Second derivatives estimate of suitable solutions to the 3D Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] S e p SECOND DERIVATIVES ESTIMATE OF SUITABLE SOLUTIONSTO THE 3D NAVIER-STOKES EQUATIONS
ALEXIS VASSEUR AND JINCHENG YANG
Abstract.
We study the second spatial derivatives of suitable weak solutionsto the incompressible Navier-Stokes equations in dimension three. We showthat it is locally L ,q for any q > , which improves from the current result L , ∞ . Similar improvements in Lorentz space are also obtained for higherderivatives of the vorticity for smooth solutions. We use a blow-up techniqueto obtain nonlinear bounds compatible with the scaling. The local study workson the vorticity equation and uses De Giorgi iteration. In this local study, wecan obtain any regularity of the vorticity without any a priori knowledge ofthe pressure. The local-to-global step uses a recently constructed maximalfunction for transport equations. Keywords.
Navier-Stokes Equation, Partial Regularity, Blow-up Techniques,De Giorgi Method, Maximal Function, Lorentz Space.
Contents
1. Introduction 22. Preliminary 52.1. Maximal Function associated with Skewed Cylinders 52.2. Lorentz Space 62.3. Helmholtz decomposition 73. Proof of the Main Results 84. Local Study: Part One, Initial Energy 124.1. Equations of v 134.2. Energy Estimate 164.3. Proof of Proposition 4.1 175. Local Study: Part Two, De Giorgi Iteration 195.1. Highest Order Nonlinear Term 215.2. Lower Order Terms 235.3. W Terms 245.4. Proof of Proposition 5.1 266. Local Study: Part Three, More Regularity 266.1. Bound Vorticity in the Energy Space 276.2. Bound Higher Derivatives in the Energy Space 306.3. Proof of the Local Theorem 34Appendix A. Suitability of Solutions 35References 38
Date : October 1, 2020.2020
Mathematics Subject Classification.
Acknowledgement . A. Vasseur was partially supported by the NSF grant: DMS 1614918. Introduction
We study the three dimensional incompressible Navier-Stokes equations, ∂ t u + u · ∇ u + ∇ P = ∆ u, div u = 0 . (1.1)Here u : (0 , T ) × R → R and P : (0 , T ) × R → R represent the velocity field andthe pressure field of a fluid in R , within a finite or infinite timespan of length T .Initial condition u (0 , · ) = u ∈ L ( R )is given by a divergence-free velocity profile u of finite energy.Leray ([Ler34]) and Hopf ([Hop51]) proved the existence of weak solutions forall time. They constructed solutions u ∈ C w (0 , ∞ ; L ( R )) ∩ L (0 , ∞ ; ˙ H ( R ))corresponding to each aforementioned initial value, and satisfying (1.1) in the senseof distribution. A weak solution is called a Leray-Hopf solution if it satisfies energyinequality 12 k u ( t ) k L ( R ) + k∇ u k L ((0 ,t ) × R ) ≤ k u k L ( R ) for every t >
0. Since then, much work has been developed in regard to theuniqueness and regularity of weak solutions. Nonuniqueness of weak solutions wasproven very recently by Buckmaster and Vicol ([BV19]) using convex integrationscheme. However, the question of the uniqueness of Leray-Hopf solutions still re-mains open. The uniqueness is related with the regularity of solutions by theLadyˇzenskaya-Prodi-Serrin criteria ([KL57, Pro59, Ser62, Ser63, FJR72]): if thevelocity belongs to any space interpolating L t L ∞ x and L ∞ t L x then it is actuallysmooth, hence unique. The endpoint case L ∞ t L x comes much later by Iskauriaza,Ser¨egin and Shverak [ISS03]. These spaces require higher spatial integrabilitythan the energy space provides, which is E = L ∞ t L x ∩ L t ˙ H x .At the level of energy space, Scheffer ([Sch76, Sch77, Sch78, Sch80]) began tostudy the partial regularity for a class of Leray-Hopf solutions, called suitable weaksolutions. These solutions exist globally and satisfy the following local energy in-equality, ∂ t | u | (cid:18) u (cid:18) | u | P (cid:19)(cid:19) + |∇ u | ≤ ∆ | u | . Scheffer showed the singular set, at which the solution is unbounded nearby, hastime-space Hausdorff dimension at most . This result was later improved byCaffarelli, Kohn and Nirenberg in [CKN82] (see also [Lin98, Vas07]), where theyshowed the 1-dimensional Hausdorff measure of the singular set is zero. We willinvestigate the regularity of suitable weak solutions. In the periodic setting, Con-stantin ([Con90]) constructed suitable weak solutions whose second derivatives havespace-time integrability L − ε for any ε >
0, provided the initial vorticities arebounded measures. This was improved by Lions ([Lio96]) to a slightly better space L , ∞ , a Lorentz space which corresponds to weak L space. These estimatesare extended to higher derivatives of smooth solutions by one of the authors andChoi ([Vas10, CV14]) using blow-up arguments: L p, ∞ loc space-time boundedness for( − ∆) α ∇ n u , where p = n + α +1 , n ≥
1, 0 ≤ α <
2. They also constructed suitableweak solutions satisfying these bounds for n + α < ECOND DERIVATIVE OF NAVIER-STOKES 3
The aim of this paper is to improve these regularity results in Lorentz space.The main result is the following. Note that the estimate does not rely on the sizeof the pressure.
Theorem 1.1.
Suppose we have a smooth solution u to the Navier-Stokes equationsin (0 , T ) × R for some < T ≤ ∞ with smooth divergence free initial data u ∈ L .Then for any integer n ≥ , for any real number q > , the vorticity ω = curl u satisfies (cid:13)(cid:13)(cid:13)(cid:13) |∇ n ω | n +2 {|∇ n ω | n +2 >C n t − } (cid:13)(cid:13)(cid:13)(cid:13) L ,q ((0 ,T ) × R ) ≤ C q,n k u k L (1.2) for some constant C n depending on n and C q,n depending only on q and n , uniformin T . The above estimate (1.2) also holds for suitable weak solutions with only L divergence free initial data in the case n = 1 . This theorem gives the following improvement on the second derivatives.
Corollary 1.2.
Let u be a suitable weak solution in (0 , ∞ ) × R with initial data u ∈ L . Then for any q > , K ⊂⊂ (0 , ∞ ) × R , there exists a constant C q,K depending on q and K such that the following holds, (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ,q ( K ) ≤ C q,K (cid:16) k u k L + 1 (cid:17) . Let us explain the main ideas of the proof. Similar as previous work on higherderivatives, the proof is also based on blow-up techniques. In particular, we blowup the equation along a trajectory, using the scaling symmetry and the Galileaninvariance of the Navier-Stokes equations. That is, if we fix an initial time t andmove the frame of reference along some X ( t ), and zoom in into ε scale, then it iseasy to verify that ˜ u ( s, y ), ˜ P ( s, y ) defined by1 ε ˜ u (cid:18) t − t ε , x − X ( t ) ε (cid:19) := u ( t, x ) − ˙ X ( t )(1.3) 1 ε ˜ P (cid:18) t − t ε , x − X ( t ) ε (cid:19) := P ( t, x ) + x · ¨ X ( t )also satisfy the Navier-Stokes equation ∂ s ˜ u + ˜ u · ∇ ˜ u + ∇ ˜ P = ∆˜ u, div ˜ u = 0 . We develop the following local theorem for ˜ u and ˜ P . Note that it needs nothingfrom the pressure. Denote B r ⊂ R to be a ball centered at the origin with radius r , and Q r = ( − r , × B r ⊂ R to be a space-time cylinder. Theorem 1.3 (Local Theorem) . There exists a universal constant η > , suchthat for any suitable weak solution u to the Navier-Stokes equations in ( − , × R satisfying ˆ B u ( t, x ) φ ( x ) d x = 0 a.e. t ∈ ( − , , (1.4) k∇ u k L p t L q x ( Q ) + k ω k L p t L q x ( Q ) ≤ η , (1.5) where φ ∈ C ∞ c ( B ) is a non-negative function with ´ φ = 1 , ω = curl u is thevorticity, ≤ p ≤ ∞ , ≤ p ≤ ∞ , ≤ q , q < satisfying p + 1 p < , q + 1 q ≤ , A. VASSEUR AND J. YANG then for any integer n ≥ , we have k∇ n ω k L ∞ ( Q − n − ) ≤ C n for some constant C n depending only on n . Let us illustrate the ideas of how to go from this local theorem towards the mainresult. We want to choose a “pivot quantity”, blow up near a point, and use thisquantity to control ∇ n ω . When we patch the local results together, we will obtaina nonlinear bound with the same scaling as the pivot quantity, so we want thepivot quantity to have the best possible scaling. The ideal pivot quantities wouldbe ´ |∇ u | d x d t and ´ |∇ P | d x d t . ´ | u | d x d t has a worse scaling and should notbe used. However, we still need to control the flux in the local theorem, so we wantto take out the mean velocity and control u by ∇ u using Poincar´e’s inequality.In order to take out the mean velocity, we choose X ( t ) to be the trajectory of themollified flow so that (1.4) can be realized. Notice that a cylinder Q r in the local( s, y ) coordinate will be transformed into a “skewed cylinder” growing along X ( t )in the global ( t, x ) coordinate. One of the authors recently constructed a maximalfunction M Q associated with these cylinders ([Yan20]), which serves as a bridgebetween the local theorem and the global result, and is one of the main reasonsfor the improvement in this paper. The idea is, if locally the vorticity gradient canbe controlled in L ∞ by the integral of something in the skewed cylinder, and theintegral in a skewed cylinder can be controlled by the maximal function M Q , thenvorticity gradient is pointwisely bounded by the maximal function.If one uses ´ |∇ u | d x d t and ´ |∇ P | d x d t as the pivot quantity, then unfor-tunately the best possible outcome would just be an L , ∞ bound, as obtained in[Yan20]. The reason is, the maximal function is bounded on L p for p >
1, but for p = 1 it is only bounded from L to L , ∞ . Unfortunately |∇ u | and |∇ P | areboth L quantities, so M Q (cid:0) |∇ u | + |∇ P | (cid:1) is only L , ∞ . We need two things toimprove from L , ∞ : replace ´ |∇ u | by ´ |∇ u | p , and drop the pressure ∇ P .Suppose we could use ( ´ |∇ u | p d x d t ) p as the pivot quantity for some p < M Q ( |∇ u | p ) p ∈ L , since p > M Q is bounded in L p . However, this poses significant difficulties in the local theorem. The nonlinearterm u · ∇ u is quadratic, and if we only have a subquadratic integrability to beginwith, we cannot treat this quadratic transport term as a source term because it isnot integrable. Observe that what we lack of is the temporal integrability ratherthan the spatial one: if p is slightly smaller than two, than u · ∇ u is still L − inspace, but L − in time. To overcome this difficulty, we write u · ∇ u as ω × u upto a gradient term, and put L − t L − x on u and L t L − x on ω . We compensate thelower integrability term by pairing with a higher integrability term to make ω × u integrable. L t L − x of ω can be interpolated between L − t L − x and L ∞ t L x , while thelatter is controlled by L t,x of ∇ u . Since L t L − x is closer to L − t L − x than to L ∞ t L x ,the pivot quantity that we use is actually δ − ν k∇ u k L p + δ k∇ u k L for ν close to 0.By using more subquadratic integrability and a tiny bit of the quadratic one, wecan complete the task by interpolation. That is why we obtain L ,q in the end: itinterpolates L bound from k∇ u k L p and L , ∞ bound from k∇ u k L . Unfortunatelywe still miss the endpoint L .The second task is more subtle and technical. Without any information on thepressure, we don’t have any control on the nonlocal effect. However, the role of the ECOND DERIVATIVE OF NAVIER-STOKES 5 pressure is not important at the vorticity level: if we take the curl of the Navier-Stokes equation, the pressure will disappear and we are left with the vorticityequation involving only local quantities: ∂ t ω + u · ∇ ω − ω · ∇ u = ∆ ω. (1.6)Inspired by Chamorro, Lemari´e-Rieusset and Mayoufi ([CLRM18]), we introduce anew velocity variable v = − curl ϕ ♯ ∆ − ϕω using only local information of vorticity( ϕ and ϕ ♯ are spatial cut-off functions), and this helps us to prove the local theorem.This is another main reason for the improvement in this paper. Consequently, thebounds we obtain in the end is on the vorticity ω rather than on the velocity u .This paper is organized as follows. In the preliminary Section 2 we introducethe analysis tools to the reader. We show how to rigorously derive the main resultsfrom the local theorem in Section 3, and then deal with technicalities of the localtheorem in the later sections. The proof of the local theorem consists of threeparts. Section 4 introduces the new variables v , and shows the smallness of v inthe energy space. Then we use De Giorgi iteration argument in Section 5 to proveboundedness of v . Finally, we inductively bound ω and all its higher derivatives inSection 6. 2. Preliminary
In this section, we introduce a few tools that we are going to use in the paper,including the maximal function, Lorentz space, and Helmholtz decomposition.2.1.
Maximal Function associated with Skewed Cylinders.
