A spatially localized LlogL estimate on the vorticity in the 3D NSE
aa r X i v : . [ m a t h . A P ] F e b A SPATIALLY LOCALIZED L log L ESTIMATE ON THE VORTICITY IN THE 3D NSE
Z. BRADSHAW AND Z. GRUJI ´CA
BSTRACT . The purpose of this note is to present a spatially localized L log L bound on the vorticityin the 3D Navier-Stokes equations, assuming a very mild, purely geometric condition. This yieldsan extra-log decay of the distribution function of the vorticity, which in turn implies breaking thecriticality in a physically, numerically, and mathematical analysis-motivated criticality scenario basedon vortex stretching and anisotropic diffusion.
1. I
NTRODUCTION
Three-dimensional Navier-Stokes equations (3D NSE) – describing a flow of 3D incompressibleviscous fluid – read u t + ( u · ∇ ) u = −∇ p + △ u, supplemented with the incompressibility condition div u = 0 , where u is the velocity of the fluid,and p is the pressure (here, the viscosity is set to 1). Applying the curl operator yields the vorticityformulation,(1) ω t + ( u · ∇ ) ω = ( ω · ∇ ) u + △ ω, where ω = curl u is the vorticity of the fluid. Taking the spatial domain to be the whole space, thevelocity can be recovered from the vorticity via the Biot-Savart Law, u ( x ) = c Z ∇ | x − y | × ω ( y ) dy ; this makes (1) a closed (non-local) system for the vorticity field alone.An a priori L -bound on the evolution of the vorticity in the 3D NSE was obtained by Constantinin [4] . This is a vorticity analogue of Leray’s a priori L -bound on the velocity, and both quantitiesscale in the same fashion. So far, there has been no a priori estimate on the weak solutions to the3D NSE breaking this scaling.The goal of this short article is to show that a very mild, purely geometric assumption yields auniform-in-time L log L bound on the vorticity, effectively breaking the aforementioned scaling.More precisely, the assumption is a uniform-in-time boundedness of the localized vorticity direction in the weighted bmo -space g bmo | log r | . An interesting feature of this space (cf. [11]) is that it allows for discontinuous functions exhibiting singularities of, e.g., sin log | log( something algebraic ) | -type. In particular, the vorticity direction can blow-up in a geometrically spectacular fashion –every point on the unit sphere being a limit point – and the L log L bound will still hold.Besides being of an independent interest, the L log L bound on the vorticity implies an extra-logdecay of the distribution function. This is significant as it transforms a recently exposed [9, 5]large-data criticality scenario for the 3D NSE into a no blow-up scenario. Shortly, creation andpersistence (in the sense of the time-average) of the axial filamentary scales comparable to the macro-scale , paired with the L -induced decay of the volume of the suitably defined region ofintense vorticity, leads to creation and persistence of the transversal micro-scales comparable tothe scale of local, anisotropic linear sparseness , enabling the anisotropic diffusion to equalize thenonlinear effects. The extra log-decay of the volume transforms the equalizing scenario into theanisotropic diffusion-win scenario.The present result is, in a way, complementary to the results obtained by the authors in [1]. Theclass of conditions leading to an L log L -bound presented in [1] can be characterized as ‘wild intime’ with a uniform spatial (e.g., algebraic) structure, while the condition presented here can becharacterized as ‘wild in space’ and uniform in time. As in [1], the proof is based on an adaptationof the method exposed in [4], the novel component being utilization of analytic cancelations in thevortex-stretching term via the Hardy space-version of the Div-Curl Lemma [2], the local version[8] of H − BM O duality [6, 7], and the intimate connection between the
BM O -norm and thelogarithm. While the argument in [1] relied on the structure of the evolution of the scalar compo-nents and the result (cf. [14, 16]) stating that the
BM O -norm of the logarithm of a polynomial isbounded independently of the coefficients , the present argument relies on sharp pointwise multipliertheorem in g bmo [11, 13, 12] and Coifman-Rochberg’s BM O -estimate on the logarithm of the max-imal function of a locally integrable function [3], the estimate being fully independent of the function and depending only on the dimension of the space.2. A
N EXCURSION TO HARMONIC ANALYSIS
In this section, we compile several results from harmonic analysis that will prove useful.
