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Vortex stretching and anisotropic diffusion in the 3D Navier-Stokes equations

The goal of this article is to present -- in a cohesive, and somewhat self-contained fashion -- several recent results revealing an experimentally, numerically, and mathematical analysis-supported \emph{geometric scenario} manifesting \emph{large data} logarithmic \emph{sub-criticality} of the 3D Navier-Stokes regularity problem. Shortly -- in this scenario -- the \emph{transversal small scales} produced by the mechanism of vortex stretching (coupled with the decay of the volume of the regions of intense vorticity) reach the threshold sufficient for the \emph{locally anisotropic diffusion} to engage and control the sup-norm of the vorticity, preventing the (possible) formation of finite time singularities.


PROLOGUE
Vortex stretching has been viewed as the principal physical mechanism responsible for the vigorous creation of small scales in turbulent fluid flows. This goes back at least to G. I. Taylor's fundamental paper "Production and dissipation of vorticity in a turbulent fluid" from 1937 [Tay37].
While the production part has been relatively well-understood (the amplification of the vorticity via the process of vortex stretching follows essentially from the conservation of the angular momentum in the incompressible fluid), the precise physics/mathematics behind the vortex stretching-induced dissipation is less transparent. For his part, Taylor inferred the thoughts on the anisotropic dissipation chiefly from the wind tunnel measurements of turbulent flow past a uniform grid, concluding the paper with the following sentence.
"It seems that the stretching of vortex filaments must be regarded as the principal mechanical cause of the high rate of dissipation which is associated with turbulent motion." Since then, it has been a grand challenge in the mathematical fluid mechanics community to try to explain/quantify the process of anisotropic dissipation in turbulent flows directly from the mathematical model -the 3D Navier-Stokes equations (NSE).
Numerical simulations (cf. [AKKG87,JWSR93,SJO91,VM94]) reveal that the regions of intense vorticity are dominated by coherent vortex structures and in particular, vortex filaments. There are two imminent morphological signatures of this geometry. One is local coherence of the vorticity direction, and the other one is local existence of sparse/thin direction(s).
Local coherence. Geometric depletion of the nonlinearity.
The pioneering work in this direction was presented by Constantin in [Co94] where he derived a singular integral representation of the stretching factor in the evolution of the vorticity magnitude featuring a geometric kernel that is depleted by local coherence of the vorticity direction, a purely geometric condition. This has been referred to as geometric depletion of the nonlinearity, and has led to the first rigorous confirmation of the phenomenon of anisotropic dissipation utilizing the 3D NSE, a theorem ( [CoFe93]) stating that as long as the vorticity direction is Lipschitzcoherent (in the regions of high vorticity), the L 2 -norm of the vorticity is controlled, and no finite time blow-up can occur. The Lipschitz-coherence condition was later scaled down to 1 2 -Hölder in [daVeigaBe02], and a full spatiotemporal localization of the 1 2 -Hölder condition was performed in [Gr09] (a different approach to localization was previously introduced in [ChKaLe07]). A family of local, hybrid geometric-analytic regularity criteria including a scaling invariant improvement of the 1 2 -Hölder condition was presented in [GrGu10-1]. The study of the coherence of the vorticity direction up to the boundary-regularity criteria in the case of the no-stress boundary conditions was presented in [daVeigaBe09], and in the case of the no-slip boundary conditions in [daVeiga07].
Essentially, an unhappy event preventing the manifestation of the anisotropic dissipation in this setting is the one of 'crossing of the vortex lines', i.e., of the vorticity direction forming even a simple spatial discontinuity -two different limit points -at a (possible) singular time.
