An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range
aa r X i v : . [ m a t h . A P ] F e b An anisotropic regularity condition for the 3D incompressibleNavier-Stokes equations for the entire exponent range
I. Kukavica, W. S. O˙za´nski
Abstract
We show that a suitable weak solution to the incompressible Navier-Stokes equations on R × ( − ,
1) is regular on R × (0 ,
1] if ∂ u belongs to a Morrey space M p/ (2 p − ,α (( − , L p ( R ))for any α > p ∈ (3 / , ∞ ). For each α > L q (( − , L p ( R )) that aresubcritical, that is for which 2 /q + 3 /p < We address conditional regularity of suitable weak solutions to the incompressible Navier-Stokesequations (NSE), u t − ∆ u + u · ∇ u + ∇ π = 0 , div u = 0 , (1)in R × (0 , T ). Our main result is the following. Theorem 1.
Suppose that ( u, π ) is a suitable weak solution to the Navier-Stokes equations on R × ( − , such that for some α > and p ∈ (3 / , ∞ ) we have k ∂ u k L p p − ( I ; L p ) ≤ C p,α (cid:18) − | I | (cid:19) α (2) for every I ⊂ ( − , with | I | < . Then u is regular on R × ( − , . Here we write L p ≡ L p ( R ), for brevity.In order to put this result in a context, we note that the study of conditional regularity of theNSE goes back to Serrin, Ladyzhenskaya, and Prodi ([S, L, P]), who proved that if u ∈ L qt L px holdswith 2 /q + 3 /p ≤
1, where p ∈ (3 , ∞ ] then the solution is regular. On the other hand, Beir˜aoda Veiga showed in [B] that the regularity holds if ∇ u ∈ L qt L px with 2 /q + 3 /p ≤ p ∈ (3 / , ∞ ).In [NP1], Neustupa and Penel proved that boundedness of only one component of the velocity(say u ) implies regularity, with the approach based on the evolution equation for ω (cf. also[NP2]). Afterwards, there have been many results [CC, H, NNP, P, PP1, SK], which approached theSerrin’s scale invariant condition in terms of one velocity component, until a recent breakthroughpaper [CW], which achieved the range of exponents with strict inequality 2 /q + 3 /p <
1. A
I. Kukavica: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA, email:[email protected]. S. O˙za´nski: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA, email:[email protected]. Kukavica was supported in part by the NSF grant DMS-1907992. W. S. O˙za´nski was supported in part by theSimons Foundation. ∂ u , Penel and Pokorn´y proved in [PP1] regularity underthe condition that ∂ u belongs to L qt L px where 2 /q + 3 /p ≤ / ≤ p ≤ ∞ . The result in [KZ]then provided a scale invariant regularity criterion 2 /q + 3 /p ≤
2, with a restricted exponent range9 / ≤ p ≤
3. The method in [KZ] was based on testing the equations for ( u , u ) with − ∆ u , ,and an identity for P i,j =1 ´ u i ∂ i u j ∆ u j in which every term contains ∂ u . The partial regularitymethods [CKN, V, O, W1] allowed localization of this condition in [KRZ]. There has been severalimprovements on the criteria since then; cf. [BG, CZ, PP2, Sk1, Sk2] for a partial list of references.In particular, in [Sk2], Skal´ak extended the range for ∂ u to 3 / < p ≤ ∂ u covering the full range of Lebesgue exponents 3 / < p < ∞ as well as all Lebesgue spaces L qt L px with sharp inequality 2 /q + 2 /p <
2. To be more precise, letting (2) be the definition of a Morreyspace M p p − ,α (( − , L p ), we immediately see that such Morrey space contains L q (( − , L p ) forevery q > p p − (that is such that 2 /q + 3 /p < k ∂ u k L p p − ( I ; L p ) ≤ k ∂ u k L q ( I ; L p ) | I | pq − q − p pq ≤ C p,q,α (cid:18) − | I | (cid:19) α for any I ⊂ ( − , | I | ≤ . Furthermore, M p p − ,α (( − , L p ) is, up to a logarithm, critical withrespect to the scaling of the equations; namely letting u λ := λu ( λx, λ t ) we have k ∂ u λ k L p p − ( I ; L p ) = k ∂ u k L p p − ( λI ; L p ) ≤ C p,α (cid:18) − | λI | (cid:19) α ≤ C p,α (cid:18) − | I | (cid:19) α O (cid:18)(cid:18) − λ (cid:19) α (cid:19) as λ → + , for every I ⊂ ( − ,
0) such that | I | ≤ .The question whether the Morrey space M p p − ,α (( − , L p ) can be replaced by a criticalLebesgue-type space, L p p − (( − , L p ), without any restriction on the range of p as in Theorem 1,remains an interesting open problem.Our approach in proving Theorem 1 is inspired by the treatment of a related regularity conditionin terms of one component of u that was recently proved by Wang et al in [WWZ], which in turndrew from a recent result of Chae and Wolf [CW], which introduced a new approach based on partialregularity and testing the local energy equality with a one-dimensional backward heat kernel. Before proceeding to the proof of our main result, we recall that ( u, π ) with E := k u k L ∞ (( − , L ) + k∇ u k L ( R × ( − , < ∞ is a suitable weak solution in R × ( − ,
1) if it satisfies the equation (1) in the distributional sense,if π = ( − ∆) − ( ∂ i u j ∂ j u i ), and the local energy inequality ˆ R | u ( t ) | φ ( t ) + 2 ˆ t − ˆ R |∇ u | φ ≤ ˆ t − ˆ R (cid:0) | u | ( ∂ t φ + ∆ φ ) + ( | u | + 2 π )( u · ∇ ) φ (cid:1) holds for all t ∈ ( − ,
1) and φ ∈ C ∞ ( R × ( − , , u , the local energy inequality can be extended to include the test functions φ thatdo not necessarily have a compact support in space. We set r n := 2 − n and U n := R × ( − r n , r n ) , Q n := U n × ( − r n , .
