess sup I n Z B n (cid:0) | u | + |∇ d | (cid:1) + Z Z Q n (cid:0) |∇ u | + |∇ d | (cid:1)(cid:21) ( . ) ≤ ess sup I n Z B n (cid:0) | u | + |∇ d | (cid:1) φ n + Z Z Q n (cid:0) |∇ u | + |∇ d | (cid:1) φ n ( . ) ≤ ¯ C (cid:26)Z Z Q (cid:2)(cid:0) | u | + |∇ d | (cid:1) | φ nt + ∆ φ n | + ( | u | + |∇ d | ) |∇ φ n | + ¯ ρ | d | |∇ d | φ n (cid:3) + Z I (cid:12)(cid:12)(cid:12)(cid:12) Z B pu · ∇ φ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) . In (3.5) as well, the values of φ for t > ¯ t are actually irrelevant. φ nt + ∆ φ n ( . ) = ψ n ( χ t + ∆ χ ) + 2 ∇ χ · ∇ ψ n ( . ) ≡ Q and hence, taking k = 2 in (4.35) and (4.36), we see that | φ nt + ∆ φ n | . r + 1 r . Q , (4.37)so that Z Z Q (cid:0) | u | + |∇ d | (cid:1) | φ nt + ∆ φ n | ( . ) . Z Z Q (cid:0) | u | + |∇ d | (cid:1) ( . ) . E / by H¨older’s inequality. Note similarly that |∇ φ n | = | χ ∇ ψ n + ψ n ∇ χ | ( . ) . |∇ ψ n | + | ψ n | in Q so that (since r n < r n ) (4.33), (4.34) and (4.35), (4.36), respectively, give |∇ φ n | . r n in Q n , |∇ φ n | . r k in Q k − \ Q k (4.38)for any n ≥ ≤ k ≤ n . Therefore n X k =2 Z Z Q k − \ Q k (cid:0) | u | + |∇ d | (cid:1) |∇ φ n | | {z } . r − k ( . ) . (cid:20) max ≤ k ≤ n − ( r k ) − q Z − Z Q k − (cid:0) | u | + |∇ d | (cid:1)(cid:21) n X k =2 ( r k ) q for any q ≥
1, and similarly
Z Z Q n (cid:0) | u | + |∇ d | (cid:1) |∇ φ n | | {z } . r − n ( . ) . (cid:20) ( r n ) − q Z − Z Q n − (cid:0) | u | + |∇ d | (cid:1)(cid:21) ( r n ) q for any q ≥
1, and we note that ∞ X k =1 ( r k ) q = ∞ X k =1 (cid:16) − q (cid:17) k = 12 q − < ∞ ∀ q ∈ [1 , ∞ ) . (4.39)Hence in view of the disjoint union Q = n [ k =2 Q k − \ Q k ! ∪ Q n (4.40)we have (taking q = 1 in (4.39)) Z Z Q (cid:0) | u | + |∇ d | (cid:1) |∇ φ n | . max ≤ k ≤ n Z − Z Q k − (cid:0) | u | + |∇ d | (cid:1) . Similarly, we have ¯ ρ Z Z Q | d | |∇ d | φ n ≤ Z Z Q | d | | {z } ≤ E + Z Z Q |∇ d | ( φ n ) uniformly, of course, over ¯ ρ ∈ (0 , Z Z Q n |∇ d | ( φ n |{z} . r − n ) ( . ) . ( r n ) − Z Z Q n |∇ d | . ( r n ) Z − Z Q k − |∇ d | n ≥ Z Z Q k \ Q k +1 |∇ d | ( φ n |{z} . r − k ) ( . ) . ( r k ) − Z Z Q k |∇ d | . ( r k ) Z − Z Q k − |∇ d | for 1 ≤ k ≤ n −