This is recentlydeveloped for incompressible flows in [Yan20]. We quote useful results here withoutproof.Suppose u ∈ L p (0 , T ; ˙ W ,p ( R ; R )) is a vector field in R . Fix φ ∈ C ∞ c ( B ) tobe a nonnegative function with ´ φ = 1 through out the paper. For ε > φ ε ( x ) = ε − φ ( − x/ε ), and let u ε ( t, · ) = u ( t, · ) ∗ φ ε be the mollified velocity. For afixed ( t, x ) we let X ( s ) solve the following initial value problem, ( ˙ X ( s ) = u ε ( s, X ( s )) ,X ( t ) = x. The skewed parabolic cylinder Q ε ( t, x ) is then defined to be Q ε ( t, x ) := (cid:8) ( t + ε s, X ( t ) + εy ) : − ≤ s ≤ , y ∈ B (cid:9) . (2.1)We use M to denote the spatial Hardy-Littlewood maximal function, which isdefined by M ( f )( t, x ) = sup r> B r ( x ) | f ( t, y ) | d y. Then we construct the space-time maximal function adapted to the flow.
Theorem 2.1 ( Q -Maximal Function) . There exists a universal constant η suchthat the following is true. We say Q ε ( t, x ) is admissible if Q ε ( t, x ) ⊂ (0 , T ) × R and ε Q ε ( t,x ) M ( |∇ u | ) d x d t ≤ η . (2.2) A. VASSEUR AND J. YANG
Define the maximal function M Q ( f )( t, x ) := sup ε> ( Q ε ( t,x ) | f ( s, y ) | d s d y : Q ε ( t, x ) is admissible ) . If u is divergence free and M ( |∇ u | ) ∈ L q for some ≤ q ≤ ∞ , then M Q is boundedfrom L ((0 , T ) × R ) to L , ∞ ((0 , T ) × R ) and from L p ((0 , T ) × R ) to itself for any p > with norm depending on p . An important consequence of the weak type (1 ,
1) bound of the Hardy-Littlewoodmaximal function is the Lebesgue differentiation theorem in R n . Similarly, we canuse the Q -maximal function to prove the Q -Lebesgue differentiation theorem. Theorem 2.2 ( Q -Lebesgue Differentiation Theorem) . Let f ∈ L ((0 , T ) × R ) .Then for almost every ( t, x ) ∈ (0 , T ) × R , lim ε → Q ε ( t,x ) | f ( s, y ) − f ( t, x ) | d s d y = 0 . In this case we say ( t, x ) is a Q -Lebesgue point of f . Lorentz Space.
Let (
X, µ ) be a measure space. Recall that for a measurablefunction f , its decreasing rearrangement is defined as f ∗ ( λ ) := inf { α > µ ( {| f | > α } ) < λ } , λ ≥ . For 0 < p < ∞ , 0 < q ≤ ∞ , Lorentz space L p,q ( X ) is defined as the set of functions f for which k f k L p,q ( X ) := k t p f ∗ k L q ( d tt ) < ∞ . Now we introduct the interpolation lemma for Lorentz spaces.
Lemma 2.3 (Interpolation of Lorentz Spaces) . Let ν > be a fixed positive number.Assume f ∈ L p ,q , f ∈ L p ,q , where < p , p < ∞ , < q , q ≤ ∞ . If f is ameasurable function satisfying | f | ≤ δf + δ − ν f ∀ δ > then f ∈ L p,q , where p = ν ν p + 11 + ν p , q = ν ν q + 11 + ν q . Proof.
Upon on decreasing rearrangement, we may assume f, f , f are nonnegativedecreasing functions on [0 , ∞ ). Set θ = ν , δ = f − θ f θ , then2 | f | ≤ f − θ f θ f + f νθ f − νθ f = 2 f − θ f θ . Then k f k L p,q = k λ p − q f ( λ ) k L q ≤ k λ − θp − − θq f − θ ( λ ) · λ θp − θq f θ ( λ ) k L q ≤ k λ − θp − − θq f − θ ( λ ) k L q − θ k λ θp − θq f θ ( λ ) k L q θ = k λ p − q f k − θL q k λ p − q f k θL q = k f k − θL p ,q k f k θL p ,q . (cid:3) ECOND DERIVATIVE OF NAVIER-STOKES 7
We would also like to mention that Riesz transform is bounded on Lorentz space.The proof can be found in [CF07]. See [Saw90] for general Lorentz spaces.
Lemma 2.4.
For < p < ∞ , ≤ q ≤ ∞ , R ij = ∂ i ∂ j ∆ − is a bounded linearoperator from L p,q ( R n ) to itself. As a spatial operator, it is also bounded in time-space from L p,q ((0 , T ) × R n ) to itself. Helmholtz decomposition.
First recall two vector calculus identities: ∇ ( u · v ) = ( u · ∇ ) v + ( v · ∇ ) u + u × curl v + v × curl u, (2.3) curl( u × v ) = u div v − v div u + ( v · ∇ ) u − ( u · ∇ ) v. (2.4)For operators A and B , denote [ A, B ] = AB − BA to be their commutator. Define P curl = − curl curl ∆ − and P ∇ = ∇ ∆ − div = Id − P curl to be the Helmholtzdecomposition. Then we compute the following commutators.[ ϕ, curl] u = −∇ ϕ × u, (2.5) [ ϕ, ∆] u = − ∇ ϕ · ∇ u − (∆ ϕ ) u = − ∇ ϕ ⊗ u ) + (∆ ϕ ) u, (2.6) [ ϕ, ∆ − ] u = ∆ − (cid:8) ∇ ϕ · ∇ ∆ − u + (∆ ϕ )∆ − u (cid:9) , (2.7) [ ϕ, P curl ] u = ∇ ϕ × curl ∆ − u + ∇ ϕ div ∆ − u − ∆ − u ∆ ϕ (2.8) + (∆ − u · ∇ ) ∇ ϕ − ( ∇ ϕ · ∇ )∆ − u + P curl (cid:8) ∇ ϕ · ∇ ∆ − u + (∆ ϕ )∆ − u (cid:9) . The first two are straightforward. The third uses[ ϕ, ∆ − ] = − ∆ − [ ϕ, ∆]∆ − , and the last one is because[ ϕ, P curl ] = [ ϕ, − curl curl ∆ − ]= − [ ϕ, curl] curl ∆ − − curl[ ϕ, curl]∆ − − curl curl[ ϕ, ∆ − ] , [ ϕ, P curl ] u = ∇ ϕ × curl ∆ − u + curl( ∇ ϕ × ∆ − u ) − curl curl ∆ − (cid:8) ∇ ϕ · ∇ ∆ − u + (∆ ϕ )∆ − u (cid:9) , and we can expand curl( ∇ ϕ × ∆ − u ) by (2.4). Lemma 2.5. ∂ i [ ϕ, P curl ] and [ ϕ, P curl ] ∂ i are both bounded linear operator from L p to L p for any < p < ∞ , i.e. k ∂ i [ ϕ, P curl ] u k L p + k [ ϕ, P curl ] ∂ i u k L p ≤ C p,ϕ k u k L p . Proof.
First, we observe that by Jacobi identity [ ϕ, P curl ] ∂ i and ∂ i [ ϕ, P curl ] differ by[[ ϕ, P curl ] , ∂ i ] = [ ϕ, [ P curl , ∂ i ]] − [ P curl , [ ϕ, ∂ i ]] = 0 − [ P curl , ∂ i ϕ ]which is bounded from L p to L p for any p , because both P curl and multiplicationby ∂ i ϕ are bounded from L p to L p , so we can complete the proof by duality. For1 < p <
3, set p ∗ = p − , from (2.8) we can see k [ ϕ, P curl ] ∂ i u k L p . k∇ ∆ − ∂ i u k L p ( R ) + C p,ϕ k ∆ − ∂ i u k L p (supp ϕ ) . k u k L p ( R ) + C p,ϕ k ∂ i ∆ − u k L p ∗ (supp ϕ ) ≤ C k u k L p ( R ) . A. VASSEUR AND J. YANG
For < p < ∞ , set 1 − p = q = q ∗ + , then 1 < p, q, q ∗ < ∞ . Take any u ∈ L p ( R )and any vector field v ∈ L q ( R ), ˆ ∂ i [ ϕ, P curl ] u · v d x = − ˆ [ ϕ, P curl ] u · ∂ i v d x = ˆ u · [ ϕ, P curl ] ∂ i v d x ≤ k u k L p ( R ) ( k v k L q ( R ) + k ∂ i ∆ − v k L q (supp ϕ ) ) ≤ k u k L p ( R ) ( k v k L q ( R ) + C q,ϕ k ∂ i ∆ − v k L q ∗ (supp ϕ ) ) ≤ C k u k L p ( R ) k v k L q ( R ) . (cid:3) Corollary 2.6. ∂ i [ ϕ, P ∇ ] and [ ϕ, P ∇ ] ∂ i are both bounded linear operator from L p to L p for any < p < ∞ .Proof. Id = P ∇ + P curl commutes with ϕ , so [ ϕ, P ∇ ] = − [ ϕ, P curl ]. (cid:3) Because of the smoothing effect of the Laplace potential, we have the following.
Lemma 2.7.
Let ϕ ∈ C ∞ c ( R ) be supported away from some openset Ω ⊂ R , thatis, dist(supp ϕ, Ω) = d > . Then for any f ∈ L ( R ) , k > , k ∆ − ( ϕf ) k C k (Ω) . k,d k f k L (supp ϕ ) . We also have k P ∇ ( ϕf ) k C k (Ω) , k P curl ( ϕf ) k C k (Ω) . k,d k f k L (supp ϕ ) . Proof of the Main Results
In this section, we show that the Local Theorem 1.3 leads to the main results.First, we show the pivot quantity is indeed enough to bound ∇ n ω . Lemma 3.1.
There exists η > such that the following holds. Let < p < , − pp − < ν ≤ p − − p . If u is a suitable solution to the Navier-Stokes equations in ( − , × R satisfying the following conditions, ˆ B u ( t, x ) φ ( x ) d x = 0 , a.e. t ∈ ( − , , (3.1) δ − ν (cid:18) ˆ Q |∇ u | p d x d t (cid:19) p ≤ η , (3.2) δ ˆ Q |∇ u | d x d t ≤ η , (3.3) for some δ ≤ η , then we have for any n ≥ , k∇ n ω k L ∞ t,x ( Q − n − ) ≤ C n . Here C n is the same constant in Theorem 1.3.Proof. First, we claim that δ k ω k L ∞ ( − , L ( B )) ≤ Cη . (3.4) ECOND DERIVATIVE OF NAVIER-STOKES 9
Formally, we can take the dot product of both sides of the vorticity equation (1.1)with ω := ω | ω | , and recalling the convexity inequality ω · ∆ ω ≤ ∆ | ω | , we have( ∂ t + u · ∇ − ∆) | ω | − ω · ∇ u · ω ≤ . (3.5)Let ψ ∈ C ∞ c (( − , × R ) be a cut-off function such that Q ≤ ψ ≤ Q . Multiply(3.5) by ψ and then integrate in space,dd t ˆ ψ | ω | d x ≤ ˆ [( ∂ t + u · ∇ + ∆) ψ ] | ω | d x + ˆ ψω · ∇ u · ω d x ≤ C ˆ B | u | + |∇ u | d x ≤ C (cid:18) ˆ B |∇ u | d x (cid:19) . for some large universal constant C >
1. The last step uses Poincar´e’s inequalityand (3.1). Integrate in time we obtain k ω k L ∞ ( − , L ( B )) ≤ C (cid:16) η δ (cid:17) ≤ C η δ . This proves the claim. A more rigorous proof can be obtained by difference quotientsame as in Constantin [Con90] or Lions [Lio96] Theorem 3.6, so we omit the details.Now we interpolate between (3.2) and (3.4). Let θ = ν , k ω k L p t L q x ( Q ) ≤ k ω k θL p ( Q ) k ω k − θL ∞ t L x ( Q ) ≤ (2 C ) − θ η δ θν + θ − ≤ η , where we choose η = η C +1 ≤ η from Theorem 1.3, and p , q are determined by1 p = θp , q = θp + 1 − θ. Combine the above with (3.2) we have k∇ u k L pt L px ( Q ) + k ω k L p t L q x ( Q ) ≤ η + 12 η = η . (3.6)By the choice of θ and the range of ν ,1 p + 1 p = 1 p + 1 p (1 + ν ) = 2 + νp (1 + ν ) < , p + 1 q = 1 p + 1 + νpp (1 + ν ) = 2 + ν + νpp (1 + ν ) ≤ . One can also easily check that p < q <
2, and thus by (3.1) and (3.6)the requirements of the Local Theorem 1.3 are satisfied with p = q = p , and itcompletes the proof of the lemma. (cid:3) Now we transform this lemma into the global coordinate. Recall that Q ε ( t, x ) isdefined by (2.1). Corollary 3.2.
There exists η > such that the following holds. If for some δ ≤ η , δ − ν Q ε ( t,x ) |∇ u | p d x d t ! p + δ Q ε ( t,x ) |∇ u | d x d t ≤ η ε − , (3.7) then |∇ n ω ( t, x ) | ≤ C n ε − n − . Proof.
