A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient
aa r X i v : . [ m a t h . A P ] J un A NEW LOCAL REGULARITY CRITERION FOR SUITABLE WEAK SOLUTIONS OF THENAVIER–STOKES EQUATIONS IN TERMS OF THE VELOCITY GRADIENT
HI JUN CHOE & JOERG WOLF & MINSUK YANGA
BSTRACT . We study the partial regularity of suitable weak solutions to the three dimensional in-compressible Navier–Stokes equations. There have been several attempts to refine the Caffarelli–Kohn–Nirenberg criterion (1982). We present an improved version of the CKN criterion with adirect method, which also provides the quantitative relation in Seregin’s criterion (2007).
1. I
NTRODUCTION
We consider the Navier–Stokes equations ( ∂ t − ∆) v + ( v · ∇ ) v + ∇ p = f in Ω × ( T ) ∇ · v = Ω × ( T ) (1)where Ω ⊂ R is a bounded domain with C boundary and T >
0. The state variables v and p denote the velocity field of the fluid and its pressure. We complete the above equations by thefollowing boundary and initial conditions v = ∂ Ω × ( T ) v = v in Ω × { } where the initial velocity v is sufficiently regular. Throughout this paper, we assume that ( v , p ) is a suitable weak solution to this problem and the definition will be given in the next section.There are a huge number of important papers that contribute to the regularity problem ofsuitable weak solutions to the Navier–Stokes equations and there are many good survey papersand books. So, we only mention a few of them. Scheffer [
8, 9 ] introduced partial regularity forthe Navier–Stokes system. Caffarelli, Kohn and Nirenberg [ ] further strengthened Scheffer’sresults. Lin [ ] gave a new short proof by an indirect argument. Neustupa [ ] and Ladyzhen-skaya and Seregin [ ] investigated partial regularity. Choe and Lewis [ ] studied singularset by using a generalized Hausdorff measure. Escauriaza, Seregin, and Šverák [ ] provedthe marginal case of the so-called Ladyzhenskaya–Prodi–Serrin condition based on the uniquecontinuation theory for parabolic equations. Gustafson, Kang, and Tsai [ ] generalize severalpreviously known criteria. Among the many important regularity conditions, the following criterion plays an importantrole because it gives better information about the possible singular points: There exists anabsolute positive constant ε such that z = ( x , t ) ∈ Ω × ( T ) is a regular point if(2) lim sup r → r − ¨ Q ( z , r ) |∇ v | d y ds < ε where Q ( z , r ) denotes the parabolic cylinder B ( x , r ) × ( t − r , t ) ⊂ R × R .There have been several attempts to refine this criterion. In particular, Seregin [ ] weakenthe above condition as follows: for each 0 < M < ∞ there exists a positive number ε ( M ) suchthat z ∈ Ω × ( T ) is a regular point iflim sup r → r − ¨ Q ( z , r ) |∇ v | d y ds ≤ M lim inf r → r − ¨ Q ( z , r ) |∇ v | d y ds < ε ( M ) .(3)The proof was done by an indirect argument, which has been widely used as an effective wayto prove such kind of regularity theorems in the field of nonlinear PDEs. The proof goes asfollows. If the theorem is false, then there should exist a sequence of suitable solutions suchthat the scaled quantity r − ¨ Q ( z , r ) |∇ v n | d y ds tends to zero on a fixed particular cylinder centered at a singular point z . One can show thatthe uniform boundedness occurs to ensure a compactness lemma and its sub-sequential limitmust be regular enough at the point z , wihch gives a contradiction to the fact that z is a singularpoint. By this argument one can know the theorem is true so that ε ( M ) should exist. However,the argument does not provide any specific information about ε ( M ) , even the quantitativedependence on M is unclear.In this paper, we shall give a new refined local regularity criterion of suitable weak solutionsto the Navier–Stokes system with a direct iteration method so that our theorem shows a reverserelation between M and ε ( M ) and gives at least a quantitative upper bound of ε ( M ) in termsof M . For simplicity we use the following notation. Definition 1.