1, we see that (4.39) with q = 2 and (4.40) again give Z Z Q |∇ d | ( φ n ) . max ≤ k ≤ n Z − Z Q k − |∇ d | . We therefore see that ¯ ρ Z Z Q | d | |∇ d | φ n . E + max ≤ k ≤ n Z − Z Q k − |∇ d | , uniformly for any ¯ ρ ∈ (0 , n ≥ L n := ess sup I n Z B n − (cid:0) | u | + |∇ d | (cid:1) + Z I n Z B n − (cid:0) |∇ u | + |∇ d | (cid:1) (4.41) . E + E / + max ≤ k ≤ n Z − Z Q k − (cid:0) | u | + |∇ d | (cid:1) + Z I (cid:12)(cid:12)(cid:12)(cid:12) Z B pu · ∇ φ n (cid:12)(cid:12)(cid:12)(cid:12) . Furthermore we claim that for 1 ≤ k ≤ n − Z I (cid:12)(cid:12)(cid:12)(cid:12) Z B pu · ∇ φ n (cid:12)(cid:12)(cid:12)(cid:12) . max k ≤ k ≤ n (cid:18) r / k Z − Z Q k − | p − ¯ p k || u | (cid:19) + k k Z Z Q | p || u | . (4.42)Assuming this for the moment and continuing, for n ≥
2, (4.41), (4.42) and Young’s convexity in-equality along with the fact that, for any k ≥
1, we can estimatemax ≤ k ≤ k Z − Z Q k − (cid:0) | u | + |∇ d | (cid:1) . k k Z Z Q (cid:0) | u | + |∇ d | (cid:1) imply (recalling (3.9)) that L n . E + E / + max k ≤ k ≤ n R k + k k Z Z Q | u | + |∇ d | + | p | / | {z } ≤ E for any k ∈ { , . . . , n − } , and hence Proposition 2 is proved.To prove (4.42), we consider additional functions χ k (so that χ k φ n = χ k χψ n ) satisfying (recall thathere Q k = Q k (0 ,
0) = B r k (0) × ( − r k , χ k will be zero in a neighborhood of the “parabolicboundary” of Q k ) χ k ∈ C ∞ ( e Q r k ) with e Q r := B r (0) × ( − r , r ) for r > ,χ k ≡ e Q r k , ≤ χ k ≤ |∇ χ k | . r k (4.43)( χ k | { t> } will again actually be irrelevant) so that in particular (as e Q r k +2 ⊂ e Q r k +1 where χ k ≡ χ k +1 ≡
1) supp ( χ k − χ k +1 ) ⊂ e Q r k \ e Q r k +2 . (4.44)29hen since Q = Q / (0 , ⊂ Q (0 ,
0) = Q r (0 , χ ≡ Q and hence for any n ≥ χ = χ n + n − X k =0 ( χ k − χ k +1 ) , for any fixed k ∈ N ∩ [1 , n −
1] and at each fixed τ ∈ I we have Z B pu · ∇ φ n ( . ) = Z B pu · ∇ [ χ φ n ]= Z B pu · ∇ [ χ n φ n ] + n − X k =0 Z B pu · ∇ [( χ k − χ k +1 ) φ n ] ( . ) , ( . ) = Z B n pu · ∇ [ χ n φ n ] + n − X k =0 Z [ B k \ B k +2 ] pu · ∇ [( χ k − χ k +1 ) φ n ] ( . ) = Z B n ( p − ¯ p n ) u · ∇ [ χ n φ n ] + k − X k =0 Z [ B k \ B k +2 ] pu · ∇ [( χ k − χ k +1 ) φ n ]+ n − X k = k Z [ B k \ B k +2 ] ( p − ¯ p k ) u · ∇ [( χ k − χ k +1 ) φ n ] , (4.45)where ¯ p k = ¯ p k ( τ ) = Z B k − p ( x, τ ) dx . Note first that (4.35), (4.36) and (4.44) imply (since r j +1 = 2 r j for any j ) that |∇ [( χ k − χ k +1 ) φ n ] | ≤ | χ k − χ k +1 ||∇ φ n | + | φ n ||∇ ( χ k − χ k +1 ) | . r − k on Q k \ Q k +2 = ( Q k \ Q k +1 ) ∪ ( Q k +1 \ Q k +2 )for any k , and similarly |∇ [ χ n φ n ] | ≤ | χ n ||∇ φ n | + | φ n ||∇ χ n | . r − n on Q n . Therefore we can estimate (recalling again (4.43) and (4.44) when integrating | (4 . | over τ ∈ I ) Z τ ∈ I (cid:12)(cid:12)(cid:12)(cid:12) Z B ×{ τ } pu · ∇ φ n (cid:12)(cid:12)(cid:12)(cid:12) . k k Z Z Q | p || u | + n X k = k r k Z − Z Q k − | p − ¯ p k || u | which, along with (4.39) with q = implies (4.42) for any k ∈ [1 , n −