Define ˜ u by (1.3). Then (3.7) implies δ − ν (cid:18) Q |∇ ˜ u | p d x d t (cid:19) p ≤ η , δ Q |∇ ˜ u | d x d t ≤ η ⇒ δ − ν (cid:18) ˆ Q |∇ ˜ u | p d x d t (cid:19) p ≤ η | Q | p , δ ˆ Q |∇ ˜ u | d x d t ≤ η | Q | . Moreover, (3.1) is satisfied by ˜ u . Therefore, if we choose η such thatmax n η | Q | p , η | Q | o = η , then by Lemma 3.1, ˜ ω := curl ˜ u has bounded derivatives at (0 , (cid:3) Then we use the maximal function to go from the local bound to a global bound.
Proof of Theorem 1.1.
First, we fix < p < − pp − < ν ≤ p − − p . Let η << < δ < ∞ . For( t, x ) ∈ (0 , T ) × R , define I ( ε ) = ε δ − ν Q ε ( t,x ) |M ( ∇ u ) | p ! p + δ Q ε ( t,x ) |M ( ∇ u ) | . If ( t, x ) is both a Q -Lebesgue point of |M ( ∇ u ) | p and of |M ( ∇ u ) | , then we claimthat there exists a positive ε = ε ( t,x ) such that one of the two cases is true: Case 1. ε ( t,x ) < t , and I ( ε ( t,x ) ) = η . Case 2. ε ( t,x ) = t , and I ( ε ( t,x ) ) ≤ η .This is because by Theorem 2.2lim ε → I ( ε ) = 0 h δ − ν ( |M ( |∇ u | ) ( t, x ) | p ) p + δ |M ( |∇ u | ) ( t, x ) | i = 0 , and I ( ε ) is a continuous function of ε .One the one hand, in both cases we have I ( ε ) ≤ η , which implies that δ − ν ε Q ε ( t,x ) |M ( ∇ u ) | p ! p ≤ √ η, δ ε Q ε ( t,x ) |M ( ∇ u ) | ! ≤ √ η. If we set η < η , then depending on δ ≥ δ ≤
1, one of the two would implyadmissibility condition (2.2) by Jensen’s inequality. Therefore Q ε ( t, x ) is admissibleand I ( ε ) ≤ ε h δ − ν M Q ( M ( ∇ u ) p ) p + δ M Q ( M ( ∇ u ) ) i , so we can combine two canses and conclude ε − t,x ) ≤ max (cid:26) η h δ − ν M Q ( M ( ∇ u ) p ) p + δ M Q ( M ( ∇ u ) ) i , t − (cid:27) . (3.8)On the other hand, if we set η < η , then in both cases I ( ε ) ≤ η . If δ ≤ η onewould have |∇ n ω ( t, x ) | ≤ C n ε − n − (3.9) ECOND DERIVATIVE OF NAVIER-STOKES 11 by Corollary 3.2. If δ > η , notice that by Jensen’s inequality, Q ε ( t,x ) |M ( ∇ u ) | p ! p ≤ Q ε ( t,x ) |M ( ∇ u ) | , so I ( ε ) ≥ ε ( δ − ν + δ − η ) Q ε ( t,x ) |M ( ∇ u ) | p ! p + η Q ε ( t,x ) |M ( ∇ u ) | ≥ ε (1 − η ) Q ε ( t,x ) |M ( ∇ u ) | p ! p + η Q ε ( t,x ) |M ( ∇ u ) | ≥ (1 − η ) η ν ε η − ν Q ε ( t,x ) |M ( ∇ u ) | p ! p + η Q ε ( t,x ) |M ( ∇ u ) | . If we require η < (1 − η ) η ν η , then ε η − ν Q ε ( t,x ) |M ( ∇ u ) | p ! p + η Q ε ( t,x ) |M ( ∇ u ) | ≤ η . again by Corollary 3.2 we would still have (3.9). In conclusion, we choose η = min (cid:8) η , (1 − η ) η ν η (cid:9) , then for any 0 < δ < ∞ one would have |∇ n ω ( t, x ) | n +2 ≤ C n +2 n max (cid:26) η h δ − ν M Q ( M ( ∇ u ) p ) p + δ M Q ( M ( ∇ u ) ) i , t − (cid:27) by putting (3.9) and (3.8) together. Denote f = |∇ n ω | n +2 , and we denote f = M Q ( M ( |∇ u | ) p ) p , f = M Q ( M ( |∇ u | ) ). Then we have almost everywhere f { f>C n t − } . n δ − ν f + δf . By Theorem 2.1, k f k L ≤ C p kM ( ∇ u ) p k p L p . C p kM ( ∇ u ) k L ≤ C p k∇ u k L , k f k L , ∞ ≤ C kM ( ∇ u ) k L ≤ C k∇ u k L . Finally, by the interpolation between Lorentz spaces Lemma 2.3, k f { f>C n t − } k L , ν . p,n k∇ u k L ((0 ,T ) × R ) ≤ k u k L ( R ) . This proves the theorem for q ≥ ν . Recall that p can be arbitrarily chosenbetween and 2, and ν can be chosen between − pp − and p − − p , so ν can bearbitrarily small, therefore we prove the theorem for any q > (cid:3) Estimates on ∇ u can be obtained by a Riesz transform of ∆ u = − curl ω . Proof of Corollary 1.2.
We can put K ⊂ ( t , T ) × B R for some t , T, R >
0. Denote Q = ( t , T ) × B R . Let ρ ∈ C ∞ c ( R ) be a smooth spatial cut-off function between B R ≤ ρ ≤ B R . Then k ∆( ρu ) k L ,q (( t ,T ) × R ) . ρ k ∆ u k L ,q ( Q ) + k∇ u k L ,q ( Q ) + k u k L ,q ( Q ) . Since ∆ u = − curl ω , the case n = 1 of Theorem 1.1 gives k ∆ u {| ∆ u | >C t − } k L ,q ((0 ,T ) × R ) ≤ C q k u k L ( R ) , so k ∆ u k L ,q ( Q ) ≤ C q k u k L ( R ) + C k t − k L ( Q ) . C q k u k L ( R ) + C (cid:18) R t (cid:19) . As for lower order terms, k∇ u k L ( Q ) . k∇ u k L ( Q ) , k u k L ( Q ) ≤ k u k L ∞ t L x ( Q ) . For Leray-Hopf solution, k∇ u k L ∞ t L x ∩ L t ˙ H x ((0 ,T ) × R ) ≤ k u k L , so k ∆( ρu ) k L ,q (( t ,T ) × R ) . q,K k u k L ( R ) + 1 + k u k L ( R ) . k u k L ( R ) + 1 . Because Riesz transform is bounded from L ,q (( t , T ) × R ) to itself by Lemma 2.4, k∇ u k L ,q ( K ) ≤ k∇ ( ρu ) k L ,q ( Q ) . q,K k u k L ( R ) + 1 . (cid:3) Remark . For smooth solutions to the Navier-Stokes equation, we have L ,q estimate for the third derivatives for any q > (cid:13)(cid:13) ∇ ω {|∇ ω | >Ct − } (cid:13)(cid:13) L ,q ((0 ,T ) × R ) ≤ C q k u k L . Local Study: Part One, Initial Energy
The following three sections are dedicated to the proof of the Local Theorem1.3. In [Vas10], the proof of the local theorem consists of the following three parts:
Step 1.
Show the velocity u is locally small in the energy space E = L ∞ t L x ∩ L t H x . Step 2.
Use De Giorgi iteration and the truncation method developed in [Vas07]to show u is locally bounded in L ∞ . Step 3.
Bootstrap to higher regularity by differentiating the original equation.In our case, directly working with u is difficult due to the lack of control on thepressure, which is nonlocal. Therefore, we would like to work on vorticity, whoseevolution is governed by (1.6) and only involves local quantities. Since ω is onederivative of u , we have less integrability to do any parabolic regularization, andwe don’t have the local energy inequality to perform De Giorgi iteration. Thismotivates us to work on minus one derivative of ω , but instead of ω we use alocalization of ω . Similar as [CLRM18], we introduce a new local quantity v := − curl ϕ ♯ ∆ − ϕ curl u = − curl ϕ ♯ ∆ − ϕω. where ϕ and ϕ ♯ are a pair of fixed smooth spatial cut-off functions, which aredefined between B ≤ ϕ ≤ B , B ≤ ϕ ♯ ≤ B . This v is divergence free andcompactly supported. It will help us get rid of the pressure P , while staying in thesame space as u : it scales the same as u , has the same regularity, inherit a localenergy inequality from u , and its evolution only depends on local information. Wewill follow the same three steps above, but we will work on v instead of u . ECOND DERIVATIVE OF NAVIER-STOKES 13
For convenience, from now on we will use η to denote a small universal constantdepending only on the smallness of η , such that lim η → η = 0. Similar as theconstant C , the value of η may change from line to line. The purpose of thissection is to obtain the smallness of v in the energy space E , which is the followingproposition. Proposition 4.1.
Under the same assumptions of the Local Theorem 1.3, we have k v k E ( Q ) = sup t ∈ ( − , ˆ B | v ( t ) | d x + ˆ Q |∇ v | d x ≤ η. (4.1)For convenience, define q , q , q by1 q = 1 q − , q = 1 q − , q = (cid:18) q − (cid:19) + . Equations of v.
We use (2.3) in (1.1) to rewrite the equation of u , then takethe curl to rewrite the equation of ω , finally apply − curl ϕ ♯ ∆ − ϕ on the vorticityequation to obtain the equation of v . ∂ t u + P curl ( ω × u ) = ∆ u,∂ t ω + curl( ω × u ) = ∆ ω,∂ t v − curl ϕ ♯ ∆ − ϕ curl( ω × u ) = − curl ϕ ♯ ∆ − ϕ ∆ ω. (4.2)The second term of (4.2) iscurl ϕ ♯ ∆ − ϕ curl( ω × u ) = B − P curl ( ϕω × u )where B denotes the quadratic commutator B := − curl(1 − ϕ ♯ )∆ − ϕ curl( ω × u ) + curl ∆ − [ ϕ, curl]( ω × u )= − curl(1 − ϕ ♯ )∆ − ϕ curl( ω × u ) + curl ∆ − ( −∇ ϕ × ( ω × u ))Here we used (2.5). The right hand side of (4.2) is − curl ϕ ♯ ∆ − ϕ ∆ ω = ∆ v + L where L denotes the linear commutator L := [ − curl ϕ ♯ ∆ − ϕ, ∆] ω = − curl[ ϕ ♯ ∆ − ϕ, ∆] ω = − curl[ ϕ ♯ , ∆]∆ − ϕω − curl ϕ ♯ ∆ − [ ϕ, ∆] ω = − curl[ ϕ ♯ , ∆]∆ − ϕω + curl ϕ ♯ ∆ − (2 div( ∇ ϕ ⊗ ω ) − (∆ ϕ ) ω ) . Here we used (2.6). Therefore we have the equation for v as the following, ∂ t v + P curl ( ϕω × u ) = B + L + ∆ v. (4.3)We observe the following localization decomposition. Lemma 4.2.
We can decompose ϕu = v + w, ϕω = curl v + ̟, where w and ̟ are harmonic inside B . Proof.
We can compute v by v = − curl ϕ ♯ ∆ − ϕ curl u = curl(1 − ϕ ♯ )∆ − ϕ curl u − curl ∆ − ϕ curl u = curl(1 − ϕ ♯ )∆ − ϕω − curl ∆ − [ ϕ, curl] u + P curl ( ϕu )= curl(1 − ϕ ♯ )∆ − ϕω + curl ∆ − ( ∇ ϕ × u ) − P ∇ ( ϕu ) + ϕu = curl(1 − ϕ ♯ )∆ − ϕω + curl ∆ − ( ∇ ϕ × u ) − ∇ ∆ − ( ∇ ϕ · u ) + ϕu using div u = 0. We denote w := − curl(1 − ϕ ♯ )∆ − ϕω − curl ∆ − ( ∇ ϕ × u ) + ∇ ∆ − ( ∇ ϕ · u ) , which implies the first decomposition ϕu = v + w . By taking the curl,curl( ϕu ) = curl v + curl w, ∇ ϕ × u + ϕω = curl v − curl curl(1 − ϕ ♯ )∆ − ϕω − curl curl ∆ − ( ∇ ϕ × u )= curl v − curl curl(1 − ϕ ♯ )∆ − ϕω + P curl ( ∇ ϕ × u ) . We denote ̟ := − curl curl(1 − ϕ ♯ )∆ − ϕω − P ∇ ( ∇ ϕ × u )= − curl curl(1 − ϕ ♯ )∆ − ϕω − ∇ ∆ − div( ∇ ϕ × u )= − curl curl(1 − ϕ ♯ )∆ − ϕω + ∇ ∆ − ( ∇ ϕ · ω )which implies the second decomposition ϕω = curl v + ̟ . We can easily see that∆ w and ∆ ̟ are both the sum of a smooth function supported outside B and theNewtonian potential of something supported inside supp( ∇ ϕ ) ⊂ B \ B , so theyare harmonic inside B . (cid:3) Using this decomposition, we can continue to expand P curl ( ϕω × u ) = ϕω × u − P ∇ ( ϕω × u )= ω × v + ω × w − P ∇ ((curl v + ̟ ) × u + ω × ( v + w ))= ω × v − P ∇ (curl v × u + ω × v ) − W , where W denotes the remainders involving w and ̟ , W := − ω × w + 12 P ∇ ( ̟ × u + ω × w ) . By subtracting (2.4) from (2.3), for divergence free u , v we havecurl v × u + curl u × v = −∇ ( u · v ) + 2 u · ∇ v + curl( u × v ) , so P curl ( ϕω × u ) = ω × v + 12 ∇ ( u · v ) − P ∇ div( u ⊗ v ) − W = ω × v + ∇ (cid:18) u · v − ∆ − div div( u ⊗ v ) (cid:19) − W . For convenience, denote the Riesz operator R = 12 tr − ∆ − div div ECOND DERIVATIVE OF NAVIER-STOKES 15
Finally, we have the equation of v as ∂ t v + ω × v + ∇ R ( u ⊗ v ) = B + L + W + ∆ v, div v = 0 . (4.4)We now check the spatial integrability of these new terms. Lemma 4.3.
