Optimality of Serrin type extension criteria to the Navier-Stokes equations

where u = (u1(x, t), · · · , un(x, t)) and π = π(x, t) denote the velocity vector eld and the pressure of the uid at the point x ∈ Rn and time t > 0, respectively, while u0 = u0(x) is the given initial vector eld for u. It is known that for every u0 ∈ Hs ≡ W s,2(Rn) (s ≥ n/2−1), there exists aunique solution u ∈ C([0, T);Hs) to (N-S) for some T > 0. Such a solution is in fact smooth in Rn × (0, T). See, for instance Fujita-Kato [9]. It is an important open question whether T may be taken as T = ∞ or T < ∞. In this direction, Giga [10] gave a Serrin type criterion, i.e., if the solution u satis es the condition


Introduction
The motion of a viscous incompressible uid in R n , n ≥ , is governed by the Navier-Stokes equations: where u = (u (x, t), · · · , un(x, t)) and π = π(x, t) denote the velocity vector eld and the pressure of the uid at the point x ∈ R n and time t > , respectively, while u = u (x) is the given initial vector eld for u.
On the other hand, Beale-Kato-Majda [1] and Beirão da Veiga [2] gave a criterion by means of the vorticity, i.e., if the solution u satis es the condition then u can be extended to a solution in the class C([ , T ′ ); H s (R n )) for some T ′ > T. Later on, the condition (1.4) was relaxed from the L p -criterion to by Kozono-Ogawa-Taniuchi [16]. Moreover, Nakao-Taniuchi [21] gave a similar type of the criterion as (1.3), instead of (1.4) and (1.5) with p = ∞ (θ = ), in such a way that Note that V β admits the following continuous embeddings in the case β = : Futhermore, the author [12] improved theḂ p,∞ -criterion (1.5) to for L r (n < r < ∞) strong solutions to (N-S). Here,V s p,q,θ is a Banach space introduced by De nition 2.1 and has a continuous embeddingḂ p,∞ ⊂V p,∞.θ . The above criteria by means of the vorticity are also important from a view point of scaling invariance. Indeed, since rot u λ = λ rot u(λx, λ t), the spaces L θ ( , ∞; L p ), L θ ( , ∞;Ḃ p,∞ ), L θ ( , ∞;V p,∞,θ ) with /θ + n/p = and L ( , ∞; V ) are scaling invariant for the vorticity with respect to (N-S).
The aim of this paper is to improve the extension criterion (1.2) to the Navier-Stokes equations by means of Banach spaces which are larger thanḂ −α ∞,∞ in the same way that the condition (1.5) was relaxed to (1.6). In fact, we prove that if the solution u to (N-S) on ( , T) satis es the condition either then u can be extended to a solution in the class C([ , T ′ ); H s (R n )) for some T ′ > T. Here,U s p,β,σ is a Banach space introduced by De nition 2.2 and has the following continuous embeddings: Hence, we see that (1.7) and (1.8) may be regarded as a weaker condition than (1.2). Moreover, note that the spaces L θ ( , ∞;U −α ∞, /θ,∞ ) with /θ + α = and L ( , ∞;V ∞,∞, ) are also scaling invariant for solutions u to (N-S). In order to obtain our extension principle, we need a logarithmic interpolation inequality by means oḟ U s p,β,σ : ) .
The present paper is organized as follows. In the next section, we shall state our main results. In section 3 and 4, proofs of our main results are established.

. Function spaces
We rst introduce some notation. Let S = S(R n ) be the set of all Schwartz functions on R n , and S ′ the set of tempered distributions. The L p -norm on R n is denoted by · p. We recall the Littlewood-Paley decomposition and use the functions ψ, ϕ j ∈ S, j ∈ Z, such that Let Z := {f ∈ S; D αf ( ) = for all α ∈ N n } and Z ′ denote the dual space of Z. We note that Z ′ can be identi ed with the quotient space S ′ /P of S ′ with respect to the space of polynomials, P. Furthermore, the homogeneous Besov spaceḂ s p,q := {f ∈ Z ′ ; f Ḃs p,q < ∞} is de ned by the norm See Bergh-Löfström [3, Chapter 6.3] and Triebel [26,Chapter 5] for details. Let C ∞ (R n ) denote the set of all C ∞ functions with compact support in R n and C ∞ ,σ := {ϕ ∈ (C ∞ (R n )) n ; div ϕ = }. Concerning Sobolev spaces we use the notation H s (R n ) for all s ∈ R. Then H s σ is the closure of C ∞ ,σ with respect to H s -norm. In Section 4 we will also use homogeneous Sobolev spacesḢ s (R n ) and note thatḢ s =Ḃ s , for all s ∈ R. We now introduce Banach spacesV s p,q,θ andU s p,β,σ which are larger than the homogeneous Besov spaceṡ B s p,q . These spaces may be regarded as modi ed versions of spaces de ned by Nakao-Taniuchi [22] and Vishik [27].
We see from the following proposition thatV s p,q,θ andU s p,β,σ are extensions ofḂ s p,q andV s p,q,θ , respectively.
Proof. We easily proveV s p,q,θ ⊂V s p,q,θ in (i) by the standard and the reverse Hölder's inequality. The others follow from the de nitions ofḂ s p,q ,V s p,q,θ andU s p,β,σ .

