Suitable weak solutions of the Navier–Stokes equations constructed by a space–time numerical discretization

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Suitable weak solutions of the Navier–Stokes equations constructed by a space–time numerical discretization Luigi C. Berselli a,∗ , Simone Fagioli b , Stefano Spirito b a

Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, I-56127 Pisa, Italy DISIM – Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, I-67100 L’Aquila, Italy b

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 October 2017 Available online xxxx

We prove that weak solutions obtained as limits of certain numerical space–time discretizations are suitable in the sense of Scheffer and Caffarelli–Kohn–Nirenberg. More precisely, in the space-periodic setting, we consider a full discretization in which the θ-method is used to discretize the time variable, while in the space variables we consider appropriate families of finite elements. The main result is the validity of the so-called local energy inequality. © 2018 Elsevier Masson SAS. All rights reserved.

MSC: 35Q30 76M10 76M20 Keywords: Navier–Stokes equations Local energy inequality Numerical schemes θ-method Finite element and finite difference methods

r é s u m é On prouve que les solutions faibles obtenues comme limites de quelques discretizations numériques en space and temps sont acceptables (suitable) dans le sens de Scheffer et Caffarelli–Kohn–Nirenberg. Plus exactement dans le torus on considère une discrétisation complète dans laquelle on utilise la méthode «theta» pour discrétiser la variable temporelle, pendant que pour celle spatiale on considère une famille appropriée des éléments finis. Le résultat principal est la validité de l’inégalité locale d’énergie. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction We consider the homogeneous incompressible 3D Navier–Stokes equations ∂t u − Δu + (u · ∇) u + ∇p = 0

in (0, T ) × T3 ,

div u = 0

in (0, T ) × T3 ,

(1.1)

* Corresponding author. E-mail addresses: [email protected] (L.C. Berselli), [email protected] (S. Fagioli), [email protected] (S. Spirito). https://doi.org/10.1016/j.matpur.2018.09.004 0021-7824/© 2018 Elsevier Masson SAS. All rights reserved.

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in the space periodic setting, with a divergence-free initial datum u|t=0 = u0

in T3 ,

(1.2)

where T > 0 is arbitrary and T3 is the three dimensional flat torus. Here, the unknowns are the vector field u and the scalar p, which are both with zero mean value. The aim of this paper is to consider a space–time discretization of the initial value problem (1.1)–(1.2) and to prove convergence (as the parameters of the discretization vanish) to a Leray–Hopf weak solution, satisfying also the local energy inequality  ∂t

|u|2 2



 + div

   2 |u|2 |u| +p u −Δ + |∇u|2 ≤ 0 in D (]0, T [×T3 ). 2 2

Solution satisfying the above inequality (and minimal assumptions on the pressure) are known in literature as suitable weak solutions and they are of fundamental importance for at least two reasons: 1) From the theoretical point of view it is known that for these solutions the possible set of singularities has vanishing 1D-parabolic Hausdorff measure, see Scheffer [1] and Caffarelli–Kohn–Nirenberg [2]; 2) The local energy inequality is a sort of entropy condition and, even if it is not enough to prove uniqueness, it seems a natural request to select physically relevant solutions; for this reason the above inequality has to be satisfied by solutions constructed by numerical methods, see Guermond et al. [3,4]. The terminology (and an existence result) for suitable weak solutions can be found in Caffarelli, Kohn, and Nirenberg [2], where a retarded-time approximation method has been used in the construction. Since uniqueness is not known in the class of weak solutions, the question of understanding which are the approximations producing suitable solutions became central, see the papers by Beirão da Veiga [5–7]. In these papers it has also been raised the question whether the “natural” Faedo–Galerkin methods will produce suitable solutions and the question remained completely unsolved for almost twenty years, when the positive answer, at least for certain finite element spaces, appeared in Guermond [8,9]. (In the above papers the space discretization is considered, while the time variable is kept continuous.) It is important to observe that, most of the known regularization procedures to construct Leray–Hopf weak solutions of the Navier–Stokes equations (hyper-viscosity, Leray, Leray-α, Voigt, artificial compressibility. . . ) seem to produce, in the limit, solutions satisfying the local energy inequality. We started a systematics study of this question and, even if technicalities could be rather different, we obtained several positive answers, see [10–12]. The technical problems related with discrete (numerical, finite dimensional) approximations are of a different nature. In particular, obtaining the local energy inequality is “formally” based on multiplying the equations by u φ, where φ is a non-negative bump function, and integrating in both space and time variables. Clearly, if u belongs to a finite dimensional space Xh (say of finite elements), then u φ is not allowed to be used as a test function and one needs to project u φ back on Xh . This is the reason why results obtained in [8,9] require the use of spaces satisfying a suitable commutator property, which is a local property, see Section 3.1. In particular, the standard Fourier–Galerkin method (which is not local, being a spectral approximation) does not satisfy the commutator property and the convergence to a suitable weak solution is still an interesting open problem, see the partial results in Craig et al. [13], with interesting links with the global energy equality. From the numerical point of view another important issue is that of considering also the time discretization, hence going from a semi-discrete scheme to a fully-discrete one. Also regarding this issue few results are available. In [14] it is proved that solutions of space-periodic Navier–Stokes constructed by semidiscretization (in the time variable, with the standard implicit Euler algorithm) are suitable. The argument has been also extended to a general domain in [15] assuming vorticity-based slip boundary conditions, which are important in the vanishing viscosity problem [16,17]. The case of Dirichlet boundary conditions has been treated in the implicit case in the last section of [18], by adapting some of the result obtained by Sohr and Von Wahl [19] in the continuous case.