For / ≤ q ≤ , we defineE q ( z , r ) = r − + q ¨ Q ( z , r ) |∇ v | q d y dsand denote E q ( z ) = lim sup r → E q ( z , r ) and E q ( z ) = lim inf r → E q ( z , r ) We omit the subscript q when q = . EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 3
Here are our main results.
Theorem 1.
Let / ≤ q < and f = . There exists a positive number ε such that z ∈ Ω × ( T ) is a regular point if E q ( z ) ( − q ) / ( q − ) E q ( z ) < ε .The range 9 / ≤ q ≤ / f = f , one caneasily show that the contribution from f is small enough so that the theorem is still true fornonzero forces f .We have a further improvement when q =
2. In this case, we treat f = Theorem 2.
Let f ∈ L r (Ω T ) for some r > / . There exists a positive number ε such that z ∈ Ω × ( T ) is a regular point if E ( z ) E ( z ) < ε .This is a quantitative version of (3): the point z ∈ Ω × ( T ) is regular if E ( z ) < ε M . Remark 3.
We shall define several scaled functionals and give various relations among them.However, the estimates of those functionals in this paper will not depend on the reference point z .So, we shall assume z = (
0, 0 ) and Q ( z , 2 ) ⊂ Ω × ( −
8, 8 ) for notational convenience. From now,we suppress z .
2. P
RELIMINARIES
We denote by L p (Ω) and W k , p (Ω) the standard Lebesgue and Sobolev spaces and we usethe boldface letters for the space of vector or tensor fields. We denote by D σ (Ω) the set of allsolenoidal vector fields φ ∈ C ∞ c (Ω) . We define L σ (Ω) to be the closure of D σ (Ω) in L (Ω) and W σ (Ω) to be the closure of D σ (Ω) in W (Ω) . Definition 2 (suitable weak solutions) . Let Ω T = Ω × ( T ) . Suppose that f ∈ L p (Ω T ) for somep > / . We say that ( v , p ) is a suitable weak solution to (1) if v ∈ L ∞ ( T ; L σ (Ω)) ∩ L ( T ; W σ (Ω)) , p ∈ L / (Ω T ) , HI JUN CHOE & JOERG WOLF & MINSUK YANG and ( v , p ) solves the Navier–Stokes equations in Ω T in the sense of distributions and satisfies thegeneralized energy inequality ˆ Ω | v ( t ) | φ ( t ) d x + ˆ t ˆ Ω |∇ v | φ d x ds ≤ ˆ t ˆ Ω | v | ( ∂ t φ + ∆ φ ) d x ds + ˆ t ˆ Ω | v | v · ∇ φ d x ds + ˆ t ˆ Ω p v · ∇ φ d x ds + ˆ t ˆ Ω f · v φ d x ds (4) for almost all t ∈ ( T ) and for all nonnegative φ ∈ C ∞ c (Ω T ) . Throughout the paper, we use the following notation.
Notation 1.
We denote the average value of g over the set E by 〈 g 〉 E = E gd µ = µ ( E ) − ˆ E gd µ . We denote A ® B if there exists a generic positive constant C such that | A | ≤ C B.
3. L
OCAL ENERGY INEQUALITIES
We shall define several scaled functionals to describe neatly various relations among them.The aim of this section is to present local Caccioppoli-type inequalities.