1] as desired. (cid:3)
In this section we prove the technical decay estimate (Proposition 3) used to prove Lemma 2. In all ofwhat follows, recall the definitions in (3.17) and (3.18) of A z , B z , C z , D z , E z , F z , G z and M z .We will require the following three claims which essentially appear in [LL96] and which generalizecertain lemmas in [CKN82]; however we include full proofs in order to clarify certain details, and tohighlight the role of G z (not utilized in [LL96]) in Claim 4 which is therefore a slightly refinedversion of what appears in [LL96]. Note that G z ( r ) . k d k ∞ uniformly in r (and z ), though in our setting we may have d / ∈ L ∞ . laim 2 (General estimates (cf. Lemmas 5.1 and 5.2 in [CKN82])) . There exist constants c , c > such that for any u and d which have the regularities in (1.9) for Ω T := Ω × (0 , T ) as in Theorem 1,the estimates C z ( γρ ) ≤ c h γ A z + γ − A z B z i ( ρ ) (4.46) and E z ( γρ ) ≤ c h C z A z B z i ( γρ ) (4.47) hold for any z ∈ R and ρ > such that Q ∗ ρ ( z ) ⊆ Ω T and any γ ∈ (0 , . Claim 3 (Estimates requiring the pressure equation (cf. Lemmas 5.3 and 5.4 in [CKN82])) . Thereexist constants c , c > 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 which satisfy the pressure equation (1.12), the estimates D z ( γρ ) ≤ c h γ ( D z + A z B z + C z ) + γ − A z B z i ( ρ ) (4.48) and F z ( γρ ) ≤ c h γ ( A z + D z + C z ) + γ − A z ( B z + B z ) i ( ρ ) . (4.49) hold for any z ∈ R and ρ > such that Q ∗ ρ ( z ) ⊆ Ω T and any γ ∈ (0 , ] . The crucial aspect of the estimates (4.46), (4.47), (4.48) and (4.49) (which control M z ( γρ )) in provingLemma 2 (through Proposition 3) is that whenever a negative power of γ appears, there is always afactor of B z as well, which will be small when proving Lemma 2. Positive powers of γ will similarly besmall; in each term evaluated at ρ (see also (4.52) below), we must have either γ α or B αz for some α > Claim 4 (Estimate requiring the local energy inequality (cf. Lemma 5.5 in [CKN82])) . There existsa constant c > 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, the estimate A z ( ρ ) ≤ c h(cid:16) G z + ρ G z (cid:17) C z + E z + F z + G z C z B z i ( ρ ) (4.50) holds for any z ∈ R and ρ > such that Q ∗ ρ ( z ) ⊆ 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 > α ∈ (0 , K ∈ { A z , B z } = ⇒ K ( αρ ) ≤ α − K ( ρ ) , K ∈ { C z , D z , E z , F z } = ⇒ K ( αρ ) ≤ α − K ( ρ ) . (4.51) Proof of Proposition 3.
