On the analysis of a geometrically selective turbulence model

Abstract: In this paper we propose some new non-uniformly-elliptic/damping regularizations of the NavierStokes equations, with particular emphasis on the behavior of the vorticity. We consider regularized systems which are inspired by the Baldwin-Lomax and by the selective Smagorinskymodel based on vorticity angles, and which can be interpreted as Large Scale methods for turbulent ows. We consider damping terms which are active at the level of the vorticity. We prove themain a priori estimates and compactness results which are needed to show existence of weak and/or strong solutions, both in velocity/pressure and velocity/vorticity formulation for various systems. We start with variants of the known ones, going later on to analyze the new proposed models.


Introduction
In this paper we consider families of Large Eddy Simulation models which are variants of the classical Smagorinsky model [1]. We follow an approach similar to the modeling done by Cottet, Jiroveanu, and Michaux [2], proposing a selective model based on the local behavior of the angle of the vorticity direction, at neighbouring points. We recall that "regularizations" of the Navier-Stokes equations with perturbation obtained by a monotone nonlinear operator A, produce widely studied models as the one below ∂ t u + (u · ∇) u − ν∆u + ∇π + A(u) = f , div u = . (1.2) We observe that we denoted by v the unknown velocity eld for a Newtonian uid described by the Navier-Stokes equations (1.1), while by u we denote the eld which is solution of (1.2). We recall that in the turbulence modeling the latter system is generally associated with the modeling or description of the large scales (eddies) of the solution. With the usual terminology if v is split into the mean ow v and the (turbulent) uctuations v = v − v, then u is an computable approximation/model for v. From the point of view of the mathematical theory, early studies of (1.2) date back to O.A. Ladyzhenskaya [3] and J.L. Lions [4] in the spirit of looking for modi cations of the Navier-Stokes equations which laxed rotational backscatter" and "relaxed rotational backscatter Baldwin Lomax" models will be done in a forthcoming paper [11], together with the comparison in the derivation of the di erent systems.
By using the Kolmogorov-Prandtl relation for the turbulent viscosity ν (here c is a constant, the mixing length, and k the kinetic energy associated to uctuations u ) one can derive the following model, where ω := curl u ∂ t u + β curl( (x)∂ t ω) + (u · ∇) u − ν∆u + ∇π + C BL curl( (x)|ω|ω) = f . (1.3) This is called the "backscatter Baldwin-Lomax" model with β and C BL non-negative parameters, while (x) is a smooth function which can is related to the mixing length. Apart the classical sources, a modern discussion with the derivation of the Baldwin-Lomax model can be found in [10][11][12].
An interesting link with near-wall-models arises when tuning the function (x) to be the distance from the boundary ∂Ω of the physical domain, hence reducing the dissipative e ect, as the point x approaches the boundary layer. A simple application to channel ows is the one implemented with the van Driest damping, which is an early approach to obtain reliable and accurate turbulence simulations near a solid at wall, see Pope [13] and also [7,Sec 3.3.1].
The analysis of the model (1.3) reveals that the rst term β curl( (x)∂ t ω) is a sort of "dispersive" operator, hence the equations (1.3) are of pseudo-parabolic type. In fact, if (x) = ∈ R + , for all x ∈ Ω then by using the fundamental identity of vector calculus curl(curl f ) = −∆f + ∇(div f ), (1.4) it holds β curl( ∂ t ω) = −β ∂ t ∆u, producing the dispersive term which characterizes the Voigt model. In this paper we focus on models dealing with the second term in (1.3), which is one of the most common in literature. Many of its variants have been already implemented and studied, hence in the sequel we will always assume that β = and C BL > , hence we consider the so called Baldwin-Lomax model. The second additional term in (1.3), namely C BL curl( (x)|ω|ω) is very similar to the dissipative term appearing in the Smagorinsky model, and we begin by considering the equations with extra dissipation given by C BL curl( (x)|ω| p− ω), with < p ≤ . We start with the simplest case of (x) = . In this case, the properties of the solutions are rather similar to those known for the classical generalized Ladyžhenskaya-Lions-Smagorinsky model. There is a critical dependency of the dispersive and smoothing operators, when the function = (x) is assigned as a multiple of the (regularized) distance from the boundary d(x, ∂Ω) of the point x ∈ Ω. The study of models with smoothing operators which are degenerate near to the boundary will be the object of a forthcoming work [12]. In Section 4 we will consider a modi ed (in terms of di erential operators) method in which is a function itself of the solution, as used in certain turbulence models.