Definition 3 (scaled functionals I) . LetA ( r ) = r − sup t − r < s < t ˆ B ( x , r ) | v | d y C ( r ) = r − ¨ Q ( r ) | v | d y dsD ( r ) = r − ¨ Q ( r ) | p − 〈 p 〉 B ( r ) | / d y dswhere 〈 p 〉 B ( r ) = ffl B ( r ) pd y . From the definition of suitable weak solution we get the next lemma. Indeed, it is a directconsequence of the inequality (4) with a standard cutoff function φ , so we omit its proof. Lemma 4 (local energy inequality I) . For < r ≤ A ( r ) + E ( r ) ® C ( r ) / + C ( r ) + C ( r ) / D ( r ) / .In terms of the following scaled functionals, we shall derive another version of a localCaccioppoli-type inequality. EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 5
Definition 4 (scaled functionals II) . LetG ( r ) = r − ˆ − r (cid:16) ˆ B ( r ) | v | d y (cid:17) / dsP ( r ) = r − inf c ∈ R ‚ ˆ − r (cid:16) ˆ B ( r ) | p − c | d y (cid:17) / ds Œ . Lemma 5 (local energy inequality II) . For < r ≤ A ( r ) + E ( r ) ® [ + E ( r )] G ( r ) + P ( r ) . Proof.
First, we fix φ ∈ C ∞ c (Ω T ) satisfying 0 ≤ φ ≤ R , φ ≡ Q ( r ) , φ ≡ R × ( −∞ , 0 ) \ Q ( r ) c and | ∂ t φ | + |∇ φ | + |∇ φ | ® r − .Then, by the definition of the suitable weak solution, we have ˆ | v ( t ) | φ d y + ¨ |∇ v | φ d y ds ® ¨ | v | ( ∂ t φ + ∆ φ ) d y ds + ¨ | v | v φ · ∇ φ d y ds + ¨ p v φ · ∇ φ d y ds = : I + I I + I I I .(5)We shall estimate each term on the right. By the Jensen inequality I = r − ¨ | v | d y ds ® r ˆ − r B ( r ) | v | d y ds ® r ˆ − r (cid:16) B ( r ) | v | d y (cid:17) / ds ® r G ( r ) .(6)Since ∇ · v =
0, we have
I I = ¨ ( | v | − |〈 v 〉 B ( r ) | ) v φ · ∇ φ d y ds .Using the Hölder inequality and then applying the Sobolev–Poincaré inequality, we obtain that I I ® r − ¨ | v − 〈 v 〉 B ( r ) || v + 〈 v 〉 B ( r ) || v | φ d y ds ® r − / ˆ − r (cid:16) ˆ B ( r ) | v − 〈 v 〉 B ( r ) | d y (cid:17) / × (cid:16) ˆ B ( r ) | v + 〈 v 〉 B ( r ) | d y (cid:17) / (cid:16) ˆ | v | φ d y (cid:17) / ds ® r − / sup s (cid:16) ˆ | v | φ d y (cid:17) / ˆ − r (cid:16) ˆ B ( r ) |∇ v | d y (cid:17) / (cid:16) ˆ B ( r ) | v | d y (cid:17) / ds ® r / sup s (cid:16) ˆ | v | φ d y (cid:17) / E ( r ) / G ( r ) / . HI JUN CHOE & JOERG WOLF & MINSUK YANG
By the Young inequality we have for some C > δ > I I ≤ δ sup s ˆ | v | φ d y + C r δ E ( r ) G ( r ) .Hölder’s inequality gives I I I = ¨ p v φ · ∇ φ d y ds ® r − ¨ | p − c || v | φ d y ds ® r − / ˆ − r (cid:16) ˆ B ( r ) | p − c | d y (cid:17) / (cid:16) ˆ | v | φ d y (cid:17) / ds ® r / sup s (cid:16) ˆ | v | φ d y (cid:17) / P ( r ) / .By the Young inequality we have for some C > δ > I I I ≤ δ sup s ˆ | v | φ d y + C r δ P ( r ) .Combining (5)–(8) with a fixed small number δ , we get the result. (cid:3) Remark 6. If f = , then we have for < r ≤ A ( r ) + E ( r ) ® [ + E ( r )] G ( r ) + P ( r ) + F ( r ) where F ( r ) = ‚ ˆ − r (cid:16) ˆ B ( r ) | f | d y (cid:17) / ds Œ / . Indeed, Hölder’s inequality gives ¨ f · v φ d y ds ® ˆ − r (cid:16) ˆ B ( r ) | f | d y (cid:17) / (cid:16) ˆ | v | φ d y (cid:17) / ds ® sup s (cid:16) ˆ | v | φ d y (cid:17) / ˆ − r (cid:16) ˆ B ( r ) | f | d y (cid:17) / ds ® r / sup s (cid:16) ˆ | v | φ d y (cid:17) / F ( r ) / . By the Young inequality we have for some C > for all δ > ¨ f · v φ d y ds ≤ δ sup s ˆ | v | φ d y + C r δ F ( r ) . As in the proof of the previous lemma, we can absorb the first term on the right by choosing small δ . We notice that F ( r ) → as r → . Remark 7.
