On a critical Leray-α model of turbulence

Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

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On a critical Leray-α model of turbulence Hani Ali ∗ IRMAR, UMR 6625, Université Rennes 1, Campus Beaulieu, 35042 Rennes cedex, France

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Article history: Received 1 September 2011 Accepted 19 October 2012 Keywords: Navier–Stokes equations Turbulence model Existence Weak solution MHD

abstract This paper aims to study a family of Leray-α models with periodic boundary conditions. These models are good approximations for the Navier–Stokes equations. We focus our attention on the critical value of regularization ‘‘θ ’’ that guarantees the global well-posedness for these models. We conjecture that θ = 41 is the critical value to obtain such results. When alpha goes to zero, we prove that the Leray-α solution, with critical regularization, gives rise to a suitable solution to the Navier–Stokes equations. We also introduce an interpolating deconvolution operator that depends on ‘‘θ ’’. Then we extend our results of existence, uniqueness and convergence to a family of regularized magnetohydrodynamics equations. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we study a Leray-α approximation for the Navier–Stokes equations subject to space-periodic boundary conditions. The Leray-α equations we are considering are the following:

 ∂u  + u · ∇ u − ν ∆u + ∇ p = f in R+ × T3 ,   ∂ t    u = α 2θ (−∆)θ u + u, in T3  ∇ · u = ∇ · u = 0, u= u = 0,    T3 T3    u(t , x + Lej ) = u(t , x), ut =0 = u0 ,   where L > 0 and ej , j = 1, 2 or 3 is the canonical basis of R3 .

(1.1)

We consider these equations on the three dimensional torus T3 = R3 /T3 where T3 = 2π Z3 /L, x ∈ T3 , and t ∈ [0, +∞[. The unknowns are the velocity vector field u and the scalar pressure p. The viscosity ν , the initial velocity vector field u0 and the external force f with ∇ · f = 0 are given. Let θ be a non-negative parameter. The nonlocal operator (−∆)θ is defined through the Fourier transform



 (−∆ )θ u(k ) = |k |2θ u(k ).



(1.2)

The fractional order Laplace operator has been used in another α -model of turbulence by Olson and Titi [1]. Motivated by their work [1], we study here the case where θ = 41 in the system (1.1). We note that the Leray-α family considered here is a particular case of the general study carried in [2] where the results do not recover the critical case

θ = 14 . ∗

Tel.: +33 6 11781229. E-mail addresses: [email protected], [email protected].

1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2012.10.019

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

The regularization critical values of the other α -models studied in [2] will be reported in a forthcoming paper. Therefore, the initial value problem we consider in particular is:

 ∂u  + u · ∇ u − ν ∆u + ∇ p = f in R+ × T3 ,   ∂t     u = α 12 (−∆) 41 u + u, in T3      u = u = 0 , u = 0, ∇ · u = ∇ ·    T3 T3   ut =0 = u0 , which is equivalent to (1.1) with θ =

1 . 4

(1.3)

One of the main results of this paper is to establish the global well-posedness of

the solution to Eqs. (1.1) in L (T3 ) for θ = 41 . We conjecture that θ = 14 is the critical value needed to get existence and uniqueness of weak solutions for all times to Eqs. (1.1) with L2 initial data. Therefore, smooth solutions of Eqs. (1.1) with θ ≥ 41 do not develop finite-time singularities. 2

When θ <

1 , 4

3

the uniqueness of weak solutions to Eqs. (1.1) with L2 initial data remains an open problem. The Hausdorff

dimension of the time singular sets for (1.1) with θ < 41 has been treated in [3]. When α = 0, Eqs. (1.1) are reduced to the usual Navier–Stokes equations for incompressible fluids. The problem of the global existence and uniqueness of the solutions of the three-dimensional Navier–Stokes equations is among the most 1

challenging problems of contemporary mathematics. We recall that H 2 (T3 )3 is a scale-invariant space for Navier–Stokes 1

equations [4]. Another result of this paper is to establish the global well-posedness of the solution to Eqs. (1.1) in H 2 (T3 )3 for θ = 14 without smallness conditions on the initial data.

We also discuss the relation between Leray-α equations with θ ≥ 41 and NSE. The first result of an α -model convergence to the Navier–Stokes equations is proved by Foias et al. [5] where the authors show the existence of a subsequence of weak solutions of the α model that converges to a Leray–Hopf weak solution of NSE. We improve their result [5] by proving an Lp convergence property in order to show the convergence of the weak solution to a suitable weak solution to the NSE. Moreover, we extend the theory to other models where we consider a deconvolution type regularization to the NSE. The deconvolution operator was introduced by Layton and Lewandowski [6]. We will define below the modified deconvolution operator in order to regularize the NSE in Section 7.1 and the MHD equations in Section 8. This modified deconvolution operator interpolates the usual deconvolution operator introduced in [6] and is given by DN =

N 

1 i (I − G− θ ),

for 0 ≤ θ ≤ 1,

(1.4)

i=0

where Gθ := I +α 2θ (−∆)θ denotes the Helmholtz operator with fractional regularization. The deconvolution regularization to the NSE that we consider consists in replacing the nonlinear term in the Navier–Stokes equations (u · ∇)u by (HN (u) · ∇)u where HN (u) = DN (u). As the Leray-α regularization, we will first study the existence and the uniqueness of the deconvolution Navier–Stokes regularization (7.1) with θ = 14 . Then, we will show an Lp convergence property to the deconvolution operator in order to deduce that the deconvolution regularization gives rise to suitable weak solutions. Moreover, we will generalize our results of existence, uniqueness and convergence to the deconvolution magnetohydrodynamics regularization (8.1). The critical value of regularization θ = 14 holds for MHD and Navier–Stokes equations. Therefore, it is natural to ask if the singularities in MHD equations are similar to the singularities in the Navier–Stokes equations. A partial answer is given in [7], where an analog of the known Caffarelli et al. result [8] is established to the MHD equations. The extension of the partial regularity result of Scheffer [9] to the MHD equation is studied in [10]. The paper is organized as follows. In Section 2, we recall some preliminary results used later in the proofs. In Section 3, we present the main results (Theorems 3.1 and 3.2 below). Sections 4 and 5 are devoted to the proof of these two results where we use the standard Galerkin approximation. In Section 6, we show an Lp convergence result in order to prove that the Leray-α equations give rise to a suitable solution to the Navier–Stokes equations. In Section 7, we introduce the interpolating deconvolution operator and we give applications to the Navier–Stokes equations. Finally, we study a deconvolution regularization to the MHD equations. 2. Functional setting and preliminaries In this section, we introduce some preliminary material and notations which are commonly used in the mathematical study of fluids, in particular in the study of the Navier–Stokes equations (NSE). For a more detailed discussion of these topics, we refer the reader to [11–15]. We denote by Lp (T3 )3 and H s (T3 )3 the usual Lebesgue and Sobolev spaces over T3 . For the L2 norm and the inner product, we write ∥·∥L2 and (·, ·)L2 . The norm in H 1 (T3 )3 is written as ∥·∥H 1 and its scalar product as (·, ·)H 1 . We denote by (H 1 (T3 )3 )′

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

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the dual space of H 1 (T3 )3 . Note that we have the continuous embeddings H 1 (T3 )3 ↩→ L2 (T3 )3 ↩→ (H 1 (T3 )3 )′ .

(2.1)

Moreover, by the Rellich–Kondrachov compactness theorem (see, e.g., [16]), these embeddings are compact. Since we work with periodic boundary conditions, we can characterize the divergence free spaces by using the Fourier series on the 3D torus T3 . We expand the velocity in Fourier series as:



u(t , x) =

ˆ k exp {ik · x} , u

k ∈I

  where I = k such that k = 2πL a , a ∈ Z 3 , a ̸= 0 , where for the vanishing space average case, we have the condition 

u = 0.

ˆ0 = u T3

Since the vector u(t , x) is real-valued, we have for every k ,

ˆ −k =  u u∗k ∗

ˆk. where uk denotes the complex conjugate of u In the Fourier space, the divergence-free condition is for all k .