Plan of the paper. In section 2 we summarize the notation and the functional setting. Next, in Section 3 we are rst proving some ne existence and regularity properties for the model with β = and a constant > , since this allows for interesting generalizations. In Section 4 we start considering a new model which introduces the smoothing in a less strong way, working directly on the vorticity equation, hence considering a model which is more related with damping than with smoothing. Finally, taking inspiration from the work of Cottet et al. [2] we propose a new version of the selective Baldwin-Lomax-type model taking into account the possible variations of the direction of the vorticity and linking the choice of the function (x) to the relative alignment of the vorticity. The approach we follow makes a stricter connections with the celebrated geometric criterion of Constantin and Fe erman [14] for the regularity of the solution and some of its variants/improvements as developed in [15,16].

Functional setting and comparison with previous results
In this section we rst introduce the notation and the precise de nitions of functions spaces we will need to deal with. We will use the customary Sobolev spaces (W s,p (Ω), . W s,p ) and we denote the L p -norm simply by .
p and since the Hilbert case plays a special role we denote the L (Ω)-norm simply by . . In most cases Ω ⊂ R will be an open bounded set with smooth boundary ∂Ω. Due to well-known technical problems arising when dealing with the vorticity equation, in some cases we will restrict to the Cauchy problem in R or with the problem in the space periodic setting. By χ A we denote the indicator function of the measurable set A.
As usual by p we denote the conjugate exponent. For (X, . ) a Banach space we will also denote the usual Bochner spaces of functions de ned on [ , T] and with values in X by (L p ( , T; X), .
L p (X) ). For the variational formulation of the Navier-Stokes equations (1.1), and more generally of all systems of partial di erential equations with the constraint of incompressibility we shall consider, we introduce the space V of smooth and divergence-free vectors elds, with compact support in Ω. We then denote the completion of V in (L (Ω)) by L σ (Ω) and the completion in (H (Ω)) by H ,σ (Ω), see also [17] for further details. Since some results are set in the Hilbert space L σ (Ω) is endowed with the natural L -norm . and inner product (·, ·), while H ,σ (Ω) with the norm ∇v and inner product ((u, v)) := (∇u, ∇v). As usual, we do not distinguish between scalar and vector valued function spaces. The dual pairing between V and V is denoted by ·, · , and the dual norm by . * .
Since we need also to consider the space-periodic setting, in this case we consider scalar and vector de ned on the torus T = R/( πZ) , which are with zero mean value, that is In that case the linear space V is made of divergence-free vector elds, π-periodic in each of the coordinate directions and with zero mean value. Its closure in (L (T )) and (H (T )) is then denoted by L σ (T ) and H σ (T ), respectively and the norms are the same as those employed in the case of a bounded domain.
In the case of a bounded domain Ω it holds the Poincaré inequality valid for all smooth enough vector elds such that at least v · n = on the boundary ∂Ω (or in the space periodic case with zero mean value).
Since we have mainly estimates on the vorticity, we also need to estimate the full gradient, by means of the vorticity, whenever it is possible. We have the following inequality (cf. Bourguignon and Brezis [18]): There exists a constant C = C(s, p, Ω) such that, for all s ≥ for all vector elds φ ∈ (W s,p (Ω)) , where n denotes the outward unit normal vector on ∂Ω. This same result has been later improved by von Wahl [19] obtaining, under geometric conditions on the domain, an estimate without lower order terms: Let Ω be such that b (Ω) = b (Ω) = , where b i (Ω) denotes the i-th Betti number, that is the the dimension of the i-th homology group H i (Ω, Z). Then, there exists C depending only on p and Ω such that for all φ ∈ (W ,p (Ω)) satisfying either (φ · n) |Γ = or (φ × n) |Γ = .