Fixing z and ρ as in Proposition 3, under the assumptions in theproposition we see that estimates (4.46), (4.47), (4.48), (4.49) and (4.50) hold for all ρ ∈ (0 , ρ ] and γ ∈ (0 , ] by Claims 2, 3 and 4.Note first that (4.46), (4.47) and (4.51) imply that E z ( γρ ) . h A z B z + γ − A z B z i ( ρ )and hence, for example, there exists some c > E z ( γρ ) ≤ c h γ A z + γ − (cid:16) A z B z + A z B z (cid:17)i ( ρ ) , (4.52)31or ρ ∈ (0 , ρ ] and γ ∈ (0 , ] (in fact, for γ ∈ (0 , ρ ≤ c ( g ) > g suchthat A z ( ρ ) ≤ c ( g ) h C z + E z + F z + C z B z i ( ρ ) (4.53)for ρ ∈ (0 , ρ ]. Now, recalling (3.18), we have from (4.53) that for some c ( g ) > g ) A z ( ρ/ ≤ c ( g ) h M z + M z B z i ( ρ ) (4.54)and, writing γρ = 2 γ · ρ for 2 γ ≤ , it follows from (4.46), (4.48), (4.49) and (4.52) and the facts that γ, B ( ρ ) ≤ M z ( γρ ) . (2 γ ) (cid:16) M z (cid:0) ρ (cid:1) + A z (cid:0) ρ (cid:1)(cid:17) + (2 γ ) − B z ( ρ ) · h A z + A z i (cid:0) ρ (cid:1) so long as γ ∈ (0 , ], after which, for such γ and for ρ ∈ (0 , ρ ], the estimate (3.20) follows from (4.54)and (4.51) (with α = ), which completes the proof of Proposition 3. (cid:3) Let us now prove the claims:
Proof of Claim 2:
For simplicity, we will suppress the dependence on z = ( x , t ) in what follows.Let us first prove (4.46). Note that for any r ≤ ρ , at any fixed t ∈ I ∗ r , taking v ∈ { u, ∇ d } wehave Z B r | v | dx ≤ Z B ρ (cid:12)(cid:12)(cid:12) | v | − | v | ρ (cid:12)(cid:12)(cid:12) dx + | B r | | v | ρ . ρ Z B ρ (cid:12)(cid:12) ∇| v | (cid:12)(cid:12) dx + (cid:18) rρ (cid:19) Z B ρ | v | dx due to Poincar´e’s inequality (4.16). Since (cid:12)(cid:12) ∇| v | (cid:12)(cid:12) ≤ | v ||∇ v | almost everywhere, H¨older’s inequalitythen implies that k v k B r . ρ k v k B ρ k∇ v k B ρ + (cid:18) rρ (cid:19) k v k B ρ . (4.55)Therefore k v k B r ( . ) . r (cid:0) k v k B r (cid:1) + k v k B r k∇ v k B r ( . ) . (cid:18) (cid:16) ρr (cid:17) (cid:19) k v k B ρ k∇ v k B ρ + 1 r (cid:18) rρ (cid:19) k v k B ρ . Summing over v ∈ { u, ∇ d } , we see that k u k B r + k∇ d k B r . (cid:18) (cid:16) ρr (cid:17) (cid:19) (cid:16) k u k B ρ + k∇ d k B ρ (cid:17) (cid:16) k∇ u k B ρ + k∇ d k B ρ (cid:17) + r ρ (cid:16) k u k B ρ + k∇ d k B ρ (cid:17) . Now integrating over t ∈ I ∗ r (where | I ∗ r | = r ), H¨older’s inequality implies that r C ( r ) . | I ∗ r | (cid:18) (cid:16) ρr (cid:17) (cid:19) (cid:13)(cid:13)(cid:13) k u k B ρ + k∇ d k B ρ (cid:13)(cid:13)(cid:13) ∞ ; I ∗ r (cid:16) k∇ u k Q ∗ ρ + k∇ d k Q ∗ ρ (cid:17) + | I ∗ r | r ρ (cid:13)(cid:13)(cid:13) k u k B ρ + k∇ d k B ρ (cid:13)(cid:13)(cid:13) ∞ ; I ∗ r . r (cid:18) (cid:16) ρr (cid:17) (cid:19) ( ρA ( ρ )) ( ρB ( ρ )) + r ρ ( ρA ( ρ )) , r , setting γ := rρ and noting that 1 ≤ γ − , precisely gives (4.46).Next, to prove (4.47), we use the Poincar´e-Sobolev inequality k g − g r k q ∗ ; B r ≤ c q k∇ g k q ; B r (the constant is independent of r due to the relationship between q and q ∗ ) corresponding to theembedding W ,q ֒ → L q ∗ for q < R ) and q ∗ = q − q . Taking q = 1, at any t ∈ I ∗ r and for v ∈ { u, ∇ d } the H¨older and Poincar´e-Sobolev inequalities give us Z B r | u | (cid:12)(cid:12)(cid:12) | v | − | v | r (cid:12)(cid:12)(cid:12) dx ≤ k u k B r k | v | − | v | r k ; B r . k u k B r k∇ ( | v | ) k B r . k u k B r k v k B r k∇ v k B r . Summing this first over v ∈ { u, ∇ d } at a fixed t and then integrating over t ∈ I ∗ r , we see that r E ( r ) . Z I ∗ r k u k B r (cid:0) k u k B r + k∇ d k B r (cid:1) (cid:0) k∇ d k B r + k∇ d k B r (cid:1) dt . k u k Q ∗ r (cid:13)(cid:13)(cid:13)(cid:0) k u k B r + k∇ d k B r (cid:1) (cid:13)(cid:13)(cid:13) I ∗ r (cid:16) k∇ u k Q ∗ r + k∇ d k Q ∗ r (cid:17) . | I ∗ r | (cid:16) k u k Q ∗ r (cid:17) (cid:13)(cid:13) k u k B r + k∇ d k B r (cid:13)(cid:13) ∞ ; I ∗ r (cid:16) k∇ u k Q ∗ r + k∇ d k Q ∗ r (cid:17) . r ( r C ( r )) ( rA ( r )) ( rB ( r )) = r [ C A B ]( r )which proves (4.47) and completes the proof of Claim 2. (cid:3) Proof of Claim 3:
As in (4.3) of Claim 1, for any t ∈ I ∗ r ( z ) ( r ≤ ρ ) we use Remark 2 to decompose Π := p ( · , t )for almost every x ∈ B ρ ( x ) using a smooth cut-off function ψ equal to one in Ω := B ρ ( x ) andsupported in Ω := B ρ ( x ), so that |∇ ψ | . ρ − and | ∆ ψ | . ρ − , (4.56)as p ( x, t ) = − Z ∇ G x · v ( t ) ψ dy | {z } =: p ( x,t ) + Z G xψ, · v ( t ) dy | {z } =: p ( x,t ) + Z G xψ, p ( · , t ) dy | {z } =: p ( x,t ) with G xψ, := − G x ∇ ψ , G xψ, := 2 ∇ G x · ∇ ψ + G x ∆ ψ and v ( t ) := [ ∇ T · ( u ⊗ u + ∇ d ⊙ ∇ d )]( · , t ) . Our goal is to estimate p ( x, t ) for x ∈ B ρ ( x ).Both p and p contain derivatives of ψ in each term so that the integrand can only be non-zerowhen | y − x | > ρ , and hence for x ∈ B ρ ( x ) one has | x − y | ≥ ρ ⇒ | G x ( y ) | . ρ − and |∇ G x ( y ) | . ρ − . (4.57)33n view of (4.56) and (4.57) and the fact that ψ is supported in B ρ ( x ), we have (omitting thedependence on t , and noting that the constants in the inequalities are independent of t as they comeonly from G x and ψ )sup x ∈ B ρ ( x ) | p ( x ) | . ρ − Z B ρ ( x ) ( | u ||∇ u | + |∇ d ||∇ d | ) dy (4.58) . ρ − Z B ρ ( x ) ( | u | + |∇ d | ) dy ! Z B ρ ( x ) ( |∇ u | + |∇ d | ) dy ! and similarly sup x ∈ B ρ ( x ) | p ( x ) | . ρ − Z B ρ ( x ) | p | dy . (4.59)For p , Young’s inequality for convolutions (where we set R := 2 ρ as in Remark 2) with2 / / /
