A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg, as a corollary, under suitable assumptions of local character on the initial data, we prove a behavior in time of the $L^\infty_{loc}$-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator. Besides its own interest, the result is a main tool to study the spatial decay estimates of a suitable weak solution, performed in paper F. Crispo and P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem (submitted).


Introduction
We deal with the Navier-Stokes Cauchy problem u t + u · ∇u + ∇π u = ∆u, ∇ · u = 0, in (0, T ) × R 3 , u(0, x) = u 0 (x) on {0} × R 3 . (1.1) In system (1.1) u is the kinetic field, π u is the pressure field, u t := ∂ ∂t u and u · ∇u := u k ∂ ∂x k u. We investigate on the partial regularity of a suitable weak solution, and we detect a new sufficient condition for the existence of a regular solution. Our results are in the wake of the ones obtained in [1] and, for small data, in [3]. As in [2,3,6], our study attempts to highlight what is possible to obtain, without extra condition, in the setting of the L 2 -theory. In this connection, although it is not our chief aim, we like to point out that our results could lead to a sort of structure theorem in the space-time cylinder. To be more precise in the claim we recall the well known Leray's structure theorem related to a weak solution. Leray's theorem claims that there exist an interval * Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", via Vivaldi 43, 81100 Caserta, Italy.
Email: francesca.crispo@unicampania.it; paolo.maremonti@unicampania.it of regularity of the kind (θ, ∞) and a sequence of intervals of regularity included in (0, θ) whose complementary set on (0, θ) is a set of zero 1 2 -Hausdorff measure. Mutatis mutandis, the results of [1] (see below Theorem 1.4) and of this note give a sort of structure theorem for a suitable weak solution related to the Cauchy problem. More precisely, under a suitable assumption for the initial data, in Theorem 1.4 it is proved that a suitable weak solution is regular for all t > 0 in the exterior of a ball with radius R 0 . In this note we prove that, almost everywhere, a point (t, x) ∈ (0, θ) × B(R 0 ) is the center of a parabolic neighborhood of regularity for a suitable weak solution. Hence in (0, θ) × B(R 0 ) there is at most a sequence of open sets of regularity, whose complementary set in (0, θ) × B(R 0 ) has at most zero 1-Hausdorff measure.
To better state the details of our main results, we split the introduction in two short subsections. In the first one we recall some definitions and notation following the ones in [1]. Then we recall two fundamental regularity results obtained in [1], and, with an alternative proof, in [12], and their consequences. In the second subsection we give the statement of our results.

Suitable weak solutions
We start by recalling the following: Definition 1.1 Let u 0 ∈ J 2 (R 3 ). A pair (u, π u ), such that u : (0, ∞) × R 3 → R 3 and π u : (0, ∞) × R 3 → R, is said a weak solution to problem (1.1) if i) for all T > 0, u ∈ L 2 (0, T ; J 1,2 (R 3 )) and π u ∈ L 5 3 ((0, iii) for all t, s ∈ (0, T ), the pair (u, π u ) satisfies the equation: In [1] in order to investigate on the regularity of a weak solution it is introduced an energy relation having a local character: is said a suitable weak solution if it is a weak solution in the sense of the Definition 1.1 and, moreover, for all t ≥ σ, for σ = 0 and a.e. in σ ≥ 0, and for all nonnegative φ ∈ C ∞ 0 (R × R 3 ).
In [1] and [7] the following existence result is proved: there exists a suitable weak solution.
As a consequence of the inequality (1.2) and of the existence theorem one gets Let us recall the definition of singular point for a weak solution.
Definition 1. 3 We say that (t, x) is a singular point for a weak solution (u, π u ) if u / ∈ L ∞ in any neighborhood of (t, x); the remaining points, where u ∈ L ∞ (I(t, x)) for some neighborhood I(t, x), are called regular.
In paper [1], in connection with the regularity of a suitable weak solution, the authors furnish two regularity criteria. The first is Proposition 1 (or Corollary 1, p.776) on p.775 : Proposition 1.1 Let (u, π u ) be a suitable weak solution in some parabolic cylinder Q r (t, x). There exist ε 1 > 0 and c 0 > 0 independent of (u, π u ) such that, if where c 1 := c 0 ε 2 3 1 . In particular, a suitable weak solution u is regular in Q r 2 (t, x).
In [1] this result is used to prove another regularity criterion, that is Proposition 2 on p.776: There is a constant ε 3 > 0 with the following property. If (u, π u ) is a suitable weak solution in some parabolic cylinder Q * r (t, x) and then (t, x) is a regular point.
The above criterion is employed to get the following two main results (respectively, Theorem B on page 772 and Theorem D on page 774 in [1]) Theorem 1.2 For any suitable weak solution the set S of singular points has onedimensional parabolic Hausdorff measure equal to zero. Theorem 1.3 There exists an absolute constant L 0 > 0 with the following property.
then there exists a suitable weak solution to (1.1) which is regular in the region There is a difference in the meaning of the above theorems. By means of Theorem 1.2 it is given a geometric measure of the possible set S of singular points. By means of Theorem 1.3 it is furnished the existence of a suitable weak solution to (1.1) having finite the following scaling invariant metric: Finally, as a corollary of the latter result, in [1] the authors prove the following (Corollary p. 820 in [1]) Theorem 1.4 Let (u, π u ) be a suitable weak solution assuming initial data u 0 . Suppose that ||∇u 0 || L 2 (|x|>R) < ∞. Then, there exists a R 0 > R such that, for all δ > 0, u ∈ L ∞ ((δ, ∞) × {x : |x| > R 0 }) .