In addition we will use the Sobolev inequality v ≤ C ∇v for all v ∈ H (Ω).
The basic compactness results for space-time functions we use is the Aubin-Lions lemma. If X , X are separable and re exive Banach spaces; if X → → X (with compact embedding) and X → X (with continuous embedding), then for < α, β < ∞ it holds that v ∈ L α ( , T; X ); ∂ t v ∈ L β ( , T; X ) → → L α ( , T; X).

On large scale models based on the vorticity
We start describing the models we want to study. In the simplest form we have the following equations with and setting ν := C BL > we consider the following initial boundary value problem where Ω ⊂ R is a smooth and bounded open set. (In this case we can allow Dirichlet boundary conditions and obviously very similar computations are also valid in the space periodic setting). For the above Baldwin-Lomax type initial boundary value problem (3.2) we can prove the following result: Theorem 1. Let be given < p ≤ , u ∈ L σ (Ω), and f ∈ L (( , T) × Ω). Then, there exists at least a weak solution to (3.2). Furthermore, if p > , then the weak solution is unique.
The proof of Theorem 1 can be obtained by an adaption of the one valid for general shear dependent uids. We will give a sketch of the proof in order to precise the functional setting, but many results are close to those reported in [20]. The result is valid also for smaller values of the exponent p, but the proof requires some technical adjustments. Here, since we consider turbulence type problems we restrict to the range < p < , but the reader can easily combine the results with those of the cited references to study the problems also a) in the shear thickening case p < ; b) for values of p larger than 2 (for technical reasons related with the embedding W , (Ω) → L p (Ω)), at least for p ≤ .
Proof of Theorem 1. One main starting point is the a priori energy estimate which is obtained by using as test function u itself and integrating over Ω. Observe that, due to the Gauss-Green formulas, we get and the boundary integral vanishes, since u = at the boundary. This shows that, for smooth enough solutions, then hence that the following a-priori estimate holds true (and it is valid for Galerkin approximate functions) ). We observe that the L p -bound for the vorticity implies a bound on the whole gradient thanks to (2.2). The estimates are justi ed by using a Galerkin approximation and in the following we will need to make use of a basis made by eigenfunctions of the Stokes operator (Spectral basis).
Once the a priori estimate is established, a fair standard application of the "monotonicity argument" (cf. [4,21] for the restricted range ≤ p ≤ ) or of Vitali's lemma ensures the existence of a weak solution. This can be done as in [20,Ch. 5], for < p ≤ As in the classical results by Lions and Ladyžhenskaya, the solution is unique if p > / . This follows by observing that in such range of exponents the solution u is smooth enough to be used as test function. The following di erential inequality holds true for the di erence U = u − u of two solutions with the same initial datum u ∈ L σ (Ω) and external force f ∈ L (( , T) × Ω) d dt Since p p− ≤ p holds true if p ≥ , then the coe cient ∇u p p− p of the term U from right-hand side belongs to L ( , T) and the Gronwall lemma implies that U ≡ if U( ) = . Details can be found in [20,Ch. 5.4].
Remark 3.1. The same computations as in (3.3) show that the boundary integral vanishes also when using the curl-based Navier-type boundary conditions These boundary conditions are particularly relevant in free-boundary problems and in turbulence modeling, see [22,23]. Hence, the same argument used in the proof of Theorem 1 applies also to the problem in the setting of Navier-type conditions. The proof can be obtained by adapting the approach outlined in [24], as in Conca [25] and Temam and Ziane [26] and does not present particular additional technical di culties, hence we do not reproduce it here.