2e also set E n ≡ E ( r n ) := sup t ∈ ( − r n , ˆ U n | u ( t ) | d x + ˆ − r n ˆ U n |∇ u | d x d s and Φ n ( x , t ) := (4 π ( r n − t )) − e − x r n − t ) , x ∈ R , t < . Note that r − k . Φ n . r − k on Q k , k = 0 , , . . . , n. (3)Let η ( x , t ) be such that supp η ⋐ ( − , ) × ( − ,
0] and η = 1 on ( − , ) × ( − , B k := k ∂ u k L p p − t L px ( Q k ) . Note that X k ≥ B k . p,α X k ≥ (cid:18) − | r k | (cid:19) α . α X k ≥ k − α . α , (4)by the assumption (2). We also set B := k ∂ u k L p p − (( − , L p ) .In order to prove the main result, Theorem 1, we need the following localization property. Proposition 2.
Let ( u, π ) be a suitable weak solution to the NSE on R × ( − , . Then r n E n . EB + n X k =0 r − k E k ( r εk B + B k ) for some ε = ε ( p ) > .Proof of Proposition 2. The local energy inequality applied with Φ n η givessup t ∈ ( − , ˆ U | u ( t ) | Φ n ( t ) η ( t )d x + 2 ˆ Q |∇ u | Φ n η . ˆ Q | u | ( ∂ t + ∆)(Φ n η ) + ˆ Q ( | u | + 2 π ) u · ∇ (Φ n η ) . (5)We show below that the right-hand side can be bounded from above by a constant multiple of E + P nk =0 r − k E k ( r εk B + B k ). This and the bound Φ n η & r − n on Q n then give the claim.For the first term on the right-hand side of (5), we have ˆ Q | u | ( ∂ t + ∆)(Φ n η ) = ˆ Q | u | (2 ∂ Φ n · ∂ η + Φ n ∂ η ) . ˆ Q | u | . E , where we used that Φ n satisfies the one-dimensional heat equation in Q in the first step, and thebounds |∇ Φ n | , | Φ n | . ∂ η ) ∩ Q in the second step.For the velocity component of the second term on the right-hand side of (5), we have12 ˆ Q | u | u · ∇ (Φ n η ) = − ˆ t − ˆ U ∂ u · u u Φ n η − ˆ t − ˆ U | u | ∂ u Φ n η . n − X k =0 ˆ Q k \ Q k +1 | ∂ u | | u | Φ n η + ˆ Q n | ∂ u | | u | Φ n η . n − X k =0 r − k ˆ Q k | ∂ u | | u | + r − n ˆ Q n | ∂ u | | u | . n X k =0 r − k k u k L p t L p ′ x ( Q k ) k ∂ u k L p p − t L px ( Q k ) . n X k =0 r − k E k B k .