Hardy spaces H and h The maximal function of a distribution f is defined as, M h f ( x ) = sup t> | f ∗ h t ( x ) | , x ∈ R n , where h is a fixed, normalized ( R h dx = 1 ) test function supported in the unit ball, and h t denotes t − n h ( · /t ) .The distribution f is in the Hardy space H if k f k H = k M h f k < ∞ .The local maximal function is defined as, m h f ( x ) = sup BM O is defined as follows, BM O = (cid:26) f ∈ L loc : sup x ∈ R ,r> Ω (cid:0) f, I ( x, r ) (cid:1) < ∞ (cid:27) , where Ω (cid:0) f, I ( x, r ) (cid:1) = 1 | I ( x, r ) | Z I ( x,r ) | f ( x ) − f I | dx is the mean oscillation of the function f withrespect to its mean f I = | I ( x,r ) | R I ( x,r ) f ( x ) dx , over the cube I ( x, r ) centered at x with the side-length r .A local version of BM O , usually denoted by bmo , is defined by finiteness of the following expres-sion, k f k bmo = sup x ∈ R , Coifman and Rochberg’s estimate on k log M f k BMO ([3])Let M denote the Hardy-Littlewood maximal operator. Coifman and Rochberg [3] obtained acharacterization of BM O in terms of images of the logarithm of the maximal function of non-negative locally integrable functions (plus a bounded part). The main ingredient in demonstratingone direction is the following estimate, k log M f k BMO ≤ c ( n ) , for any locally integrable function f . (The bound is completely independent of f .)This estimate remains valid if we replace M f with M f = (cid:0) M p | f | (cid:1) (cf. [10]); the advantage ofworking with M is that the L -maximal theorem implies the following estimate,(2) kM f k ≤ c ( n ) k f k , a bound that does not hold for the original maximal operator M .3. S ETTING AND THE RESULT Consider a weak (distributional) Leray solution u on R . The vorticity analogue of the Leray’s apriori bound on the energy was presented in [4]: assuming that the initial vorticity is in L (or,more generally, a bounded measure), the L -norm of the vorticity remains bounded on any finitetime-interval.Our goal is to obtain a spatially localized L log L bound on the vorticity under a suitable assump-tion on the structural blow-up of the vorticity direction ξ = ω | ω | .Fix a spatial ball B (0 , R ) , and consider a test function ψ supported in B = B (0 , R ) such that ψ = 1 on B (0 , R ) , and |∇ ψ ( x ) | ≤ c R ψ δ ( x ) for some δ > .Let w = p | ω | . We aim to control the evolution of ψ w log w ; by the Stein’s lemma [15], this isessentially equivalent to controlling the L -norm of M w .For simplicity of the exposition, we assume that the initial vorticity is also in L , and that T > isthe first (possible) blow-up time. This way, the solution in view is smooth on (0 , T ) , and we canfocus on obtaining a sup t ∈ (0 ,T ) -bound. Alternatively, one can employ the retarded mollifiers. Theorem 1. Let u be a Leray solution to the 3D NSE. Assume that the initial vorticity ω is in L ∩ L ,and that T > is the first (possible) blow-up time. Suppose that sup t ∈ (0 ,T ) k ( ψξ )( · , t ) k g bmo | log r | < ∞ . Then, sup t ∈ (0 ,T ) Z ψ ( x ) w ( x, t ) log w ( x, t ) dx < ∞ . Remark . Since ω is in L , in addition to the Leray’s a priori bounds on u , sup t k u ( · , t ) k L < ∞ and Z t Z x |∇ u ( x, t ) | dx dt < ∞ , ORTICITY IN L log L the following a priori bounds on ω are also at our disposal [4], sup t k ω ( · , t ) k L < ∞ and Z t Z x |∇ ω ( x, t ) | ǫ dx dt < ∞ . Proof. Setting q ( y ) = p | y | , the evolution of w = p | ω | satisfies the following partialdifferential inequality ([4]),(3) ∂ t w − △ w + ( u · ∇ ) w ≤ ω · ∇ u · ωw . Since our goal is to control the evolution of ψ w log w , it will prove convenient to multiply (3) by ψ (1 + log w ) . Here is the calculus corresponding to each of the four terms. time-derivative ∂ t w × ψ (1 + log w ) = ∂ t ( ψ w log w ) . Laplacian −△ w × ψ (1 + log w )= −△ ( ψ w log w ) + △ ψ w log w + ψ w X i ( ∂ i w ) + 2 X i ∂ i ψ ∂ i w (1 + log w ) . advection ( u · ∇ ) w × ψ (1 + log w )= X i u i ∂ i w ψ (1 + log w )= X i (cid:0) ∂ i ( u i w ψ (1 + log w )) − u i w ∂ i ψ (1 + log w ) − u i ψ ∂ i w (cid:1) = X i (cid:0) ∂ i ( u i w ψ (1 + log w )) − u i w ∂ i ψ (1 + log w ) − ∂ i ( u i ψw ) + ( u i ∂ i ψ w ) (cid:1) . vortex-stretching ω · ∇ u · ωw × ψ (1 + log w ) = ω · ∇ u · ψ ω | ω | (1 + log w ) + ω · ∇ u · ψ (cid:18) ωw − ω | ω | (cid:19) (1 + log w ) . Z. BRADSHAW AND Z. GRUJI ´C Integrating over the space-time, the above representation yields (dropping the zero and the pos-itive terms, and estimating the remaining terms in the straightforward fashion via Hölder andSobolev), I ( τ ) ≡ Z ψ ( x ) w ( x, τ ) log w ( x, τ ) dx ≤ I (0) + c Z τ Z x ω · ∇ u · ψ ξ log w dx dt + a priori bounded , for any τ in [0 , T ) .In order to take the advantage of the Coifman-Rochberg’s estimate, we decompose the logarithmicfactor as log w = log w M w + log M w. Denoting Z τ Z x ω · ∇ u · ψ ξ log w dx dt by J , this yields J = J + J where J = Z τ Z x ω · ∇ u · ψ ξ log w M w dx dt and J = Z τ Z x ω · ∇ u · ψ ξ log M w dx dt. For J , we use the pointwise inequality w log w M w ≤ M w − w (a consequence of the pointwise inequality M f ≥ f , and the inequality e x − ≥ x for x ≥ ).This leads to J ≤ Z τ Z x |∇ u | (cid:16) M w − w (cid:17) ψ dx dt which is a priori bounded by the Cauchy-Schwarz and the L -maximal theorem.For J , we have the following string of inequalities, ORTICITY IN L log L J ≤ c Z τ k ω · ∇ u k h k ψ ξ log M w k bmo dt ≤ c Z τ k ω · ∇ u k H k ψ ξ log M w k g bmo dt ≤ c Z τ k ω k k∇ u k (cid:16) k ψ ξ k ∞ + k ψ ξ k g bmo | log r | (cid:17)(cid:16) k log M w k BMO + k log M w k (cid:17) dt ≤ c sup t ∈ (0 ,T ) (cid:26)(cid:16) k ψ ξ k g bmo | log r | (cid:17)(cid:16) k log M w k BMO + k log M w k (cid:17)(cid:27) Z t Z x |∇ u | dx dt ≤ c (cid:16) t ∈ (0 ,T ) k ω k (cid:17) (cid:16) t ∈ (0 ,T ) k ψ ξ k g bmo | log r | (cid:17) Z t Z x |∇ u | dx dt by h − bmo duality, the Div-Curl Lemma, the pointwise g bmo -multiplier theorem, the Coifman-Rochberg’s estimate, and the bound (2) combined with a couple of elementary inequalities. Thiscompletes the proof of the L log L estimate. (cid:3) Acknowledgements. Z.B. acknowledges the support of the Virginia Space Grant Consortium via theGraduate Research Fellowship; Z.G. acknowledges the support of the Research Council of Norway via the grant 213474/F20, and the National Science Foundation via the grant DMS 1212023. We aregrateful to Luong Dang Ky for the comments on the previous version of the paper.R EFERENCES [1] Z. Bradshaw and Z. Gruji´c. Blow-up scenarios for 3D NSE exhibiting sub-criticality with respect to the scaling ofone-dimensional local sparseness. To appear in J. Math. Fluid Mech., DOI 10.1007/s00021-013-0155-0, 2013.[2] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes. Compensated compactness and Hardy spaces. J. Math. PuresAppl. (9) , 72(3):247–286, 1993.[3] R. R. Coifman and R. Rochberg. Another characterization of BMO . Proc. Amer. Math. Soc. , 79(2):249–254, 1980.[4] P. Constantin. Navier-Stokes equations and area of interfaces. Comm. Math. 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