ˆk · k = 0 u

For s ∈ R, the usual Sobolev spaces H s (T3 )3 with zero space average can be represented as

 Hs =

 u=



ˆ k exp {ik · x} , uˆ −k =  u u∗k , ∥u∥2H s < ∞ ,

k ∈I

where



∥u∥2H s =

|k |2s |uˆ k |2 .

k ∈I

Let

ˆ k · k = 0}, V s = {u ∈ H s , u ′

we identify the continuous dual space of V s as V −s with the pairing given by

(u, v )V s =



|k |2s uˆ k · vˆ −k ,

k ∈I

and the action of V −s on V s is denoted by ⟨·, ·⟩V −s ,V s . We denote V 0 by H and V 1 by V , where the norm ∥ · ∥2H ≡ ∥ · ∥2L2 and the scalar product as (·, ·)H ≡ (·, ·)L2 . The norm

in V is written as ∥ · ∥2V ≡ ∥ · ∥2H 1 and its scalar product as (·, ·)H 1 . We denote by Pσ the Leray–Helmholtz projection operator and define the Stokes operator A := −Pσ △ with domain D (A) := H 2 (T3 )3 ∩ V . For u ∈ D (A), we have the norm equivalence ∥Au∥L2 ∼ = ∥u∥V 2 . Furthermore, in our case it is known that A = −△ due to the periodic boundary conditions (see, e.g., [11,13]). It is natural to define the powers As of the Stokes operator in the periodic case as



As u =

|k |2s uˆ k exp {ik · x} .

k ∈I s/2

For u ∈ D (A ), we have the norm equivalence ∥As/2 u∥L2 ∼ = ∥u∥V s . In particular for s = 1 we recover the space V , hence D (A1/2 ) = V , (see, e.g., [13]). Note also that H s+ϵ is compactly embedded in H s (resp. V s+ϵ is compactly embedded in V s ) for any ϵ > 0, and we have the following Sobolev embedding theorem (see [17]). Theorem 2.1. The space H 1/2 is embedded in L3 (T3 )3 and the space L3/2 (T3 )3 is embedded in H −1/2 . The following result deals with some interpolation inequalities [18]. Lemma 2.1. Let T > 0, 1 ≤ p1 < p < p2 ≤ ∞, s1 < s < s2 and η ∈]0, 1[ such that 1 p Let u ∈

= 2

η p1

i=1

+

1−η p2

and

s = ηs1 + (1 − η)s2 .

Lpi ([0, T ], H si ), then u ∈ Lp ([0, T ], H s ) and η

1−η

∥u∥Lp ([0,T ],H s ) ≤ C ∥u∥Lp1 ([0,T ],H s1 ) ∥u∥Lp2 ([0,T ],H s2 ) . The result holds true when we work with the spaces Lpi ([0, T ], V si ).

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

The regularization effect of the nonlocal operator involved in the relation between u and u is described by the following lemma. Lemma 2.2. Let θ ∈ R+ , 0 ≤ β ≤ 2θ , s ∈ R and assume that u ∈ H s . Then u ∈ H s+β and

∥u∥rH s+β ≤

1

∥u∥rH s .

αr β

(2.2)

In particular, when β = 2θ we get u ∈ H s+2θ and

∥u∥H s+2θ ≤ Proof. When u = u=

1

α 2θ 

∥u∥H s .

k ∈I

ˆ k exp {ik · x}, then u

ˆk u



(2.3)

1 + α 2θ |k |2θ k ∈I

exp {ik · x} .

(2.4)

Formula (2.4) easily yields the estimate

 2r |k |2β ∥u∥rH s 2θ 2θ 2 k ∈I (1 + α |k | )  2  2r  α β |k |β 1 sup ∥u∥rH s ≤ α 2β k ∈I 1 + α 2θ |k |2θ

∥u∥rH s+β ≤

≤



sup

1

αr β

∥u∥rH s ,

(2.5)

where we have used the fact that

 sup k ∈I

α β |k |β 1 + α 2θ |k |2θ

Remark 2.1. For θ =

1 , 4

2

≤ 1,

for all 0 ≤ β ≤ 2θ .

1

u ∈ H s+ 2 and ∥u∥

(2.6)



1

H

s+ 1 2

≤ α − 2 ∥u∥H s .

Following the notations of the Navier–Stokes equations, we set B(u, v ) := Pσ (u · ∇ v )

(2.7)

for any u, v in V . We list several important properties of B which can be found for example in [13]. Lemma 2.3. The operator B defined in (2.7) is a bilinear form which can be extended as a continuous map B : V × V → V ′ . For u, v, w ∈ V ,

⟨B(u, v ), w ⟩V −1 = − ⟨B(u, w ), v ⟩V −1 ,

(2.8)

and therefore

⟨B(u, v ), v ⟩V −1 = 0.

(2.9)

The following result holds true. Lemma 2.4. The bilinear form B defined in (2.7) satisfies the following. 1

(i) Assume that u ∈ V 2 , v and w ∈ V . Then the following inequality holds

|(B(u, v ), w )| ≤ C ∥u∥

1

V2

∥v ∥V ∥w ∥V .

(2.10) 1

3

1

1

3

(ii) B can be extended as a continuous map B : V 2 × V 2 → V − 2 . In particular, for every u ∈ V 2 , v ∈ V 2 the bilinear form B satisfies the following inequalities:

∥B(u, v )∥

V

1 −2

≤ C ∥u ∥

1

V2

∥v ∥

3

V2

.

(2.11)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

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Proof. (i) We have

    |(B(u, v ), w )| ≤  u ⊗ v : ∇ w  .

(2.12)

T3

By the Hölder inequality combined with the Sobolev embedding theorem we get

|(B(u, v ), w )| ≤ ∥u ⊗ v ∥L2 ∥w ∥H 1 ≤ ∥u∥L3 ∥v ∥L6 ∥w ∥H 1 ≤ C ∥u∥

1

V2

∥v ∥V ∥w ∥V .

(2.13)

(ii) We have

∥B(u, v )∥

V

1 −2

≤ c ∥u∥L3 ∥∇ v ∥L3 ≤ c ∥u ∥

1

V2

∥v ∥

3

V2

,

(2.14)

where we have used the Hölder inequality and the Sobolev injection given in Theorem 2.1. The following result is given for θ = Lemma 2.5. Let θ = L2 ([0, T ]; V −1 ).

1 . 4

1 4



and for any u ∈ L ([0, T ]; H ) ∩ L ([0, T ]; V ). ∞

2

1

3

If u ∈ L∞ ([0, T ]; H ) ∩ L2 ([0, T ]; V ), then u ∈ L∞ ([0, T ]; V 2 ) ∩ L2 ([0, T ]; V 2 ) and B(u, u) ∈

Proof. The regularity on u is obtained from Lemma 2.2. Then we use Lemma 2.4 to obtain

∥(B(u, u))∥V −1 ≤ C ∥u∥

1

V2

∥u ∥V .

Now,

∥u ∥

1

V2

∈ L∞ (0, T ) and ∥u∥V ∈ L2 (0, T ),

imply that ∥u∥

1

1 V2

∥u∥V ∈ L2 (0, T ) (that is L∞ ([0, T ]; V 2 ) ∩ L2 ([0, T ]; V ) ⊆ L2 ([0, T ]; H )).

We remove the pressure from further consideration by projecting (1.1) with θ = the space V s . Thus, we obtain the infinite dimensional evolution equation

1 4



by Pσ and searching for solutions in

 du   + ν Au + B(u, u) = f , dt

1

(2.15)

1

 u = α 2 A 4 u + u, ut =0 = u0 . Note that the pressure may be reconstructed from u and u by solving the elliptic equation

∆p = ∇ · (u · ∇ u). Later in Sections 4 and 5, we will prove that Eqs. (1.1) with θ = 41 have unique regular solution (u, p) such that the velocity part u solves Eqs. (2.15). We conclude this section with the interpolation lemma of Lions [19] and a compactness theorem; see [20]. Lemma 2.6. Let s > 0 and suppose that u ∈ L2 ([0, T ]; V s )

and

du dt

∈ L2 ([0, T ], V −s ).

Then (i) u ∈ C ([0, T ]; H ), with



   du   sup ∥u(t )∥H ≤ c ∥u∥L2 ([0,T ];V s ) +   dt  2 t ∈[0,T ]



L ([0,T ],V −s )

(ii)

and   d ∥u∥2H = 2 du , u V −s ,V s . dt dt

To set more regularity for the unique solution, we also need the following similar result.

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

Corollary 2.1. Assume that for some k ≥ 0 and s > 0, u ∈ L2 ([0, T ]; V k+s ) and

du dt

∈ L2 ([0, T ], V k−s ).