3s for the part involving the pressure π = ( − ∆) − ( ∂ i u j ∂ j u i ), we choose χ k ( x , t ) ∈ C ∞ ( B ( r k ) × ( − r k , , χ k = 1 on B r k +1 × ( − r k +1 ,
0] and set φ j := ( χ j − χ j +1 ; j = 0 , . . . , n − ,χ n ; j = n. Then we may write12 ∂ π = ( − ∆) − ∂ i ∂ l ( u l ∂ u i ) = ( − ∆) − ∂ i ∂ l ( u l ∂ u i χ ) + ( − ∆) − ∂ i ∂ l ( u l ∂ u i (1 − χ ))= n X j =0 ( − ∆) − ∂ i ∂ l ( u l ∂ u i φ j ) + ( − ∆) − ∂ i ∂ l ( u l ∂ u i (1 − χ )) =: n X j =0 p j + q, (6)and thus the pressure term may be decomposed as − ˆ t − ˆ U π u · ∇ (Φ n η ) = 12 ˆ t − ˆ U ∂ π u Φ n η + 12 ˆ t − ˆ U π∂ u Φ n η. (7)Using the notation in (6), we rewrite the first term as12 ˆ t − ˆ U ∂ π u Φ n η = n X j =0 ˆ t − ˆ U p j u Φ n η + ˆ t − ˆ U q u Φ n η = n X k =0 n X j =max { ,k − } ˆ Q k p j u Φ n φ k η + n − X j =0 n X k = j +4 ˆ Q k p j u Φ n φ k η + ˆ t − ˆ U q u Φ n η =: I + I + I . For I , we note that P nj = k − p j = ( − ∆) − ∂ i ∂ l ( u l ∂ u i χ k − ) for k ≥
3, which gives | I | . n X k =0 r − k ˆ Q k (cid:12)(cid:12)(cid:12)(cid:12) n X j =max { ,k − } p j u (cid:12)(cid:12)(cid:12)(cid:12) . n X k =0 r − k k u ∂ u k L p p − t L pp +1 x ( Q k ) k u k L p t L p ′ x ( Q k ) . n X k =0 r − k k ∂ u k L p p − t L px ( Q k ) k u k L p t L p ′ x ( Q k ) . n X k =0 r − k E k B k , as required. For I , we note that p j is harmonic with respect to the spatial variables in Q j +2 ,and thus using the anisotropic interior estimates for harmonic functions (cf. [CW, Lemma A.2]) weobtain k p j k L q ( R × ( − r k ,r k )) . r q k r q − l j k p j k L l ( R × ( − r j +2 ,r j +2 )) for all l ≥ q ≥ l . Fix ε := 14 p and l > (cid:18) − l (cid:19) < p . Note that then − (cid:18) − l (cid:19) + 12 p − ε > p + 32 l − ε > , (8)4hich gives | I | . n − X j =0 n X k = j +4 r − k ˆ Q k | p j u | . n − X j =0 X k ≥ j +4 r − + p k r p − l j k p j k L lp lp − l − pt L lx ( Q j +2 ) k u k L lp − lp +3 l +3 pt L p ′ x ( Q k ) . n − X j =0 n X k = j +4 r − + p k r p − l j k u ∂ u k L lp lp − l − pt L lx ( Q j ) k u k L p t L p ′ x ( Q k ) r − + l k . n − X j =0 n X k = j +4 r − p + l − ε k r p − l j k u k L lp lp − p + l ) t L lpp − lx ( Q j ) k ∂ u k L p p − t L px ( Q j ) r ε k E k , and thus | I | . n X m =0 r − εm E m ! n − X j =0 r p − l j E j k ∂ u k L p p − t L px ( Q j ) X k ≥ j +4 r − + p + l − ε k . n X m =0 r − εm E m ! n − X j =0 r − + p + l − ε j E j B j . B n X m =0 r − εm E m , where, in the last step, we used the Cauchy-Schwarz inequality and P j ≥ r p + l − εj .
1, which inturn is a consequence of (8). The condition (8) was also used to sum the infinite series in k in thesecond inequality.The estimate on I is analogous to the one for I above, but does not require summation in j .Indeed, q is harmonic in (supp η ) ∩ ( U × ( − , t )), and so we perform the same estimate as inthe first four inequalities above, but with Q j +2 and Q j replaced by R × ( − ,
0) and without thesummation in j . We obtain | I | . k u k L lp lp − p + l ) (cid:18) ( − , L lpp − l (cid:19) k ∂ u k L p p − (( − , L p ) n X k =0 r − p + l k E k . E B n − X m =0 r − εm E m ! . BE + B n X m =0 r − εm E m , as required, where we used − p + l − ε > π (ratherthan to ∂ π ) to obtain π = n X j =0 ( − ∆) − ∂ i ∂ l ( u l u i φ j ) + ( − ∆) − ∂ i ∂ l ( u l u i (1 − χ )) =: n X j =0 ˜ p j + ˜ q, ˆ t − ˆ U π∂ u (Φ n η ) . n X k =0 r − k ˆ Q k | ( − ∆) − ∂ i ∂ m ( u i u m χ max { ,k − } ) | | ∂ u | + n − X j =0 n X k = j +4 r − k ˆ Q k | ˜ p j ∂ u | + n X k =0 r k ˆ Q k | ˜ q∂ u | . n X k =0 r − k k u k L p t L p ′ x ( Q k ) B k + n − X j =0 n X k = j +4 r k k ˜ p j k L q ′ t L p ′ x ( Q k ) k ∂ u k L qt L px ( Q k ) + n X k =0 r k k ˜ q k L q ′ t L p ′ x ( Q k ) k ∂ u k L qt L px ( Q k ) , where q ∈ (1 , p ′ ). Therefore, ˆ t − ˆ U π∂ u (Φ n η ) . n X k =0 r − k k u k L p t L p ′ x ( Q k ) B k + n − X j =0 n X k = j +4 r − p k r p ′ − q j k ˜ p j k L q ′ t L qx ( Q j +2 ) r q − p k B k + k ˜ q k L q ′ t L qx ( Q ) n X k =0 r − p k r q − p k B k . n X k =0 r − k E k B k + n − X j =0 r p ′ − q j k u k L q ′ t L qx ( Q j ) B j X k ≥ j r (cid:16) q − p ′ (cid:17) k + EB . n X k =0 r − k E k B k + EB, where we used H¨older’s inequality k f k L q ′ ( − r j , ≤ k f k L l ′ ( − r j , r (cid:16) q ′ − l (cid:17) j in the last estimate. Proof of Theorem 1.