12 gives k p k ; B ρ ( x ) . (cid:13)(cid:13)(cid:13)(cid:13) | · | (cid:13)(cid:13)(cid:13)(cid:13) ; B ρ (0) (cid:13)(cid:13) ( | u | + |∇ d | )( |∇ u | + |∇ d | ) (cid:13)(cid:13) ; B ρ ( x ) . ρ (cid:13)(cid:13) ( | u | + |∇ d | )( |∇ u | + |∇ d | ) (cid:13)(cid:13) ; B ρ ( x ) and then H¨older’s inequality with 11 /
12 = 1 / / / k p k ; B ρ ( x ) . (cid:18) ρ (cid:13)(cid:13)(cid:13) ( | u | + |∇ d | ) (cid:13)(cid:13)(cid:13) B ρ ( x ) (cid:13)(cid:13)(cid:13) ( | u | + |∇ d | ) (cid:13)(cid:13)(cid:13) B ρ ( x ) (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) B ρ ( x ) (cid:19) . ρ ( ρA ( ρ )) k | u | + |∇ d | k B ρ ( x ) (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) B ρ ( x ) . (4.60)For the following, we fix now any r ∈ (0 , ρ ], and omit the dependence on x , t and z in B r ( x ), B ρ ( x ), I ∗ ( t ), A z , B z , C z and D z (we will retain z in the notation for F z to distinguish it from F = ∇ f ).To first prove (4.48), we note that (4.58) implies (since r ≤ ρ ) that Z B r | p | dx . r ρ − Z B ρ ( | u | + |∇ d | ) dy ! Z B ρ ( |∇ u | + |∇ d | ) dy ! ≤ r ρ − ( ρA ( ρ )) Z B ρ ( |∇ u | + |∇ d | ) dy ! so that, integrating over t ∈ I ∗ r and using H¨older’s inequality, we have r − Z Z Q ∗ r | p | dz . r − r ρ − A ( ρ ) · | I ∗ ρ | ( ρB ( ρ )) = rρ · [( AB ) ]( ρ ) , (4.61)and that (4.59) similarly implies that r − Z Z Q ∗ r | p | dz . rρ − Z I ∗ r Z B ρ | p | dy ! . rρ · D ( ρ ) . (4.62)Finally, integrating (4.60) over t ∈ I ∗ r , H¨older with 1 = 1 / / r − k p k ; Q ∗ r . r − ρ A ( ρ ) k | u | + |∇ d | k Q ∗ ρ (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) Q ∗ ρ . r − ρ A ( ρ ) (cid:0) ρ C ( ρ ) (cid:1) ( ρB ( ρ )) = (cid:16) C ( ρ ) (cid:17) · (cid:18)(cid:16) rρ (cid:17) − A ( ρ ) B ( ρ ) (cid:19) . r/ρ ) α for any α ∈ R , Cauchy’s inequality gives r − k p k ; Q ∗ r . (cid:18) rρ (cid:19) α C ( ρ ) + (cid:18) rρ (cid:19) − α − A ( ρ ) B ( ρ ) . (4.63)Since we want a positive power of γ = r/ρ in the first term and a negative one on the second (becauseit contains B which will be small), we want to take α >