The implied constants of the estimates in this section are all absolute.
EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 7
4. P
RESSURE INEQUALITIES
In this section we present pressure inequalities, Lemma 8 and Lemma 13, which are used tocomplete iteration schemes.
Lemma 8 (pressure inequality I) . For < r ≤ and < θ < / D ( θ r ) ® θ D ( r ) + θ − e C ( r ) . Proof.
We may assume r =
1. In the sense of distributions we have − ∆ p = ∂ j ∂ k ( v j v k ) .Let e v = v − 〈 v 〉 B ( ) and let p satisfy the equation − ∆ p = ∂ j ∂ k ( e v j e v k φ ) where φ is a cutoff function which equals 1 in Q ( / ) and vanishes outside of Q ( ) . By theCalderon–Zygmund inequality(9) θ − ¨ Q ( θ ) | p | / d y ds ® θ − e C ( r ) .Since p : = p − p is harmonic in B ( / ) , we have by the mean value property θ − ¨ Q ( θ ) | p | / d y ds ® θ ¨ Q ( / ) | p | / d y ds ® θ D ( ) + θ ¨ Q ( ) | p | / d y ds (10)Since we have D ( θ ) ® θ − ¨ Q ( θ ) | p | / + | p | / d y ds ,combining the two estimates (9) and (10) yields the result. (cid:3) Now, we recall a decomposition of Lebesgue spaces.
Definition 5.
For < p < ∞ define A p (Ω) = n ∆ v : v ∈ W p (Ω) o , B p (Ω) = n p h ∈ L p (Ω) ∩ C ∞ (Ω) : ∆ p h = o . Lemma 9.
Let < p < ∞ and Ω ⊂ R n be a bounded C -domain. ThenL p (Ω) = A p (Ω) ⊕ B p (Ω) . Proof.
The proof can be found in [ ] . (cid:3) HI JUN CHOE & JOERG WOLF & MINSUK YANG
Remark 10.
Denote L p (Ω) = { f ∈ L p (Ω) : 〈 f 〉 Ω = } and B p (Ω) = B p (Ω) ∩ L p (Ω) . Since A p (Ω) ⊂ L p (Ω) , Lemma 9 implies thatL p (Ω) = A p (Ω) ⊕ B p (Ω) . Lemma 11.