For any < p < ∞ , k v k L p , k∇ w k L p , k ̟ k L p . k ω k L ( B ) + k u k L p ( B ) , k∇ v k L p , k∇ ̟ k L p . k ω k L p ( B ) , k∇ w k L p . k u k W ,p ( B ) . If we denote q = min { , p + } − , then k B k L p ( B ) . k ω × u k L q ( B ) , k L k L p ( B ) . k ω k L p ( B ) , k W k L p ( B ) . k ω × w k L p ( B ) + k ̟ × u k L p ( B ) . Proof. v, w, ̟ are all supported inside B , so k v k L p ≤ k ϕu k L p + k w k L p . k u k L p ( B ) + k∇ w k L p , k∇ w k L p ≤ k∇ curl(1 − ϕ ♯ )∆ − ϕω k L p ( B ) + k∇ curl ∆ − ( ∇ ϕ × u ) k L p + k∇ ∆ − ( ∇ ϕ · u ) k L p . k (1 − ϕ ♯ )∆ − ϕω k C + k∇ ϕ × u k L p + k∇ ϕ · u k L p . k ω k L ( B ) + k u k L p ( B ) , k ̟ k L p ≤ k curl curl(1 − ϕ ♯ )∆ − ϕω k L p ( B ) + k∇ ∆ − ( ∇ ϕ · ω ) k L p ≤ k (1 − ϕ ♯ )∆ − ϕω k C + k∇ ∆ − div( ∇ ϕ × u ) k L p ≤ k ω k L ( B ) + k∇ ϕ × u k L p ≤ k ω k L ( B ) + k u k L p ( B ) . Here we used Lemma 2.7 since ϕ and 1 − ϕ ♯ are supported away from each other, andwe also used the boundedness of Riesz transform by Lemma 2.4. Their derivativesare bounded by k∇ v k L p = k∇ curl ϕ ♯ ∆ − ϕω k L p ≤ k∇ curl ∆ − ϕω k L p + k∇ curl(1 − ϕ ♯ )∆ − ϕω k L p ( B ) . k ω k L p ( B ) + k ω k L ( B ) . k ω k L p ( B ) , k∇ w k L p ≤ k∇ curl(1 − ϕ ♯ )∆ − ϕω k L p ( B ) + k∇ curl ∆ − ( ∇ ϕ × u ) k L p + k∇ ∆ − ( ∇ ϕ · u ) k L p . k ω k L ( B ) + k u k W ,p ( B ) . k u k W ,p ( B ) , k∇ ̟ k L p ≤ k∇ curl curl(1 − ϕ ♯ )∆ − ϕω k L p ( B ) + k∇ ∆ − ( ∇ ϕ · ω ) k L p . k ω k L ( B ) + k ω k L p ( B ) . k ω k L p ( B ) . The proof for B , L , W are similar so we omit here. (cid:3) Since u ∈ E , it can be seen from the above lemma that v, ∇ w, ̟ ∈ E , thus k B k L ( B ) . k ω × u k L ( B ) ∈ L t , k L k L ( B ) . k ω k L ( B ) ∈ L t , k W k L ( B ) . k ω × w k L ( B ) + k ̟ × u k L ( B ) ∈ L t , therefore B , L , W ∈ L t L loc ,x . In the appendix we prove the suitability for v : itsatisfies the following local energy inequality, ∂ t | v | |∇ v | + div [ v R ( u ⊗ v )] ≤ ∆ | v | v · ( B + L + W ) . (4.5)4.2. Energy Estimate.
Multiply (4.5) by ϕ then integrate over R yieldsdd t ˆ ϕ | v | x + ˆ ϕ |∇ v | d x ≤ ˆ | v | ϕ d x + ˆ ( v · ∇ ϕ ) R ( u ⊗ v ) d x + ˆ ϕ v · B d x + ˆ ϕ v · L d x + ˆ ϕ v · W d x. Let us discuss these terms. For the first four terms on the right hand side, I ∆ := ˆ | v | ϕ d x ≤ C k ϕ v k L k v k L , (4.6) I R := ˆ ( v · ∇ ϕ ) R ( u ⊗ v ) d x ≤ C k ϕ v k L k R ( u ⊗ v ) k L (4.7) ≤ C k ϕ v k L k u ⊗ v k L ,I B := ˆ ϕ v · B d x ≤ k ϕ v k L k ϕ B k L (4.8) ≤ C k ϕ v k L k ω × u k L ( B ) ,I L := ˆ ϕ v · L d x ≤ k ϕ | v | k L k| v | k L q k ϕ L k L q (4.9) ≤ k ϕ v k L k v k L q k ω k L q ( B ) . Here we use H¨older’s inequality, ϕ is compactly supported in B and q + q + ≤ W term, I W := ˆ ϕ v · W d x = − ˆ ϕ v · ω × w d x + 12 ˆ ϕ v · P ∇ ( ̟ × u + ω × w ) d x = − I W + 12 I W . For the first one, we break it as I W = ˆ ϕ v · ω × w d x = ˆ ϕ v × curl v · w d x + ˆ ϕ v · ̟ × w d x. ECOND DERIVATIVE OF NAVIER-STOKES 17
Using (2.3), v × curl v = 12 ∇| v | − ( v · ∇ ) v, we have ˆ ϕ v × curl v · w d x = − ˆ | v | div( ϕ w ) d x + ˆ v · ∇ ( ϕ w ) · v d x ≤ C k ϕ v k L ( k∇ w ⊗ v k L + k w ⊗ v k L ) . The remaining is of lower order, ˆ ϕ v · ̟ × w d x ≤ C k ϕ v k L k ̟ × w k L For the second one, I W = ˆ P ∇ ( ϕ v ) · ( ̟ × u + ω × w ) d x ≤ k P ∇ ( ϕ v ) k L k ̟ × u + ω × w k L where k P ∇ ( ϕ v ) k L = k [ P ∇ , ϕ ] ϕ v k L . k ϕ v k L . So I W can be bounded by I W ≤ C k ϕ v k L (cid:16) k∇ w ⊗ v k L + k ̟ × w k L + k ̟ × u + ω × w k L (cid:17) . (4.10)In summary, we conclude that for − ≤ t ≤ t ˆ ϕ | v | x + ˆ ϕ |∇ v | d x ≤ I ∆ + I R + I B + I L + I W (4.11)with good estimates on each of the term on the right.4.3. Proof of Proposition 4.1.
First we check the integrability of each terms.
Lemma 4.4 (Integrability) . Given conditions (1.4) and (1.5) , we have k u k L p t L q x ( Q ) ≤ η, k ϕω k L p t L q x (( − , × R ) ≤ η, k ϕω k L p t L q x (( − , × R ) ≤ η, k∇ v k L p t L q x (( − , × R ) ≤ η, k∇ v k L p t L q x (( − , × R ) ≤ η, k v k L p t L q x (( − , × R ) ≤ η, k v k L p t L q x (( − , × R ) ≤ η, k∇ w k L p t L q x (( − , × R ) ≤ η, k w k L p t L q x (( − , × R ) ≤ η, (4.12) k ̟ k L p t L q x (( − , × R ) ≤ η, k ̟ k L p t L q x (( − , × R ) ≤ η. (4.13) Proof.
Integrability of u is obtained by Sobolev embedding and that ϕu has average0. Integrability of ϕω is given. The remaining are consequences of Lemma 4.3 andSobolev embedding. (cid:3) Proof of Proposition 4.1.
We prove Proposition 4.1 using a Gr¨onwall argument.Multiply (4.11) by an increasing smooth function ψ ( t ) with ψ ( t ) = 0 for t ≤ − ψ ( t ) = 1 for t ≥
0, we havedd t (cid:18) ψ ( t ) ˆ ϕ | v | x (cid:19) + ψ ( t ) ˆ ϕ |∇ v | d x = ψ ′ ( t ) ˆ ϕ | v | x + ψ ( t ) ( I ∆ + I R + I B + I L + I W ) . Integrate from − t < ψ ( t ) ˆ ϕ | v | x + ˆ t − ψ ( s ) ˆ ϕ |∇ v | d x = ˆ t − ψ ′ ( s ) ˆ ϕ | v | x d t + ˆ t − ψ ( s ) ( I ∆ , R , B , L , W ) d t. Because (4.6), (4.7), (4.8), (4.9), (4.10), and k ϕ v k L ( B ) , k ϕ v k L ( B ) ≤ C (cid:18) ˆ ϕ | v | d x (cid:19) , we can conclude thatdd t (cid:18) ψ ( t ) ˆ ϕ | v | x (cid:19) + ψ ( t ) ˆ ϕ |∇ v | d x ≤ C Φ( t ) (cid:18) ψ ( t ) ˆ ϕ | v | x (cid:19) , where Φ( t ) = ψ ′ ( t ) ˆ ϕ | v | x + k v k L + k u ⊗ v k L + k ω × u k L ( B ) + k v k L q x k ω k L q x ( B ) + k∇ w ⊗ v k L + k ̟ × w k L + k ̟ × u + ω × w k L ≤ k v k L + k v k L + k v k L q x k u k L q x ( B ) + k ω k L q x ( B ) k u k L q x ( B ) + k v k L q x k ω k L q x ( B ) + k∇ w k L q x k v k L q x + k ̟ k L q x k w k L q x + k ̟ k L q x k u k L q x + k ω k L q x k w k L q x ≤ (cid:16) k v k L q x + k v k L q x + k u k L q x ( B ) + k∇ w k L q x + k w k L q x (cid:17) × (cid:16) k v k L q x + k v k L q x + k ω k L q x + k ̟ k L q x (cid:17) Here we used interpolation for k v k L ≤ k v k L q x k v k L q x . Therefore k Φ k L t . (cid:13)(cid:13)(cid:13)(cid:16) k v k L q x + k v k L q x + k u k L q x ( B ) + k∇ w k L q x + k w k L q x (cid:17)(cid:13)(cid:13)(cid:13) L p t × (cid:13)(cid:13)(cid:13)(cid:16) k v k L q x + k v k L q x + k ω k L q x + k ̟ k L q x (cid:17)(cid:13)(cid:13)(cid:13) L p t ≤ η. ECOND DERIVATIVE OF NAVIER-STOKES 19
By a Gr¨onwall’s lemma, we conclude that for every − ≤ t ≤ ψ ( t ) ˆ ϕ | v | x + ˆ t − ψ ( t ) ˆ ϕ |∇ v | d x ≤ e ´ t − C Φ( s ) d s ≤ e Cη . Therefore by taking the sup over − ≤ t ≤ t = 0 respectively, we concludesup − ≤ t ≤ ˆ | v ( t ) | d x ≤ η, ˆ Q |∇ v | d x d t ≤ η. (cid:3) Local Study: Part Two, De Giorgi Iteration
In this section, we derive the boundedness of v in Q which is the following. Proposition 5.1.
Let v solves (4.4) . If (4.1) holds for sufficiently small η , andwe have integrability bounds in Lemma 4.4, then we have k v k L ∞ ( Q ) = sup t ∈ ( − , k v ( t ) k L ∞ ( B ) ≤ . The proof uses De Giorgi technique and the truncation method. First, we setdyadically shrinking radius, r ♭k = 12 (1 + 8 − k ) , r ♮k = 12 (1 + 2 × − k ) , r ♯k = 12 (1 + 4 × − k ) . Then we define dyadically shrinking cylinder Q k ’s, T ♭k = r ♭k , B ♭k = B r ♭k (0) , Q ♭k = ( − T ♭k , × B ♭k ,T ♮k = r ♮k , B ♮k = B r ♮k (0) , Q ♮k = ( − T ♮k , × B ♮k ,T ♯k = r ♯k , B ♯k = B r ♯k (0) , Q ♯k = ( − T ♯k , × B ♯k . We also introduce positive smooth space-time cut-off functions ρ k and ρ ♯k with Q ♭k ≤ ρ k ≤ Q ♮k , Q ♯k ≤ ρ ♯k ≤ Q ♭k − . Then, let c k denote a sequence of rising energy level, c k = 1 − − k , v k = ( | v | − c k ) + , β k = v k | v | , Ω k = { v k > } , k = Ω k , α k = 1 − β k . We define analogous of vector derivative d k and energy quantity U k : d k = k (cid:0) α k |∇| v || + β k |∇ v | (cid:1) ,U k = k v k k L ∞ ( − T ♭k , L ( B ♭k )) + k d k k L ( Q ♭k ) . We have the following truncation estimates.