Then (i) u ∈ C ([0, T ]; V k ), and (ii)

d dt

  ∥Ak/2 u∥2H = 2 Ak/2 du , Ak/2 u V −s ,V s . dt

The result holds true in the space H s . Theorem 2.2. Let X b H ⊂ Y be Banach spaces, where X is reflexive. Suppose that {un }n∈N is a sequence such that it is uniformly n bounded in L2 ([0, T ]; X ), and { du } is uniformly bounded in L2 ([0, T ], Y ). Then there is a subsequence which converges dt n∈N strongly in L2 ([0, T ], H ). The above theorem is a special case of the following more general result [20]. Theorem 2.3 (Aubin–Lions). Let X b H ⊂ Y be Banach spaces, where X is reflexive. Suppose that {un }n∈N is a sequence such n } is uniformly bounded in Lq ([0, T ], Y ), (1 ≤ q ≤ ∞). that it is uniformly bounded in Lp ([0, T ]; X ), (1 < p < ∞), and { du dt n∈N Then {un }n∈N is relatively compact in Lp ([0, T ], H ). 3. Main results In the following, we will assume that α > 0, T > 0. One of the aims of this paper is to establish the global well-posedness of the solution to Eqs. (1.1) in L2 (T3 )3 for θ = It must be mentioned that the result holds true for θ ≥

1 . 4

Theorem 3.1. For any T > 0, let f ∈ L2 ([0, T ], V −1 ) and u0 ∈ H . Assume that θ =

1 . 4

solution (u, p) := (uα , pα ) to Eqs. (1.1) that satisfies u ∈ C ([0, T ]; H ) ∩ L ([0, T ]; V ) and p ∈ L2 ([0, T ], L2 (T3 )) such that u verifies 2



du dt

+ ν Au + B(u, u) − f , φ

1 . 4

Then there exists a unique du dt

∈ L2 ([0, T ]; V −1 ) and

 =0 V −1 ,V 1

for every φ ∈ V and almost every t ∈ (0, T ). Moreover, this solution depends continuously on the initial data u0 in the L2 norm. Therefore, smooth solutions of Eqs. (1.1) with θ ≥

1 4

do not develop finite-time singularities. 1

We also prove the global well-posedness of the solution to Eqs. (1.1) in H 2 (T3 )3 for θ = conditions on the initial data. 1

1 4

and without smallness

1

Theorem 3.2. For any T > 0, let f ∈ L2 ([0, T ], V − 2 ) and u0 ∈ V 2 . Let (u, p) be the unique solution to problem (1.1) with 1 2

3 2

θ = 41 , then u ∈ C ([0, T ]; V ) ∩ L2 ([0, T ]; V ),

du dt

∈ L2 ([0, T ], V

− 12

1

) and p ∈ L2 ([0, T ], H 2 (T3 )).

We note that the above result holds true for any s > 0 and without smallness conditions on the initial data. 4. Proof of Theorem 3.1 The proof is divided into five steps. 4.1. Galerkin approximation Let us define the finite dimensional space Hm ≡ span {exp {ik · x} : |k | ≤ m} . We look at the projection of the equations over Hm . In order to use classical tools for systems of ordinary differential equations we need that f belongs to C ([0, T ], V −1 ). To do so, we extend f outside [0, T ] by zero and we set fϵ = ρϵ ∗ f  where ρϵ (t ) = 1ϵ ρ( ϵt ), 0 ≤ ρ(s) ≤ 1, ρ(s) = 0 for |s| ≥ 1, and R ρ = 1. So the approximate sequence {fϵ }ϵ>0 is very smooth with respect to time for ρ smooth and converges to f in the sense that fϵ → f

strongly in L2 ([0, T ]; V−1 ) when ϵ → 0.

(4.1)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

The Galerkin approximation of Eqs. (2.15) with θ =

1 4

1569

is given by

 dum  + ν Pm Aum + Pm B(Pm um , Pm um ) = Pm f 1 ,  dt

m

1

(4.2)

1

m m m  u m = α 2 A 4 u + u , u (0) = Pm u0

where for some m ∈ N and all −1 ≤ s ≤ 2, Pm (w) ≡ ˆ k exp {ik · x} : V s → Hm is the orthogonal projector onto |k |≤m w the first m Fourier modes that verifies (see in [21] for more details):



∥Pm ∥L(V s ,V s ) ≤ 1,

v ∈ V s : Pm v → v strongly in V s when m → ∞.

and for all

(4.3) 1

m

 The classical theory of ordinary differential equations implies that Eqs. (4.2) have a unique C solution u = ˆ k (t ) exp {ik · x} for a given time interval that, a priori, depends on m, where for all v ∈ Hm and for all t ∈ [0, Tm ] |k |≤m u ˆ k (t ) solve the following system the coefficients u    m     du , v + ν ∇ um : ∇ vdx + (um · ∇)um vdx = f 1 , v , m dt (4.4) T3 T3  m u (0) = Pm u0 . Our goal is to show that the solution remains finite for all positive times, which implies that Tm = ∞. In the next subsection, we find uniform energy estimates for this solution with respect to m. 4.2. Energy estimates in H We follow here a similar method to the one used for the Navier–Stokes equations (see [15]). Lemma 4.1. Let f ∈ L2 ([0, T ], V −1 ) and u0 ∈ H , there exist K1 (T ) and K2 (T ) independent of m such that the solution um to the Galerkin truncation (4.2) satisfies

∥um ∥2L2 ([0,T ],V ) ≤ K1 (T ),

for all T ≥ 0, where K1 (T ) =

1

ν



∥u0 ∥2H +

1

ν

∥f ∥2L2 ([0,T ],V −1 )

 (4.5)

and

∥um ∥2L∞ ([0,T ],H ) ≤ K2 (T ),

for all T ≥ 0, where K2 (T ) = ν(K1 (T )).

(4.6)

Proof. Taking the L2 -inner product of the first equation of (4.2) with u and integrating by parts, using (2.9), the incompressibility of the velocity field and the duality relation we obtain 1 d 2 dt

∥um ∥2H + ν∥∇ um ∥2H =



Pm f 1 · um dx ≤ ∥f ∥V −1 ∥um ∥V 1 . T3

(4.7)

m

Using the Young inequality we get d dt

∥um ∥2H + ν∥∇ um ∥2H ≤

1

ν

∥f ∥2V −1 .

(4.8)

In particular, this leads to the estimate sup ∥um ∥2H ≤ ∥u0 ∥2H +

t ∈[0,Tm )

1

ν

∥f ∥2L2 ([0,T ],V −1 ) .

(4.9)

This implies that Tm = T . Indeed, consider [0, Tmmax ) the maximal interval of existence. Either Tmmax = T and we are done, or Tmmax < T and we have lim supt →(Tmmax )− ∥um (t )∥2H = ∞ a contradiction to (4.9). Hence we have global existence of um for f ∈ L2 ([0, ∞), V −1 ) and u0 ∈ H and hereafter we take an arbitrary interval [0, T ] and we assume that f ∈ L2 ([0, T ], V −1 ). Integrating (4.8) with respect to time gives the desired estimate (4.5) for all t ∈ [0, T ].  m

Now we use the regularization effect described in Lemma 2.2 to get the following estimates on u . Lemma 4.2. Under the notation of Lemma 4.1, we have

∥u m ∥2

3 L2 ([0,T ],V 2

  ≤ )

1

α

K1 (T ),

for all T ≥ 0,

(4.10)

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

and

 

m 2

∥u ∥

1

≤

1

α

L∞ ([0,T ],V 2 )

K2 (T ),

for all T ≥ 0.

(4.11) 1

3

By interpolation (see Lemma 2.1) between L2 ([0, T ], V 2 ) and L∞ ([0, T ], V 2 ), we can deduce the following result. Corollary 4.1. Under the notation of Lemma 4.1, we deduce

∥um ∥2L4 ([0,T ],V ) ≤

  1

α

K1 (T )1/2 K2 (T )1/2 ,

for all T ≥ 0.

(4.12)

Our next result provides an estimate on the time derivative of um . Lemma 4.3. Let f ∈ L2 ([0, T ], V −1 ) and u0 ∈ H , there exists K3 (T ) independent of m such that the time derivative of the solution um to the Galerkin truncation (4.2) satisfies

 m 2  du     dt  2

L ([0,T ],V −1 )

≤ K3 (T ),

(4.13)

where K3 (T ) = 4 ν 2 + C (1/α) K2 (T ) K1 (T ) + 2∥f ∥2L2 ([0,T ],V −1 ) .





Proof. Taking the H −1 norm of (4.2), one obtains:

 m  du     dt 

≤ ν∥um ∥V 1 + ∥B(um , um )∥V −1 + ∥f ∥V −1

(4.14)

V −1

where we note that

  ∥B(um , um )∥V −1 = sup (B(um , um ), w); w ∈ V 1 , ∥w∥V 1 ≤ 1 . It remains to show that ∥B(u , um )∥V −1 is bounded. Indeed, we have from Lemma 2.4 that m

∥B(um , um )∥V −1 ≤ C ∥um ∥

1

V2

∥um ∥V .

(4.15)

It follows that

 m 2  du  m 2 2 m 2 2    dt  −1 ≤ 4(ν + C ∥u ∥V 12 )∥u ∥V + 2∥f ∥V −1 .