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The aim of this paper is to extend the results from [8] and [14,18] to a general space–time numerical discretization, with a general θ-method in the time-variable and finite elements in the space variables. The extension of the results regarding only on the space or only on the time discretization presents some additional difficulties and it is not just a combination of the previous ones. In particular, the main core of the proof is obtaining appropriate a-priori estimates and using compactness results. Contrary to [8,14] results obtained here require a more subtle compactness argument for space–time discrete functions and the main theorem is obtained by using a technique borrowed from the treatment of non-homogeneous and compressible fluids and resembling the compensated compactness arguments, see P.L. Lions [20, Lemma 5.1]. Observe that in the present paper the simple convergence in the sense of distributions is not enough, contrary to the case of the product density/velocity in the weak formulation of the Navier–Stokes equations with variable density. We also observe that, at present, the extension to the Dirichlet problem (as in [9]) seems a challenging problem; the estimates in the negative spaces obtained in [9] look not enough to handle the discretization in time made with step functions, which cannot be in fractional Sobolev spaces with order larger than one-half. 1 To set the problem we consider, as in [8], two sequences of discrete approximation spaces {Xh }h ⊂ H# and 1 {Mh }h ⊂ H# which satisfy – among other properties described in Section 3 – an appropriate commutator property, see Definition 3.1. Then, given a net tm := m Δt we consider the following space–time discretization of the problem (1.1)–(1.2): Set u0h = Ph (uh0 ), where Ph is the projection over Xh . For any m = 1, ..., N and m given um−1 ∈ Xh and pm−1 ∈ Mh , find um h ∈ Xh and p ∈ Mh such that h h m,θ m,θ m,θ m (dt um h , vh ) + (∇uh , ∇vh ) + bh (uh , uh , vh ) − (ph, , div vh ) = 0,

(div um h , qh ) = 0,

(1.3)

um −um−1

m,θ m−1 where um,θ := θ um , dt um := h Δth , and bh (um,θ h + (1 − θ) uh h h , uh , vh ) is a suitable discrete approximation of the non-linear term; see also Quarteroni and Valli [21] and Thomée [22] for general properties of θ-schemes for parabolic equations. We also remark that the scheme with the non-linear term of the type m,θ m−1 bh (um , um,θ h , uh , vh ) or bh (uh h , vh ) could be probably treated with similar methods but it seems to require some non trivial modifications of the proofs, this will be left for future studies, hoping to find a way to unify the treatment and have a general abstract argument. However, we point out that the fact that in the m,θ second entry of the tri-linear term bh ( . , um,θ is crucial in order to exploit the cancellation h , . ) there is uh of the non-linear term in the basic energy estimate, see Lemma 4.1. For further notations, definitions, and properties regarding the scheme (1.3) we refer to Sections 2–3. As usual in time-discrete problem (see for instance [23]), in order to study the convergence to the solutions of the continuous problem it is useful to rewrite (1.3) on (0, T ) as follows:

 Δt     Δt Δt   Δt  ∂t vh , wh + ∇uΔt h , ∇wh + bh uh , uh , wh − ph , div qh = 0,   div uΔt h , wh = 0,

(1.4)

N Δt Δt where vhΔt is the linear interpolation of {um h }m=1 (over the net tm = mΔt), while uh and ph are the m,θ m time-step functions which on the interval [tm−1 , tm ) are equal to uh and ph , respectively, see (3.12). Δt Notice that (vhΔt , uΔt h , ph ) depends implicitly on θ, which is fixed in [0, 1]. However, in order to avoid heavy notations, we do not use denoting this dependence in an explicit way. The main result of the paper is the following, and we refer to Section 2 for further details on the notation.

Theorem 1.1. Let the finite element space (Xh , Mh ) satisfy the discrete commutation property and the tech1 Δt nical conditions described in Section 3.1. Let u0 ∈ Hdiv and θ ∈ (1/2, 1]. Let {(vhΔt , uΔt h , ph )}Δt,h be a sequence of solutions of (1.4) computed by solving (1.3). Then, there exists

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1 (u, p) ∈ L∞ (0, T ; L2div ) ∩ L2 (0, T ; Hdiv ) × L4/3 (0, T ; L2# ),

such that, up to a sub-sequence, as (Δt, h) → (0, 0), vhΔt → u strongly in L2 ((0, T ) × T3 ), 2 3 uΔt h → u strongly in L ((0, T ) × T ), 2 3 ∇uΔt h  ∇u weakly in L ((0, T ) × T ), 4

2 3 pΔt h  p weakly in L (0, T ; L# ).

Moreover, the couple (u, p) is a suitable weak solution of (1.1)–(1.2) in the sense of Definition 2.2. Remark 1.2. Theorem 1.1 is not limited to decaying flows, but the result holds also in the presence of an external force f satisfying suitable bounds. For example, f ∈ L2 (0, T ; L2 (T3 )) is enough. For simplicity we consider the viscosity equal to 1, but clearly the result is valid also for any positive viscosity. The proof of Theorem 1.1 is given in Section 5 and it is based on a compactness argument. We briefly explain the main novelty in the proof: First from the standard discrete energy inequality (Lemma 4.1) only an H 1 -bound in space on uΔt h is available but no compactness in time. Note that this is not enough in general to deduce strong convergence of uΔt h , which turns out to be necessary to prove even the convergence to a Leray–Hopf weak solution. It is also relevant to observe that in many references (e.g. as in [23,21]) authors focus on the order of the convergence between the numerical and continuous solution in the L2-norm. 1 Nevertheless, in our case it is very relevant to obtain the uniform l2 (Hdiv ) bound on the numerical solution, since this is requested to show convergence towards a weak solution, considered in the genuine sense of Leray and Hopf. This explains the limitations on θ, which in this paper are not due to classical stability issues. Hence, in the proof of the main result also the Step 1 (that of proving that the numerical solutions converge to Leray–Hopf weak solutions) is rather original, or at least we did not find this explicitly proved in any reference (For instance in [23] a partial analysis of this point, valid only for certain schemes, is provided). On the other hand, from the equations (1.4) it is possible to prove some mild time regularity on vhΔt ; this 1 3 will be enough to ensure that the product vhΔt uΔt h is weakly convergent in L (T ) to v u, where v and u Δt Δt are the weak limits of vh and uh , respectively. This is the technical point where results à la compensated compactness are used. Finally, this additional information combined again with a discrete energy inequality 2 3 allows us to infer that u = v and that uΔt h is strongly convergent in L ((0, T ) × T ). Plan of the paper. In Section 2 we fix the notation that we use in the paper and we recall the main definitions and tools used. In Section 3 we introduce and give some details about the space–time discretization methods. Finally, in Section 4 we prove the main a priori estimates needed to study the convergence and finally in Section 5 we prove Theorem 1.1. 2. Notations and preliminaries In this section we declare the notation we will use in the paper, we recall the main definitions concerning weak solutions of incompressible Navier–Stokes equations and also a compactness result. 2.1. Notations We introduce the notations typical of space-periodic problems. The flat three-dimensional torus T3 is defined by (R/2πZ)3 . In the sequel we will use the customary Lebesgue spaces Lp (T3 ) and Sobolev spaces W k,p (T3 ) and we will denote their norms by  · p and  · W k,p We will not distinguish between scalar

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and vector valued functions, since it will be clear from the context which one has to be considered. In the case p = 2, the L2 (T3 ) scalar product is denoted by (·, ·), we use the notation H s (T3 ) := W s,2 (T3 ), and we define, for s > 0, the dual spaces H −s (T3 ) = (H s (T3 )) . Moreover, we will consider always subspaces of functions with zero mean value and these will be denoted by Lp# :=

⎧ ⎨ ⎩

⎫ ⎬

w ∈ Lp (T3 ) :

w dx = 0 T3

⎭

1 ≤ p < +∞,

and also s H# := H s (T3 ) ∩ L2# .