It follows by Proposition 2.3 (i) and (iii) thaṫ
for s ∈ R and ≤ θ < ∞. We observe from the following examples that the continuous embeddings (2.1) are proper if s > −n and ≤ θ < ∞, which is important in terms of Theorem 2.9.
Indeed, sincef ∈ L ∞ holds, we obtain f ∈ Z ′ . We easily see that On the other hand, for any N = , , · · · , there exists k N ∈ N such that k θ+ where C is dependent only on n, s and θ. Thus, it follows that Indeed, sinceĝ ∈ L ∞ holds, we obtain g ∈ Z ′ . We easily see that For any N = , , · · · , we take k N ∈ N such that k θ+ where C is dependent only on n and θ. Hence, we have On the other hand, it follows that max |j|≤N js ϕ j * g ∞ ≤ max where C is dependent only on n and s. Thus, we obtain (ii) Let s ∈ R, ≤ β < ∞ and ≤ p, σ ≤ ∞, and let X be a normed space of distributions on Z. Assume that X satis es the following conditions: (C1) X → Z ′ ; (C2) there exists a constant K > such that f (· − y) X ≤ K f X for all f ∈ X and all y ∈ R n ; (C3) there exists a constant K > such that ρ * f X ≤ K ρ f X for all ρ ∈ Z and all f ∈ X; (C4) there exist s , s ∈ R satisfy s < s < s and K > such that Then, X →U s p,β,σ holds.

Remark 2.6.
(1) In the rst part of Theorem 2.5, the assumption s < s < s is essential. If either of s or s tends to s , then the constant C appearing on the right hand side diverges to in nity.

De nition 2.7. Let s > n/ − and let u ∈ H s σ . A measurable function u on R n × ( , T) is called a strong solution to (N-S) in the class CLs( , T) if
then the estimate (2.8) is easily obtained, so that the solution can be extended beyond t = T.
(2) From Example 2.4, the proper embeddingsḂ ∞,∞ ⊂V ∞,∞, ⊂U ∞, / ,∞ hold. Hence, Theorem 2.9 (ii) may be regarded as an extension of theḂ ∞,∞ -criterion given by Kozono-Ogawa-Taniuchi [16] for s > n/ . On the other hand, it seems to be di cult to obtain the same result as in Theorem 2.9 (ii) under the condition This stems from inapplicability of Lemma 4.1 with α = .
As an immediate consequence of the above Theorem 2.9, we have the following blow-up criteria of strong solutions:

and let u ∈ H s σ . Assume that u is a strong solution to (N-S) in the class CLs( , T). If T is maximal, i.e., u cannot be extended in the class CLs( , T ′ ) for any T ′ > T, then it holds that
In particular, we have lim sup t→T u(t) U −α ∞, /θ,∞ = ∞.

Proof of Theorem 2.5 (i).
We rst consider the case ≤ σ < ∞. By the de nition of the Besov space, we obtain where C is dependent only on s and s . For S , in the same way as (3.2), we have where C is dependent only on s and s . We nally estimate S . By De nition 2.2, it clearly follows that Therefore, using the same argument as in the previous case ≤ σ < ∞, we get (2.6).
In order to prove the second part of Thorem 2.5, we use the following Lemma.
Lemma 3.1. Let ρ ∈ Z and Let X be a normed space. Assume that X satis es conditions (C1) and (C2) given in Theorem 2.5 (ii). Then, it holds that ρ * g ∈ L ∞ for all g ∈ X. (3.7) Proof. By (C1), we get that for all ϕ ∈ Z, there exists a constant C = C(ϕ) > such that |g(ϕ)| ≤ C g X for all g ∈ X. (3.8) Assume that (3.8) does not hold. Then, there is ϕ ∈ Z with the following property: for each positive integer N, there is a g N ∈ X such that |g N (ϕ )| > N g N X . (3.9) Letting h N := g N N g N X (∈ X), we obtain h N X = N − → as N → ∞, which implies h N → in X. By (C1), this convergence holds in Z ′ . On the other hand, by (3.9), which contradicts h N → in Z ′ . Thus we get (3.8).
We are now in position to prove the second part of Theorem 2.5 and follow arguments given by Nakao-Taniuchi [21] and the author [12].
We rst consider the case ≤ σ < ∞. The left-hand side of (3.12) can be estimated from below as follows. Noting that suppΦ (3.13) Concerning the second term on the right-hand side of (3.13), we obtain j=N,N+ js σ ϕ j * Φ N * g σ p ≥ −|s |σ Ns σ j=N,N+ (3.14) As in (3.14), similar estimates hold when replacing N and N + by −N and −N − , respectively. Summarizing (3.13), (3.14) we obtain that Next, we estimate the rst term on the right-hand side of (3.12). From Young's inequality and Hölder's inequality, it holds that where C depends only on n and s . In the same way as (3.16), we have where C depends only on n and s . In the end, from (3.16) and (3.17), we get that This implies g U s p,β,σ ≤ C g X for all g ∈ X, i.e., the embedding X →U s p,β,σ . In the case σ = ∞, we obtain, instead of (3.13), Therefore, by using the same argument as in the case ≤ σ < ∞, we get This proves Theorem 2.5 (ii).