For < s < ∞ the operator T s : A s (Ω) → W − s (Ω) defined by 〈 T s p , v 〉 = ˆ Ω p ∆ v , v ∈ W s ′ (Ω) . is an isomorphism.Proof. Let p ∈ A s (Ω) and set q = | p | s − p ∈ L s ′ (Ω) .By Lemma 9 there exist unique q ∈ A s ′ (Ω) and q h ∈ B s ′ (Ω) such that q = q + q h .In particular, q = ∆ v for some v ∈ W s ′ (Ω) . Hence k p k sL s (Ω) = ˆ Ω p q = ˆ Ω p ∆ v ≤ k T s p k W − s (Ω) k v k W s ′ (Ω) ® k T s p k W − s (Ω) k q k L s ′ (Ω) ® k T s p k W − s (Ω) k p k s − L s (Ω) .This implies that k p k L s (Ω) ® k T s p k W − s (Ω) and the operator T s has closed range. Furthermore, Lemma 9 implies also that if T s p =
0, then p ∈ A s (Ω) ∩ B s (Ω) = { } . Hence T s is injective and the result follows from the closed rangetheorem. (cid:3) Remark 12. (1)
Let f ∈ L s (Ω ; R n × n ) , < s < ∞ . Then by Lemma 11 there exists a uniquep ∈ A s (Ω) such that (11) ∆ p = ∇ · ∇ · f in Ω in the sense of distributions. Morevoer, there holds the estimate (12) k p k L s (Ω) ® k f k L s (Ω) .(2) Let g ∈ L s (Ω ; R n ) , < s < n. Then by means of Sobolev’s embedding theorem ∇ · g ∈ W − s (Ω) , → W − s ∗ (Ω) where s ∗ = ns / ( n − s ) . Thus, there exists a unique p ∈ A s ∗ (Ω) such that (13) ∆ p = ∇ · g EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 9 in Ω in the sense of distributions. By the definition of A s ∗ (Ω) there exist v ∈ W s ∗ (Ω) with ∆ v = p , and there holds ∆ v = ∇ · g in Ω in the sense of distributions. By meansof elliptic regularity we find v ∈ W s (Ω) together with the estimate (14) k∇ p k L s (Ω) ® k v k W s (Ω) ® k g k L s (Ω) .(3) Let p ∈ L s (Ω) . In view of Lemma 9 we have p = p + p h with unique p ∈ A s (Ω) andp h ∈ B s (Ω) . Observing that p − 〈 p 〉 Ω = p + ( p h − 〈 p h 〉 Ω ) and appealing to Remark 10 itfollows that (15) k p h − 〈 p h 〉 Ω k L s (Ω) ® k p − 〈 p 〉 Ω k L s (Ω) .(4) The implied constant in (12) , (14) and (15) depend only on s and Ω . When Ω equals aball, these constants depend on s but not on the radius of the ball. Lemma 13 (pressure inequality II) . For < r ≤ and < θ ≤ / P ( θ r ) ® θ P ( r ) + θ − E ( r ) + θ − F ( r ) . Proof.
We may assume r = B = B ( ) and Q = Q ( ) . By Lemma 9 we maydecompose for a. e. t ∈ I R p = p + p h where p ∈ A ( B ) and p h ∈ B ( B ) is harmonic. By Remark 12 we may decompose p = p + p where p ∈ A ( B ) is the unique weak solution to ∆ p = −∇ · ∇ · (( v − 〈 v 〉 B ) ⊗ ( v − 〈 v 〉 B )) in B in the sense of distributions, while p ∈ A ( B ) is the unique weak solution to ∆ p = ∇ · f in B in the sense of distributions for a. e. t ∈ I ( r ) : = ( − r , 0 ) .By the aid of (12) and (13) along with Sobolev-Poincaré’s inequality, we find that for a. e. t ∈ I R k p ( t ) k L ( B ) ® k v ( t ) − 〈 v 〉 B ( t ) k L ( B ) ® k∇ v ( t ) k L ( B ) , k p ( t ) k L ( B ) ® k∇ p ( t ) k L / ( B ) ® k f ( t ) k L / ( B ) . Integrating in time, we get ˆ I k p k L ( B ) ds ® ˆ I k∇ v k L ( B ) ds = E ( ) ,(16) ˆ I k p k L ( B ) ds ® ˆ I k f k L / ( B ) ds = F ( ) / .(17)On the other hand, employing (15), we see that p h − 〈 p h 〉 B ∈ L ( I ; L ( B )) and ˆ I k p h − 〈 p h 〉 B k L ( B ) ds ® ˆ I k p − 〈 p h 〉 B k L ( B ) ds .Applying the Poincaré-type inequality and using the mean value property of harmonic functions,we obtain that ˆ I ( θ ) k p h − 〈 p h 〉 B θ R k L ( B ( θ )) ds ® θ ˆ I ( / ) k∇ p h k L ∞ ( B ( / )) ds ® θ ˆ I k p − 〈 p 〉 B k L ( B ) ds .(18)Combining (16), (17), and (18), we get P ( θ ) / ® θ − ˆ I ( θ ) k p − 〈 p 〉 B ( θ ) k L ( B ( θ )) ds ® θ − ˆ I ( θ ) k p h − 〈 p h 〉 B ( θ ) k L ( B ( θ )) ds + θ − ˆ I k p k L ( B ) ds + θ − ˆ I k p k L ( B ) ds ® θ ˆ I k p − 〈 p 〉 B k L ( B ) ds + θ − E ( ) + θ − F ( ) / and the result follows. (cid:3) Remark 14.