As usual in fluid mechanics one has to consider spaces of divergence-free vector fields, defined as follows

L2div := w ∈ (L2# )3 :



div w = 0

and

s s 3 Hdiv := (H# ) ∩ L2div .

Finally, given a Banach space (X,  . X ), we denote by Lp (0, T ; X) the classical Bochner space of X-valued functions, endowed with its natural norm, denoted by  ·Lp (X) . We denote by lp (X) the discrete counterpart for X-valued sequences {xm }m , defined on the net {mΔt}, with weighted norm given by xplp (X) := M Δt m=0 xm pX . 2.2. Weak solutions and suitable weak solutions We start by recalling the notion of weak solution (as introduced by Leray and Hopf) for the space periodic setting. Definition 2.1. The vector field u is a Leray–Hopf weak solution of (1.1)–(1.2) if 1 u ∈ L∞ (0, T ; L2div ) ∩ L2 (0, T ; Hdiv ),

and if u satisfies the Navier–Stokes equations (1.1)–(1.2) in the weak sense, namely the integral equality T



 (u, ∂t φ) − (∇u, ∇φ) − ((u · ∇) u, φ) dt + (u0 , φ(0)) = 0,

(2.1)

0

holds true for all φ ∈ Cc∞ ([0, T ); C ∞ (T3 )3 ) smooth, periodic, and divergence-free functions such that  φ dx = 0. Moreover, the initial datum is attained in the strong L2 -sense, that is T3 lim u(t) − u0 2 = 0,

t→0+

and the following global energy inequality holds 1 u(t)22 + 2

t ∇u(s)22 ds ≤

1 u0 22 , 2

for all t ∈ [0, T ].

(2.2)

0

Suitable weak solutions are a particular subclass of Leray–Hopf weak solutions and the definition is the following.

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Definition 2.2. A pair (u, p) is a suitable weak solution to the Navier–Stokes equation (1.1) if u is a Leray– 4 Hopf weak solution, p ∈ L 3 (0, T ; L2# ), and the local energy inequality T 

T |∇u| φ dxdt ≤ 2

0

T3

0

T3

|u|2 (∂t φ + Δφ) + 2



  |u|2 + p u · ∇φ dxdt 2

(2.3)

holds for all smooth scalar functions φ ∈ C0∞ (0, T ; C ∞ (T3 )) such that φ ≥ 0, 5

Remark 2.3. The definition of suitable weak solution is usually stated with p ∈ L 3 ((0, T ) × T3 ) while in 4 Definition 2.2 we require just p ∈ L 3 (0, T ; L2 (T3 )). This is not an issue since we have a bit less integrability in time but we gain the L2 -integrability in the space variables. We stress that the main property of suitable weak solutions is the fact that they satisfy the local energy inequality (2.3) and weakening the request on pressure does not influence too much the validity of local regularity results, see for instance discussion in Vasseur [24]. 2.3. A compactness lemma In this subsection we recall the main compactness lemma, which allows us to prove the strong convergence of the approximations. We remark that it is a particular case of a more general lemma, whose statement and proof can be found in [20, Lemma 5.1]. For sake of completeness we give a proof adapted to the case we are interested in. Lemma 2.4. Let {fn }n∈N and {gn }n∈N be sequences uniformly bounded in L∞ (0, T ; L2 (T3 )) and let be given f, g ∈ L∞ (0, T ; L2 (T3 )) such that fn  f

weakly in L2 ((0, T ) × T3 ),

gn  g

weakly in L2 ((0, T ) × T3 ).

(2.4)

Let p ≥ 1 and assume that {∂t fn }n ⊂ Lp (0, T ; H −1 (T3 ))

and

{gn }n ⊂ L2 (0, T ; H 1 (T3 )),

(2.5)

with uniform (with respect to n ∈ N) bounds on the norms. Then, fn gn  f g

weakly in L1 ((0, T ) × T3 ).

(2.6)

Proof. By using (2.4), (2.5), and the fact that L2 (T3 ) is compactly embedded in H −1 (T3 ) it follows from the Banach space version of Arzelà–Ascoli theorem that fn → f

strongly in C(0, T ; H −1 (T3 )).

(2.7)

weakly in L2 (0, T ; H 1 (T3 )).

(2.8)

From (2.5) it follows that gn  g Then, (2.7) and (2.8) easily imply that fn gn → f g

in the sense of distribution on (0, T ) × T3 .

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The L1 -weak convergence in (2.6) follows by noting that the bounds in (2.5) imply that the sequence {fn gn }n is equi-integrable. 2 3. Setting of the numerical approximation In this section we introduce the time and space discretization of the initial value problem (1.1)–(1.2). We start by introducing the space discretization. 3.1. Space discretization 1 For the space discretization we strictly follow the setting considered in [8]. Let {Xh }h>0 ⊂ H# be the 2 discrete space for approximate velocity and {Mh }h>0 ⊂ L# be that of approximate pressure. To avoid 1 further technicalities, we assume as in [8], that Mh ⊂ H# . We make the following (technical) assumptions on the spaces Xh and Mh : 1 1. For any v ∈ H# and for any q ∈ L2# there exists {vh }h and {qh }h with vh ∈ Xh and qh ∈ Mh such that

vh → v

1 strongly in H#

qh → q

strongly in L2#

as h → 0, as h → 0;

(3.1)

2. Let πh : L2 (T3 ) → Xh be the L2 -projection onto Xh . Then, there exists c > 0 independent of h such that, πh (∇qh ) 2 ≥ c qh 2

∀ qh ∈ Mh ;

(3.2)

1 3. There is c > 0, independent of h, such that for all v ∈ H#

v − πh (v)2 =

inf v − wh 2 ≤ c hvH 1 ,

wh ∈Xh

πh (v)H 1 ≤ c vH 1 ; 4. There exists c > 0, independent of h, such that vh H 1 ≤ c h−1 vh 2

∀ vh ∈ Xh .

