aa r X i v : . [ m a t h . A P ] S e p SMALL ˙ B − ∞ , ∞ IMPLIES REGULARITY
TAOUFIK HMIDI AND DONG LIA bstract . We show that smallness of ˙ B − ∞ , ∞ norm of solution to d -dimensional ( d ≥
3) incompressibleNavier-Stokes prevents blowups.
1. I ntroduction
In recent [9], Farhat, Gruji´c and Leitmeyer proved that any unique L ∞ mild solution to 3D Navier-Stokes equation cannot develop finite-time blowups if the B − ∞ , ∞ norm is su ffi ciently small (near first pos-sible blowup time). This result is perhaps a bit surprising in view of the illposedness result of Bourgain-Pavlovi´c [3]. The proof in [9] has a strong geometric flavor, and in particular relies on a geometricregularity criteria and characterization of the super-level sets developed in the series of works [6, 11, 10].We refer the readers to the introduction in [9] and the references therein (see also [1]–[13]) for moredetails on these techniques and also related developments. The purpose of this note is to revisit thisproblem from the point of view of Littlewood-Paley calculus. In particular we will give a streamlinedproof for all dimensions d ≥ d -dimensional Navier-Stokes Equation (NSE): ∂ t v + ( v · ∇ ) v = ∆ v − ∇ p , ( t , x ) ∈ (0 , ∞ ) × R d , ∇ · v = , v (cid:12)(cid:12)(cid:12)(cid:12) t = = v . (1.1) Theorem 1.1.
Let d ≥ . Suppose v is a smooth solution to (1.1) and let T > be the first possibleblow-up time. There exists a positive constant m depending only on the dimension d, such that if thesolution v satisfies sup t ∈ ( T − ǫ, T ) k v ( t ) k ˙ B − ∞ , ∞ ≤ m , for some < ǫ < T , then T is not a blow-up time, and the solution can be continued past T .Remark . Here to allow some generality we do not specify the particular class of smooth solution.As an example one can consider as in [9] the unique mild solution emanating from L ∞ initial data. Bysmoothing (cf. [7]) the solution is immediately in W k , ∞ for all k . Other classes of solutions can also beconsidered and we will not dwell on this issue here.We gather below some notation used in this note. Notation.
For any two quantities X and Y , we denote X . Y if X ≤ CY for some constant C >
0. Thedependence of the constant C on other parameters or constants are usually clear from the context and wewill often suppress this dependence.We will need to use the Littlewood–Paley (LP) frequency projection operators. To fix the notation, let φ ∈ C ∞ c ( R n ) and satisfy0 ≤ φ ≤ , φ ( ξ ) = | ξ | ≤ , φ ( ξ ) = | ξ | ≥ / . Let φ ( ξ ) : = φ ( ξ ) − φ (2 ξ ) which is supported in ≤ | ξ | ≤ . For any f ∈ S ( R n ), j ∈ Z , define [ P ≤ j f ( ξ ) = φ (2 − j ξ ) ˆ f ( ξ ) , d P j f ( ξ ) = φ (2 − j ξ ) ˆ f ( ξ ) , ξ ∈ R n . Sometimes for simplicity we write f j = P j f , f ≤ j = P ≤ j f . Note that by using the support property of φ ,we have P j P j ′ = | j − j ′ | >
1. The Bony paraproduct for a pair of functions f , g take the form f g = X i ∈ Z f i ˜ g i + X i ∈ Z f i g ≤ i − + X i ∈ Z g i f ≤ i − , where ˜ g i = g i − + g i + g i + . For s ∈ R , 1 ≤ p ≤ ∞ , the homogeneous Besov ˙ B s ∞ , ∞ norm is given by k f k ˙ B s ∞ , ∞ = sup j ∈ Z (cid:0) js k P j f k ∞ (cid:1) . We will use without explicit mentioning the simple estimate: k e t ∆ P j f k L ∞ ( R d ) . e − c j t k P j f k L ∞ ( R d ) , ∀ t > , where c > d .2. P roof of T heorem Lemma 2.1.