The next step in the analysis of the model regards the existence of strong solutions and the vorticity behavior, especially when p < / . The balance equation for the curl is obtained by taking the curl of (3.1) hence getting the following equation As usual, dealing with the equation satis ed by ω, requires to deal with the problem without boundaries (Cauchy or periodic) or the curl-type Navier conditions (3.5). Here, in order to avoid inessential details, let us assume that curl f = and observe that the system satis ed by the vorticity ω is the following initial value problem (we used (1.4)) Let us analyze the space periodic case and then we will explain changes needed to handle the other cases.
In the space periodic case we can prove in a slightly simpler and alternate way the results stating that the critical exponent to have global existence of strong solution is p = / . Earlier results in this direction can be found in [20,27].
Theorem 2. Let p ≥ / and let be given u ∈ H σ (T ). Then, there exists a unique solution ω of (3.6) Proof. The proof is mainly based on estimate obtained multiplying (3.6) by ω and integrating by parts. This can be transferred to Galerkin-Fourier approximate equations, with the basis of complex exponential, to prove existence in a standard way. With this technique one obtains for the extra-nonlinear term the following equality where we used that T ∇q · ω dx = and classical integration by parts as developed for -at least for the velocity eld-the NSE in [28], but applied to the vorticity. In this way, we can apply the Sobolev embedding (All calculations are also valid in the whole space case R , by assuming su cient decay at in nity of the integrated elds.) Next, we recall that by integration by parts T (u · ∇) ω · ω dx = , while, by using Hölder inequality and explicit integral representation formulas we get since the gradient can be written as a proper Calderón-Zygmund singular integral in terms of the vorticity, by using (1.4) and di erentiating the Biot-Savart formula. The constant C CZ from the right-hand-side is the one coming from the estimate for singular integrals ∇u ≤ C CZ ω , and C CZ depends (being p = xed) only on the fact that we are working in three space dimensions.
In this way we arrive to the di erential inequality d dt We now use the following two convex-interpolation inequalities We split the term from the right-hand side of the di erential inequality as ω ( −α) ω α , with α = p( p− ) (p− ) . With this splitting and the two above inequalities we get and by Young inequality with δ = − p and δ = p− (all calculations are justi ed for p > / ) we nally show that for all ϵ > exists Cϵ > such that p− < if and only if p > / , with equality in the limit case. Consequently, for p ≥ / , we have proved the di erential inequality Being T ω(s) p p ds < ∞ from the a-priori estimate valid for weak solutions, the basic theory of di erential inequalities shows that in this case, where the constant C depends only on the data of the problem. The above a priori estimates can be used in a spectral Galerkin method to prove existence of strong solutions. The uniqueness of strong solutions, follows in the same manner as before in the larger range p ≥ / , but now for solutions starting with the smoother initial datum in H σ (T ).  [23,). The rst one is valid for all smooth enough vector elds, while the following one is related with the vorticity eld − ∂ω ∂n with summation over repeated indices. Since ω × n = , it holds ω = (ω · n) n at the boundary and we obtain that (ω · ∇) ω = (ω · n) ∂ω ∂n . This shows the following two inequalities are valid By using the trace theorem and the embedding H / (Γ) ⊂ L (Γ), applied to ω and to |ω| p/ , we can easily see that the two surface integrals can be bounded with the terms from the left hand side, obtaining the following di erential inequality (cf. [24,Eq. (23)] and [23,Eq. (2.3) for a constant C depending only on p and Ω. At this point the same reasoning as before can be applied to prove results of existence and uniqueness, by using a functional setting with the Hilbert space H σ (Ω) := (H (Ω)) ∩ L σ (Ω), which replaces the space H ,σ (Ω), due to the fact that the usual Sobolev machinery works also in this setting, as explained in the cited references.