Lemma 5.2. α k v ≤ c k ≤ , k β k v k L ∞ t L x ∩ L t ˙ H x ( Q ♭k − ) ≤ U k − , k k k L ∞ t L x ∩ L t L x ( Q ♭k − ) ≤ C k U k − . Proof.
The first estimate follows from the definition. By Lemma 4 in [Vas07], wehave |∇ v k | ≤ d k and |∇ ( β k v ) | ≤ d k . Moreover, since |∇| v || ≤ |∇ v | , we see d k ≤ d k − , as v k and β k are monotonously decreasing. So k∇ ( β k v ) k L ( Q ♭k − ) ≤ k d k k L ( Q ♭k − ) ≤ k d k − k L ( Q ♭k − ) . Moreover, the truncation gives | β k v | + 2 − k k = v k + 2 − k k = k v k − , so k β k v k L ∞ t L x ( Q ♭k − ) ≤ k v k − k L ∞ t L x ( Q ♭k − ) , − k k k k L ∞ t L x ( Q ♭k − ) ≤ k v k − k L ∞ t L x ( Q ♭k − ) , − k k k k L t L x ( Q ♭k − ) ≤ k v k − k L t L x ( Q ♭k − ) ≤ k v k − k L ∞ t L x ( Q ♭k − ) + k∇ v k − k L ( Q ♭k − ) ≤ k v k − k L ∞ t L x ( Q ♭k − ) + k d k − k L ( Q ♭k − ) . (cid:3) Corollary 5.3 (Nonlinearization) . If f ∈ L pt L qx ( Q k − ) , with p + γ (cid:18) θ − θ ∞ (cid:19) = 1 , q + γ (cid:18) θ − θ (cid:19) = 1 , for some ≤ θ ≤ , < σ ≤ γ , then uniformly in σ , ˆ Q ♭k − | β k v | σ | f | d x d t ≤ C k k f k L pt L qx ( Q k − ) U γ k − . Proof.
By interpolation, k β k v k , k k k L pθt L qθx ( Q k − ) ≤ U k − , where 1 p θ = θ − θ ∞ , q θ = θ − θ . Therefore, using H¨older’s inequality, ˆ Q k − | β k v | σ | f | d x d t ≤ k f k L pt L qx k β k v k σL pθt L qθx k k k γ − σL pθt L qθx ≤ k f k L pt L qx U γ k − . (cid:3) First, we recall the following identities from [Vas07]. α k v · ∂ • v = ∂ • (cid:18) | v | − v k (cid:19) , (5.1) α k v · ∆ v = ∆ (cid:18) | v | − v k (cid:19) + d k − |∇ v | . (5.2)Since α k v is bounded, we can multiply equation (4.4) by α k v and obtain ∂ t (cid:18) | v | − v k (cid:19) + α k v · ∇ R ( u ⊗ v )(5.3) = ∆ (cid:18) | v | − v k (cid:19) + d k − |∇ v | + α k v · ( B + L + W ) . ECOND DERIVATIVE OF NAVIER-STOKES 21 using (5.1) and (5.2). Denote C v = B + L + W . Subtracting (5.3) from (4.5), wehave ∂ t v k d k + div( v R ( u ⊗ v )) − α k v · ∇ R ( u ⊗ v ) ≤ ∆ v k β k v · C v . Multiply by ρ k , then integrate in space and from σ to τ in time, (cid:20) ˆ ρ k v k x (cid:21) τσ + ˆ τσ ˆ ρ k d k d x d t ≤ ˆ τσ ˆ ( ∂ t ρ k + ∆ ρ k ) v k x d t − ˆ τσ ˆ ρ k div( v R ( u ⊗ v )) d x d t + ˆ τσ ˆ ρ k α k v · ∇ R ( u ⊗ v ) d x d t + ˆ τσ ˆ ρ k β k v · C v d x d t. Take the sup over τ > − T ♭k , and set σ < − T ♭k − , we obtain U k ≤ sup τ ∈ ( − T ♭k , ˆ ρ k v k x + ˆ − T ♭k − ˆ ρ k d k d x d t (5.4) ≤ C k ˆ Q ♮k v k d x d t + sup τ ∈ ( − T ♭k , (cid:26) ˆ τ − T ♮k ˆ ρ k α k v · ∇ R ( u ⊗ v ) d x d t − ˆ τ − T ♮k ˆ ρ k div( v R ( u ⊗ v )) d x d t + ˆ τ − T ♮k ˆ ρ k β k v · C v d x d t (cid:27) . Using Corollary 5.3, the first one is bounded by ˆ Q ♮k v k d x d s ≤ ˆ Q ♭k − | β k v | d x d s ≤ U k − . (5.5)Now let’s deal with the last few terms. For simplicity, we use ˜ d x d t to denote ´ τ − T ♮k ´ R d x d t in the rest of this section.5.1. Highest Order Nonlinear Term.
Define three trilinear forms, T ◦ [ v , v , v ] = ¨ ρ k div( v R ( v ⊗ v )) d x d t, T ∇ [ v , v , v ] = ¨ ρ k v · ∇ R ( v ⊗ v ) d x d t, T div [ v , v , v ] = ¨ ρ k div v R ( v ⊗ v ) d x d t. They are symmetric on v , v positions. When we have enough integrability, thatis, when |∇ v || v || v | , | v ||∇ v || v | , | v || v ||∇ v | ∈ L t,x , we have Leibniz rule T ◦ = T ∇ + T div . The goal is to estimate the first two double integrals in (5.4), ¨ ρ k α k v · ∇ R ( u ⊗ v ) d x d t − ¨ ρ k div( v R ( u ⊗ v )) d x d t = T ∇ [ α k v, u, v ] − T ◦ [ v, u, v ] . We first separate w ⊗ v from u ⊗ v , and we will have T ∇ [ α k v, w, v ] − T ◦ [ v, w, v ] = T ∇ [ α k v, w, v ] − T ∇ [ v, w, v ] − T div [ v, w, v ]= − T ∇ [ β k v, w, v ]= − ¨ ρ k β k v · ∇ R ( w ⊗ v ) d x d t. Denote −∇ R ( w ⊗ v ) =: W and we will deal with it later. The remaining ( u − w ) ⊗ v can be separated into interior part and exterior part,( u − w ) ⊗ v = ρ ♯k v ⊗ v + (1 − ρ ♯k )( u − w ) ⊗ v. The exterior part is bounded and smooth in space over the support of ρ k . k ρ k R ((1 − ρ ♯k )( u − w ) ⊗ v ) k L p t C ∞ x ≤ C k ( u − w ) ⊗ v k L p t L x ≤ C k u − w k L p t L q x ( Q ) k v k L p t L q x ≤ η. Here, we denote 1 p = 1 p + 1 p < . Therefore we can use Leibniz rule similar as w and T ∇ [ α k v, (1 − ρ ♯k )( u − w ) , v ] − T ◦ [ v, (1 − ρ ♯k )( u − w ) , v ]= T ∇ [ α k v, (1 − ρ ♯k )( u − w ) , v ] − T ∇ [ v, (1 − ρ ♯k )( u − w ) , v ]= − T ∇ [ β k v, (1 − ρ ♯k )( u − w ) , v ]= − ¨ ρ k β k v · ∇ R ( v ⊗ (1 − ρ ♯k )( u − w )) d x d t ≤ C k U − p k − by nonlinearization Corollary 5.3. The interior part is T ∇ [ α k v, ρ ♯k v, v ] − T ◦ [ v, ρ ♯k v, v ]= T ∇ [ α k v, ρ ♯k β k v, β k v ] + 2 T ∇ [ α k v, ρ ♯k α k v, β k v ]+ T ∇ [ α k v, ρ ♯k α k v, α k v ] − T ◦ [ v, ρ ♯k v, v ]= T ∇ [ α k v, ρ ♯k β k v, β k v ]+ 2 T ◦ [ α k v, ρ ♯k α k v, β k v ] − T div [ α k v, ρ ♯k α k v, β k v ]+ T ◦ [ α k v, ρ ♯k α k v, α k v ] − T div [ α k v, ρ ♯k α k v, α k v ] − T ◦ [ v, ρ ♯k v, v ]= T ∇ [ α k v, ρ ♯k β k v, β k v ]+ 2 T div [ β k v, ρ ♯k α k v, β k v ] + T div [ β k v, ρ ♯k α k v, α k v ]+ 2 T ◦ [ α k v, ρ ♯k α k v, β k v ] + T ◦ [ α k v, ρ ♯k α k v, α k v ] − T ◦ [ v, ρ ♯k v, v ]= T ∇ [ α k v, ρ ♯k β k v, β k v ] + T div [ β k v, ρ ♯k α k v, ( β k + 1) v ] − T ◦ [ α k v, ρ ♯k β k v, β k v ] − T ◦ [ β k v, ρ ♯k v, v ] . Notice that the boundedness of α k v guarantees enough integrability to switch be-tween trilinear forms. Then | T ∇ [ α k v, ρ ♯k β k v, β k v ] | , | T div [ β k v, ρ ♯k α k v, ( β k + 1) v ] | . k∇ ( β k v ) k L ( Q k − ) U k − ≤ U k − , | T ◦ [ α k v, ρ ♯k β k v, β k v ] | , | T ◦ [ β k v, ρ ♯k v, v ] | . U k − . In conclusion, (cid:12)(cid:12)(cid:12)(cid:12) ¨ ρ k α k v · ∇ R ( u ⊗ v ) d x d t − ¨ ρ k div( v R ( u ⊗ v )) d x d t (5.6) − ¨ ρ k β k v · W d x d t (cid:12)(cid:12)(cid:12)(cid:12) . C k U min { , − p } k − . Lower Order Terms.
For the bilinear and linear term, recall that inside B , B = − curl ∆ − ( ∇ ϕ × ( ω × u )) , L = curl ∆ − (2 div( ∇ ϕ ⊗ ω ) − (∆ ϕ ) ω ) . Therefore, k ρ k B k L p t L ∞ x ≤ k ω × u k L p t L x ( Q ) ≤ k u k L p t L q x k ω k L p t L q x ≤ η, k ρ k L k L p t L ∞ x ≤ k ω k L p t L q x ( Q ) ≤ η, Thus ¨ B · ρ k β k v d x d t ≤ C k U − p k − , (5.7) ¨ L · ρ k β k v d x d t ≤ C k U − p k − . (5.8) W Terms.
Finally, let us deal with W + W = − ω × w + 12 P ∇ ( ̟ × u + ω × w ) − ∇ R ( w ⊗ v ) . Here ∇ R = ∇ tr − P ∇ div, so ∇ R ( w ⊗ v ) = 12 ∇ ( w · v ) − P ∇ div( v ⊗ w )= 12 ( w · ∇ v + v · ∇ w + w × curl v + v × curl w ) − P ∇ ( v · ∇ w )= 12 ( w · ∇ v − v · ∇ w ) + P curl ( v · ∇ w )+ 12 ( w × curl v + v × curl w ) , ∇ R ( w ⊗ v ) = P ∇ ( ∇ R ( w ⊗ v ))= 12 P ∇ ( w · ∇ v − v · ∇ w ) + 12 P ∇ ( w × curl v + v × curl w )= 12 P ∇ (curl( v × w ) − v div w + w div v )+ 12 P ∇ ( w × curl v + v × curl w )= − P ∇ ( v ( u · ∇ ϕ )) + 12 P ∇ ( w × curl v + v × curl w ) . Hence W + W = − ω × w + 12 P ∇ ( v ( u · ∇ ϕ ))+ 12 P ∇ ( ̟ × u + ω × w + curl v × w + curl w × v ) . Again, we separate W + W into exterior and interior part, with W + W = W ext + W intECOND DERIVATIVE OF NAVIER-STOKES 25 where W ext = − (1 − ρ ♯k ) ω × w + 12 P ∇ ( v ( u · ∇ ϕ ))+ 12 P ∇ (cid:16) (1 − ρ ♯k ) ( ̟ × u + ω × w + curl v × w + curl w × v ) (cid:17) , W int = − ρ ♯k ω × w + 12 P ∇ (cid:16) ρ ♯k ( ̟ × u + ω × w + curl v × w + curl w × v ) (cid:17) = − ρ ♯k curl v × w − ρ ♯k ̟ × w + 12 P ∇ (cid:16) ρ ♯k ( ̟ × u + curl w × v + ̟ × w ) (cid:17) + 12 P ∇ (cid:16) ρ ♯k ( ω × w + curl v × w − ̟ × w ) (cid:17) = − ρ ♯k curl v × w − ρ ♯k ̟ × w + P ∇ (cid:16) ρ ♯k ̟ × u (cid:17) + P ∇ (cid:16) ρ ♯k curl v × w (cid:17) = − P curl ( ρ ♯k curl v × w ) − P curl ( ρ ♯k ̟ × w ) + P ∇ (cid:16) ρ ♯k ̟ × v (cid:17) . Similar as bilinear terms, ρ k W ext is small in L p t L ∞ x . Among the three terms in W int , ρ ♯k ̟ × w is bounded in L p t L ∞ x , and ρ ♯k ̟ is in L p t L ∞ x . Finally, for the firstterm, P curl (curl v × ρ ♯k w ) = − P curl (curl ρ ♯k w × v ) + P curl ( v · ∇ ρ ♯k w + ρ ♯k w · ∇ v ) , P curl ( ρ ♯k w · ∇ v ) = P curl (curl( v × ρ ♯k w ) + v · ∇ ρ ♯k w − v div ρ ♯k w )= curl( v × ρ ♯k w ) + P curl ( v · ∇ ρ ♯k w − v div ρ ♯k w ) , curl( v × ρ ♯k w ) = v div ρ ♯k w + ρ ♯k w · ∇ v − v · ∇ ρ ♯k w. Every term is a product of v and ∇ ρ ♯k w (possibly with a Riesz transform) except ρ ♯k w · ∇ v . Because in Ω k , ∇| v | = ∇ v k are the same, we have ˆ ρ k β k v · ( ρ ♯k w · ∇ ) v d x = ˆ ρ k β k ( w · ∇ ) | v | x = ˆ ρ k β k | v | ( w · ∇ ) | v | d x = ˆ ρ k v k ( w · ∇ ) v k d x = ˆ ρ k ( w · ∇ ) v k x = − ˆ v k ρ k w ) d x. Therefore, every term of P curl (curl v × ρ ♯k w ) is a product of v and ∇ ρ k w or ∇ ρ ♯k w .Inside B , w ∈ L p t C ∞ x . In conclusion, ¨ ρ k β k v · W ext d x d t ≤ C k U − p k − , ¨ ρ k β k v · P curl ( ρ ♯k curl v × w ) d x d t ≤ C k U − p k − , ¨ ρ k β k v · P curl ( ρ ♯k ̟ × w ) d x d t ≤ C k U − p k − , ¨ ρ k β k v · P ∇ ( ρ ♯k ̟ × v ) d x d t ≤ C k U − p k − . So the sum is bounded in ¨ ρ k β k v · ( W + W ) d x d t = ¨ ρ k β k v · ( W int + W ext ) d x d t ≤ C k U − p k − (5.9)provided U k − < Proof of Proposition 5.1.