(4.16)

V

m

Integrating with respect to time, and recalling that u Lemmas 4.1 and 4.2 that

1

∈ L∞ ([0, T ], V 2 ) and um ∈ L2 ([0, T ], V ), it follows from

 t  m 2  du  2 2    dt  −1 ≤ 4(ν + C (1/α)K2 (T ))K1 (T ) + 2∥f ∥L2 ([0,T ],V −1 ) .  0

(4.17)

V

4.3. Passing to the limit m → +∞ From Lemmas 4.1 and 4.3 and from the reflexivity of the appearing Banach spaces we can extract a subsequence of m

m

{um }m∈N and { dudt }m∈N such that um converges weakly to some u in L2 ([0, T ], V 1 ) and dudt converges weakly to some du in dt 2 −1 L ([0, T ], V ) respectively. Now the interpolation lemma of Lions and Magenes (Lemma 2.6) implies that u ∈ C ([0, T ]; H ). In order to show the convergence of um to u in C ([0, T ], H )∩ L2 ([0, T ], V ), we need to show that for u0 ∈ H , the sequence {um }m∈N is a Cauchy sequence in C ([0, T ], H ) ∩ L2 ([0, T ], V ) with respect to the norm ∥ · ∥L∞ ([0,T ],H ) ∩ ∥ · ∥L2 ([0,T ],V ) . We know that um ∈ C ([0, T ], H ) ∩ L2 ([0, T ], V ) for m ∈ N. The difference um+n − um , for m, n ∈ N, satisfies d(um+n − um ) dt

= Pm+n f

1 m+n

+ Pm B((um+n − um ), um ) + Pm B(um+n , (um+n − um )) + ν A(um+n − um ) − Pm f 1 . m

(4.18)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

1571

By taking um+n − um as a test function in (4.18) we get 1 d 2 dt

∥um+n − um ∥2H + ν∥um+n − um ∥2V ≤ |(B((um+n − um ), um ), um+n − um )|   + Pm+n f 1 − Pm f 1 , um+n − um −1 m+n

.

(4.19)

∥um+n − um ∥2V .

(4.20)

m

V

,V 1

Lemma 2.4 combined with the Young inequality gives

|(B((um+n − um ), um ), um+n − um )| ≤

1

ν

∥um+n − um ∥2V 1 ∥um ∥2V + 2

ν 4

The duality norm combined with the Young inequality gives



Pm+n f

1 m+n

− Pm f 1 , um+n − um

 V −1 ,V 1

m

≤

1

ν

 Pm+n f

1 m+n

2 ν  − Pm f 1  −1 + ∥um+n − um ∥2V . m

(4.21)

4

V

Thus we get d

∥um+n − um ∥2H + ν∥um+n − um ∥2V ≤ C α −1 ν −1 ∥um+n − um ∥2H ∥um ∥2V +

dt By Grönwall’s inequality we get

∥u

m+n



um 2H

m +n

∥ ≤ ∥u

−

(0) − u (0)∥ + m

2 H

1

T



ν

0

  Pm+n f

1 m+n

1

ν

 Pm+n f

 2   − Pm f 1  −1 exp m

V

2  − Pm f 1  −1 .

(4.22)

α −1 ν −1 ∥um ∥2V dt .

(4.23)

1 m+n

m

V

T 0

We know that um ∈ L2 ([0, T ], V ), thus there exists C (α, ν) ≥ 0 such that T



α −1 ν −1 ∥um ∥2 dt ≤ C (α, ν).

exp 0

We observe that T

 0

  Pm+n f

1 m+n

 T  T 2 2 2      − Pm f 1  −1 ≤ Pm+n f 1 − Pm f 1  −1 + Pm f 1 − Pm f 1  −1 . m V m+n m+n V m+n m V 0   0   I

(4.24)

II

For I we have

  Pm+n f

1 m+n

− Pm f

1 m+n

2   −1 → 0 V

∈ L ([0, T ], V −1 ) and m+n  2     Pm+n f 1 − Pm f 1  −1 ≤ Pm+n f

because f

2

1

m+n

m+n

V

1 m+n

2   −1 ≤ ∥f ∥2V −1 ∈ L1 [0, T ]. V

Using the dominate convergence theorem, we conclude that I tends to zero when m tends to ∞. Similarly by using the dominate convergence theorem combined with the fact that

  Pm f

1 m+n

2  − Pm f 1  −1 → 0 m

V

and

  Pm f

1 m+n

2    − Pm f 1  −1 ≤ f m

V

1 m+n

2  − f 1  −1 ≤ ∥f ∥2V −1 ∈ L1 [0, T ], m

V

we obtain that II tends to zero when m tends to ∞. Since u0 ∈ H then ∥um+n (0) − um (0)∥2H converges to zero when m goes to ∞. By integrating (4.22), we deduce that m+n u − um tends to zero in C ([0, T ], H ) ∩ L2 ([0, T ], V ). This implies that {um }m∈N is a Cauchy sequence in C ([0, T ], H ) ∩ L2 ([0, T ], V ). Concerning the initial data, we can check that u(0) = u0 . Thanks to the result above we have um converges to u in C ([0, T ]; H ), in particular um (0) converges to u(0) in H . On the other hand, we have that um (0) = Pm u0 and um (0) converges to u0 in H . The unicity of the limit in H allows us to deduce the result. m It is obvious from Lemma 2.2 that u converges to u in C ([0, T ]; V 1/2 ). Moreover, T



(Au , φ)dt = − m

0

for all φ ∈ L2 ([0, T ], V 1 ).

T



1/2 m

(A 0

1/2

u ,A

φ)dt → −

T



(A 0

1/2

1/2

u, A

φ)dt =

 0

T

(Au, φ)dt ,

1572

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

Thus, Aum converges weakly to Au in L2 ([0, T ], V −1 ) as m → ∞. m We finish by showing that the non-linear term B(u , um ) converges weakly to B(u, u) in L2 ([0, T ], V −1 ) as m → ∞. From the properties of the trilinear form we have

 T    T    (B(um , um ), φ)dt − ≤ ( B ( u , u ), φ) dt   0

0

T

  (B(um − u, um ), φ) dt + 0

T



  (B(u, um − u), φ) dt , 0

and by using Lemma 2.4 combined with the Hölder inequality we get T



  (B(um − u, um ), φ) dt ≤ ∥um − u∥L∞ ([0,T ],V 1/2 ) ∥um ∥L2 ([0,T ],V 1 ) ∥φ∥L2 ([0,T ],V 1 ) ,

0

and T



  (B(u, um − u), φ) dt ≤ ∥u∥L∞ ([0,T ],V 1/2 ) ∥um − u∥L2 ([0,T ],V 1 ) ∥φ∥L2 ([0,T ],V 1 ) .

0

Thus B(u , um ) converges weakly to B(u, u) in L2 (([0, T ], V −1 )) as m → ∞. This implies that Pm B(u , um ) converges weakly to B(u, u) in L2 (([0, T ], V −1 )) as m → ∞. The convergence of Pm f 1 to f in L2 ([0, T ], V −1 ) is obvious because f ∈ L2 ([0, T ], V −1 ). We have shown that u satisfies m

m

m

(2.15) viewed as a functional equality in V −1 . 4.4. Uniqueness The solution constructed above is unique. Next, we will show the continuous dependence of the solutions on the initial data and in particular the uniqueness. Let u and v be any two solutions of (2.15) on the interval [0, T ], with initial values u0 ∈ H and v0 ∈ H , respectively. Let us denote by w = u − v and w = u − v. Then, we can write the evolution equation for w as an equality in V −1 given by d dt

w + ν Aw + B(w , u) + B(v , w ) = 0.

(4.25)

We take the inner product of (4.25) with w. Applying Lemma 2.6 of Lions–Magenes and by the properties of the trilinear form we get 1 d 2 dt

∥w ∥2H + ν∥∇ w ∥2H + (B(w , u), w ) = 0.

(4.26)

Using Lemma 2.4 combined with the Young inequality and Lemma 2.2, we obtain d dt

∥w ∥2H + ν∥∇ w ∥2H ≤

14

αν

∥w ∥2H ∥u∥2V .

(4.27)

Using Grönwall’s inequality we conclude that

∥w ∥2L∞ ([0,T ],H ) ≤ ∥w0 ∥2H exp

14

αν

K1 .