A couple of new turbulence models
>From the results of the previous section it turns out that the relevant case p = produces a unique strong solution but it is also well-known that the dissipation/stabilization introduced by the Smagorinsky like term is generally too strong and, in particular, there is an extremely arti cial stabilizing e ect on the boundary layer. The Baldwin-Lomax model associated with a function of the distance from the boundary implies -by results on weighted spaces-a control of a less strong dissipation norm. This observation comes from the use of Poincaré type estimate ∂Ω), see Hurri-Syrjänen [29]. This result, coupled with the machinery developed as in Kufner [30] and with other several technical tools, can be used to produce similar results, at least for a model with a nonlinear perturbation given by for an appropriate exponent δ > . Considering the operator depending only on the deformation tensor or the curl version as in Baldwin-Lomax, requires also to use appropriate variants also of other tools as the Korn inequality and the way to pass to the limit in the approximate system. This will be addressed in a forthcoming work, which is out of the scopes of the present one. Here, we consider a di erent approach, based not on the knowledge and enough smoothness of the function (x). We study a weakly-dissipative model, based on an anisotropic and selective choice of the dissipation term, producing a non uniformly elliptic regularization, but with an approach more oriented towards computations. Since the Smagorinsky term is too strong and the model associated does not well-reproduce smaller scales a single universal constant C S for di erent turbulent elds in rotating or sheared ows, near solid walls, or in transitional regimes is not likely to be determined; the approach of Lilly (and its variants in [31]) seem to work only in the homogeneous and isotropic case of fully developed turbulence; several methods has been designed to overcome this fact. Apart Obukhov approach and the already cited Van Driest damping in the channel ow setting, an early method is the dynamic one introduced in Germano, Piomelli, Moin, and Cabot [32]. Several modi cations, leading also the multiscale variational methods have been proposed, see [7] but here we focus on a method which seems to be the closest to the analysis coming from the Partial Di erential Equations framework. It has also strong connections with the use of a geometric approach in the study of the regularity of the weak solutions to the Navier-Stokes equations [33].
One important modeling idea in many variations of the eddy viscosity models is that the eddy viscosity terms need to be active only in regions where the solution is not regular, or where there is a strong generation of small scales.
In the "dynamic model" introduced in [32], the Smagorinsky model's "parameter" C S is not anymore a constant but it is chosen locally in space and time, so to make the Smagorinsky model to agree (in a least squares sense) as closely as possible with the Bardina scale similarity model. In the selective model of Cottet, Jiroveanu, and Michaux [2] the parameter C S is not obtained by a solution of a local minimization problem, but by detecting the regions where vorticity is active as measured by the nonalignment of the direction of ω. This can be achieved in a computational way by considering a function lter Ψ(t, x) which is the indicator function of the points where the vorticity direction is badly behaved, that is in regions where the angle of vorticity is not a Lipschitz or even Hölder function of the space position. Hence, the model studied in the cited reference has the eddy viscosity term leading to the LES system (1.2), which is considered also in a bounded domain for numerical tests. Both these methods need to be calibrated and their implementations are well documented, but in any case they su er from some limitations in the mathematical formulation. In particular, the selective term acts at the level of the velocity eld, but the localization regards the vorticity activity, and the two facts are not treated with full mathematical rigour in the aforementioned papers, while the computational aspects and the results of implementation are instead well-documented. Here, we mainly propose a model which is more amenable to a precise treatment, which do not produce strong dissipation at the level of the velocity. Instead our approach produces a model acting directly on the vorticity balance equation and which gives a precise bound on the growth of the vorticity magnitude.

. A new model, with an eddy viscosity damping the vorticity
Here, we consider the NSE in the velocity/vorticity formulation and we analyze the enstrophy behavior for the following model which is much weaker in terms of dissipation than the Baldwin-Lomax type previously studied. In the Cauchy setting, or in the space periodic, or even with curl-Navier condition we consider the above system where C S,ω > is a given constant and u is formally obtained from the vorticity ω after solving u = curl − ω. More precisely u is obtained by ω through the Biot-Savart law.