Let γ > . Then for any j ∈ Z , we have k P j (( v · ∇ ) v ) k ∞ . j (2 − γ ) k v k ˙ B − ∞ , ∞ k v k ˙ B γ ∞ , ∞ . Proof of Lemma 2.1.
Although this is utterly standard we give a proof for completeness. By paraproductdecomposition, we have( v · ∇ ) v = X l ∈ Z ( v ≤ l − · ∇ ) v l + X l ∈ Z ( v l · ∇ ) v ≤ l − + X l ∈ Z ( v l · ∇ )˜ v l = : A + B + C , where ˜ v l = v l − + v l + v l + . Then by frequency localization, we have k P j ( A ) k ∞ . X | l − j |≤ k v ≤ l − · ∇ v l k ∞ . j k v k ˙ B − ∞ , ∞ · j (1 − γ ) k v k ˙ B γ ∞ , ∞ . Similar estimate hold for B . Now for the estimate of C , note that by using divergence-free property wecan write ( v l · ∇ )˜ v l = ∇ · ( v l ⊗ ˜ v l ) and this gives k P j ( C ) k ∞ . j X l ≥ j − − l k v l k ∞ · k ˜ v l k ∞ · γ l · − l ( γ − . j (2 − γ ) k v k ˙ B − ∞ , ∞ k v k ˙ B γ ∞ , ∞ . Here we used the assumption γ > (cid:3) Lemma 2.2.
Suppose v = v ( t ) is a smooth solution to (1.1) on some time interval [0 , T ] with smoothinitial data v . Let γ > . There exists constants C > , δ > which depend only on ( γ, d ) , such that if sup ≤ t ≤ T k v ( t ) k ˙ B − ∞ , ∞ ≤ δ , then max ≤ t ≤ T k v ( t ) k ˙ B γ ∞ , ∞ ≤ C k v k ˙ B γ ∞ , ∞ . Proof of Lemma 2.2.
Write v j = P j v . Then ∂ t v j − ∆ v j = − P j (cid:0) Π (( v · ∇ ) v ) (cid:1) , where Π is the usual Leray projection operator. Then for any t >
0, by using Lemma 2.1, we have k v j ( t ) k ∞ . e − c j t k v j (0) k ∞ + Z t e − c · j ( t − s ) j (2 − γ ) k v ( s ) k ˙ B − ∞ , ∞ k v ( s ) k ˙ B γ ∞ , ∞ ds . e − c j t k v j (0) k ∞ + (1 − e − c j t ) · − j γ · sup ≤ s ≤ t k v ( s ) k ˙ B − ∞ , ∞ · max ≤ s ≤ t k v ( s ) k ˙ B γ ∞ , ∞ . ESOV REGULARITY 3
This implies that for some constants ˜ C >
0, ˜ C > γ, d ),max ≤ t ≤ T k v ( t ) k ˙ B γ ∞ , ∞ ≤ ˜ C k v k ˙ B γ ∞ , ∞ + ˜ C · sup ≤ t ≤ T k v ( t ) k ˙ B − ∞ , ∞ · max ≤ t ≤ T k v ( t ) k ˙ B γ ∞ , ∞ . The result obviously follows. (cid:3)
Proof of Theorem 1.1.
Choose γ = / m = δ as specified in Lemma 2.2. Consider the solution v = v ( t ) on the time interval [ T − ǫ, T − η ], where η > v ( T − ǫ )as initial data), we then obtain uniform estimate on k v k ˙ B γ ∞ , ∞ independent of η . A standard argument thenimplies that v must be regular beyond T . (cid:3) A cknowledgements D. Li was supported by an Nserc grant. T. Hmidi was partially supported by the ANR project DyficoltiANR-13-BS01-0003- 01. R eferences [1] K. Abe and Y. Giga, Y, Analyticity of the Stokes semigroup in spaces of bounded functions,
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63, no. 1 (1934), 193–248.(T. Hmidi) IRMAR, U niversit ´ e de R ennes
1, C ampus de B eaulieu , 35 042 R ennes cedex , F rance E-mail address : [email protected] (D. Li) D epartment of M athematics , U niversity of B ritish C olumbia , V ancouver BC C anada
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