We have shown the continuous dependence of the solutions on the initial data in the L∞ ([0, T ], H ) norm. In particular, if w0 = 0 then w = 0 and the solutions are unique for all t ∈ [0, T ]. Remark 4.1. Since T > 0 is arbitrary, the solution above may be uniquely extended for all times. 4.5. About the pressure The pressure is absent in Eqs. (2.15) once we need to recover it by using the de Rham theorem [22]. To do so, we recall that for almost every t ∈ [0, T ]

F=

∂u + u · ∇ u − ν ∆u − f ∂t

is orthogonal to divergence-free vector fields in V 1 and the de Rham theorem applies. Thus, we deduce that for each Lebesgue point t of F, there is a scalar function p ∈ L2 (T3 ), such that F = ∇ p. This yields the following equation, satisfied in the sense of the distributions:

∂u + u · ∇ u − ν ∆u + ∇ p = f . ∂t

(4.28)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

1573

In order to check the regularity of p, we take the divergence of (4.28), this yields the following equation for the pressure

− ∆p = ∇ · (∇ · uu),

(4.29)

where uu is the tensor (u u )1≤i,j≤3 . We recall that for any divergence-free field u, we have i j

(u · ∇)u = ∇ · (uu). Since (u · ∇)u ∈ L2 ([0, T ], H −1 ). We conclude from the classical elliptic theory that p is bounded in L2 ([0, T ], L2 (T3 )). It remains to check the uniqueness of the pressure. We have the following lemma. Lemma 4.4 (Uniqueness of the Pressure). The pressure is uniquely determined by the velocity. Proof. Let (u, p) and (u, q) be two weak solutions of (1.1) (with θ ≥

1 ) 4

such that

+

 T3

p=

 T3

q = 0. Then for every φ ∈ H 1

and for almost every t ∈ R we have

(∇(p − q), φ) = 0, 



combined with the condition T p = T q = 0 gives p = q. 3 3 This concludes the proof of Theorem 3.1.  5. Regularity result In this section, we give the proof of Theorem 3.2. The existence and uniqueness of solutions is established in Theorem 3.1. We prove only the higher-order regularity. 1

5.1. Energy estimates in V 2 For w =



k ∈I

(v , w )

1

V2

ˆ k exp {ik · x}, we define |∇|w = w

1



k ∈I

ˆ k exp {ik · x}, thus the scalar product on V 2 is given by |k |w

= (v , |∇|w )H . 1

Let us write the energy inequality in the space V 2 . Taking |∇|um ∈ Hm as a test function in (4.2) it easily yields 1 d 2 dt

∥um ∥2 1 + ν∥∇ um ∥2 1 = (Pm B(um , um ), um ) V2

1

V2

V2

+ (Pm f 1 , um ) m

1

V2

.

(5.1)

The Sobolev embedding in Theorem 2.1 together with a Hölder estimate gives

  (B(um , um ), |∇|um )L2  ≤ c ∥B(um , um )∥V −1/2 ∥∇ um ∥V 1/2 ≤ c ∥B(um , um )∥L3/2 ∥∇ um ∥V 1/2 ≤ c ∥um ∥L6 ∥∇ um ∥L2 ∥∇ um ∥V 1/2 ≤ c ∥um ∥V ∥um ∥V ∥∇ um ∥V 1/2 .

(5.2)

By the interpolation inequality between V 1/2 and V 3/2 we get

  (B(um , um ), |∇|um )L2  ≤ ∥um ∥V ∥um ∥1/12/2 ∥∇ um ∥3/12/2 . V

(5.3)

V

Using the fact that

(Pm f 1 , um )V 1/2 ≤ ∥f ∥V −1/2 ∥∇ um ∥V 1/2

(5.4)

m

combined with the Young inequality, we infer d

∥um ∥2V 1/2 + ν∥∇ um ∥2V 1/2 ≤

9

2

∥f ∥2V −1/2 . dt 2ν ν The following lemmas are crucial to the proof of Theorem 3.2. 3

∥um ∥4V ∥um ∥2V 1/2 +

1

(5.5)

1

Lemma 5.1. Let f ∈ L2 ([0, T ], V − 2 ) and u0 ∈ V 2 , there exists M1 (T ) and M2 (T ) independent of m such that the solution um to the Galerkin truncation (4.2) satisfies

∥um ∥2L2 ([0,T ],V 3/2 ) ≤ M1 (T ),

for all T ≥ 0,

(5.6)

∥um ∥2L∞ ([0,T ],V 1/2 ) ≤ M2 (T ),

for all t ≥ 0.

(5.7)

and

1574

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

Proof. We proceed by two steps. First step. From (5.5) and by using Grönwall’s inequality we conclude that

∥um ∥2L∞ ([0,T ],V 1/2 ) ≤ M2 (T ),

(5.8)

where M2 (T ) is given by M2 (T ) =

  2 9 ∥u0 ∥2V 1/2 + ∥f ∥2L2 ([0,T ],V −1/2 ) exp 3 K1 (T )1/2 K2 (T )1/2 . ν 2ν α

Second step. Integrate the original inequality (5.5) on [0, T ]. We get

∥um (T , x)∥2V 1/2 + ν

T

 0

∥∇ um (t , x)∥2V 1/2 dt ≤ ∥u0 ∥2V 1/2 + +

9 2ν 3 α

2

ν

∥f ∥2L2 ([0,T ],V −1/2 )

M2 (T )K1 (T )1/2 K2 (T )1/2 .

(5.9)

We set M1 (T ) = ∥u0 ∥2V 1/2 +

2

ν

∥f ∥2L2 ([0,T ],V −1/2 ) +

9 2ν 3 α

M2 (T )K1 (T )1/2 K2 (T )1/2 .

Thus um ∈ L∞ ([0, T ], V 1/2 ) ∩ L2 ([0, T ], V 3/2 ) for all T > 0.



We deduce from Lemma 2.2 that

∥um ∥2L2 ([0,T ],V 2 ) ≤ (1/α)M1 (T ),

for all T ≥ 0

(5.10)

and

∥um ∥2L∞ ([0,T ],V 1 ) ≤ (1/α)M2 (T ),

for all T ≥ 0.

(5.11) m

Our next result provides an estimate on the time derivative of u . Lemma 5.2. Let f ∈ L2 ([0, T ], V −1/2 ) and u0 ∈ V 1/2 , there exists M3 (T ) independent of m such that the time derivative of the solution um to the Galerkin truncation (4.2) satisfies

 m 2  du     dt  2

L ([0,T ],V −1/2 )

≤ M3 (T ),

for all T ≥ 0,

(5.12)

where





M3 (T ) = 4(ν 2 + CM1 (T ))M2 (T ) + 2∥f ∥2L2 ([0,T ],V −1/2 ) .

(5.13)

Proof. Taking the V −1/2 norm of (4.2), one obtains:

 m  du     dt 

V −1/2

≤ ν∥um ∥V 1/2 + ∥B(um , um )∥V −1/2 + ∥f ∥V −1/2

(5.14)

where we note that

  ∥B(um , um )∥V −1/2 = sup (B(um , um ), w); w ∈ V 1/2 , ∥w∥1/2 ≤ 1 . It remains to show that ∥B(u , um )∥V −1/2 is bounded. Indeed, we have from Lemma 2.4 that m

∥B(um , um )∥V −1/2 ≤ C ∥um ∥V 1/2 ∥um ∥V 3/2 ≤ C ∥um ∥V 1/2 ∥um ∥V 3/2 .

(5.15)

It follows that

 m 2  du  2 m 2 m 2 2    dt  −1/2 ≤ 4(ν + C ∥u ∥V 3/2 )∥u ∥V 1/2 + 2∥f ∥V −1/2 .

(5.16)

V

Integrating with respect to time, it follows from Lemma 5.1 that T

 0

 m 2  du  2 2    dt  −1/2 ≤ 4(ν + CM1 (T ))M2 (T ) + 2∥f ∥L2 ([0,T ],V −1/2 ) V

yields (5.12) and the proof is complete.



(5.17)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

1575

5.2. Passing to the limit We are now ready to take limits along subsequences of {um }m∈N to show the higher-order regularity of the solution. From Lemmas 5.1 and 5.2 combined with the Alaoglu compactness theorem we can extract a subsequence of {um }m∈N and 3

m

1

m

{ dudt }m∈N such that um converges weakly to some v in L2 ([0, T ], V 2 ) and dudt converges weakly to some dv in L2 ([0, T ], V − 2 ) dt respectively.