An alternative mathematical description of the anisotropic dissipation in the 3D incompressible viscous flows was recently exposed in [Gr13], and is based on the concept of anisotropic diffusion. Taken at face value, the 3D NSE diffusion -generated by the Laplacian -is isotropic. The (isotropic) diffusion is then utilized via sharp, local-in-time spatial analyticity properties of solutions in L ∞ which provide an ambient amenable to the application of the harmonic measure majorization principle. More precisely, as long as the region of intense vorticity, defined to be the region in which the vorticity magnitude exceeds a fraction of the L ∞ norm, exhibits the property of local existence of a sparse/thin direction at a scale comparable to the radius of spatial analyticity (essentially, 1 where ω denotes the vorticity of the fluid), an argument relying on the translational and rotational invariance of the equations and certain geometric properties of the harmonic measure (this is what introduces anisotropy), and the harmonic measure maximum principle, shows that the L ∞ norm of the vorticity is controlled, and no finite-time blow up can occur. It is worth mentioning that it suffices to assume the aforementioned sparseness property intermittently in time.
Of course, a key question is whether there is any evidence, either numerical, or mathematical, that the scale of local linear sparseness/thinness needed for triggering the mechanism of anisotropic diffusion is in fact achieved in a turbulent flow. Thinking in terms of vortex filaments, the scale we are interested in is essentially the length scale of the diameters of the cross-sections, i.e., the transversal scale of the filament. It appears easier -both numerically and analytically -to estimate the axial length scale of the filaments instead. This, coupled with a suitable estimate on the volume of the region of the intense vorticity, provides an (indirect) estimate on the desired length scale.
Direct numerical simulations suggest that -intermittently in time/in the time average -the axial lengths of the filaments are essentially comparable to the macro scale (e.g., the side length L in the case of the L-periodic boundary conditions). On the other hand, the a priori L 1 -estimate on the vorticity [Co90] implies that the volume of the region of intense vorticity is bounded by C ω(t) −1 ∞ . Hence (intermittently in time), the transversal micro scale of the filament is bounded by ∞ ; in other words, the NSE regularity problem in this scenario becomes critical.
In addition to the numerical evidence, a very recent work [DaGr12-3] presented a mathematical evidence of creation and persistence (in the time average) of the macro scale-long vortex filaments. More precisely, exploiting a dynamic, spatial multi-scale ensemble averaging process designed to detect sign-fluctuations of an a priori sign varying physical quantity across scales, it was shown that there exists a range of scales -extending from a suitable micro scale to the macro scale -at which the vortex stretching term is essentially positive. (The averaging procedure utilized had been previously developed in a recent series of papers [DaGr11-1, DaGr11-2, DaGr12-1, DaGr12-2] as a mathematical framework for the study of turbulent cascades in physical scales of 3D incompressible fluid flows.) The aforementioned ruminations offer a physically, numerically, and mathematical analysis-supported large data criticality scenario for the 3D NSE. The NSE themselves are (still) super-critical; regardless of the functional setup, there has been a 'scaling gap' between a regularity criterion in view and the corresponding a priori bound. An instructive example is given by the regularity condition u ∈ L ∞ t L 3 x obtained by Escauriaza, Seregin and Sverak in [ESS03], to be contrasted to Leray's a priori bound u ∈ L ∞ t L 2 x ([Le34]). A natural question to ask is whether it is possible to break the criticality in this setting; i.e., whether the intricate interplay between the vortex stretching and the anisotropic diffusion results in preventing the formation of singularities, rather than in a critical blow-up scenario. In a very recent article [BrGr13-2], it was shown that a very mild, purely geometric assumption yields a uniform-in-time L log L bound on the vorticity; this in turn implies an extra-log decay of the vorticity distribution function, i.e., of the volume of the region of intense vorticity, breaking the scaling, and transforming the aforementioned criticality scenario into an anisotropic diffusion-win scenario (no singularities). More precisely, the assumption is a uniformin-time boundedness of the localized vorticity direction in a suitable, logarithmically weighted, local space of bounded mean oscillations (BM O). An interesting feature of this space is that it allows for discontinuous functions exhibiting singularities of, e.g., sin log | log( something algebraic )|-type. Hence, the vorticity direction can form a singularity in a geometrically spectacular fashion -every point on the unit sphere being a limit point -and the L log L bound will still hold (in particular, a simple 'crossing of the vortex lines' is not an obstruction).