(3.3)

Moreover, we assume that Xh and Mh satisfy the following discrete commutator property. Definition 3.1. We say that Xh (resp. Mh ) has the discrete commutator property if there exists an operator Ph ∈ L(H 1 ; Xh ) (resp. Qh ∈ L(L2 ; Mh )) such that for all φ ∈ W 2,∞ (resp. φ ∈ W 1,∞ ) and all vh ∈ Xh (resp. qh ∈ Mh ) vh φ − Ph (vh φ)H l ≤ c h1+m−l vh H m φW m+1,∞ ,

(3.4)

qh φ − Qh (qh φ)2 ≤ c hqh 2 φW 1,∞ ,

(3.5)

for all 0 ≤ l ≤ m ≤ 1. Remark 3.2. Explicit examples of admissible couples of finite element spaces satisfying all the above requirements are the MINI and Hood-Taylor elements, with quasi-uniform mesh and in the periodic setting, see [8]. Probably more general meshes could be considered by using the results in [25] and references therein.

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Remark 3.3. We want to stress that in the case of spectral methods, e.g., the Faedo–Galerkin methods based on Fourier expansion on the torus, the discrete commutator property fails. This is one of the main obstacles in proving the local energy inequality for weak solutions of (1.1) constructed by the Fourier–Galerkin method. 2 We recall (see also [8]) that the coercivity hypothesis (3.2) allows us to define the map ψh : H# → Mh 2 such that, for all q ∈ H# , the function ψh (q) is the unique solution to the variational problem:

(πh (∇ψh (q)), ∇rh ) = (∇q, ∇rh ) .

(3.6)

2 This map has the following properties: there exists c > 0 independent of h such that for all q ∈ H# ,

∇(ψh (q) − q)2 ≤ c hqH 2 , πh ∇ψh (q)H 1 ≤ c qH 2 .

(3.7)

Let us introduce the discrete divergence-free finite element spaces as follows

Vh = vh ∈ Xh : (div vh , qh ) = 0

 ∀qh ∈ L2 (Ω) .

To have the basic energy estimate we need to modify the non-linear term, since Vh is not a subspace of 1 Hdiv . Let us define the following operator 1 nlh (u, v) := (u · ∇) v + v div u. 2

(3.8)

Then, nlh ( . , . ) is a bi-linear operator 1 1 nlh : H# × H# → H −1 , 1  where H −1 := (H# ) . Moreover, the following estimate holds true

nlh (u, v)H −1 ≤ uL3 vH 1

1 ∀ u, v ∈ H# .

(3.9)

Finally, by defining bh (u, v, w) := nlh (u, v), w H −1 ×H#1 , it follows that bh (u, v, v) = 0

1 ∀ u, v ∈ Hdiv + Vh .

Then, the space discretization of (1.1)–(1.2) reads as follows: Find uh ∈ C(0, T ; Xh ) with ∂t uh ∈ L2 (0, T ; Xh ) and ph ∈ L2 (0, T ; Mh ) such that, for all vh ∈ Xh and qh ∈ Mh : (∂t uh , vh ) + bh (uh , uh , vh ) − (ph , div vh ) + (∇uh , ∇vh ) = 0, (div uh , qh ) = 0, with the initial datum uh |t=0 = uh0 , where uh0 is an approximation of u0 such that uh0 ∈ Xh , and uh0 → u0

strongly in L2div

as h → 0.

(3.10)

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3.2. Time discretization For the time variable t we define the mesh as follows: Given N ∈ N the time-step 0 < Δt ≤ T is defined as Δt := T /N . Accordingly, we define the corresponding net {tm }N m=1 by t0 := 0

and

tm := m Δt,

m = 1, . . . , N.

To discretize in time the semi-discrete problem (3.10) we consider the following θ-method (cf. [21, § 5.6.2]) for θ ∈ [0, 1]: 1 m Set u0h = u0 ∈ Hdiv . For any m = 1, ..., N , given um−1 ∈ Xh and pm−1 ∈ Mh , find um h ∈ Xh and ph ∈ Mh h h such that m,θ m,θ m,θ m (dt um h , vh ) + (∇uh , ∇vh ) + bh (uh , uh , vh ) − (ph, , div vh ) = 0,

(div um h , qh ) = 0,

(3.11)

for all vh ∈ Xh and for all qh ∈ Mh . We recall that dt um is the backward finite-difference approximation for the time-derivative in the interval (tm−1 , tm ) ∂t uh ∼ dt um :=

m−1 um h − uh , Δt

m−1 m+1 and um,θ := θum is a convex combination of um . With a slight abuse of notation h + (1 − θ)uh h and uh h we consider Δt = T /N and h, instead of (N, h), as the indexes of the sequences for which we prove the convergence. Then, the convergence will be proved in the limit of (Δt, h) both going to zero. We stress that this does not affect the proofs since all the convergences are proved up to sub-sequences. Once (3.11) is solved, we consider a continuous version, which turns out to be useful to study the m m convergence. To this end, we associate to the triple (um,θ h , uh , ph ) the functions Δt 3 3 3 (vhΔt , uΔt h , ph ) : [0, T ] × T → R × R × R,

defined as follows: ⎧ m−1 ⎨ um−1 + t − tm−1 (um ) for t ∈ [tm−1 , tm ), h − uh h Δt Δt vh (t) := ⎩ N for t = tN , uh  m,θ uh for t ∈ [tm−1 , tm ), uΔt h (t) := for t = tN , uN,θ h  m for t ∈ [tm−1 , tm ), ph pΔt h (t) := N for t = tN . ph

(3.12)

Then, the discrete equations (3.11) can be rephrased as follows: 

    Δt   Δt  Δt ∂t vhΔt , wh + bh uΔt h , uh , wh + ∇uh , ∇wh − ph , div wh = 0,   div uΔt h , qh = 0,

(3.13)

for all wh ∈ Ls (0, T ; Xh ) (with s ≥ 4) and for all qh ∈ L2 (0, T ; Mh ). We notice that the divergence-free condition comes from the fact that um h is such that (div um h , qh ) = 0

for m = 1, ..., N,

∀ qh ∈ Mh .

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4. A priori estimates In this section we prove the a priori estimates that we need to study the convergence of solutions of (3.13) to suitable weak solutions of (1.1)–(1.2). We start with the following discrete energy equality. Lemma 4.1. For any 1/2 < θ ≤ 1, N ∈ N, and m = 1, .., N the following discrete energy-type equality holds true: (2θ − 1) m 1 m−1 2 2 2 (um uh − um−1 2 ) + 22 + Δt ∇um,θ h 2 − uh h h 2 = 0. 2 2

(4.1)

Proof. For any m = 1, ..., N take wh = χ[tm−1 ,tm ) um,θ ∈ L∞ (0, T ; Xh ) in (3.13). Then, h 

m−1 um h − uh , um,θ h Δt

 2 + ∇um,θ h 2 = 0,

because since um,θ ∈ Xh and pm h ∈ Mh it follows that h m,θ m,θ bh (um,θ h , uh , uh ) = 0

and

m,θ (pm h , div uh ) = 0.