Proof of Proposition 5.1.
Coming back to (5.4), by estimates (5.5) on the first term,(5.6) on the trilinear terms, (5.7), (5.8) on the B , L terms and (5.9) on the W terms,we conclude that U k ≤ C k U min { − p , } k − provided U k − <
1. Here p > U = sup t ∈ ( − , ˆ | v | d x + ˆ − ˆ B d d x d t = sup t ∈ ( − , ˆ | v | d x + ˆ − ˆ B |∇ v | d x d t ≤ η by Proposition 4.1, we know that if η is small enough, U k → k → ∞ . So in Q , | v | ≤ (cid:3) Local Study: Part Three, More Regularity
In this section, we will show that the vorticity ω is smooth in space. We will onlywork with the vorticity equation from now on. After the previous two steps, in B we should always decompose u = v + w , because v is bounded and w is harmonic.For convenience, given a vector ω , we denote ω := ω | ω | , ω α := | ω | α ω , α ∈ R . ECOND DERIVATIVE OF NAVIER-STOKES 27
Let ∂ • be the partial derivative in any space direction or time, then we have ∂ • ( | ω | α ) = αω α − · ∂ • ω,∂ • ( ω α ) = | ω | α − ∂ • ω + ( α − ω α − · ∂ • ω ) ω, α ∂ • ∂ • ( | ω | α ) = | ω | α − | ∂ • ω | + ( α − ω α − · ∂ • ω ) + ω α − · ∂ • ∂ • ω ≥ ( α − ω α − · ∂ • ω ) + ω α − · ∂ • ∂ • ω = 4( α − α (cid:12)(cid:12) ∂ • ω α (cid:12)(cid:12) + ω α − · ∂ • ∂ • ω. Bound Vorticity in the Energy Space.
We will first show ω is boundedin the energy space. Proposition 6.1. If u = v + w in Q , where v, w are bounded in k v k L ∞ ( Q ) + k∇ v k L ( Q ) ≤ , (6.1) k curl w k L t L x ( Q ) + k w k L t Lip x ( Q ) ≤ , (6.2) ω = curl u solves the vorticity equation (1.6) , then (a) k ω k E ( Q ) ≤ C , (b) k ω k E ( Q ) ≤ C ,Proof of Proposition 6.1 (a). We fix a pair of smooth space-time cut-off functions ̺ and ς which satisfy Q ≤ ς ≤ Q ≤ ̺ ≤ Q . Take the dot product of the vorticity equation (1.6) with ω :32 ω · ∂ t ω = ∂ t ( | ω | ) , ω · ( u · ∇ ) ω = ( u · ∇ )( | ω | ) , ω · ∆ ω ≤ ∆( | ω | ) − |∇ ω | . Therefore, ( ∂ t + u · ∇ − ∆)( | ω | ) + 32 ω · ∇ u · ω + 43 |∇ ω | ≤ . Multiply by ̺ then integrate over space, ˆ ̺ ( ∂ t + u · ∇ − ∆)( | ω | ) d x + 43 ˆ ̺ |∇ ω | d x ≤ − ˆ ̺ ω · ∇ u · ω d x. (6.3)For the left hand side, we can integrate by part, ˆ ̺ ( ∂ t + u · ∇ − ∆)( | ω | ) d x (6.4) = dd t ˆ ̺ | ω | d x − ˆ (cid:0) ( ∂ t + u · ∇ + ∆) ̺ (cid:1) | ω | d x, where the latter can be controlled by ˆ (cid:0) ( ∂ t + u · ∇ + ∆) ̺ (cid:1) | ω | d x ≤ C (cid:16) k u k L ∞ ( B ) (cid:17) ˆ ̺ | ω | d x. (6.5)For the right hand side, using u = v + w over the support of ̺ we can separate ˆ ̺ ω · ∇ u · ω d x = ˆ ̺ ω · ∇ v · ω d x + ˆ ̺ ω · ∇ w · ω d x, (6.6)The ∇ v term can be controlled by ˆ ̺ ω · ∇ v · ω d x = − ˆ ω · ∇ ( ̺ ω ) · v d x (6.7) = − ˆ ̺ ω · ∇ ( ω ) · v d x − ˆ ω · ( ω ⊗ ∇ ̺ ) · v d x, where ω · ∇ ( ω ) = | ω | − ω · ∇ ω −
12 ( ω · ∇ ω · ω − ) ω = ω · ∇ ω −
12 ( ω · ∇ ω · ω ) ω ⇒ | ω · ∇ ( ω ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ω · ∇ ω (cid:12)(cid:12)(cid:12)(cid:12) = 2 | ω | (cid:12)(cid:12)(cid:12)(cid:12) ω − · ∇ ω (cid:12)(cid:12)(cid:12)(cid:12) = 2 | ω | (cid:12)(cid:12)(cid:12) ∇| ω | (cid:12)(cid:12)(cid:12) ≤ | ω | |∇ ω | ≤ | ω | + |∇ ω | . Here the second to the last inequality is due to ∂ i | ω | = ∂ i ω · ω . Since | v | ≤ ̺ , ˆ ̺ ω · ∇ ( ω ) · v d x ≤ ˆ ̺ | ω | d x + ˆ ̺ |∇ ω | d x. (6.8)By using (6.4)-(6.8) in (6.3), we concludedd t ˆ ̺ | ω | d x + 43 ˆ ̺ |∇ ω | d x ≤ ˆ (cid:2) ( ∂ t + u · ∇ + ∆) ̺ (cid:3) | ω | d x + ˆ ̺ ω · ∇ w · ω d x + ˆ ω · ( ω ⊗ ∇ ̺ ) · v d x + ˆ ̺ | ω | d x + ˆ ̺ |∇ ω | d x dd t ˆ ̺ | ω | d x + 13 ˆ ̺ |∇ ω | d x ≤ C (cid:16) k u ( t ) k L ∞ ( B ) + k∇ w ( t ) k L ∞ ( B ) (cid:17) ˆ ̺ | ω | d x. By H¨older’s inequality, ˆ ̺ | ω | d x ≤ k ω ( t ) k L ( B ) (cid:18) ˆ ̺ | ω | d x (cid:19) . Therefore we can writedd t ˆ ̺ | ω | d x + 13 ˆ ̺ |∇ ω | d x ≤ C Φ( t ) (cid:18) ˆ ̺ | ω | d x (cid:19) , ECOND DERIVATIVE OF NAVIER-STOKES 29 where Φ( t ) = (cid:16) k u ( t ) k L ∞ ( B ) + k∇ w ( t ) k L ∞ ( B ) (cid:17) k ω ( t ) k L ( B ) ≤ (cid:16) k w ( t ) k L ∞ ( B ) + k∇ w ( t ) k L ∞ ( B ) (cid:17) × k curl v ( t ) k L ( B ) + k curl w ( t ) k L ( B ) ! since u = w + v , and | v | ≤ B . By (6.1), ˆ − Φ( t ) d t . k w k L t Lip x ( Q ) ! k∇ v k L ( Q ) + k curl w ( t ) k L t L x ( Q ) ! ≤ C. So by Gr¨onwall’s inequality, k ω k L ∞ t L x ∩ L t ˙ H x ( Q ) ≤ e C − . (cid:3) Proof of Proposition 6.1 (b).
From Proposition 6.1 (a) and Sobolev embedding, k ω k L ∞ t L x ∩ L t L x ( Q ) ≤ C, this interpolates the space k ω k L t L x ( Q ) ≤ C. Multiply the vorticity equation (1.6) by ς ω then integrate over R ,dd t ˆ ς | ω | x + ˆ ς |∇ ω | d x = ˆ ( ∂ t ς + ∆ ς ) | ω | x − ˆ ( u · ∇ ω ) · ς ω d x + ˆ ( ω · ∇ u ) · ς ω d x. The first integral is L in time because ω ∈ L t L x . For the second, ˆ ( u · ∇ ω ) · ς ω d x = ˆ ς u · ∇ | ω | x = − ˆ | ω | u · ∇ ς d x = − ˆ ς | ω | u · ∇ ς ≤ k ςω k L k u · ∇ ς | ω |k L , the latter is bounded L in time, by u ∈ L t L ∞ x and ω ∈ L t L x . For the thirdintegral, ˆ ( ω · ∇ u ) · ς ω d x = ˆ ( ω · ∇ v ) · ς ω d x + ˆ ( ω · ∇ w ) · ς ω d x. w is bounded in L t Lip x , and for v , ˆ ( ω · ∇ v ) · ς ω d x = ˆ v · ( ω · ∇ )( ς ω ) d x = ˆ v · ω ( ω · ∇ ς ) d x + ˆ v · ( ς ω · ∇ ω ) d x. The former is L in time, while the latter can be bounded by Cauchy-Schwartz, ˆ v · ( ς ω · ∇ ω ) d x ≤ ˆ | v ⊗ ςω | d x + 12 ˆ ς |∇ ω | d x. In conclusion,dd t ˆ ς | ω | x + 12 ˆ ς |∇ ω | d x ≤ C k ω ( t ) k L ( B ) + C k u ( t ) k L ∞ ( B ) k ω ( t ) k L ( B ) k ςω ( t ) k L + C k∇ w k L ∞ ( B ) k ςω ( t ) k L ≤ C Φ( t ) (cid:18) ˆ ς | ω | x (cid:19) where Φ( t ) = k ω ( t ) k L ( B ) + k u ( t ) k L ∞ ( B ) k ω ( t ) k L ( B ) + k∇ w ( t ) k L ∞ ( B ) , whose integral is bounded using (6.1), ˆ − Φ( t ) d t ≤ k ω k L ( Q ) + k u k L t L ∞ x ( Q ) k ω k L t L x ( Q ) + k∇ w k L t L ∞ x ( Q ) ≤ C. By a Gr¨onwall argument, we have k ω k L ∞ t L x ∩ L t ˙ H x ( Q ) ≤ e C − . (cid:3) Bound Higher Derivatives in the Energy Space.
Now we iterativelyshow higher derivatives of vorticity by induction.
Proposition 6.2.
For any n ≥ , if u = v + w in Q − n , where v, w are bounded in k v k L ∞ ( Q − n/ ) + k v k L t H n +1 x ( Q − n/ ) ≤ c n , (6.9) k w k L t C n +1 x ( Q − n/ ) ≤ c n , (6.10) for some constant c n , ω = curl u solves the vorticity equation (1.6) , and is boundedin k ω k L ∞ t H n − x ∩ L t H nx ( Q − n/ ) ≤ c n , (6.11) then for any multiindex α with | α | = n , (a) k∇ α ω k E ( Q − n/ ) ≤ C n (b) k∇ α ω k E ( Q − n − ) ≤ C n for some C n depending on c n and n . ECOND DERIVATIVE OF NAVIER-STOKES 31
Proof of Proposition 6.2 (a).