The proof is based on an adaptation of the method utilized in [Co90], the novel components being exploiting analytic cancelations in the vortex-stretching term via In summary, the papers [Gr13, DaGr12-3, BrGr13-2] can be viewed as providing a rigorous mathematical framework (directly from the 3D NSE) for justification of Taylor's view on vortex stretching as the principal mechanical cause for the high rate of dissipation in turbulent flows. Incidentally, they also point to a possible new direction in the study of the 3D NSE regularity problem.

ANISOTROPIC DIFFUSION
3D Navier-Stokes equations (NSE) -describing a flow of 3D incompressible viscous fluid -read supplemented with the incompressibility condition div u = 0, where u is the velocity of the fluid, p is the pressure, and ν is the viscosity. Taking the curl yields the vorticity formulation, where ω = curl u is the vorticity. Utilization of the identity △u = − curl ω leads to the Biot-Savart law, closing the system for the vorticity field.
Computational simulations of 3D homogeneous turbulence reveal that the regions of intense vorticity organize in coherent vortex structures, and in particular, in elongated vortex tubes/filaments, cf. [S81, AKKG87, SJO91, JWSR93, VM94]. An indepth analysis of creation and dynamics of vortex tubes in 3D turbulent flows was presented in [CPS95]; in particular, a suitably defined dynamical scale of coherence of the vorticity direction field was estimated. The current body of work containing analytical, as well as analytical and numerical results on the dynamics of coherent vortex structures includes [GGH97,GFD99,Oh09,Hou09].
In what follows, we will focus on sparseness. Denote by Ω t (M ) the vorticity super-level set at time t; more precisely, Then the following manifestation of anisotropic diffusion holds ( [Gr13]).
Theorem 2.1. Suppose that a solution u is regular on an interval (0, T * ).
Assume that either , and there exists ǫ in (0, T * ) such that for The quantity (d 2 0 ω(t) ∞ ) −1 is the time step in the local-in-time well-posedness scheme in L ∞ initiated at t. The scheme can be complexified [GrKu98,Gu10]. For any s ∈ t, t + (d 2 0 ω(t) ∞ ) −1 , ω(s) is a restriction of the function holomorphic in the region {x + iy ∈ C 3 : |y| < 1/c 1 √ s}; moreover, the sup-norm of the complexified solution is controlled by c 2 ω(t) ∞ .
The idea of the proof is as follows. Let x 0 be a spatial point, and d = d(x 0 ) a sparse direction within the region of intense vorticity, at the scale comparable to the uniform lower bound on the radius of spatial analyticity. By the translational invariance of the equations, we can send x 0 to the origin, and by the rotational invariance, we can align d with one of the coordinate directions. The (real) coordinate is then embedded in the complex plane, and the harmonic measure maximum principle applied with respect to the disk centered at the origin -with the radius comparable to the analyticity radius -is utilized to exploit the sparseness condition resulting in a "self-improving" bound on the sup-norm of the complexified vorticity, preventing the finite time blow-up.
The main engine behind the argument is local-in-time spatially analytic smoothing in L ∞ , a strong manifestation of the (isotropic) diffusion generated by ∂ t −△; a locally anisotropic diffusion effect is a consequence of the translational and rotational invariance of the equations, and geometric properties of the harmonic measure.
Remark 2.1. It suffices to assume the sparseness condition at (suitably chosen) finitely many times/intermittently in time.

A POSSIBLE ROAD TO CRITICALITY.
Adopting the notation introduced in the preceding section, define the region of intense vorticity to be the set Ω s(t) 1 c 1 ω(t) ∞ for an appropriate c 1 > 1. Let R 0 be a suitable macro scale associated with the flow. Computational simulations indicate that (intermittently-in-time) dominant geometry in the region of intense vorticity is the one of R 0 -long vortex filaments; in order to estimate the transversal microscale of the filaments, it suffices to have a good estimate on the rate of the decrease of the volume of the vorticity super-level sets.