0. Choosing α = 1 purely to make thefollowing expression simpler, since p = p + p + p , we see from (4.61), (4.62) and (4.63) that D ( r ) . rρ · [ D + ( AB ) + C ]( ρ ) + (cid:18) rρ (cid:19) − h A B i ( ρ )which implies (4.48) for γ := rρ ≤ .To prove (4.49), we note that F z ( r ) ≤ F ( r ) + F ( r ) + F ( r ), where we set F j ( r ) := 1 r Z Z Q r | p j || u | dz . To estimate F we use H¨older and (4.60) to see that (in fact, for r ≤ ρ ) Z B r | p || u | dx ≤ k u k B ρ k p k ; B ρ . k u k B ρ · ρ ( ρA ( ρ )) k | u | + |∇ d | k B ρ (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) B ρ ≤ ρ A ( ρ ) k | u | + |∇ d | k B ρ (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) B ρ and hence Cauchy-Schwarz in time gives F ( r ) . r − ρ A ( ρ ) k | u | + |∇ d | k Q ∗ ρ (cid:13)(cid:13) |∇ u | + |∇ d | (cid:13)(cid:13) Q ∗ ρ . r − ρ A ( ρ )( ρ C ( ρ )) ( ρB ( ρ )) = (cid:18)(cid:18) rρ (cid:19) α C ( ρ ) (cid:19) · (cid:18) rρ (cid:19) − − α [ A B ]( ρ ) ! . (cid:18)(cid:18) rρ (cid:19) α C ( ρ ) (cid:19) + (cid:18) rρ (cid:19) − − α [ A B ]( ρ ) ! for any α ∈ R . Taking, say, α = , we have F ( r ) . (cid:18) rρ (cid:19) C ( ρ ) + (cid:18) rρ (cid:19) − [ AB ]( ρ ) . (4.64)Now for F note that, using (4.58), we have (since r ≤ ρ ) Z B r | p || u | dx . ρ − Z B ρ ( | u ||∇ u | + |∇ d ||∇ d | ) dy Z B r | u | dx . ρ − k | u | + |∇ d | k B ρ k |∇ u | + |∇ d | k B ρ ( r ) k u k B r . ρ − r ( ρA ( ρ )) k |∇ u | + |∇ d | k B ρ so that integrating over t ∈ I ∗ r and using H¨older in time we have F ( r ) . r r ρ ( ρA ( ρ ))( ρB ( ρ )) ( r ) = (cid:18) rρ (cid:19) [ AB ]( ρ ) . (4.65)35or F , using (4.59) and H¨older, we see that1 r Z B r | p || u | dx ≤ r ρ Z B ρ | p | dy ! (cid:18)Z B r | u | dx (cid:19) ≤ r ρ Z B ρ | p | dx ! ( ρ ) (cid:18)Z B r ( | u | ) dx (cid:19) (cid:18)Z B r ( | u | ) dx (cid:19) ( r ) which gives us (setting γ := rρ ) F ( r ) . r ρ ( rA ( r )) Z Z Q ∗ ρ | p | dx ! Z Z Q ∗ r | u | dx ! ( r ) ≤ r ρ ( rA ( r )) (cid:0) ρ D ( ρ ) (cid:1) (cid:0) r C ( r ) (cid:1) ( r ) ≤ (cid:18) rρ (cid:19) ( γ − A ) ( ρ ) D ( ρ )( γ − C ) ( ρ ) = (cid:18) rρ (cid:19) A ( ρ ) D ( ρ ) C ( ρ )by (4.51). Hence Young’s inequality implies F ( r ) . (cid:18) rρ (cid:19) (cid:16) A ( ρ ) + D ( ρ ) + C ( ρ ) (cid:17) . (4.66)Adding (4.64), (4.65) and (4.66) and passing to the smallest powers of γ = rρ ( <
1) we see that F z ( r ) . (cid:18) rρ (cid:19) (cid:16) A + D + C (cid:17) ( ρ ) + (cid:18) rρ (cid:19) − [ A ( B + B )]( ρ )which implies (4.49), and completes the proof of Claim 3. (cid:3) Proof of Claim 4:
We will again omit the dependence on z (except in F z ).To estimate A ( ρ ), we use the local energy inequality (1.13) with a non-negative cut-off function φ ∈ C ∞ ( Q ∗ ρ ) which is equal to 1 in Q ∗ ρ , with |∇ φ | . ρ − and | φ t | , |∇ φ | . ρ − . We’ll need to estimate terms which control those that appear on the right-hand side of the local energyinequality (1.13), which we’ll call I - V as follows: I := Z Z Q ∗ ρ ( | u | + |∇ d | ) | φ t + ∆ φ | dz . ρ − k | u | + |∇ d | k ; Q ∗ ρ ( ρ ) . ρ − ( ρ C ( ρ )) ( ρ ) = ρC ( ρ ) . (4.67)Using the assumption (1.11) that ∇ · u = 0 weakly and indicating by g ρ the average of a function g in B ρ , we have II := Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ρ ( | u | + |∇ d | ) u · ∇ φ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ρ h ( | u | − | u | ρ ) + ( |∇ d | − |∇ d | ρ ) i u · ∇ φ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt II . ρ − ( ρ E ( ρ )) = ρE ( ρ ) . (4.68)Clearly we have III := Z Z Q ∗ ρ | pu · ∇ φ | dz . ρ − ( ρ F z ( ρ )) = ρF z ( ρ ) . (4.69)Using the weak divergence-free condition ∇ · u = 0 in (1.11) to write (see (1.2))( u · ∇ ) d = ∇ T · ( d ⊗ u )(at almost every x ) and integrating by parts we have IV := Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ρ u ⊗ ∇ φ : ∇ d ⊙ ∇ d dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ρ [( u · ∇ ) d ] · [( ∇ φ · ∇ ) d ] dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ρ [ ∇ T · ( d ⊗ u )] · [( ∇ φ · ∇ ) d ] dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z I ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z B ρ d ⊗ u : ∇ T [( ∇ φ · ∇ ) d ] dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt , and clearly |∇ T [( ∇ φ · ∇ ) d ] | . |∇ φ ||∇ d | + |∇ φ ||∇ d | . Therefore IV . ρ − Z Z Q ∗ ρ | d || u ||∇ d | dz + ρ − Z Z Q ∗ ρ | d || u ||∇ d | dz . ρ − k d k Q ∗ ρ k u k Q ∗ ρ k∇ d k Q ∗ ρ + ρ − k d k Q ∗ ρ k u k Q ∗ ρ k∇ d k Q ∗ ρ . ρ − k d k Q ∗ ρ ( ρ C ( ρ )) + ρ − k d k Q ∗ ρ ( ρ C ( ρ )) ( ρB ( ρ )) = ρ n ρ − k d k Q ∗ ρ C ( ρ ) + ρ − k d k Q ∗ ρ C ( ρ ) B ( ρ ) o . ρ (cid:16) ρ − k d k Q ∗ ρ (cid:17) n C ( ρ ) + C ( ρ ) B ( ρ ) o , and hence (recalling (3.17)) IV . ρ [ G ( C + C B )]( ρ ) . (4.70)Finally, we have V := Z Z Q ∗ ρ | d | |∇ d | φ dz . k d k Q ∗ ρ k∇ d k Q ∗ ρ ≤ k d k Q ∗ ρ (cid:0) ρ C ( ρ ) (cid:1) = ρ (cid:16) ρ − k d k Q ∗ ρ (cid:17) C ( ρ ) . Therefore, again recalling the definition (3.17), we see that V . ρ [ G C ]( ρ ) . (4.71)Finally, using (4.67) - (4.71), the local energy inequality gives ρ A ( ρ ) . I + II + III + IV + V . ρ [ C + E + F z + G ( C + C B ) + ρ G C ]( ρ )which implies (4.50) and proves Claim 4. (cid:3) eferences [CKN82] L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of theNavier-Stokes equations. Comm. Pure Appl. Math. , 35(6):771–831, 1982.[DHW19] H. Du, X. Hu, and C. Wang. Suitable weak solutions for the co-rotational Beris-Edwardssystem in dimension three. arXiv:1905.08440 (preprint), 2019.[GT01] D. Gilbarg and N. S. Trudinger.
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