By using the elementary algebraic identity (a − b, a) =

|b|2 |a − b|2 |a|2 − + , 2 2 2

the term involving the discrete derivative reads as follows: m−1 m−1 m−1 m (um , um,θ , θum ) h − uh h + (1 − θ)uh h ) = (uh − uh m−1 m−1 m−1 = θ(um , um − um ) h − uh h ) + (1 − θ)(uh h , uh

θ m−1 2 m−1 2 2 (um  + um 2 ) h 2 − uh h − uh 2 (1 − θ) m−1 2 2 m − (um−1 22 − um 2 ) h  + uh − uh h 2 1 (2θ − 1) m m−1 2 2 = (um uh − um−1 2 ) + 22 . h 2 − uh h 2 2 =

Then, multiplying by Δt > 0, Eq. (4.1) holds true. In addition, summing over m we also get N N  1 0 2 1 N 2 (2θ − 1)  m 2 2 uh 2 + uh − um−1  + Δt ∇um,θ 2 h h 2 = uh 2 , 2 2 2 m=0 m=0 1 which proves the l∞ (L2# ) ∩ l2 (H# ) uniform bound for the sequence {um h }.

2

Remark 4.2. The requirement θ > 1/2 with strict inequality is not required for the proof of Lemma 4.1. However, it is needed in order to deduce the crucial a priori estimates since it makes the coefficient of the second term from the left-hand side of (4.1) strictly positive. Moreover, since we actually need that term in the convergence proof to a weak solution, we cannot consider the endpoint case θ = 1/2.

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The next lemma concerns the regularity of the discrete pressure. We follow the argument in [8] and we notice that we are essentially solving the standard discrete Poisson problem associated to the pressure. Lemma 4.3. There exists a constant c > 0, independent of Δt and of h, but eventually depending on θ, such that   m,θ m,θ m,θ 1 3 1 pm  ≤ c u  + u  u  for m = 1, . . . , N. (4.2) H L H h 2 h h h 2 Proof. Let q m ∈ H# be the unique solution of the following weak Poisson problem:

(∇q m , ∇φ) = (pm h , φ)

1 ∀ φ ∈ H# .

(4.3)

Standard elliptic estimates imply there exists an absolute constant c > 0 such that q m H 2 ≤ c pm h 2 . Let us consider πh ∇(ψh (q m )) ∈ Xh as test function in (3.11). Then, we get m (dt um , πh ∇(ψh (q m ))) + (∇um,θ h , ∇πh ∇(ψh (q ))) m,θ m m m + bh (um,θ h , uh , πh ∇(ψh (q ))) − (ph , div πh ∇(ψh (q ))) = 0.

First, by using (3.6) and (4.3) we prove m m m − (pm h , div πh ∇(ψh (q ))) = (∇ph , πh ∇(ψh (q ))) m m m m 2 = (∇pm h , ∇q ) = (ph , ph ) = ph 2 .

Then, we get m,θ 2 m m m pm h 2 = − (∇uh , ∇πh ∇(ψh (q ))) − (dt u , πh ∇(ψh (q ))) m m − bh (um h , uh , πh ∇(ψh (q ))).

By using (3.7) and (4.4) we have     m,θ m,θ  ∇uh , ∇πh ∇(ψh (q m ))  ≤ ∇uh 2 πh ∇(ψh (q m ))H 1 m ≤ C∇um,θ h 2 q H 2 m ≤ C∇um,θ h 2 ph 2 .

Concerning the term involving the discrete time-derivative we have     m m−1 , πh ∇(ψh (q m )) = um , ∇(ψh (q m )) uh − um−1 h − uh h   m−1 = − div(um ), ψh (q m ) = 0. h − uh Finally, regarding the non-linear term by using (3.9) and (3.7) we have     m,θ m,θ m,θ m m |bh (um,θ , u , π ∇(ψ (q )))| ≤ nl (u , u ), π ∇(ψ (q ))   h h h h h h h h h m,θ m ≤ Cum,θ h L3 uh H 1 ph 2 .

(4.4)

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Then, by collecting all the estimates, we get   m,θ m,θ m,θ 2 3 1 1 pm  ≤ c u  u  + u  pm L H H h 2 h 2 , h h h and (4.2) readily follows. 2 We are now in position to prove the main a priori estimates on the approximate solutions of (3.13). Proposition 4.4. Let be given u0 ∈ L2div and 1/2 < θ ≤ 1. Then, there exists a constant c > 0, independent of Δt and of h, such that a) b) c) d)

vhΔt L∞ (L2 ) ≤ c, uΔt h L∞ (L2 )∩L2 (H 1 ) ≤ c, pΔt h L4/3 (L2 ) ≤ c, ∂t vhΔt L4/3 (H −1 ) ≤ c.

Moreover, we also have the following estimate 

T vhΔt

−

2 uΔt h 2

dt ≤ Δt

 N

1 − θ + θ2 3

m−1 2 um 2 . h − uh

(4.5)

m=1

0

1 Proof. The bound in L∞ (0, T ; L2# ) ∩ L2 (0, T ; H# ) for uΔt h follows from (3.12) and Lemma 4.1, as well as the bound on vhΔt in a). The bound on the pressure pΔt follows again from (3.12) and Lemma 4.3. Finally, the h Δt bound on the time derivative of vh follows by (3.13) and a standard comparison argument. Concerning (4.5), by using the definitions in (3.12) we get for t ∈ [tm−1 , tm ) m−1 vhΔt − vhΔt = θ um − um−1 − h + (1 − θ) uh h    t − tm−1  m = θ− uh − um−1 . h Δt

t − tm−1 m (uh − um−1 ) h Δt

Then, computing explicitly the integrals, we have T vhΔt

−

0

vhΔt 22

dt =

N 

tm  um h

−

m=1

 ≤ Δt

t − tm−1 θ− Δt

um−1 22 h

2 dt

tm−1

1 − θ + θ2 3

 N

m−1 2 um 2 , h − uh

m=1

ending the proof. 2 5. Proof of the main theorem In this section we prove Theorem 1.1. We split the proof in two main steps. Proof of Theorem 1.1. We first prove the convergence of the numerical sequence to a Leray–Hopf weak solution and then we prove that the weak solution constructed is suitable, namely that it satisfies a certain estimate on the pressure and the local energy inequality (2.3).