Similarly we fix smooth cut-off functions ̺ n and ς n which satisfy Q − n − ≤ ς n ≤ Q − n/ ≤ ̺ n ≤ Q − n/ . Differentiate (1.6) by ∇ α , ∂ t ∇ α ω + u · ∇∇ α ω − ∇ α ω · ∇ u + P α = ∆ ∇ α ω, (6.12)where P α = X β<α (cid:18) αβ (cid:19) curl (cid:0) ∇ β ω × ∇ α − β u (cid:1) . Multiply (6.12) by ̺ n ( ∇ α ω ) then integrate in space,dd t ˆ ̺ n |∇ α ω | d x + 43 ˆ ̺ n |∇∇ α ω | d x ≤ ˆ (cid:2) ( ∂ t + u · ∇ + ∆) ̺ n (cid:3) |∇ α ω | d x + ˆ ̺ n ∇ α ω · ∇ w · ( ∇ α ω ) d x + ˆ ∇ α ω · (( ∇ α ω ) ⊗ ∇ ̺ n ) · v d x + k v k L ∞ ( Q − n ) ˆ ̺ n |∇ α ω | d x + ˆ ̺ n |∇∇ α ω | d x + 32 ˆ ̺ n ( ∇ α ω ) · P α d x same as in the proof of Proposition 6.1 (a). Sodd t ˆ ̺ n |∇ α ω | d x + 13 ˆ ̺ n |∇∇ α ω | d x ≤ C (cid:16) k u ( t ) k L ∞ ( B − n ) + k∇ w ( t ) k L ∞ ( B − n ) (cid:17) ˆ ̺ n |∇ α ω | d x + 32 ˆ ̺ n ( ∇ α ω ) · P α d x. Terms other than P α are dealt with by the same way as in Proposition 6.1: ˆ ̺ n |∇ α ω | d x ≤ k∇ α ω ( t ) k L ( B − n ) (cid:18) ˆ ̺ n |∇ α ω | d x (cid:19) . The induction condition (6.11) ensures that k∇ α ω k L ( Q − n ) ≤ c n . Therefore ˆ − − n (cid:16) k u ( t ) k L ∞ ( B − n ) + k∇ w ( t ) k L ∞ ( B − n ) (cid:17) k∇ α ω ( t ) k L ( B − n ) d t . (cid:18) k v k L ∞ ( B − n ) + k w k L t C x ( B − n ) (cid:19) k∇ α ω k L ( Q − n ) ≤ C n . Now let’s focus on P α . | P α | . n X k =0 |∇ k ω ||∇ n − k +1 u | ≤ n X k =0 |∇ k ω ||∇ n − k +1 v | + n X k =0 |∇ k ω ||∇ n − k +1 w | . We denote P v,k = |∇ k ω ||∇ n − k +1 v | , P w,k = |∇ k ω ||∇ n − k +1 w | . First we estimate P v,k . By (6.9) and (6.11), when k = 0, k P v, k L t L x ( Q − n ) ≤ k ω k L t L x ( Q − n ) k∇ n +1 v k L t L x ( Q − n ) ≤ C n , and when 0 < k ≤ n , k P v,k k L t L x ( Q − n ) ≤ k∇ k ω k L t L x ( Q − n ) k∇ n +1 − k v k L t L x ( Q − n ) ≤ C n . Next we estimate P w,k . When 0 ≤ k < n , k P w,k k L t L x ( Q − n ) ≤ k∇ k ω k L ∞ t L x ( Q − n ) k∇ n +1 − k w k L t L ∞ x ( Q − n ) ≤ C n . Finally, when k = n , | P w,n | L x ( B − n ) ≤ |∇ n ω ||∇ w | . Therefore, ˆ ̺ n ( ∇ α ω ) · P α d x ≤ (cid:18) ˆ ̺ n |∇ α ω | d x (cid:19) × n X k =0 k P v,k k L x ( B − n ) + n − X k =0 k P w,k k L x ( B − n ) + k∇ w k L ∞ x ( B − n ) ! In conclusion, we have shown thatdd t ˆ ̺ n |∇ α ω | d x + 13 ˆ ̺ n |∇∇ α ω | d x ≤ C Φ( t ) (cid:18) ˆ ̺ n |∇ n ω | d x (cid:19) , whereΦ( t ) = (cid:16) k u ( t ) k L ∞ ( B − n ) + k∇ w ( t ) k L ∞ ( B − n ) (cid:17) k∇ α ω ( t ) k L ( B − n ) + n X k =0 k P v,k k L x ( B − n ) + n − X k =0 k P w,k k L x ( B − n ) + k∇ w k L ∞ x ( B − n ) with integral ˆ − − n / Φ( t ) d t ≤ C n . Taking the sum over all multi-index α with size | α | = n , we havedd t ˆ ̺ n |∇ n ω | d x + 13 ˆ ̺ n |∇ n +1 ω | d x ≤ C Φ( t ) (cid:18) ˆ ̺ n |∇ n +1 ω | d x (cid:19) , Finally, Gr¨onwall inequality gives k|∇ n +1 ω | k L ∞ t L x ∩ L t ˙ H x ( Q − n/ ) ≤ C n . (cid:3) ECOND DERIVATIVE OF NAVIER-STOKES 33
Proof of Proposition 6.2 (b).
Now we multiply (6.12) by ς n ∇ α ω then integrate over R , dd t ˆ ς n |∇ α ω | x + ˆ ς n |∇∇ α ω | d x = ˆ ( ∂ t ς n + ∆ ς n ) |∇ α ω | x − ˆ ( u · ∇∇ α ω ) · ς n ∇ α ω d x + ˆ ( ∇ α ω · ∇ u ) · ς n ∇ α ω d x + ˆ ς n ∇ α ω · P α d x For the same reason, the only term that we need to take care of is P α term, andthe others are dealt the same as in Proposition 6.1 (b): ˆ ( ∂ t ς n + ∆ ς n ) |∇ α ω | x − ˆ ( u · ∇∇ α ω ) · ς n ∇ α ω d x + ˆ ( ∇ α ω · ∇ u ) · ς n ∇ α ω d x . n k∇ α ω k L ( Q − n/ ) + k u k L ∞ ( Q − n/ ) k∇ α ω k L ( Q − n/ ) (cid:18) ˆ ς n |∇ α ω | x (cid:19) + k∇ w k L ∞ ( Q − n/ ) ˆ ς n |∇ α ω | x + k v k L ∞ ( Q − n/ ) k∇ α ω k L ( Q − n/ ) + 1 ε k v k L ∞ ( Q − n/ ) ˆ ς n |∇ α ω | x + ε ˆ ς n |∇∇ α ω | d x. The last term can be absorbed into the left, and we will use Gr¨onwall on theremaining terms.Now we shall focus on the P α term. From Proposition 6.2 (a), we have k∇ n ω k L ∞ t L x ∩ L x L t ( Q − n/ ) ≤ C n . (6.13)Again by interpolation, k∇ n ω k L t L x ( Q − n/ ) ≤ C n , k∇ n ω k L t L x ( Q − n/ ) ≤ C n , First we estimate P w,k . In this case, for any 0 ≤ k ≤ n , k P w,k k L t L x ( Q − n/ ) ≤ k∇ k ω k L t L x ( Q − n/ ) k∇ n +1 − k w k L t L ∞ x ( Q − n ) ≤ C n . Then we estimate P v,k . When 0 < k ≤ n , k P v,k k L t L x ( Q − n ) ≤ k∇ k ω k L t L x ( Q − n ) k∇ n +1 − k v k L t L x ( Q − n ) ≤ C n . For the case k = 0 of the v term, we put the curl on ∇ α ω , ˆ ς n ∇ α ω · curl ( ω × ∇ α v ) d x = ˆ ( ω × ∇ α v ) · curl( ς n ∇ α ω ) d x ≤ ˆ ς n | ω ||∇ α v ||∇∇ α ω | + ς n |∇ ς n || ω ||∇ α v ||∇ α ω | d x ≤ ˆ ς n | ω | |∇ α v | d x + ε ˆ ς n |∇∇ α ω | d x + 1 ε ˆ |∇ ς n | |∇ α ω | d x. where |∇∇ α ω | term can be absorbed to the left. By (6.13) and Sobolev embedding, k ω k L ∞ t L x ( Q − n/ ) ≤ C n . Therefore ¨ ς n | ω | |∇ α v | d x d t ≤ k ω k L ∞ t L x ( Q − n/ ) k∇ α v k L t L x ( Q − n/ ) ≤ C n . In conclusion,dd t ˆ ς n |∇ α ω | x + ˆ ς n |∇∇ α ω | d x ≤ C Φ( t ) (cid:18) ˆ ς n |∇ α ω | x (cid:19) , whereΦ( t ) = k∇ α ω ( t ) k L ( B − n/ ) + k u k L ∞ ( B − n/ ) k∇ α ω k L ( B − n/ ) + k∇ w k L ∞ ( B − n/ ) + k v k L ∞ ( B − n/ ) k∇ α ω k L ( B − n/ ) + 1 ε k v k L ∞ ( B − n/ ) + n X k =0 k P w,k k L ( B − n/ ) + n − X k =0 k P v,k k L ( B − n/ ) + k ω k L ( B − n/ ) k∇ α v k L ( B − n/ ) + 1 ε k∇ α ω ( t ) k L ( B − n/ ) has integral ´ − − n / Φ( t ) d t ≤ C n . Finally Gr¨onwall inequality gives k∇ α ω k L ∞ t L x ∩ L t ˙ H x ( Q − n − ) ≤ C n +1 . (cid:3) Proof of the Local Theorem.
Proof of the Local Theorem 1.3.
First, Proposition 4.1 gives k v k E ( Q ) ≤ η where η can be chosen arbitrarily small if we pick η small. Next, by Proposition5.1, we know k v k L ∞ ( Q ) ≤ . These two steps implies (6.1). As for (6.2), curl w = ̟ in B , so we use interpolationin (4.13): k curl w k L t L x ( Q ) ≤ k ̟ k L t L x ≤ k ̟ k L p t L q x k ̟ k L p t L q x ≤ ηw is harmonic inside B , therefore k w k L t C nx ( Q ) . n k w k L t L x ( Q ) ≤ η due to (4.12) and p ≥ . Therefore, we can use Proposition 6.1 to obtain k ω k E ( Q ) ≤ C. The next step is to use Proposition 6.2 iteratively. Suppose for n ≥ k∇ n − ω k E ( Q − n ) ≤ c n ECOND DERIVATIVE OF NAVIER-STOKES 35 which is equivalent to (6.11). Let ϕ n and ϕ ♯n be a pair of smooth spatial cut-offfunctions, with B n +4 ≤ ϕ n ≤ B n +3 , B n +2 ≤ ϕ ♯n ≤ B n +1 , and set v n := − curl ϕ ♯n ∆ − ϕ n ω, w n = ϕ n u − v n . On the one hand, ∇ v n is a Riesz transform of ϕ n ω up to lower order terms, so bythe boundedness of Riesz transform we know k∇ n +1 v n k L ( Q − n/ ) ≤ k∇ n ω k L ( Q − n ) ≤ c n − . On the other hand, we have similar boundedness estimates following Proposition5.1 as before, k v n k L ∞ ( Q − n/ ) ≤ .w n is harmonic in B n +4 , so we also have k w n k L t C n +1 x ( Q − n/ ) . n k w n k L t L x ( Q n +4 ) ≤ η. Therefore, by Proposition 6.2 k∇ n ω k E ( Q − n − ) ≤ C n . By induction, we have k∇ n ω k L ∞ t L x ∩ L t ˙ H x ( Q − n − ) ≤ C n for any n . By Sobolev embedding, this implies for any n , k∇ n ω k L ∞ ( Q − n − ) . k∇ n ω k L ∞ t L x ( Q − n − ) + k∇ n +2 ω k L ∞ t L x ( Q − n − ) ≤ C n . (cid:3) Appendix A. Suitability of Solutions
Theorem A.1.
Let u be a suitable weak solution to the Navier-Stokes equation in R . That is, u ∈ L ∞ t L x ∩ L t ˙ H x solves the following equation ∂ t u + u · ∇ u + ∇ P = ∆ u, div u = 0(A.1) where P is the pressure, and u satisfies the following local energy inequality, ∂ t | u | (cid:18) u (cid:18) | u | P (cid:19)(cid:19) + |∇ u | ≤ ∆ | u | . (A.2) Suppose v ∈ L ∞ t L x ∩ L t ˙ H x is compactly supported in space and solves the followingequation, ∂ t v + ω × v + ∇ R ( u ⊗ v ) = ∆ v + C v , div v = 0(A.3) where ω = curl u is the vorticity, C v ∈ L t L loc ,x is a force term, and R = 12 tr − ∆ − div div is a symmetric Riesz operator. Moreover, suppose v differs from ϕu by ϕu − v = w ∈ L ∞ t H x ∩ L t H x for some fixed ϕ ∈ C ∞ c ( R ) . Then v satisfies the following local energy inequality, ∂ t | v | v R ( u ⊗ v )) + |∇ v | ≤ ∆ | v | v · C v . (A.4) Proof.