We deal primary with the space-periodic case. The interpretation of this model is that we are adding a zeroth order damping term directly at the vorticity level, which is needed to balance in a sharp form the vortex-stretching term T (ω · ∇) u · ω dx, which behaves, roughly speaking, as the integral of |ω| . In fact, by multiplying by ω and integrating over the physical domain, one gets d dt ω + ν ∇ω + C S,ω ω p p ≤ C CZ ω . The constant C CZ from the right-hand-side is the one coming from the standard estimate with singular integrals. Hence, if p = and if C S,ω ≥ C CZ , then we get the di erential inequality d dt The choice of the parameter p = seems obliged in this setting, but the above estimate allows us to show global existence of a solution, once the constant C S,ω is chosen large enough, but in an universal (not depending on the solution) way. The life-span of the solution turns out to be independent of the size of the initial datum. We recall that without the damping term (that is when C S,ω = ) one can prove results which are local or that are global only for unrealistic extremely small initial data. In this case there are not other a priori estimates immediately available even at the level of the velocity, this explains the fact that smaller values of p seem not treatable. Hence, also results as those of Zhou [34] with smaller critical exponents p, but with the damping |u| p− u at the level of velocity, are not available in this case.
Remark 4.1. The above model (4.1) in the velocity/vorticity formulation can be considered also in the velocity/pressure formulation. To write this we need a proper right-inverse of the curl operator, to write "formally" system (4.1) as the curl of the following one The linear operator " curl − " is de ned in and it is a right inverse of the curl modulo a gradient term, which is nevertheless inessential in the dynamic equation for the enstrophy. Observe that the di erential operators we are using are invariant by rotations, hence the equations we obtain are invariant by change of reference frame and are physically meaningful. We need to apply curl − to f = |ω|ω and in the case of the torus, in order to invert the Laplace operator, the zero mean value is needed. Observe that clearly T ω dx = , but in general T |ω|ω dx ̸ = . To conclude, the model with constant damping (4.1) should be written in the velocity/pressure formulation as follows In terms of vorticity we obtain, in fact, taking the curl or, with obvious simpli cations, The introduction of the velocity/pressure formulation does not change the enstrophy balance. In fact, with this system we have the following identity, obtained by integration by parts, where we used again (1.4) and the fact that gradients and divergence-free vector elds are orthogonal, together with the fact that ω (as well as ∇u and Du) have zero mean value. [35] show the following "positivity lemma"

Remark 4.2. Concerning the velocity/pressure formulation, we observe that results on the Fractional Laplacian in Córdoba and Córdoba
for all values of p ≥ . In our case the energy inequality for weak solutions of the model (4.4) will hold if one would be able to prove the following inequality which correspond to the negative value α = − in (4.5). From the latter estimate in fact one could show which derives immediately from integration by parts. At present, the validity of (4.6) represents an open problem, and we are forced to restrict to work only in the setting of velocity/vorticity formulation. We are then considering a system for which the energy inequality does not follow directly: Some of the basic machinery and estimates available for the 3D Navier-Stokes equations cannot be applied directly, and a slightly di erent treatment for system (4.1) is needed.
The main result of this section is the following Theorem 3. Let be given ω ∈ L σ (T ) and let C S,ω large enough, but independent of the data of the problem. Then, there exists a unique weak¹ solution to the following initial value problem

7)
Proof. We start by observing that natural a-priori estimate for system (4.7) is (4.2), and if C S,ω ≥ C CZ , then it follows that ω ∈ L ∞ ( , T; L σ (T )) ∩ L ( , T; H σ (T )) ∩ L ( , T; (L (T )) ), which is enough to ensure the existence through a Fourier-Galerkin approximation. In fact, the bound on ω proves by comparison also that ∂ t ω ∈ L ( , T; (H σ ) ). This is obtained by multiplying by a periodic and with zero mean value φ ∈ V it follows that showing the requested property by using the already obtained bound. This implies, by using the classical Aubin-Lions lemma, that from the approximating sequences we can extract a sub-sequence converging strongly in L ( , T; (L (T )) ), cf. (2.3). Next by interpolation and by using the a priori bounds this shows also strong convergence in L ( , T; (L (T )) ). This is enough to pass to the limit in all terms and to show that the limit is a solution of the problem. We do not give details for this wellknown step, which can be found in [17], simply translating the results on the velocity to those for the vorticity.