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Step 1: Convergence towards a Leray–Hopf weak solution We start by observing that from a simple density 1 argument, the test functions considered in (2.1) can be chosen in the space Ls (0, T ; Hdiv ) ∩ C 1 (0, T ; L2div ), s 1 1 2 with s ≥ 4. In particular, by using (3.1) for any w ∈ L (0, T ; Hdiv ) ∩ C (0, T ; Ldiv ) such that w(T, x) = 0 1 ) ∩ C 1 (0, T ; L2# ) such that we can find a sequence {wh }h ⊂ Ls (0, T ; H# wh → w

as h → 0,

1 strongly in Ls (0, T ; H# )

wh (0) → w(0)

strongly in

2

∂t wh  ∂t w

weakly in L

as h → 0,

L2#

(0, T ; L2# )

(5.1)

as h → 0.

Let {(vhΔt , vhΔt , pΔt h )}(Δt,h) , defined as in (3.12), be a family of solutions of (3.13). By Proposition 4.4-a) we have that

vhΔt

 (Δt,h)

⊂ L∞ (0, T ; L2# )

with uniform bounds on the norms.

Then, by standard compactness arguments there exists v ∈ L∞ (0, T ; L2# ), such that (up to a sub-sequence) vhΔt  v

as (Δt, h) → (0, 0).

weakly in L2 (0, T ; L2# )

(5.2)

Again by using Proposition 4.4 b), there exists u ∈ L∞ (0, T ; L2# ) such that (up to a sub-sequence) uΔt h u uΔt h

as (Δt, h) → (0, 0),

weakly in L2 (0, T ; L2# ) 2

u

weakly in L

as (Δt, h) → (0, 0).

1 (0, T ; H# )

(5.3)

Moreover, by using (3.1), for any q ∈ L2 (0, T ; L2# ) we can find a sequence {qh }h ⊂ L2 (0, T ; L2# ) such that qh ∈ L2 (0, T ; Mh ) and qh → q

strongly in L2 (0, T ; L2# )

as h → 0.

Then, by using (5.3) and (3.13) we have that T 0=



div uΔt h , qh



T dt →

0

as (Δt, h) → (0, 0),

(div u, q) dt 0

1 hence u is divergence-free, since it belongs to Hdiv . Let us consider (4.5), then



T 2 vhΔt − uΔt h 2 dt ≤ Δt

1 − θ + θ2 3

0

 N

m−1 2 um 2 ≤ c Δt, h − uh

(5.4)

m=1

T 2 where in the last inequality we used Proposition 4.4. Hence, the integral 0 vhΔt − uΔt h 2 dt vanishes as Δt → 0. Then, by using (5.2) and (5.3) it easily follows that v = u. At this point we note that Proposition 4.4 b) and d) imply that (with uniform bounds) {∂t vhΔt }(Δt,h) ⊂ L 3 (0, T ; H −1 ) 4

and

2 1 {uΔt h }(Δt,h) ⊂ L (0, T ; H# ).

Then, by using Lemma 2.4 and the fact that u = v we get that Δt 2 uΔt h vh  |u|

weakly in L1 ((0, T ) × T3 )

as (Δt, h) → (0, 0).

(5.5)

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In particular, by using (5.4) and (5.5) we have that vhΔt → u uΔt h

as (Δt, h) → (0, 0),

strongly in L2 (0, T ; L2# )

→u

2

strongly in L

as (Δt, h) → (0, 0).

(0, T ; L2# )

(5.6)

Concerning the pressure term, the uniform bound in Proposition 4.4 d) ensures the existence of p ∈ 4 L 3 (0, T ; L2# ) such that (up to a sub-sequence) 4

pΔt h p

as (Δt, h) → (0, 0).

weakly in L 3 (0, T ; L2# )

(5.7)

Then, by using (5.1) and (5.2) we have that T (∂t vhΔt , wh ) dt

lim

(Δt,h)→(0,0) 0

=

⎛

lim

(Δt,h)→(0,0)

⎝−

T

⎞ (vhΔt , ∂t wh ) dt − (u0 , wh (0))⎠

0

T =−

(u, ∂t w) dt − (u0 , w(0)). 0

Next, by using (5.3) we also get T

T (∇uΔt h , ∇wh ) dt

lim

(Δt,h)→(0,0) 0

(∇u, ∇w) dt.

= 0

By (5.1), (5.7), and the fact that w is (weakly) divergence-free we obtain T (pΔt h , div wh ) dt → 0

as (Δt, h) → (0, 0).

0

Concerning the non-linear term, let s ≥ 4 and s and s∗ be real numbers such that 1 1 +  =1 s s

1 1 1 + ∗ = . s s 2

and

(5.8)

By using (5.6), (5.3), a standard interpolation argument, and Proposition 4.4 b) it follows that uΔt h →u

∗

strongly in Ls (0, T ; L3# ) as (Δt, h) → (0, 0),

and by (3.8), with a standard compactness argument,   Δt nlh uΔt  (u · ∇) u h , uh



weakly in Ls (0, T ; H −1 )

as (Δt, h) → (0, 0).

Then, by using also (5.1) it follows that T

T Δt bh (uΔt h , uh , wh ) dt →

0

0



 (u · ∇) u, w dt

as (Δt, h) → (0, 0).

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Finally, the energy inequality follows by Lemma 4.1, by using the lower semi-continuity of the L2 -norm with respect to the weak convergence. Remark 5.1. Notice that in order to prove the global energy inequality (2.2) it seems crucial to obtain the estimate (4.1) on the discrete solution. Indeed, even if the bounds in Proposition 4.4 are given, then it is possible to prove the convergence to a distributional solution, but the global energy inequality (2.2) would be very complicated to show when it is not satisfied at the approximation level. The regularity of weak solutions is in fact not enough to perform the natural integration by parts and energy estimates; the energy inequality can proved by lower semi-continuity results along approximating sequences, see for instance [23] for a discussion on this classical problem. Step 2: Proof of the local energy inequality In order to conclude the proof of Theorem 1.1 we need to prove that the Leray–Hopf weak solution constructed in Step 1 is suitable. According to Definition 2.2 this requires just to prove the local energy inequality. To this end, let us consider a smooth and periodic in the space variable function φ ≥ 0, vanishing for t = 0, T , and use as test function Ph (uΔt h φ) in the momentum equation in (3.13). We first consider the term involving the time derivative, which we handle as follows: T





∂t vhΔt , Ph (uΔt h φ)

T dt =

0



 Δt Δt ∂t vhΔt , Ph (uΔt h φ) − uh φ + uh φ dt

0

T =



 ∂t vhΔt , uΔt h φ dt +

0

T

 Δt  Δt ∂t vh , Ph (uΔt h φ) − uh φ dt =: I1 + I2 .