Let (0, T ) be an interval of interest. In [Co90], provided the initial vorticity is a bounded measure (and the initial velocity is of finite energy), Constantin showed that a corresponding weak solution satisfies sup

Chebyshev's inequality then implies
Vol which -in turn -yields the decrease of the transversal micro-scale of the filaments of at least at least . This is precisely the scale of local, linear sparseness needed to trigger the mechanism of anisotropic diffusion exposed in the previous section, i.e., we arrive at criticality.
It is instructive to check the scaling in the geometrically worst case scenario -no sparseness -the super level set being clumped in a ball. In this case, the criticality would require (In fact, this is precisely the vorticity analogue of the velocity scaling gap -L ∞ t L 3 x vs. L ∞ t L 2 x .) Summarizing -in this scenario -the vortex stretching acts as the mechanism bridging (literally) the scaling gap in the regularity problem.

MATHEMATICAL EVIDENCE OF CRITICALITY.
In this section, we identify the range of scales of positivity of the vortex-stretching term Sω · ω (S denotes the symmetric part of the gradient of u); this corresponds to the range of scales of creation and persistence of vortex filaments.
To this end, we exploit a spatial multi-scale averaging method designed to detect sign fluctuations of a quantity of interest across physical scales recently introduced in the study of turbulent transport rates in 3D incompressible fluid flows [DaGr11-1, DaGr11-2, DaGr12-1, DaGr12-2].
Let B(0, R 0 ) be a macro-scale domain. A physical scale R, 0 < R ≤ R 0 , is realized via suitable ensemble averaging of the localized quantities with respect to '(K 1 , K 2 )-covers at scale R'. Definition 4.1. Let K 1 and K 2 be two positive integers, and 0 < R ≤ R 0 ; a cover and any point x in B(0, R 0 ) is covered by at most K 2 balls B(x i , 2R).
The parameters K 1 and K 2 are the maximal global and local multiplicities, respectively.
For a physical density of interest f , consider -localized to the cover elements B(x i , R) (per unit mass) -local quantitiesf xi,R , for some 0 < δ ≤ 1. The smooth cut-off functions ψ i = ψ xi,R are equal to 1 on B(x i , R), vanish outside of B(x i , 2R), and satisfy for a suitably chosen ρ, 1 2 < ρ < 1. Denote by ψ 0 the cut-off corresponding to the macro-scale domain B(0, R 0 ). The cut-offs associated with 'boundary elements', i.e., the cover elements B(x i , R) intersecting the boundary of the macro-scale domain, are modified to satisfy certain compatibility relations with the global cut-off ψ 0 ; for technical details see, e.g., [DaGr11-1].
Denote by F R the ensemble average given by The key feature of { F R } 0<R≤R0 is that F R being stable -i.e., nearly-independent on a particular choice of the cover (with the fixed local multiplicity K 2 ) -indicates there are no significant sign fluctuations at scales comparable or greater than R. On the other hand, if f does exhibit significant sign fluctuations at scales comparable or greater than R, suitable rearrangements of the cover elements up to the maximal multiplicity -emphasizing first the positive and then the negative parts of f -will result in F R experiencing a wide range of values, from positive through zero to negative, respectively (the larger K 2 , the finer detection).
For a non-negative density f , the ensemble averages are all comparable to each other throughout the full range of scales, 0 < R ≤ R 0 ; in particular, they are all comparable to the simple average over the macro-scale domain, Back to vortex stretching. Denote the time-averaged localized vortex-stretching terms per unit mass associated to the cover element B(x i , R) by V ST xi,R,t , where η is a smooth function on the timeinterval of interest (0, T ) satisfying for a suitable κ, 0 < κ < 1. The quantity of interest is the ensemble average of B(x i , R)-localized enstrophy level dynamics is then as follows, for any t in (2/3 T, T ), and 1 ≤ i ≤ n. The hope is that the ensemble-averaging of the right-hand side detects a dynamic range of scales of positivity of the vortex stretching term Sω · ω.