The implied constants of the estimates in this section are all absolute.
5. I
NTERPOLATION INEQUALITIES
In this section we give a few interpolation inequalities. We shall use one more scaled func-tional, e C ( r ) = r − ¨ Q ( r ) | v − 〈 v 〉 B ( r ) | d y ds . Lemma 15.
For < r ≤ and < θ ≤ C ( θ r ) ® θ C ( r ) + θ − e C ( r ) and (19) C ( θ r ) ® θ A ( r ) / + θ − e C ( r ) . EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 11
Proof.
We may assume r = B = B ( ) and 〈 v 〉 B = ffl B v d y . By subtracting theaverage 〈 v 〉 B we have ˆ B ( θ ) | v | d y ® θ |〈 v 〉 B | + ˆ B ( θ ) | v − 〈 v 〉 B | d y .Integrating in time and using Jensen’s inequality we get the result. (cid:3) Lemma 16 (interpolation inequality I) . Let (20) 95 ≤ q ≤
2, 3 − q q − ≤ k ≤ − q Then for < r ≤ e C ( r ) ® A ( r ) ( − q − qk ) / ( − q ) E q ( r ) k / ( − q ) . Proof.
By scaling we may assume r = B = B ( ) . By the Sobolev-Poincaré inequal-ity ˆ B | v − 〈 v 〉 B | d y ® (cid:16) ˆ B | v | d y (cid:17) ( − kq ∗ ) / (cid:16) ˆ B | v − 〈 v 〉 B | q ∗ d y (cid:17) k ® A ( ) ( − kq ∗ ) / (cid:16) ˆ B |∇ v | q d y (cid:17) kq ∗ / q where q ∗ = q / ( − q ) . Note that from (20) we have 0 < ( − kq ∗ ) / <
1, 0 < k <
1, and0 < ( − kq ∗ ) / + k ≤
1, 0 < kq ∗ / q ≤ ˆ − ˆ B | v − ( v ) B | d y ds ® A ( ) ( − kq ∗ ) / ˆ − (cid:16) ˆ B |∇ v | q d y (cid:17) kq ∗ / q ds ® A ( ) ( − kq ∗ ) / E q ( ) kq ∗ / q .A calculation shows ( − kq ∗ ) / = ( − q − qk ) / ( − q ) and kq ∗ / q = k / ( − q ) . (cid:3) Remark 17.
If we choose q = and k = / , then the estimate (21) becomes the well-knownestimate e C ( r ) ® A ( r ) / E ( r ) / . If we choose k = ( − q ) / , then the estimate (21) becomes (22) e C ( r ) ® A ( r ) ( − q ) / E q ( r ) . Lemma 18.
Let X ( r ) : = C ( r ) + D ( r ) . If ≤ q ≤ and − q q − ≤ k ≤ − q , then for < r ≤ and < θ < X ( θ r ) ® θ X ( r ) + θ − A ( r ) ( − q − qk ) / ( − q ) E q ( r ) k / ( − q ) . Proof.
It follows from combining Lemma 15, 8, and 16. (cid:3)
Lemma 19 (interpolation inequality II) . For < r ≤ and < θ ≤ G ( θ r ) ® θ − E ( r ) + θ A ( r ) . Proof.