The aims of this note.
We work in the setting of the results of Theorem 1.3 and Theorem 1.7 (below) already proved in [3]. Both these theorems work with a scaling invariant norm that leads to (1.11) provided that at the initial instant the weighted norm, that is (1.10), is small in a suitable sense. The consequence of the smallness is the existence of a regular solution global in time.
In this note we study the existence of a suitable weak solution that at least locally in time satisfies the regularity criterion of Proposition 1.1 and, as a consequence, is locally a regular solution. Also in this case the result follows from the assumption that the weighted norm (1.12) of the initial data is finite, but, contrary to Theorem 1.3 and Theorem 1.7, we do not require smallness. As a consequence we are able to deduce the regularity only locally in time.
where the constant c is independent of u 0 , x and v 0 . Then there exists a t(x) > 0 such that (1.14) with c independent of τ .
We give some comments. Firstly we observe that Theorem 1.5 seems similar to Theorem 1.3. The difference is in the fact that we do not require condition (1.10) to the initial data, but the weaker condition (1.13), that is almost everywhere satisfied by means of u 0 ∈ J 2 (Ω). The theorem establishes a result of local regularity for a suitable weak solution of (1.1). The local character is expressed in (1.14) either by the fact that the solution is L ∞ just on the parabolic cylinder, and by the fact that the height of the cylinder depends on x, through t(x).
In the way specified below, the set E represents the new aspect of our result of local regularity stated with an initial data in J 2 (R 3 ). Actually, if we consider u 0 ∈ J 2 (R 3 ), then the Riesz potential is well posed a.e. in x ∈ R 3 . This claim is consequence of the fact that, by the Hardy-Littlewood-Sobolev theorem, the following transformation is well defined: Then, for all q ∈ [1, 3) and for any compact Hence it is almost everywhere finite. Having premise that, denoting by {u k 0 } a sequence of smooth functions converging to u 0 in L 2 (R 3 ), for example the mollified of u 0 , for x ∈ R 3 and k ∈ N we define the sequence By Hardy-Littlewood-Sobolev theorem (see Lemma 2.6), it is easy to verify that the sequence {ψ k } converges to zero almost everywhere in x ∈ E ⊆ R 3 . This makes satisfied almost everywhere in x the assumption (1.13) and E is the set indicated in Corollary 1.2. We prove that for any x ∈ E there exists a t(x) > 0 such that M( 7 6 s, x, r) ≤ ε 1 for suitable r and for any s ∈ (0, t(x)). This result, by means of Proposition 1.1, ensures the regularity in Q r 2 ( 7 6 s, x), for any s ∈ (0, t(x)). Therefore, if we denote by S x the projection onto R 3 of the set S of singular points given in Theorem 1.2 (whose one-dimensional Hausdorff measure is zero from the same theorem), throughout Corollary 1.5 we can claim that S ⊆ R 3 \ E. This last claim makes clear that we do not improve the regularity exhibited in [1], but we investigate on the existence of a size, as function of x belonging to E, of the parabolic neighborhood of regularity of a weak solution. In Corollary 1.2 it is claimed a dependence on σ of the set E: this is due to the fact that we have to employ both (1.2) and the continuity on the right in L 2 -norm of the weak solution.
We observe that if Ω ≡ R 3 then Theorem 1.6 gives the existence of a regular solution (u, π u ) on (0, T 0 ) × R 3 .