1

Now the Lions–Magenes Lemma 2.6, implies that v ∈ C ([0, T ]; V 2 ) and um converges to v in C ([0, T ]; H ) and in 1

particular we have v (0) = v0 . Thus is obvious that u converges to v in C ([0, T ]; V 2 ). m

1

On the other hand, we already know that um converges to u in C ([0, T ]; H ). Therefore u = v ∈ C ([0, T ]; V 2 ) ∩ 3 2

L2 ([0, T ], V ). 5.3. Regularity of the pressure The pressure is absent in Eqs. (2.15) once we can reconstruct it from u and u; see Section 4.5. To check the regularity of the pressure, we take the divergence of (1.1). This yields the following equation for the pressure

− ∆p = ∇ · (∇ · uu)

(5.18)

where uu is the tensor (u u )1≤i,j≤3 . We recall that for any field u with divergence equal to zero, we have i j

(u · ∇)u = ∇ · (uu). Since (u · ∇)u ∈ L2 ([0, T ], H −1/2 ), one concludes from the classical elliptic theory that p is bounded in L2 ([0, T ], H 1/2 (T3 )). This finishes the proof of Theorem 3.2. 6. Relation between NSE and Leray-α: construction of suitable weak solutions The regularized solution constructed above with θ = 14 is unique. So the solution is a weak suitable solution. In this section we will construct a suitable weak solution to the Navier–Stokes equations by taking the limit when α tends to zero to the regularized solution. We start by the following lemma. Lemma 6.1. Let (uα , pα ) be the unique solution of (1.1) with θ = 2ν



T

0



|∇ uα |2 φ dxdt =

T





0

T3

1 4

then (uα , pα ) verifies the following local inequality

|uα |2 (φt + ν ∆φ) T3

  + |uα |2 uα + 2puα · ∇φ dxdt + 2

T

 0



fuφ dxdt ,

(6.1)

T3

for all T ∈ (0, +∞] and for all non negative functions φ ∈ C ∞ and supp φ b T3 × (0, T ). Proof. We take 2φ uα as a test function in (1.1). We note that the condition θ = 1/4 ensures that all the terms are well defined. In particular the integral T





uα ∇ uα · uα φ dxdt

2 0

T3

is finite by using the fact that uα ∇ uα ∈ L2 ([0, T ]; H −1 ) and 2φ uα ∈ L2 ([0, T ]; H 1 ). An integration by part combined with φ(T , ·) = φ(0, ·) = 0 and the following identity



uα ∇ uα · uα φ dx =

2 T3



uα |uα |2 · ∇φ dx

(6.2)

T3

yield that (uα , pα ) verifies (6.1). In order to take the limits α → 0 over (6.1), we need first to show that for all uα ∈ Lp (0, T ; Lp (T3 )3 ), 2 ≤ p < 10/3, we have: uα → u strongly in Lp (0, T ; Lp (T3 )3 ) for all 2 ≤ p < 10/3.  This is the aim of the following lemma.

(6.3)

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

Lemma 6.2. Assume uα belongs to L∞ ([0, T ]; H ) ∩ L2 ([0, T ]; V ) ∩ L Navier–Stokes equations, then for any θ fixed in R+ we have uα → u strongly in Lp (0, T ; Lp (T3 )3 ) for all 2 ≤ p < 3p−6 2p

Proof. From the Sobolev injection H

10 3

10 3

10

([0, T ]; L 3 (T3 )3 ), the energy space of solutions of the

.

(6.4)

↩→ Lp (T3 )3 , it is sufficient to show that

T



∥uα − uα ∥p 3p−6 dt −→ 0,

when α → 0.

(6.5)

H 2p

0

From the relation between uα and uα we have

∥uα − uα ∥p 3p−6 ≤ α 2θ p ∥uα ∥p 3p−6

+2θ H 2p

H 2p

.

(6.6)

Lemma 2.2 implies that T



∥uα − uα ∥ 0

p 3p−6 H 2p

10−3p 2

dt ≤ α



T

∥uα ∥p 2 dt .

(6.7)

Hp

0

Recall that T



∥uα ∥p 2 dt < ∞ for any 2 ≤ p ≤ ∞.

(6.8)

Hp

0

This yields the desired result for any 2 ≤ p < 10/3.



In order to show that (uα , pα ) gives rise to a suitable weak solution of the Navier–Stokes equations, it is necessary to take the limit α → 0. We have the following theorem. Theorem 6.1. Let (uα , pα ) be the solution of (1.1) with θ = the Navier–Stokes equations.

1 , 4

in the α → 0 limit, (uα , pα ) tends to a suitable weak solution to

Proof. By a classical compactness argument [5], we deduce that uα approaches u strongly in L2 ([0, T ], H ) for all T > 0 when α tends to zero, where u is a weak solution to the Navier–Stokes equations. It remains to show that (u, p) verifies the following local energy inequality 2ν

T





0

|∇ u|2 φ dxdt ≤

T

 0

T3



  |u|2 (φt + ν ∆φ) + |u|2 u + 2pu · ∇φ dxdt + 2

 0

T3

T



fuφ dxdt .

(6.9)

T3

To do so, we need to find estimates that are independent from α . Using the fact that (uα , pα ) belong to the energy space: 10

10

L∞ ([0, T ]; H ) ∩ L2 ([0, T ]; V ) ∩ L 3 ([0, T ]; L 3 (T3 )3 ) and from the Aubin–Lions compactness lemma (the same arguments as in Section 4) we can find a not relabeled subsequence {(uα , pα )}α>0 and (u, p) such that when α tends to zero we have: uα ⇀∗ u weakly∗ in L∞ ([0, T ]; H ), uα ⇀ u weakly in L ([0, T ]; V ) ∩ L 2 5 3

(6.10) 10 3

10 3

([0, T ]; L (T3 ) ), 3

(6.11)

5 3

pα ⇀ p weakly in L ([0, T ]; L (T3 )),

(6.12)

uα → u strongly in Lp ([0, T ]; Lp (T3 )3 ) for all 2 ≤ p < uα → u strongly in Lp ([0, T ]; Lp (T3 )3 ) for all 2 ≤ p < pα ⇀ p strongly in Lp ([0, T ]; Lp (T3 )) for all 1 < p <

10 3 10 3

,

(6.13)

,

(6.14)

5

. (6.15) 3 These convergence results allow us to take the limit in (6.1) in order to obtain, by using the weak lower semicontinuity of the norm in L2 ([0, T ]; V ) T



∥∇ uα ∥2H dt ≥

lim inf α→0

0

T



∥∇ u∥2H dt , 0

that (u, p) verifies the local energy inequality (6.9).



Remark 6.1. We recall that Lemma 6.2 holds true for any θ ∈ R+ thus we deduce that the above result holds true for any

θ ≥ 14 .

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

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7. The deconvolution case 7.1. The modified deconvolution operator In this section, we collect few results regarding the modified deconvolution operator which interpolates the usual deconvolution operator introduced in [6]. These results are proved in [6,23] with θ = 1.

• Assume u ∈ H s . Then HN (u) ∈ H s+2θ and ∥HN (u)∥H s+2θ ≤ C (N , α)∥u∥H s , where C (N , α) blows up when α goes to zero and/or N goes to infinity. This is due to the fact HN





 

uk e

i k·x

=



 1−

k∈I3

k∈I3

α 2θ |k|2θ 1 + α 2θ |k|2θ

 N +1 

uk ei k·x ,

and



 1−

α 2θ |k|2θ 1 + α 2θ |k|2θ

 N +1  ≈

N +1 α 2θ |k|2θ

as |k|∞ → ∞.

• The operator HN maps continuously H s into H s and ∥HN ∥L(H s ,H s ) = 1. • Assume u ∈ H s . Then the sequence {uα }α>0 converges strongly to u in the space H s . • Let u ∈ H s . Then the sequence {HN (u)}N ∈N converges strongly to u in H s when N goes to infinity. 7.2. The deconvolution model: well-posedness results We consider here a family of equations interpolating between the Navier–Stokes equations [24] and the Leraydeconvolution model [6] with periodic boundary conditions.

 ∂u  + HN (u) · ∇ u − ν ∆u + ∇ p = f    ∂t       ∇ · u = ∇ · HN (u) = 0, u= HN (u) = 0, T3 T3      u(t , x + Lej ) = u(t , x),    ut =0 = u0

in R+ × T3 , (7.1)

where HN (u) is an interpolating deconvolution operator introduced above. When N = 0, we obtain Eqs. (1.1). Similarly, we obtain the same results of well-posedness for the interpolating model (7.1). Theorem 7.1. For any T > 0, let f ∈ L2 ([0, T ], V −1 ) and u0 ∈ H . Assume that θ = 1/4. Then there exists a unique solution (u, p) := (uα,N , pα,N ) to (7.1) that satisfies u ∈ C ([0, T ]; H )∩L2 ([0, T ]; V ) and du ∈ L2 ([0, T ]; V −1 ) and p ∈ L2 ([0, T ], L2 (T3 )) dt such that u verifies



du dt

+ ν Au + B(HN (u), u) − f , φ

 =0 V −1 ,V 1

for every φ ∈ V and almost every t ∈ (0, T ). Moreover, this solution depends continuously on the initial data u0 in the L2 norm. 1

1

Theorem 7.2. For any T > 0, let f ∈ L2 ([0, T ], V − 2 ) and u0 ∈ V 2 . Let (u, p) := (uα,N , pα,N ) be the unique solution to Eqs. (7.1) with θ =

1 4

1

3

then (u, p) satisfies u ∈ C ([0, T ]; V 2 )∩ L2 ([0, T ]; V 2 ),

du dt

1

1

∈ L2 ([0, T ], V − 2 ) and p ∈ L2 ([0, T ], H 2 (T3 )).