0

Concerning the term I1 we have that T (∂t vhΔt , (vhΔt − vhΔt + uΔt h ) φ) dt

I1 = 0

T

T (∂t vhΔt , vhΔt ) φ dt

= 0

Δt (∂t vhN , (uΔt h − vh ) φ) dt

+ 0

=: I11 + I12 . Let us first consider I11 . By splitting the integral over [0, T ] as the sum of integrals over [tm−1 , tm ] and, by integrating by parts, we get T (∂t vhΔt , vhΔt φ) dt

=

tm N  m=1t m−1

0

(∂t vhΔt , vhΔt φ) dt

 tm  N  1 ∂t |vhΔt |2 , φ dt = 2 m=1 tm−1

 tm  N N   m−1 2   1  m2 1 Δt 2 |vh | , ∂t φ dt, = |uh | , φ(tm , x) − |uh | , φ(tm−1 , x) − 2 m=1 2 m=1 tm−1

m−1 um −uh , Δt

where we used that ∂t vhΔt (t) = h compact support in (0, T ) we get

for t ∈ [tm−1 , tm [. Next, since the sum telescopes and φ is with

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T

T (∂t vhΔt , vhΔt φ) dt

  1 Δt 2 |v | , ∂t φ dt. 2 h

=−

0

0

By the strong convergence of vhΔt → u in L2 (0, T ; L2# ) we can conclude that T

T (∂t vhΔt , vhΔt φ) dt

lim

(Δt,h)→(0,0)

=−

0

1 2  |u| , ∂t φ dt. 2

0

Then, we consider the term I12 . Since uΔt h is constant on the interval [tm−1 , tm [ we can write T

tm N   Δt Δt    Δt Δt Δt ∂t vh , (uh − vh φ) dt = − ∂t (vhΔt − uΔt h ), (vh − uh ) φ dt m=1t m−1

0

=−

tm  N  m=1t m−1

 2 |vhΔt − uΔt h | , φ dt ∂t 2

 tm  Δt N 2  |vh − uΔt h | = , ∂t φ dt 2 m=1 tm−1

−

N  m=1





2 |vhΔt (tm ) − uΔt h (tm )| φ(tm ) 2

 −



2 |vhΔt (tm−1 ) − uΔt h (tm−1 )| , φ(tm−1 ) 2

 tm  Δt N 2  |vh − uΔt h | , ∂t φ dt, = 2 m=1 tm−1

where in the last line we have used the fact we do not have boundary terms at t = 0 and t = tN since φ has Δt 2 2 compact support in (0, T ) and the sum telescopes. Then, since uΔt h −vh vanishes (strongly) in L (0, T ; L# ), we get that I12 → 0 as (Δt, h) → (0, 0). We have that the term I2 → 0 as (Δt, h) → (0, 0). Indeed, by the discrete commutator property (3.4), Proposition 4.4, and the inverse inequality (3.3) we can infer that  T         Δt  Δt Δt I2  =  ∂t vh , Ph (uh φ) − uh φ dt    0

T ≤

Δt ∂t vhΔt H −1 Ph (uΔt h φ) − uh φH 1 dt 0

≤ c h∂t vhΔt 

4

L 3 (H −1 )

uΔt h L4 (H 1 ) 1

1

≤ c h 2 ∂t vhΔt 

4

L 3 (H −1 )

1

1

Δt 2 2 2 uΔt h L∞ (L2 ) uh L2 (H 1 ) ≤ c h .

Hence, also this term vanishes as h → 0 and this concludes the considerations for the term involving the time-derivative.

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Concerning the viscous term, by adding and subtracting ∇uΔt h φ we obtain the following three terms: Δt Δt Δt Δt Δt Δt (∇uΔt h , ∇Ph (uh φ)) = (∇uh , ∇(uh φ)) + (∇uh , ∇(Ph (uh φ) − uh φ))   1 Δt 2 2 |u | , Δφ + Rvisc , = (|∇uΔt h | , φ) − 2 h

where the “viscous remainder” Rvisc is defined as follows   Δt Δt Rvisc := ∇uΔt h , ∇[Ph (uh φ) − uh φ] . 2 1 2 2 Since uΔt h converges to u weakly in L (0, T ; H# ) and strongly in L (0, T ; L# ), by integrating over (0, T ) we can infer the following two results:

T

T 2 (|∇uΔt h | , φ) dt

lim inf

(Δt,h)→(0,0)

≥

0

T lim

(Δt,h)→(0,0)

(|∇u|2 , φ) dt, 0

1 ( |uΔt |2 , Δφ) dt = 2 h

0

T

1 ( |u|2 , Δφ) dt, 2

0

where in the first inequality we used that φ is non-negative. For the remainder Rvisc , by using again the discrete commutator property from Definition 3.1, we have that  T    T   2  Rvisc dt ≤ c h ∇uΔt h  dt → 0     0

as (Δt, h) → (0, 0).

0

We consider now the nonlinear term. We have Δt Δt Δt Δt N Δt Δt Δt Δt bh (uΔt h , uh , Ph (uh φ)) = bh (uh , uh , uh φ) + bh (uh , uh , Ph (uh φ) − uh φ) Δt Δt = bh (uΔt h , uh , uh φ) + Rnl .

(5.9)

Δt Δt Δt The “nonlinear remainder” Rnl := bh (uΔt h , uh , Ph (uh φ) − uh φ) can be estimated by using (3.9), the discrete commutator property, and (3.3). Indeed, we have Δt Δt Δt |Rnl | ≤ nlh (uΔt h , uh )H −1 Ph (uh φ) − uh φH 1 Δt Δt ≤ c huΔt h 3 uh H 1 uh H 1 1/2

1/2

Δt Δt 2 ≤ c huΔt h 2 uh H 1 uh H 1 1

Δt 2 ≤ c h 2 uΔt h 2 uh H 1 ,

hence, by integrating in time T Rnl dt → 0

as (Δt, h) → (0, 0).