Corollary 1.3 Let u(t, x) be a suitable weak solution. For any B(R) and for any
Theorem 1.7 Let u(t, x) be a suitable weak solution, and assume also that ess sup is sufficiently small. Then, (u, π u ) is regular for all t > 0 and it is unique up to a function c(t) for the pressure field.
The last theorems are the regular solutions counterpart of Theorem 1.5 and Corollary 1.2, provided that the assumptions on the data are stronger than the simple assumption u 0 ∈ J 2 (R 3 ). The theorems work in the light of the scaling invariant weighted norm (1.18). Theorem 1.6 establishes a local existence result stated by requiring a "suitable closeness", in the weighted norm (1.18), of the initial data u 0 ∈ L 2 (R 3 ) to a smooth function v 0 . As the existence is achieved on the element v 0 of the approximation which is close to u 0 in the metric (1.18), we are not able to give a size of T 0 by means of u 0 , but (0, T 0 ) is just (a priori) a subinterval of existence of the smooth solution (v, π v ) corresponding to v 0 . In this connection we point out that the above question on the size of T 0 is the same that we meet assuming the data u 0 in J 3 (Ω) or in L 3 (Ω) ⊂ L(3, ∞), respectively completion of C 0 (Ω) in L 3 (Ω) and in L(3, ∞)(Ω). Both these spaces are scaling invariant and in order to prove the existence local in time we need an auxiliary function, say, u 0 which is close to u 0 in the metric of L 3 or L(3, ∞) and u 0 ∈ X, where X is a function space adequate to ensure the existence of a regular solution on some interval (0, T 0 ). This is an aspect developed with details in [5]. We conclude that in the statement of Theorem 1.6 we can substitute J 1,2 with any space X which is suitable to ensure the existence of a regular solution corresponding to v 0 . Corollary 1.3 makes operational condition (1.21) on a suitable subdomain. Indeed the existence of the domain Ω ε ⊆ B(R) follows from the construction of a sequence {ψ k } almost everywhere converging to zero and the Severini-Egorov theorem. Theorem 1.7 furnishes a global existence result just requiring a smallness condition. It is also an immediate consequence of our previous result in [3].

Preliminaries
Below we recall some results which are fundamental for our aims.
Lemma 2.1 Suppose that |x| β u ∈ L 2 (R 3 ) and |x| α ∇u ∈ L 2 (R 3 ). Also Then, with a constant c independent of u, the following inequality holds: Lemma 2.2 Assume that K is a singular bounded transformation from L p into L p , p ∈ (1, ∞), of Calderón-Zigmund kind. Then, K is also a bounded transformation from L p into L p with respect to the measure (µ + |x|) α dx, µ ≥ 0, provided that α ∈ (−n, n(p − 1)).

Lemma 2.3
Assume that (u, π u ) is a suitable weak solution. Then the pressure field admits the following representation formula

2)
and the following holds:
Proof. The result is due to Leray, see [9].
For µ ≥ 0 we define the functionals When no confusion arises, we omit some or all the dependences on (v, t, x, µ). For µ ≥ 0, we call weighted energy.
Lemma 2.5 Let (v, π v ) be the regular the solution of Lemma 2.4. Then, for all µ > 0, the following weighted energy relation and weighted energy inequality hold:
Proof. Identity (2.7) can be formally obtained by multiplying equation (1.1) 1 by vp and integrating by parts on (0, t) × R 3 . Let us show that it is well posed for any µ > 0. We start by remarking that in our hypotheses on v 0 we get E (0, x, µ) < ∞ for all x ∈ R 3 and µ ≥ 0. By multiplying equation (1.1) 1 by vp and integrating by parts on (0, t) × R 3 , we obtain Let us show that the right-hand side is well defined. Applying Hölder's inequality and inequality (2.1), we get From the representation formula (2.2), after integrating by parts, we get Hence, applying Hölder's inequality and employing Lemma 2.2, we deduce Applying again Hölder's inequality and subsequently (2.1), we deduce the following estimate: Hence from (2.9) and via estimates for terms J 1 and J 2 we obtain the integral inequality (2.8), from which, thanks to the regularity of v, it is easy to deduce that (2.7) holds for all µ > 0 and for all t ∈ [0, T ).
Lemma 2.6 -Let u 0 ∈ J 2 (R 3 ). There exists a set E such that R 3 − E has zero Lebesgue measure, and for all x ∈ E and for all η > 0 there exists a u 0 ∈ J 1,2 (R 3 ) such that