Since the proofs of Theorems 7.1 and 7.2 are similar to that of Theorems 3.1 and 3.2 we omit their proofs here. Remark 7.1. Theorems 7.1 and 7.2 hold true for any θ ≥

1 4

.

7.3. Convergence to a suitable weak solution to the NSE By a similar argument to that in [6], it is possible to prove that up to a subsequence the solution for the deconvolution model (with θ = 14 ) converges when α → 0 or/and N → +∞ to a weak Leray solution to the Navier–Stokes equations. Next, we will show that the deconvolution regularization gives rise to a suitable weak solution to the Navier–Stokes equations. We need first the following.

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

Lemma 7.1. Let (uα,N , pα,N ) be the unique solution of (7.1) with θ ≥ 2ν

T 

 0



|∇ uα,N |2 φ dxdt =

T 

0

T3

1 4

then (uα,N , pα,N ) verifies the following local inequality

|uα,N |2 (φt + ν ∆φ) dxdt T3



T



  |uα,N |2 HN (uα,N ) + 2pα,N uα,N · ∇φ dxdt

+ 0

T3 T





fuα,N φ dxdt ,

+2 0

(7.2)

T3

for all T ∈ (0, +∞] and for all non negative functions φ ∈ C ∞ and supp φ b T3 × (0, T ). Proof. See Lemma 6.1.



Then we have the following. Lemma 7.2. Assume uα,N belong to the energy space of the solutions of the Navier–Stokes equations, then HN (uα,N ) → u

strongly in Lp (0, T ; Lp (T3 )3 ) for all 2 ≤ p <

Proof. From the Sobolev injection H

3p−6 2p

10 3

.

(7.3)

↩→ Lp (T3 )3 , it is sufficient to show that

T



∥HN (uα,N ) − uα,N ∥p 3p−6 dt −→ 0,

when α → 0 or/and N → ∞.

(7.4)

H 2p

0

From the relation between HN (uα,N ) and uα,N we have

 ∥HN (uα,N ) − uα,N ∥

p 3p−6 H 2p

=



|k |

3p−6 p



k ∈I



 sup |k |

≤

3p−10 p

α 2θ k 2θ 1 + α 2θ k 2θ 

k ∈I

α 2θ |k |2θ 1 + α 2θ |k |2θ

Using the fact that lm ↩→ l∞ for any m ≥ 1 we get for p <

 ∥HN (uα,N ) − uα,N ∥

p 3p−6 H 2p

Now, we choose m such that m

≤



1

 k ∈I

10−3p p

|k |

m

10−3p p

2(N +1)



10 3

 2p |uˆ k |2

2(N +1)  2p

∥u∥p 2 .

(7.5)

Hp

that

α 2θ |k |2θ 1 + α 2θ |k |2θ

2m(N +1)  2p

∥u∥p 2 .

(7.6)

Hp

> 3, and recall that

T



∥uα,N ∥p 2 dt < ∞ for any 2 ≤ p ≤ ∞,

(7.7)

Hp

0

and α→0 α 2θ |k |2θ −→ 0 1 + α 2θ |k |2θ

α 2θ |k |2θ sup 1 + α 2θ |k |2θ k ∈I 

or/and

2m

≤ 1.

(7.8)

Using the dominate convergence theorem, we conclude the desired result for any 2 ≤ p < 10/3.



Remark 7.2. Note that the above result holds true for p = 10/3. The above Lp convergence combined with the fact that uα,N belong to the energy space of solutions of the Navier–Stokes equations and the Aubin–Lions compactness lemma allows us to take the limit α → 0 or/and N → ∞ in (7.2) and to deduce the following theorem. Theorem 7.3. Let (uα,N , pα,N ) be the solution of (7.1), Then when α → 0 and/or N → ∞, (uα,N , pα,N ) tends to a suitable weak 10

10

solution (u, p) to the Navier–Stokes equations. The convergence to u is weak in L2 ([0, T ]; V ) ∩ L 3 ([0, T ]; L 3 (T3 )3 ) and strong in Lq ([0, T ]; Lq (T3 )3 ) for all 2 ≤ q < 10 and the convergence to p is strong in Lq ([0, T ]; Lq (T3 )) for all 1 < q < 53 and weak in 3 5

5

L 3 ([0, T ]; L 3 (T3 )).

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

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8. The MHD case In this section, we consider a deconvolution-type regularization of the magneto-hydrodynamic (MHD) equations, given by 1

∂t u − ν1 ∆u + (HN (u) · ∇)u − (HN (B ) · ∇)B + ∇ p + ∇|B |2 = 0,

(8.1a)

∂t B − ν2 △B + (HN (u) · ∇)B − (HN (B ) · ∇)u = 0, ∇ · B = ∇ · HN (B ) = ∇ · u = ∇ · HN (u) = 0, B (0) = B0 , u(0) = u0 ,

(8.1b)

2

(8.1c) (8.1d)

where the boundary conditions are taken to be periodic, and we also assume as before that T u dx = T B dx = 0. 3 3 Here, the unknowns are the fluid velocity field u(t , x), the fluid pressure p(t , x), and the magnetic field B (t , x). Note that when α = 0, we formally retrieve the MHD equations and the MHD-Leray-α equations are obtained when N = 0 and θ = 1. Existence and uniqueness results for MHD equations are established by Duvaut and Lions in [25]. These results are completed by Sermange and Temam in [26]. They showed that the classical properties of the Navier–Stokes equations can be extended to the MHD system. The use of Leray-α regularization to the MHD equations has received many studies; see [27–31]. The idea to use the deconvolution operator from [6] to MHD equations is a new feature for the present work.





8.1. Existence, unicity and convergence results First, we establish the global existence and uniqueness of solutions for the MHD-Deconvolution equations (8.1) for

θ = 1/4. Similar results concerning a general family of regularized MHD equations have been studied in [2]. In particular,

it has been shown there the existence and the uniqueness of solutions to the zeroth order MHD-Deconvolution equations (8.1) with θ > 14 . We have the following theorem. Theorem 8.1. For θ = 1/4, assume u0 ∈ H and B0 ∈ H . Then for any T > 0, (8.1) has a unique solution (u, B , p) := (uα,N , Bα,N , pα,N ) such that, u, B ∈ C ((0, T ], H ) ∩ L2 ([0, T ], V 1 ), and p ∈ L2 ([0, T ], L2 (T3 )). Furthermore the solution verifies 2ν

T





0

  |∇ u|2 + |∇ B |2 φ dxdt = T3

T





0

|u|2 (φt + ν1 ∆φ) + |B |2 (φt + ν2 ∆φ) dxdt T3

T







+ 0

  |u|2 + |B |2 HN (u) + 2pu · ∇φ dxdt

T3 T





(uB ) HN (B ) · ∇φ dxdt

−4 0

(8.2)

T3

for all T ∈ (0, +∞] and for all non negative functions φ ∈ C ∞ and supp φ b T3 × (0, T ). Proof. We only sketch the proof since it is similar to the Navier–Stokes equations case. The proof is obtained by taking the inner product of (8.1a) with u, (8.1b) with B and then adding them, the existence of a solution to Problem (8.1) can be derived thanks to the Galerkin method. Notice that u, B satisfy the following estimates 1 d 

   ∥u(t , x)∥2H + ∥B (t , x)∥2H + min(ν1 , ν2 ) ∥∇ u(t , x)∥2H + ∥∇ B (t , x)∥2H ≤ 0.

2 dt

(8.3)

The pressure p is reconstructed from u, HN (u), B , HN (B ) (as we work with periodic boundary conditions) and its regularity results from the fact that (HN (u) · ∇)u, (HN (B ) · ∇)B ∈ L2 ([0, T ], H −1 ). It remains to prove the uniqueness. Let (u1 , B1 , p1 ) and (u2 , B2 , p2 ), be two solutions, δ u = u2 − u1 , δ B = B2 − B1 , δ p = p2 − p1 . Then one has

∂t δ u + (HN (u1 )∇)δ u − (HN (B1 )∇)δ B − ν1 ∆δ u + ∇δ p = −(HN (δ u)∇)u2 + (HN (δ B )∇)B2 , ∂t δ B + (HN (u1 )∇)δ B − (HN (B1 )∇)δ u − ν2 ∆δ B = (HN (δ B )∇)u2 − (HN (δ u)∇)B2 ,

(8.4a) (8.4b)

and δ u = 0, δ B = 0 at initial time. One can take δ u ∈ L ((0, T ], H ) ∩ L ([0, T ], V ) as test in (8.4a) and δ B ∈ L∞ ((0, T ], H ) ∩ L2 ([0, T ], V ) as test in (8.4b). Since HN (u1 ) is divergence-free, one has ∞

T

 0



(HN (u1 )∇)δ u · δ u = 0, T3

2

(8.5a)

1580

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584 T 

 0

(HN (u1 )∇)δ B · δ B = 0.