0

The definition of nlh in (3.8) allows us to handle the first term on the right hand side in (5.9) as follows

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 Δt  1  Δt  Δt Δt Δt Δt Δt u div uΔt bh (uΔt h , uh , uh φ) = (uh · ∇) uh , uh φ + h , uh φ 2 h   1 Δt 2 1 Δt 2 Δt Δt = (uh · ∇) |uh | + |uh | div uh , φ 2 2       1 Δt 2 Δt 1 Δt 2 |u |u | | , ∇φ . , φ = − u = div uΔt h h 2 h 2 h Then, for 4 < s ≤ 6 and s∗ as in (5.8) it follows that 1 1 Δt 2 |u | → u |u|2 uΔt h 2 h 2

strongly in L

s∗ 3

as (Δt, h) → (0, 0),

(0, T ; L1 )

and one shows that T Δt Δt bh (uΔt h , uh , uh φ) dt 0

→−

T 

 1 u |u|2 , ∇φ dt 2

(Δt, h) → (0, 0).

as

0

The last term we consider is that involving the pressure. By integrating by parts we have Δt Δt Δt (pΔt h , div Ph (uh φ)) = (ph uh , ∇φ) + Rp1 + Rp2 ,

where the two remainders are defined as follows   Δt Δt Rp1 := pΔt h , div(Ph (uh φ) − uh φ)

and

  Δt Rp2 := φ pΔt h , div uh .

By (3.3) and by using again the discrete commutator property (3.5), we easily get Δt Δt |Rp1 | ≤ pΔt h 2 Ph (uh φ) − uh φH 1 Δt ≤ c hpΔt h 2 uh H 1 .

Then, by integrating in time and using (3.3) we have  T    1 1   Δt 2 Δt 2  Rp1 dt ≤ c h 12 pΔt h L 43 (L2 ) uh L2 (H 1 ) uh L∞ (L2 ) ,     0

which implies T Rp1 dt → 0

as (Δt, h) → (0, 0).

0

The term Rp2 can be treated in the same way but now using the discrete commutation property for the projector over Qh Δt Δt |Rp2 | ≤ cQh (pΔt h φ) − φph 2 uh φH 1 1

1

≤ c h 2 pΔt h  and finally this implies that

4

L 3 (L2 )

1

Δt 2 2 uΔt h L2 (H 1 ) uh L∞ (L2 ) ,

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T Rp2 dt → 0

as (Δt, h) → (0, 0).

0

It remains to prove that T

T (pΔt h

0

uΔt h , ∇φ)

→

(p u, ∇φ)

as (Δt, h) → (0, 0),

0

which is an easy consequence of (5.6), (5.7), and Proposition 4.4 b). In conclusion, collecting all terms we have finally proved the local energy inequality (2.3). 2 References [1] V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Commun. Math. Phys. 55 (2) (1977) 97–112. [2] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. 35 (6) (1982) 771–831. [3] J.L. Guermond, On the use of the notion of suitable weak solutions in CFD, Int. J. Numer. Methods Fluids 57 (9) (2008) 1153–1170. [4] J.L. Guermond, J.T. Oden, S. Prudhomme, Mathematical perspectives on large eddy simulation models for turbulent flows, J. Math. Fluid Mech. 6 (2) (2004) 194–248. [5] H. Beirao da Veiga, On the suitable weak solutions to the Navier–Stokes equations in the whole space, J. Math. Pures Appl. 64 (1) (1985) 77–86. [6] H. Beirao da Veiga, On the construction of suitable weak solutions to the Navier–Stokes equations via a general approximation theorem, J. Math. Pures Appl. 64 (3) (1985) 321–334. [7] H. Beirao da Veiga, Local energy inequality and singular set of weak solutions of the boundary non-homogeneous Navier– Stokes problem, in: Y. Ohya, K. Kasahara, N. Shimakura (Eds.), Current Topics in Partial Differential Equations, Kinokuniya Company Ltd., Tokyo, 1986, Papers dedicated to Professor Sigeru Mizohata on the occasion of his sixtieth birthday. [8] J.L. Guermond, Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the threedimensional torus are suitable, J. Math. Pures Appl. 85 (3) (2006) 451–464. [9] J.L. Guermond, Faedo–Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable, J. Math. Pures Appl. 88 (1) (2007) 87–106. [10] L.C. Berselli, S. Spirito, On the construction of suitable weak solutions to the 3d Navier–Stokes equations in a bounded domain by an artificial compressibility method, Commun. Contemp. Math. 20 (1) (2018) 1650064, 16 pp. [11] L.C. Berselli, S. Spirito, Suitable weak solutions to the 3d Navier–Stokes equations are constructed with the Voigt approximation, J. Differ. Equ. 262 (5) (2017) 3285–3316. [12] D. Donatelli, S. Spirito, Weak solutions of Navier–Stokes equations constructed by artificial compressibility method are suitable, J. Hyperbolic Differ. Equ. 8 (1) (2011) 101–113. [13] A. Biryuk, W. Craig, S. Ibrahim, Construction of suitable weak solutions of the Navier–Stokes equations, in: Stochastic Analysis and Partial Differential Equations, in: Contemp. Math., vol. 429, Amer. Math. Soc., 2007, pp. 1–18. [14] L.C. Berselli, S. Spirito, Weak solutions to the Navier–Stokes equations constructed by semi-discretization are suitable, in: Recent Advances in Partial Differential Equations and Applications, in: Contemp. Math., vol. 666, Amer. Math. Soc., 2016, pp. 85–97. [15] L.C. Berselli, Weak solution constructed by semi-discretization are suitable: the case of slip boundary conditions, Int. J. Numer. Anal. Model. 15 (4–5) (2018) 479–491. [16] L.C. Berselli, S. Spirito, On the vanishing viscosity limit for the Navier–Stokes equations under slip boundary conditions in general domains, Commun. Math. Phys. 316 (1) (2012) 171–198. [17] L.C. Berselli, S. Spirito, An elementary approach to inviscid limits for the 3d Navier–Stokes equations with slip boundary conditions and applications to the 3d Boussinesq equations, Nonlinear Differ. Equ. Appl. 21 (2) (2014) 149–166. [18] N. Gigli, S.J.N. Mosconi, A variational approach to the Navier–Stokes equations, Bull. Sci. Math. 136 (3) (2012) 256–276. [19] H. Sohr, W. von Wahl, On the regularity of the pressure of weak solutions of Navier–Stokes equations, Arch. Math. (Basel) 46 (5) (1986) 428–439. [20] P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, Lecture Series in Mathematics and Its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. [21] A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. [22] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. [23] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and Its Applications, vol. 2, North-Holland Publishing Co., Amsterdam–New York–Oxford, 1977.

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[24] A.F. Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations, Nonlinear Differ. Equ. Appl. 14 (5–6) (2007) 753–785. [25] J.H. Bramble, J.E. Pasciak, O. Steinbach, On the stability of the L2 -projection in H 1 (Ω), Math. Comput. 72 (237) (2002) 147–156.