10)
Moreover, for all R > 0 and ε > 0 there exists Ω ε ⊆ E such that meas(B(R) − Ω ε ) < ε and Proof. We denote by {u k } the mollified functions of u 0 . It is known that By the Hardy-Littlewood-Sobolev theorem we get, for r ∈ [1, 3), Hence, the sequence {ψ k } converges to zero in L r (K), for all r ∈ [1, 3). In particular, there exists a subsequence {ψ k j } which converges to zero almost everywhere in x ∈ K. We denote by {K ν } a sequence of compact sets such that K ν ⊂ K ν+1 and ∪ ν∈N K ν = R 3 . By virtue of the above convergence, we denote E ν ⊆ K ν the set of the convergence almost everywhere of the sequence {ψ k j }. Then, by means of Cantor's diagonal method, we construct a sequence {ψ ℓ } which converges to 0 for all x ∈ E := ∪ ν∈N E ν . Hence for all x ∈ E and η > 0 there exists a ψ ℓ ∈ {ψ ℓ } such that u 0 verifies (2.10). Property (2.11) is a consequence of the above construction and of the Severino-Egorov theorem. The lemma is completely proved.

Local in time weighted energy inequality for a suitable weak solution
In this section we prove that any suitable weak solution admits at least locally in time a weighted energy inequality with µ = 0. Actually, the following lemma holds with E (u, t, x) and D(u, τ, x) defined in (2.5).
Proof. The proof of estimate (3.1) reproduces in a suitable way an idea employed in [2]. This idea follows the Leray-Serrin arguments employed for the proof of the energy inequality in strong form. The proof is achieved by means of five steps. We set w := u − v and π w := π u − π v , where (u, π u ) is the suitable weak solution and (v, π v ) the regular solution corresponding to v 0 and furnished by Lemma 2.4. The first four steps are devoted to prove the following inequality Step 1. -We start proving that for all t > 0 In the energy inequality (1.2) we set φ(τ, y) := (|x−y| 2 +µ 2 ) − 1 2 h R (y)k(τ ) ∈ C ∞ 0 (R×R 3 ), with h R and k such that We get Since π u , u 2 ∈ L 5 3 (0, T ; L 5 3 (R 3 )), applying Hölder's inequality and employing the decay of ∇h R , ∆h R , for all t > 0, we get F (t, R) = o(R). We estimate the terms I i , i = 1, 2. Since µ > 0, by virtue of the integrability properties of a suitable weak solution, applying Lemma 2.1 we get For I 2 , applying the Hölder's inequality and Lemma 2.2, we obtain Hence, as in the previous case, applying Lemma 2.1, we get Employing the estimates obtained for I i , i = 1, 2, via the Lebesgue dominated convergence theorem, in the limit as R → ∞, for all t > 0 we deduce the inequality (3.3).
Step 3. -Setting w := u − v and π w := π u − π v , let us derive the following estimate We remark that from the representation formula (2.2) and regularity of v we get that π w = π 1 + π 2 , We sum estimates (2.7) and (3.3), then we add twice formula (3.7). written for s = 0.
Recalling the definition of (w, π w ) and formula (3.9), after a straightforward computation we get The term F 1 admits the same estimate as I 1 and I 2 given in Step 1, hence we get For term F 2 we estimate the first two terms in a different way from the last. Taking the representation formula of π 2 into account, we get Hence, applying the same arguments employed in Lemma 2.5 to estimate J 1 and J 2 , we get For the last term in F 2 , applying Hölder's inequality, we get By virtue of estimate (2.1), applying Young's inequality we deduce: Hence, we obtain Finally, applying Young's inequality, we get Estimates for F 1 , F 2 and (3.10) furnish the integral inequality (3.8).