Before stating the result, several macro-scale quantities need to be introduced. Denote by E 0,t time-averaged enstrophy per unit mass associated with the macro scale domain B(0, 2R 0 ) × (0, t), by P 0,t a modified time-averaged palinstrophy per unit mass, (the modification is due to the shape of the temporal cut-off η), and by σ 0,t a corresponding Kraichnan-type scale, Then the following holds [DaGr12-3].
A couple of remarks.
(i) Suppose that T is the first (possible) singular time, and that the macro-scale domain contains some of the spatial singularities (at time T ). This, paired with the assumption that u is a global-in-time local Leray solution implies hence, the condition (4.5) in the theorem is automatically satisfied for any t sufficiently close to the singular time T .
(iii) The power of 1 2 on σ 0,t is a correction originating in the need for a suitable control of the localized transport term.

LOGARITHMIC SUB-CRITICALITY
The purpose of this section is to show how a very mild, purely geometric, condition transforms the criticality scenario exposed in the previous sections into a log subcritical scenario, preventing the possible formation of singularities.
The idea is to get a uniform-in-time L log L-bound on w, where w = 1 + |ω| 2 ; this would impose an extra decay on the distribution function of the vorticity, breaking the criticality.
Suppose that the solution in view is smooth on (0, T ), and focus on some macroscale spatial domain, e.g., B(0, R 0 ). The evolution of w satisfies the following partial differential inequality ([Co90]), Since our goal is to control the evolution of w log w over B(0, R 0 ), it is convenient to multiply (5.1) by ψ (1 + log w) where ψ = ψ 0 (a smooth cut-off associated with the macro-scale domain as in the previous section). After a fair amount of calculation, the following bound transpires, for any τ in [0, T ); ξ denotes the vorticity direction.
Note that with respect to the scaling of a priori bounded quantities, the integral on the right-hand side is log super-critical. The strategy to overcome this is as follows. Exploit the analytic cancellations in the vortex-stretching term ω · ∇u utilizing a version of the Div-Curl Lemma in the Hardy space H 1 , and then transform the gain via H 1 − BM O duality into some sort of an oscillation condition on ξ. To do this in an efficient manner, we will need a sharp pointwise multiplier theorem in a version of local BM O, and a result quantifying the intimate relationship between log and BM O.
In order to keep the exposition self-contained, a number of relevant definitions and results from harmonic analysis are listed below; for more details, as well as the references, see [BrGr13-2].
Let f be a distribution. The maximal function of f is defined as where h is a fixed, normalized test function supported in the unit ball, and h t denotes t −n h(·/t). A distribution f is in the Hardy space Div-Curl Lemma Suppose that E and B are L 2 -vector fields satisfying div E = curl B = 0 (in the sense of distributions). Then, H 1 * = BM O and h 1 * = bmo; the duality is realized via integration.
Of special interest will be the spaces bmo = bmo 1 , and bmo 1 | log r| Let h be in bmo, and g in L ∞ ∩ bmo 1 | log r| . Then, g h bmo ≤ c(n) g ∞ + g bmo 1 | log r| h bmo .
Let M denote the Hardy-Littlewood maximal operator, and f be a locally integrable function. Then, log M f BMO ≤ c(n) (the bound is completely independent of f ). The estimate remains valid if we replace M f with Mf = M |f | 2 ; the advantage of working with M is that the L 2 -maximal theorem implies the following estimate Mf 1 ≤ c(n) f 1 .
Back to J ≡ τ 0 x ω · ∇u · ψ ξ log w dx dt. Decomposing log w as log w = log w Mw + log Mw, induces a decomposition of the integral; denote this decomposition by J = J 1 +J 2 .
The first integral can be bounded without paying much attention to oscillations, utilizing a couple of elementary inequalities and the L 2 -maximal theorem.
For the second integral, we have the following string of inequalities, Since the L 1 -norm of the vorticity is a priori bounded, we arrive at the following theorem [BrGr13-2]. ψ(x) w(x, t) log w(x, t) dx < ∞.