We may assume r = B = B ( ) and 〈 v 〉 B = ffl B v d y . By the Sobolev-Poincaréinequality ˆ B ( θ ) | v | d y ® ˆ B ( θ ) | v − 〈 v 〉 B | d y + ˆ B ( θ ) |〈 v 〉 B | d y ® (cid:16) ˆ B |∇ v | d y (cid:17) + ( θ r ) |〈 v 〉 B | .Thus, we have G ( θ ) = θ − ˆ − θ (cid:16) ˆ B ( θ ) | v | d y (cid:17) / ds ® θ − E ( r ) + ˆ − θ |〈 v 〉 B | ds ,and the result follows. (cid:3) Lemma 20. (cid:16) r − ¨ Q ( r ) | v | d y ds (cid:17) / ® A ( r ) + E ( r ) . Proof.
By scaling we may assume r = B = B ( ) and Q = Q ( ) . By the Hölderinequality ¨ Q | v | d y ds ≤ ˆ − (cid:16) ˆ B | v | d y (cid:17) / (cid:16) ˆ B | v | d y (cid:17) / ds .By the Young inequality (cid:16) ¨ Q | v | d y ds (cid:17) / ® A ( ) / G ( ) / ≤ A ( ) + G ( ) . EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 13
By Lemma 19 with θ = (cid:3) Remark 21.
The implied constants of the estimates in this section are all absolute.
6. C
ONTROL OF LOCAL KINETIC ENERGY AND PRESSURE
The aim of this section is to prove that the scaled quantities of local kinetic energy andpressure are controlled by the velocity gradient.
Lemma 22.
Let / ≤ q ≤ . There exists an absolute positive constant γ such that if < E q < ∞ ,then (23) lim sup r → [ A ( r ) + D ( r )] ≤ γ E / ( q − ) q . Remark 23.
We assume E q > for convenience. Indeed, we may consider the case that E q has apositive lower bound because of the criterion (2) .Proof. Fix q and denote M = E q . There is R < < r < RE q ( r ) ≤ M .From the local energy inequality I in Section 3, we have for 0 < r < R and 0 < θ ≤ A ( θ r ) ® + X ( θ r ) where X ( r ) = C ( r ) + D ( r ) . If we set Y ( r ) : = A ( r ) + X ( r ) ,then, by using the trivial estimate X ( θ r ) ≤ X ( θ r ) , we get(24) Y ( θ r ) ® + X ( θ r ) .Using Lemma 18 with k = ( − q ) / < r < R and 0 < θ < / X ( θ r ) ® θ X ( r ) + θ − A ( r ) ( − q ) / M ® θ Y ( r ) + θ − ( − q ) / ( q − ) M / ( q − ) .(25)Thus, combining (24) and (25) yields that for some positive constant β ≥ Y ( θ r ) ≤ β θ Y ( r ) + β θ − ( − q ) / ( q − ) M / ( q − ) + β ≤ β θ Y ( r ) + β θ − ( − q ) / ( q − ) M / ( q − ) .If we fix θ = ( β ) − , then the last inequality becomes Y ( θ r ) ≤ Y ( r ) + ( β ) / ( q − ) M / ( q − ) . By the standard iteration argument we getlim sup r → Y ( r ) ≤ γ M / ( q − ) where γ = ( β ) / ( q − ) . This completes the proof. (cid:3) Lemma 24.
There exists an absolute positive constant γ such that if E < ∞ , then lim sup r → P ( r ) ≤ γ E . Proof.
