Navier–Stokes Cauchy Problem with | v 0 ( x ) | 2 Lying in the Kato Class K 3

: We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum v 0 is weakly divergence free, sup R 3 (cid:82) R 3 | v 0 ( y ) | 2 | x − y | dy < ∞ and | v 0 ( y ) | 2 ∈ K 3 , where K 3 denotes the Kato class. The existence is local for arbitrary data and global if sup R 3 (cid:82) R 3 | v 0 ( y ) | 2 | x − y | dy is small. Regularity and uniqueness also hold.


Introduction
We consider the Navier-Stokes Cauchy problem: In System (1) v is the fluid velocity field,π v is the pressure field of an incompressible viscous fluid, v t := ∂ ∂t v and v · ∇v := v k ∂ ∂x k v. We set |U(y)| |x − y| dy = 0 } .
In a different context, the set K 3 was introduced by Kato in [1]. In [2], Simon studies and develops properties related to the elements of K 3 (see also [3]). For ρ > 0, we set |u(y)| 2 |x − y| dy, and ||u|| 2 wt := sup By the symbol K we mean the set K := {u ∈ L 2 wt , weakly divergence free and |u| 2 ∈ K 3 }.
We set T(v 0 ) := sup ρ>0 t(ρ), where t(ρ) is such that where c is an absolute constant. The definition of T(v 0 ) is well posed for all v 0 ∈ K. Indeed, observing that |v 0 | 2 ∈ K 3 and taking into account that, for fixed ρ, h(t, ρ) → 0 as t → 0 (see Remark 1), one can choose ρ and subsequently t in such a way that (4) is satisfied. We are interested in the following result.

Proposition 1 (Weighted energy relation).
Let v be a solution to (1) enjoying properties (5) and (6). Then the following weighted energy relation holds: for all s < t in (0, T(v 0 )).

Proposition 2 (Uniqueness).
For all v 0 ∈ K a solution of Theorem 1 is unique.
To better explain the aim of the previous theorem, we recall a result by Caffarelli, Kohn, and Nirenberg in [4] and another, based on [4], achieved in [5,6]. In [4], Proposition 1 is a criterion of regularity for a suitable weak solution. As a consequence, one understands that if a suitable weak solution v(t, x), corresponding to an initial datum v 0 ∈ L 2 (R 3 ), admits a possible singularity in (t 0 , x 0 ), then in a neighborhood of the point (t 0 , x 0 ) there is the behavior (see (1.18) in [4]) Instead, in [5,6], recognizing that, via the Hardy-Littlewood-Sobolev inequality, v(t, y) ∈ L 2 (R 3 ) ensures that the Newtonian potential ψ(t, x) := R 3 |v(t,y)| 2 |x−y| dy is finite almost everywhere in x ∈ R 3 , by means of the quoted criterion of regularity the authors prove that such points (t, x) are the center of suitable parabolic cylinders of regularity Q(t, x) for a suitable weak solution v (we stress that in [7], under a suitable assumption of smallness, the parabolic cylinder Q(t, x) of the partial regularity is of the kind (0, ∞) × R 3 − B(0, R), see also [4]). Therefore, the Kato class K 3 appears of some interest since, in connection with (8), the class K 3 has the potential to preserve the solution from the singularity of the kind (8), and in connection with results of [5,6], the properties of K 3 make the potential ψ(t, x) a continuous function of x ∈ R 3 . We point out that, from (4), the existence interval (0, T(v 0 )) in Theorem 1is determined by means of properties of the elements of K. We can iterate the arguments, achieving a )} of existence intervals that by the construction collapse to the empty set. The collapse is due to the fact that ρ m → 0. Hence, in the limit on m one loses the opportunity to take advantage of the smallness related to ||v m 0 || K ρ . Concerning estimate (6) 1 for the pressure field, it is actually stronger, in the sense that one obtains ||π v (t)|| wt ≤ c||v(t)|| ∞ ||v(t)|| wt for all t ∈ (0, T(v 0 )).
We point out that in the case of small data in L 2 wt the existence can be given relaxing the assumptions on the initial datum to v 0 divergence free and belonging to L 2 wt . The result is weaker. In fact, the properties (6) 3,4 do not hold.The initial datum is assumed in weak form, the uniqueness can be discussed following the duality approach furnished in [8], but, as made in [6] We remark that in the hypothesis v 0 ∈ K ∩ L 2 , the results of existence and regularity of Theorem 1, coupled with the uniqueness theorems proved in [6], can be employed to deduce a structure theorem for suitable weak solutions. The set L 2 wt is enclosed in BMO −1 , space detected in [9] as the widest scale-invariant space where problem (1) is well posed (Ḣ [9], or BMO −1 0 (for this space see [10]). In fact, w 0 ∈ BMO −1 0 means w 0 ∈ BMO −1 and where w is the heat solution corresponding to the distribution w 0 . For w 0 ∈ K, by virtue of Lemmas 4 and 6 in Section 3, we get where for all ρ > 0, we have h(t, ρ) → 0 letting t → 0. Hence, we easily get Hence, in connection with these spaces our result does not add a new statement. On the other hand, in the case of scale-invariant norms, to date, a functional dependence between the dimensionless size of the initial datum and the dimensionless size of the existence interval (0, T) is not known and, to the best of our knowledge, the one exhibited in (4) is the first. Actually, in the setting of the scaling of the Navier-Stokes equations, one can consider the dimensionless ratio t 1 2 /ρ, as we make in (4). In (4) this ratio a priori cannot be constant. The ratio changes by means of the size of the other quantities, which are all dimensionless, and they realize the size of the existence interval. In the proof of global existence for small data, being possible a constant ratio t 1 2 /ρ (which follows from the smallness of ||v 0 || wt ), we achieve t 1 2 → ∞ choosing proportionally ρ . So, a priori, one cannot compare the interval of existence given in [9] and the one given in Theorem 1. However, as remarked by the authors of [9], one of the interesting aspects of these results is the strict connection between the metrics employed in the existence theorems and the regularity criteria, such as the ones given in [4], which could be useful for an improvement of the partial regularity.
It is natural to inquire about the connection between K and the scale-invariant spaces In fact, no comparison is possible. This is a consequence of the fact that L 3 is We conclude claiming that an existence theorem of weak solutions corresponding to a datum v 0 ∈ K holds. Roughly speaking, for this goal it is enough to follow the argument lines given in [11,12]. That is, to look for a weak solution v to problem (1) as the sum of two fields u and w. The field w is a smooth solution to a linear problem. Instead, in the ordinary L 2 setting, u is a weak solution to a suitable perturbed Navier-Stokes system. If the goal is the well posedness for the Navier-Stokes Cauchy problem, such an existence result of weak solutions appears of little interest since, by introducing the field u, one loses the advantage of v 0 ∈K. This is why we do not give the proof. The question is different in the context of stability of motions, such as in [13], where thanks to the result in [11] it is possible to show a transition from a stationary regime in Finn's class to a non-stationary one in L(3, ∞) a.e. in t, or to show the converse transition. This paper is organized as follows. In Section 2 we recall some preliminary results. In Section 3 we furnish a priori estimates in the functionals that define ||| · ||| (t,ρ) in (2). In Section 4 we achieve the proof of Theorem 1, Propositions 1 and 2.