We show now that the obtained regularity is enough to show uniqueness. In fact the duality between ∂ t ω and ω is well-de ned. We take two solutions ω , ω corresponding to the same initial datum ω (x) and, multiplying by the di erence Ω = ω − ω , we obtain d dt and we observe that by integrating by parts and T (u · ∇) Ω · Ω dx = , while by usual Hölder and Sobolev estimates Hence, with Young inequality we get d dt and due to the available information on the two weak solutions it follows that ( ∇ω / + ω + ω ) ∈ L ( , T), hence we get Ω ≡ . The exponent p = represents the natural one in order to control the growth of the nonlinear term. Observe that having ω ∈ L t,x would be enough as additional assumption if applied to the NSE. In our setting due to the fact that we have a nonlinear additional term and lack of the energy inequality, this is not enough to ensure a direct result of global existence, unless not some restrictions on the size of the damping coe cient C S,ω is added, cf. Remark 4.3. In this section we use a rather di erent approach, which is based on a selective damping, where selection is based on the criterion of adding damping of the vorticity only in regions where there is a large vortex stretching. We now propose an alternate (with respect to the ones already present in literature, cf. especially [2,37]) selective method with a non-constant turbulent coe cient, in such a way to ensure damping only at the level of the vorticity, and for which the mathematical theory can be formulated in a rigorous way.
To detect the regions where the vorticity is highly active we formulate a criterion based not on the relative size of the vorticity, but by considering a turbulent viscosity functions which is a multiple of the indicator of the region with intense vortex activity; this is detected by considering the local behavior of the direction of the vorticity itself in the neighborhood of the point (t, x) ∈ ( , T) × Ω.
We observe that a similar model has been already considered by Cottet, Jiroveanu, and Michaux [2], but in their case they worked directly with the equation for the velocity, proposing the following LES model with Dirichlet conditions, where The constant β is a threshold angle, while (4.9) and ω λ is the average over a the surface of ball or radius λ > where the length λ is related to the mesh size and the integral can be numerically estimated by averaging over the six closest neighboring points (in a cubic uniform mesh). The interpretation of the rationale behind the criterion (see also Guermond and Prudhomme [38]) is based on the fact that if the angle between the vorticity is well behaved, then weak solutions of the Navier-Stokes equations are smooth, cf. [14,15]. Hence, the lack of (anti) alignment of the vorticity is a measure of the potential singular behavior of the solutions. The main result is the theorem stating that if where θ(t, x, y) := ∠(ξ (t, x), ξ (t, y)), then weak solutions of the Navier-Stokes equations are smooth, see [14][15][16]39].
Here, we consider and rigorously analyze a variation of the model (4.8), in such a way that the a priori estimate can be put in a precise quantitative relation with the growth of the enstrophy and the behavior of the vorticity direction. We are in fact introducing a new model with the aim of being able to perform some quantitative estimates, missing in the cited references [2,37]) which were more focused on the implementation and on the e ective numerical results).
The new model we propose and for which we are able to prove quantitative a priori estimates is the following: in the velocity/vorticity formulation we consider with the scalar function Ψ de ned as follows: where "jumps" of the vorticity direction are large, elsewhere.
The precise quantitative notion of "large jumps" will be speci ed later in the formulation of Theorem 4. We remark that the expected smoothness of Ψ is very low (nothing more than L ∞ can be inferred) and not enough to establish probably good regularity properties, nevertheless it can be at least used to prove a priori bounds. The fact that the operator is with non-constant coe cients makes the treatment more complex than in [20] and in addition the damping is not uniform, producing additional L loc estimates which are not uniform in the whole spatial domain.