From Lemma 13 we have for all r < < θ ≤ / P ( θ r ) ® θ P ( r ) + θ − E ( r ) + θ − F ( r ) .Since lim r → F ( r ) =
0, we initially start from a small number r = R and then perform a standarditeration argument to get the result. (cid:3)
7. P
ROOF OF T HEOREM q and denote M = E q and m = E q .Suppose 1 < M < ∞ for convenience. Lemma 22 implies that there is a positive number R suchthat for all 0 < r ≤ R (26) A ( r ) ® M / ( q − ) and D ( r ) ® M / ( q − ) .On the other hand, there exists a sequence of positive numbers r n such that r n < R andlim n →∞ r n = n →∞ E q ( r n ) = m .Combining (19) and (22), we have for all n and 0 < θ ≤ C ( θ r n ) ® θ A ( r n ) / + θ − A ( r n ) ( − q ) / E q ( r n ) .Hence from (26) we obtain that for some β > C ( θ r n ) ≤ β θ M / ( q − ) + β θ − M ( − q ) / ( q − ) E q ( r n ) .If 0 < m , then we take θ = [ M − q / ( q − ) m ] / so that C ( θ r n ) ≤ β θ M / ( q − ) + β θ − M ( − q ) / ( q − ) E q ( r n ) ≤ β (cid:16) M ( − q ) / ( q − ) m (cid:17) / (cid:16) + m − E q ( r n ) (cid:17) .Since lim n →∞ m − E q ( r n ) = EGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS 15 we have for all large n C ( θ r n ) ≤ β ε / .If ε is small, then we take R = θ r N and a large natural number N so that z is a regular point.If m =
0, then theorem is trivially true. Indeed, we can choose θ so that β θ M / ( q − ) issmall enough and lim n →∞ β θ − M ( − q ) / ( q − ) E q ( r n ) = z is a regular point. This completes the proof of Theorem 1.8. P ROOF OF T HEOREM θ r < R and0 < θ < / A ( θ r ) + E ( θ r ) ® [ + E ( θ r )] G ( θ r ) + P ( θ r ) + F ( R ) where R will be determined later. From Lemma 13, we have for 0 < θ < / P ( θ r ) ® θ P ( r ) + θ − E ( r ) + θ − F ( R ) .From Lemma 19 G ( θ r ) ® θ − E ( r ) + θ A ( r ) .We also have E ( θ r ) ≤ ( θ ) − E ( r ) by the definition. Combining all the above estimates, we conclude that for 2 θ r < R and 0 <θ < / A ( θ r ) + E ( θ r ) ® θ A ( r ) E ( r ) + θ A ( r ) + θ P ( r ) + θ − E ( r ) + θ − E ( r ) + θ − F ( R ) .(27)Let us denote M = E and m = E .If m =
0, then theorem is trivially true. We may consider the case 0 < m and 1 ≤ M < ∞ .Lemma 22 with q = R such that for all 0 < r ≤ R (28) A ( r ) ® M .Lemma 24 implies that there is a positive number R such that for all 0 < r ≤ R (29) P ( r ) ® M .Since lim r → F ( r ) =
0, there is a positive number R such that for all 0 < r ≤ R (30) F ( r ) ≤ M − ε . We also have for some R and for all 0 < r ≤ R E ( r ) ≤ M .We can take R = min { R , R , R , R } and fix a sequence r n such that r n < R ,lim n →∞ r n = n →∞ E ( r n ) = m .Combining (27)–(30), we have for all sufficiently large n and for all 0 < θ < / A ( θ r n ) + E ( θ r n ) ® θ M E ( r n ) + θ M + θ − E ( r n ) + θ − E ( r n ) + θ − M − ε ® θ M m + θ M + θ − m + θ − m + θ − M − ε .Since M m < ε and ε < /
16, we can take θ = ε / M − < / A ( θ r n ) + E ( θ r n ) ® ε / + ε + ε / ® ε / .As it has been proved in [ ] there exists an absolute constant ε such that if D ( r ) ≤ ε that z is a regular point (cf. [ ] ). This together with Lemma 20 shows that there exists a positiveconstant ε such that z is a regular point if for some r > A ( r ) + E ( r ) < ε .Due to (29) and (31), we conclude that the reference point z is regular for the case m >
0. Thiscompletes the proof of Theorem 2. A
CKNOWLEDGEMENT
H. J. Choe has been supported by the National Reserch Foundation of Korea(NRF) grant,funded by the Korea government(MSIP) (No. 20151009350). J. Wolf has been supported bythe German Research Foundation (DFG) through the project WO1988 / EFERENCES [ ] Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equa-tions,
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On the local regularity of suitable weak solutions to the generalized Navier–Stokes equations , Annalidella Universita Ferrara , 149–171 (2015)H. J. C HOE : D
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