Preliminaries
We set H * a(t, x) := H(t, x − y)a(y)dy, We look for a solution to the integral equation where H(t, z) := (4πt) − 3 2 exp[−|z| 2 /4t] is the fundamental solution of the heat equation and E(s, z) is the Oseen tensor, fundamental solution of the Stokes system, with components where E is the fundamental solution of the Laplace equation. For the Oseen tensor the following estimates hold (cf. [14], estimates (VI) on page 215 and (VIII) on page 216, or [15]): for all θ ∈ (0, 1), uniformly in (s, z) where D h z is the symbol of the partial derivatives with respect to z i -variable α i times, i = 1, 2, 3, and h = α 1 + α 2 + α 3 .
We use the method of successive approximations. The lemmas of this and the following section ensure boundedness and convergence of the approximating sequence of velocity fields {v m }. Finally the pressure with the corresponding estimates are recovered by solving a suitable Poisson equation and applying Lemma 3.
Following [16] we state: Proof. The proof is immediate.

Lemma 2.
There exist constants c independent of u such that i.
Let us consider the equation For problem (15) we recall the following result: Lemma 3. Let a and u be divergence free in (15). For a solution π to problem (15) there exist constants c independent of u such that i.
Proof. We note that under our assumptions the following identity holds: ∇a · ∇u T = ∇ · ∇ · (a ⊗ u). Hence, from the representation formula for solutions of (15) and the theory of singular integrals (in particular [17] with weight), one deduces the result.

Lemmas Related to the Functionals Defined on L ∞ , K 3 , L 2
wt Lemma 4. Let a ∈ L 2 wt . Then for the convolution product H * a we get Proof. By the definition of the heat kernel and applying Hölder's inequality, we get where for functions h 0 and h one easily proves the values claimed in (18).
Then there exists a constant c independent of a and b such that for all ρ > 0 for all t ∈ (0, T) .

By our hypotheses we get
Via estimates for I 1 and I 2 we arrive at (19).
Proof. The proof is analogous to the one of the previous lemma. Hence it is omitted.
Then there exists a constant c independent of a(t, x) and b(t, x) such that Proof. Setting y − z = ξ in the convolution product, we have to estimate the integral Employing the Minkowski inequality, we get By virtue of our hypotheses and estimate (10) for the Oseen tensor, we find Hence one easily arrives at (22).

Lemma 9.
Let sup (0,T) t 1 2 ||a(t)|| ∞ + ||b(t)|| wt < ∞. Then there exists a constant c independent of a(t, x) and b(t, x) such that Proof. The proof is analogous to the one of the previous lemma. Hence it is omitted.

Lemma 10.
In the hypotheses of Lemmas 4 and 5 the convolution products H * a and ∇E * (a ⊗ b) are Hölder continuous functions, with exponent θ ∈ [0, 1), Proof. Following the classical proof of the Hölder property for solutions to the heat equation, and the estimate given in Lemma 4, one obtains (24) 1 . We omit further details, as the computations are similar to the following ones for ∇E. Taking estimates (11) into account, applying Hölder's inequality, for the term ∇E * (a ⊗ b)(t, x) we easily get the following estimate, θ ∈ [0, 1), Employing the arguments developed in the proof of Lemma 5, one easily arrives to the estimate Analogously, one proves the Hölder property with respect to time for the term ∇E * (a ⊗ b). Hence, for all η > 0, in (η, T) × R 3 , we have (24) 2 .
Proof. Recalling that |v 0 | 2 ∈ K 3 and Remark 1 for h, we choose ρ and subsequently t in such a way that We denote by T(v 0 ) the supremum of t(ρ) for which (31) holds. Then, by virtue of (27) and Lemma 1, t ∈ [0, T(v 0 )) and uniformly in m ∈ N, we get |||v m ||| (t,ρ) ≤ 2 (1/(2 Employing again the definition given in (3), we also have the following immediate property: P: for any sequence {t p } → 0, one constructs a sequence {ρ p } → 0 such that for all p ∈ N (31) holds and the right-hand side of (32) tends to zero.