Note that Proposition 2, the fact that the B k ’s are summable (recall (4)), andthe discrete Gronwall inequality give that r − n E n . C for all n ≥
0, where C = C ( B, E ) > r − k u k L ( B r (0) × ( − r , + r − E ( r ) . C, r ∈ (0 , , where we used an interpolation inequality for the L norm (cf. [O, Lemma 2.1], for example). Since k ∂ u k L p p − (( − r , L p ( B r )) → r → , the next lemma gives that (0 ,
0) is a regular point of u . Regularity at any other point in R × ( − , U n , Q n , A n , and E n , which concludes the proof ofTheorem 1 once we establish the next lemma. Lemma 3 (conditional local regularity) . Given
M > there exists ε ( M ) > with the followingproperty: If ( u, π ) is a suitable weak solution in Q such that sup r ∈ (0 , r sup t ∈ ( − r , ˆ B r | u ( t ) | d x + ˆ Q r |∇ u | ! ≤ M < ∞ then (0 , is a regular point provided r − p − q k ∂ u k L q (( − r , L p ( B r )) ≤ ε ( M ) for some r ∈ (0 , C ( u, π )) , p ≥ / , and q ≥ . u, p )we denote by Q r := B r × ( − r , r , and we set P ( π, r ) := 1 r ˆ Q r | π | , C ( u, r ) := 1 r ˆ Q r | u | ,A ( u, r ) := 1 r sup t ∈ ( − r , ˆ B r | u ( t ) | d x, E ( u, r ) := 1 r ˆ Q r |∇ u | . Proof of Lemma 3.
Note that, by interpolation, the assumption gives C ( u, r ) . M for every r ∈ (0 , C ( u, π ) := min (cid:26) , (cid:0) C ( u,
1) + P ( π, (cid:1) − (cid:27) . Suppose that the claim does not hold. Then there exists a sequence ( u k , π k ) and r k ∈ (0 , C ( u k , π k ))such that C ( u k , r ) . M , r − p − q k k ∂ u k k L qt L px ( Q rk ) ≤ k and (0 ,
0) is a singular point of u k for every k . Using [WZ1, Lemma A.2], we obtain A ( u k , r ) + E ( u k , r ) + P ( π k , r ) ≤ C ( M ) , for all r ∈ (0 , r k ). In order to relax the restriction on the range of r we apply the rescaling v k ( x, t ) := r k u k ( r k x, r k t ) , q k ( x, t ) := r k π k ( r k x, r k t )to obtain A ( v k , r ) + E ( v k , r ) + P ( q k , r ) + C ( v k , r ) + k k ∂ v k k L qt L px ( Q ) ≤ C ( M )for all r ∈ (0 , v k , q k ) together with the Aubin-Lions Lemma (cf. [RRS,Theorem 4.12], for example) is sufficient to extract a subsequence, which we relabel, such that v k → v in L ( Q / ) , q k ⇀ q in L ( Q / ) , and ∂ v k → L qt L px ( Q ) , where ( v, q ) is a suitable weak solution to the Navier-Stokes equations on Q / such that ∂ v = 0.It follows that v and ∇ v are bounded functions in Q / due to the localized regularity conditionon ∂ v of [KRZ]. Since also ´ Q / | q | / < ∞ , we get, using the elliptic regularity on the equation − ∆ q = ∂ i v j ∂ j v i ∈ L ∞ ( Q / ) at almost every time t ∈ ( − / , k q k W ,p ( B / ) . k∇ v k L p ( B / ) + k q k L / ( B / ) for every p ∈ (1 , ∞ ) and almost every t ∈ ( − / ,
0] [GT, Theorem 9.11]. Usingthis statement with p sufficiently large, we obtain q ∈ L / t L ∞ x ( Q / ). This immediately implies r − ´ Q r | q | / < ∞ for r ∈ (0 , / r ∈ (0 , /
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