It is well-known that the pressure P can be recovered from u by P = − ∆ − div div( u ⊗ u ) . Since u · ∇ u + ∇ P = ∇ | u | ω × u − ∇ ∆ − div div( u ⊗ u )= ω × u + ∇ R ( u ⊗ u ) , The Navier-Stokes equation (A.1) can be rewritten as ∂ t u + ω × u + ∇ R ( u ⊗ u ) = ∆ u, (A.5)and local energy inequality (A.2) can be rewritten as ∂ t | u | u R ( u ⊗ u )) + |∇ u | ≤ ∆ | u | , (A.6)First, multiply (A.5) by ϕ , ∂ t ϕu + ω × ϕu + ∇ R ( u ⊗ ϕu ) = ∆( ϕu ) + [ ∇ R , ϕ ]( u ⊗ u ) + [ ϕ, ∆] u. Denote C u = [ ∇ R , ϕ ]( u ⊗ u ) + [ ϕ, ∆] u for these commutator terms. Subtracting the equation of v from this equation of ϕu , we will have the equation for w . In summary, ∂ t ϕu + ω × ϕu + ∇ R ( u ⊗ ϕu ) = ∆( ϕu ) + C u , (A.7) ∂ t v + ω × v + ∇ R ( u ⊗ v ) = ∆ v + C v , (A.8) ∂ t w + ω × w + ∇ R ( u ⊗ w ) = ∆ w + C u − C v . (A.9)Recall from [Vas10] that ∆ u ∈ L − ε loc( t,x ) . Since ∆ w ∈ L t,x , we have ∆ v ∈ L − ε loc( t,x ) .Therefore, we can multiply (A.7) and (A.8) by w , and (A.9) by ϕu and v , w · ∂ t ( ϕu ) + w · ω × ϕu + w · ∇ R ( u ⊗ ϕu ) = w · ∆( ϕu ) + w · C u , (A.10) w · ∂ t v + w · ω × v + w · ∇ R ( u ⊗ v ) = w · ∆ v + w · C v (A.11) ϕu · ∂ t w + ϕu · ω × w + ϕu · ∇ R ( u ⊗ w ) = ϕu · ∆ w + ϕu · ( C u − C v ) . (A.12) v · ∂ t w + v · ω × w + v · ∇ R ( u ⊗ w ) = v · ∆ w + v · ( C u − C v ) . (A.13)Now take the sum of (A.10)-(A.13). ∂ t terms are ϕu · ∂ t w + w · ∂ t ( ϕu ) + v · ∂ t w + w · ∂ t v = ∂ t ( ϕu · w ) + ∂ t ( w · v )= ∂ t ( | ϕu | − | v | ) .ω × terms are w · ω × ϕu + ϕu · ω × w + w · ω × v + v · ω × w = 0 . ECOND DERIVATIVE OF NAVIER-STOKES 37 ∇ R terms are w · ∇ R ( u ⊗ ϕu ) + v · ∇ R ( u ⊗ w )+ ϕu · ∇ R ( u ⊗ w ) + w · ∇ R ( u ⊗ v )= div( w R ( u ⊗ ϕu )) + div( v R ( u ⊗ w ))+ div( ϕu R ( u ⊗ w )) + div( w R ( u ⊗ v )) − div( w ) ∇ R ( u ⊗ ϕu ) − div( v ) ∇ R ( u ⊗ w ) − div( ϕu ) ∇ R ( u ⊗ w ) − div( ϕ ) ∇ R ( u ⊗ v )= 2 div( ϕu R ( u ⊗ ϕu ) − v R ( u ⊗ v )) − ( u · ∇ ϕ ) ( ∇ R ( u ⊗ ϕu ) + ∇ R ( u ⊗ w ) + ∇ R ( u ⊗ v ))= 2 div( ϕu R ( u ⊗ ϕu ) − v R ( u ⊗ v )) − u · ∇ ϕ ) R ( u ⊗ ϕu ) . Here we use div v = 0 , div( ϕu ) = div w = u · ∇ ϕ . ∆ terms are ϕu · ∆ w + w · ∆( ϕu ) + v · ∆ w + w · ∆ v = ∆( u · w ) − ∇ ( ϕu ) : ∇ w + ∆( v · w ) − ∇ v : ∇ w = ∆( | ϕu | − | v | ) − |∇ ( ϕu ) | − |∇ v | ) . Commutator terms are w · C u + ϕu · ( C u − C v ) + w · C v + v · ( C u − C v ) = 2 ϕu · C u − v · C v . In summary, half the sum of these four identities (A.10)-(A.13) gives ∂ t | ϕu | − | v | ϕu R ( u ⊗ ϕu ) − v R ( u ⊗ v )) + |∇ ( ϕu ) | − |∇ v | (A.14) = ∆ | ϕu | − | v | ϕu · C u − v · C v + ( u · ∇ ϕ ) R ( ϕu ⊗ u ) . Next, multiply local energy inequality of u (A.6) by ϕ , ∂ t | ϕu | | ϕ ∇ u | + div (cid:0) ϕ u R ( u ⊗ u ) (cid:1) ≤ ∆ | ϕu | ϕ , ∆] | u | , ϕ ] ( u R ( u ⊗ u )) ,∂ t | ϕu | |∇ ( ϕu ) | + div ( ϕu R ( u ⊗ ϕu )) ≤ ∆ | ϕu | ϕ , ∆] | u | | u ⊗ ∇ ϕ | + 2( u ⊗ ∇ ϕ ) : ( ϕ ∇ u )(A.15) + [div , ϕ ] ( u R ( u ⊗ u )) + div( ϕu [ R , ϕ ]( u ⊗ u )) . The quadratic commutator terms in (A.15) are[ ϕ , ∆] | u | | u ⊗ ∇ ϕ | + 2( u ⊗ ∇ ϕ ) : ( ϕ ∇ u )= [ ϕ , ∆] | u | | u | |∇ ϕ | + 2 ∇ ϕ · ϕ ∇ u · u = − ∇ ( ϕ ) · ∇ | u | − ∆( ϕ ) | u | | u | |∇ ϕ | + 2 ∇ ϕ · ∇ u · ϕu = − ϕ ∇ ϕ · ∇ | u | −
12 ∆( ϕ ) | u | + | u | |∇ ϕ | + 2 ∇ ϕ · ∇ u · ϕu = − ϕ ∇ ϕ · ∇ u · u − ϕ ∆ ϕ | u | = ϕu · ( − ∇ ϕ · ∇ u − (∆ ϕ ) u )= ϕu · [ ϕ, ∆] u. The cubic commutator terms in (A.15) are[div , ϕ ] ( u R ( u ⊗ u )) + div( ϕu [ R , ϕ ]( u ⊗ u ))= 2 ϕ ∇ ϕ · u R ( u ⊗ u ) + ϕu · ∇ [ R , ϕ ]( u ⊗ u ) + div( ϕu )[ R , ϕ ]( u ⊗ u )= 2 ϕ ( u · ∇ ϕ ) R ( u ⊗ u ) + ϕu · ∇ [ R , ϕ ]( u ⊗ u ) + ( u · ∇ ϕ )[ R , ϕ ]( u ⊗ u )= 2 ϕ ( u · ∇ ϕ ) R ( u ⊗ u ) + ϕu · ∇ [ R , ϕ ]( u ⊗ u )+ ( u · ∇ ϕ ) R ( ϕu ⊗ u ) − ( u · ∇ ϕ ) ϕ R ( u ⊗ u )= ϕu · ∇ ϕ R ( u ⊗ u ) + ϕu · ∇ [ R , ϕ ]( u ⊗ u ) + ( u · ∇ ϕ ) R ( ϕu ⊗ u )= ϕu · [ ∇ , ϕ ] R ( u ⊗ u ) + ϕu · ∇ [ R , ϕ ]( u ⊗ u ) + ( u · ∇ ϕ ) R ( ϕu ⊗ u )= ϕu · ([ ∇ , ϕ ] R − ∇ [ ϕ, R ]) ( u ⊗ u ) + ( u · ∇ ϕ ) R ( ϕu ⊗ u )= ϕu · [ ∇ R , ϕ ]( u ⊗ u ) + ( u · ∇ ϕ ) R ( ϕu ⊗ u ) . Therefore, local energy inequality for ϕu can be simplified as ∂ t | ϕu | |∇ ( ϕu ) | + div ( ϕu R ( u ⊗ ϕu )) ≤ ∆ | ϕu | ϕu · C u + ( u · ∇ ϕ ) R ( ϕu ⊗ u ) . Subtracting (A.14) from this, we obtain (A.4). (cid:3)
References [BV19] Tristan Buckmaster and Vlad Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equation.
Ann. of Math. (2) , 189(1):101–144, 2019.[CF07] Jos´e A. Carrillo and Lucas C. F. Ferreira. Self-similar solutions and large time asymp-totics for the dissipative quasi-geostrophic equation.
Monatsh. Math. , 151(2):111–142,2007.[CKN82] Luis Caffarelli, Robert Kohn, and Louis Nirenberg. Partial regularity of suitable weaksolutions of the Navier-Stokes equations.
Comm. Pure Appl. Math. , 35(6):771–831,1982.[CLRM18] Diego Chamorro, Pierre-Gilles Lemari´e-Rieusset, and Kawther Mayoufi. The role ofthe pressure in the partial regularity theory for weak solutions of the Navier-Stokesequations.
Arch. Ration. Mech. Anal. , 228(1):237–277, 2018.[Con90] Peter Constantin. Navier-Stokes equations and area of interfaces.
Comm. Math. Phys. ,129(2):241–266, 1990.
ECOND DERIVATIVE OF NAVIER-STOKES 39 [CV14] Kyudong Choi and Alexis F. Vasseur. Estimates on fractional higher derivatives ofweak solutions for the Navier-Stokes equations.
Ann. Inst. H. Poincar´e Anal. NonLin´eaire , 31(5):899–945, 2014.[FJR72] Eugene Fabes, B. Frank Jones, and N´estor Rivi`ere. The initial value problem for theNavier-Stokes equations with data in L p . Arch. Rational Mech. Anal. , 45:222–240,1972.[Hop51] Eberhard Hopf. ¨Uber die Anfangswertaufgabe f¨ur die hydrodynamischen Grundgle-ichungen.
Math. Nachr. , 4:213–231, 1951.[ISS03] Luis Iskauriaza, Gregory Ser¨egin, and Vladim´ır Shverak. L , ∞ -solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk , 58(2(350)):3–44, 2003.[KL57] A. A. Kiselev and Olga Ladyˇzenskaya. On the existence and uniqueness of the solutionof the nonstationary problem for a viscous, incompressible fluid.
Izv. Akad. NaukSSSR. Ser. Mat. , 21:655–680, 1957.[Ler34] Jean Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace.
Acta Math. ,63(1):193–248, 1934.[Lin98] Fanghua Lin. A new proof of the Caffarelli-Kohn-Nirenberg theorem.
Comm. PureAppl. Math. , 51(3):241–257, 1998.[Lio96] Pierre-Louis Lions.
Mathematical topics in fluid mechanics. Vol. 1 , volume 3 of
OxfordLecture Series in Mathematics and its Applications . The Clarendon Press, Oxford Uni-versity Press, New York, 1996. Incompressible models, Oxford Science Publications.[Pro59] Giovanni Prodi. Un teorema di unicit`a per le equazioni di Navier-Stokes.
Ann. Mat.Pura Appl. (4) , 48:173–182, 1959.[Saw90] Eric Sawyer. Boundedness of classical operators on classical Lorentz spaces.
StudiaMath. , 96(2):145–158, 1990.[Sch76] Vladimir Scheffer. Partial regularity of solutions to the Navier-Stokes equations.
PacificJ. Math. , 66(2):535–552, 1976.[Sch77] Vladimir Scheffer. Hausdorff measure and the Navier-Stokes equations.
Comm. Math.Phys. , 55(2):97–112, 1977.[Sch78] Vladimir Scheffer. The Navier-Stokes equations in space dimension four.
Comm. Math.Phys. , 61(1):41–68, 1978.[Sch80] Vladimir Scheffer. The Navier-Stokes equations on a bounded domain.
Comm. Math.Phys. , 73(1):1–42, 1980.[Ser62] James Serrin. On the interior regularity of weak solutions of the Navier-Stokes equa-tions.
Arch. Rational Mech. Anal. , 9:187–195, 1962.[Ser63] James Serrin. The initial value problem for the Navier-Stokes equations. In
NonlinearProblems (Proc. Sympos., Madison, Wis., 1962) , pages 69–98. Univ. of WisconsinPress, Madison, Wis., 1963.[Vas07] Alexis F. Vasseur. A new proof of partial regularity of solutions to Navier-Stokesequations.
NoDEA Nonlinear Differential Equations Appl. , 14(5-6):753–785, 2007.[Vas10] Alexis Vasseur. Higher derivatives estimate for the 3D Navier-Stokes equation.
Ann.Inst. H. Poincar´e Anal. Non Lin´eaire , 27(5):1189–1204, 2010.[Yan20] Jincheng Yang. Construction of Maximal Functions associated with Skewed Cylin-ders Generated by Incompressible Flows and Applications. arXiv e-prints , pagearXiv:2008.05588, August 2020.
Department of Mathematics, The University of Texas at Austin, 2515 Speedway StopC1200 Austin, TX 78712, USA
E-mail address : [email protected] Department of Mathematics, The University of Texas at Austin, 2515 Speedway StopC1200 Austin, TX 78712, USA
E-mail address ::