The main original point is to work directly with the vorticity balance equation, as in the model of the previous Section 4.1 and to establish or identify a "good set" Ω λ ⊂ ( , T) × R such that if Ψ(t, x) = χ Ω λ (t, x), then it follows that the degenerate damping term in the equations for the vorticity (4.11) is enough to control the growth of the enstrophy. The results in [14] and further developments in [15,24] are devoted to the analysis of the weaker possible alignment of ξ able to ensure the regularity of weak solution. By studying a special setting with "type I singularities," Giga and Miura [39] have been able to study also a case of mere continuity, without any other further requirement (Lipschitz or Hölder) as in the above references. Nevertheless, the modulus of continuity is not easily computable in numerical tests, since the uniform control requires rather extensive computations. The LES model, as introduced with the determination of the threshold angle (4.9), is based on this rationale from condition (4.10). On the other hand, we consider here a criterion built up on the results from Ref. [16] which states regularity for the Navier-Stokes equations if the possible jumps of angles between the vorticity at neighboring points is small enough. This is more t in the framework of the determination of the criterion, since averaging is not needed.

Theorem 4.
Let us assume ² that |ω(t, x)| ≥ M for all (t, x) ∈ R × ( , T), for some M > . Let us x the two constant < ϵ << and λ > and de ne the following set Then, if ω ∈ L σ (R ), if ϵ is small enough, and if λ is large enough, then there exists a family of uniformly bounded and global in time approximate solutions to model (4.11).
Proof. As usual the main part of the proof is to properly estimate the vortex stretching term. We work in the whole space where explicit formulas are neat, but a similar treatment can be easily done also in the spaceperiodic setting. We split the term responsible of the vortex stretching into two parts as follows where [a×b] j = ϵ ijk a k b l is the exterior product (with the totally anti-symmetric Ricci tensor ϵ ijk ) and y = y/|y|. The integral in dy is split into two parts: the inner where |y| < λ and the outer one, by means of a smooth cut-o non-increasing function ρ : R + → R + which equals 1 for < s < λ/ and zero for s ≥ λ/ . In particular, we set We use now the Hardy-Littlewood-Sobolev inequality to infer R |ω(x + y)| dy |y| / L ≤ c HLS ω L , with c HLS = / π / .
By the above formula and the Hölder inequality in Ω λ with exponents / and , with the usual interpolation and Young's inequality, we get The other term is the one such that the singular integral is not truncated by the cut-o function but we can use on it the smallness condition on the angle to show Ω λ |ω(x)| P.V . π R ρ(|y|)F(dx, dy) dxdy ≤ ϵ ω ,Ω λ ω ≤ ϵ C ω ∇ω + ω .
Next, we consider the integral over the set Ω λ , where do not have control on the direction of the vorticity. First we use the Hölder inequality and the norm of S[ω](x) is a proper singular integral on the whole space, this proves the following inequality Ω λ S[ω](x) ω(x) · ω(x) dx ≤ C ω ,Ω λ ω ,R , and the constant C is independent of the data of the problem (depends only on the space dimension). Next, we use the usual Sobolev and convex interpolation inequalities (with constants independent of the solution) in the whole space and the Hölder inequality with exponents / , , and to prove Hence, by collecting all the inequalities, we get d dt ω + ν − ϵ + λ ω / ∇ω + C S,ω − Cν − ω / ω ,Ω λ ≤ ω . This shows that by an appropriate choice of the constant C S,ω large enough, then the following a priori estimate holds true ω ∈ L ∞ ( , T; L (R )) ∩ L ( , T; H (R )), provided ϵ is chosen small enough and λ large enough.

Remark 4.5.
In the de nition of the set Ω λ there are two parameters, one related with the relative nonalignment of the vorticity and the other with the size of the set where the condition has to be checked. In some sense both depend on the viscosity, since the a priori estimate requires the smallness of ϵ and the largeness of λ in order to have non negative terms. In this way we quantitatively link the amount of damping on the vorticity with the size of the set where this has to be ful lled. In practice, controlling the vorticity direction in a small set imposes as counterpart to have larger constants for the dissipative term.