(8.5b)

T3

Therefore,



d 2dt

|δ u|2 + ν1



|∇δ u|2 −





(HN (δ u)∇)u2 · δ u +

=−

(HN (B1 )∇)δ B · δ u T3

T3

T3



(HN (δ B )∇)B2 · δ u,

(8.6)

T3

T3

and



d 2dt

|δ B |2 + ν2





(HN (B1 )∇)δ u · δ B

T3

T3

T3



(HN (δ B )∇)u2 · δ B −

=



|∇δ B |2 −

T3

(HN (δ u)∇)B2 · δ B .

(8.7)

T3

One has by integration by parts,





(HN (B1 )∇)δ B · δ u =

− T3

(HN (B1 )∇)δ u · δ B .

(8.8)

T3

One has by adding (8.6), (8.7) and using (8.8) d



2dt

|δ u|2 + T3



d



2dt

|δ B |2 + ν1 T3



T3





|∇δ B |2 T3

(HN (δ B )∇)B2 · δ u

T3

(HN (δ B )∇)u2 · δ B −

+

|∇δ u|2 + ν2 T3

(HN (δ u)∇)u2 · δ u +

=−



T3



(HN (δ u)∇)B2 · δ B .

(8.9)

T3

One has by integration by parts,



(HN (δ u)∇)u2 · δ u =

−



T3



HN (δ u) ⊗ u2 : ∇δ u,

(HN (δ B )∇)B2 · δ u = −



T3



(8.10a)

T3

HN (δ B ) ⊗ B2 : ∇δ u,

(8.10b)

HN (δ B ) ⊗ u2 : ∇δ B ,

(8.10c)

HN (δ u) ⊗ B2 : ∇δ B .

(8.10d)

T3

(HN (δ B )∇)u2 · δ B = −



T3

T3



(HN (δ u)∇)B2 · δ B =

− T3

 T3

By the Young inequality,

      1 2  (HN (δ u)∇)u2 · δ u ≤ ν1 |∇δ u | + |HN (δ u)|2 |u2 |2 ,   4 T3 ν1 T3 T3       1 2  (HN (δ B )∇)B2 · δ u ≤ ν1 |∇δ u | + |HN (δ B )|2 |B2 |2 ,   4 T3 ν1 T3 T3       1 2  (HN (δ B )∇)u2 · δ B  ≤ ν2 |∇δ B | + |HN (δ B )|2 |u2 |2 ,   4 T3 ν2 T3 T3       1 2  (HN (δ u)∇)B2 · δ B  ≤ ν2 |∇δ B | + |HN (δ u)|2 |B2 |2 .   4 T3 ν2 T3 T3

(8.11a)

(8.11b)

(8.11c)

(8.11d)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

1581

By the Hölder inequality combined with the Sobolev injection



1

ν1

|HN (δ u)|2 |u2 |2 ≤ T3

≤ 

1

ν1

|HN (δ B )|2 |B2 |2 ≤ T3

≤ 

1

ν2

|HN (δ B )|2 |u2 |2 ≤ T3

≤ 

1

ν2

|HN (δ u)|2 |B2 |2 ≤ T3

≤

1

ν1 1

ν1 1

ν1 1

ν1 1

ν2 1

ν2 1

ν2 1

ν2

∥HN (δ u)∥2L3 ∥u2 ∥2L6 ∥HN (δ u)∥2 1 ∥u2 ∥2V ,

(8.12a)

V2

∥HN (δ B )∥2L3 ∥B2 ∥2L6 ∥HN (δ B )∥2 1 ∥B2 ∥2V ,

(8.12b)

V2

∥HN (δ B )∥2L3 ∥u2 ∥2L6 ∥HN (δ B )∥2 1 ∥u2 ∥2V ,

(8.12c)

V2

∥HN (δ u)∥2L3 ∥B2 ∥2L6 ∥HN (δ u)∥2 1 ∥B2 ∥2V .

(8.12d)

V2

Hence, d



2dt

≤ ≤

|δ u|2 + T3

d



2dt

1 min(ν1 , ν2 ) C (α, N ) min(ν1 , ν2 )

|δ B |2 + T3

ν1 2



|∇δ u|2 +

ν2

T3



2

|∇δ B |2

T3

    ∥HN (δ u)∥2 1 + ∥HN (δ B )∥2 1 ∥u2 ∥2V + ∥B2 ∥2V V2

V2

   ∥u2 ∥2V + ∥B2 ∥2V ∥δ u∥2H + ∥δ B ∥2H .

(8.13)

Therefore, d  2dt

   ∥δ u∥2H + ∥δ B ∥2H ≤ C (t ) ∥δ u∥2H + ∥δ B ∥2H ,

where C (t ) =

C (α,N ) min(ν1 ,ν2 )

  ∥u2 ∥2V + ∥B2 ∥2V ∈ L1 ([0, T ]). We conclude that δ u = δ B = 0 thanks to Grönwall’s lemma.



In order to deduce the local energy equality (8.2), we multiply (8.1a) with 2uφ , Eq. (8.1b) with 2B φ, for all T ∈ (0, +∞] and for all non negative functions φ ∈ C ∞ with supp φ b T3 × (0, T ), and then adding them. The rest can be done in the exact way as in Lemma 6.1, so we omit the details. With smooth initial data we may also prove the following theorem. 1

1

Theorem 8.2. For any T > 0, assume u0 ∈ V 2 and u0 ∈ V 2 . Let (u, B , p) := (uα,N , Bα,N , pα,N ) be the unique solution to 1 3 1 Eqs. (8.1) with θ = 14 then (u, B , p) satisfies u, B ∈ C ([0, T ], V 2 ) ∩ L2 ([0, T ], V 2 ), and p ∈ L2 ([0, T ], H 2 (T3 )). Proof. We only sketch the proof since it is similar to the one for Navier–Stokes equations. The proof is obtained by taking the inner product of (8.1a) with |∇|u, (8.1b) with |∇|B and then adding them, the existence of a solution to Problem (8.1) can be derived thanks to the Galerkin method. Notice that u, B satisfy the following estimates 1 d









∥u∥ 1 + ∥B ∥ 1 + min(ν1 , ν2 ) ∥∇ u∥ 1 + ∥∇ B ∥ V2 V2 V2         ≤  HN (u)∇ u|∇|udx +  HN (B )∇ B |∇|udx T3 T3         α,N α,N    + HN (u)∇ B |∇|B dx +  HN (B )∇ u|∇|B dx . 2

2

2 dt

T3

T3

2

2 V 12

(8.14)

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H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

The first non-linear term is estimated by

     HN (u)∇ u|∇|udx ≤ ∥HN (u)∇ u∥ − 1 ∥∇ u∥V 12  V 2 T3

≤ ∥HN (u)∇ u∥ 3 ∥∇ u∥ L2

1

V2

≤ ∥HN (u)∥L6 ∥∇ u∥L2 ∥∇ u∥ ≤ ∥HN (u)∥V ∥∇ u∥H ∥∇ u∥

1

V2 1

V2

1

3

V2

V2

≤ ∥HN (u)∥V ∥u∥ 2 1 ∥∇ u∥ 2 1

(8.15)

where we have used the Sobolev embedding in Theorem 2.1 together with the Hölder estimate and the interpolation 1

3

inequality between V 2 and V 2 . Similarly, we can estimate the third term by

  3 1     ≤ ∥HN (u)∥ ∥B ∥ 2 1 ∥∇ B ∥ 2 1 . H ( u )∇ B |∇| B dx N   V2 V2

(8.16)

T3

The second non-linear term is estimated by

      ≤ ∥HN (B )∇ B ∥ 1 ∥∇ u∥ 1 H ( B )∇ B |∇| udx N −   V 2 V2 T3

≤ ∥HN (B )∇ B ∥ 3 ∥∇ u∥ L2

1

V2

≤ ∥HN (B )∥L6 ∥∇ B ∥L2 ∥∇ u∥ ≤ ∥HN (B )∥V ∥∇ B ∥H ∥∇ u∥ ≤ C (α, N )∥B ∥

1

V2

1

V2 1

V2

∥B ∥V ∥∇ u∥

.

1

V2

(8.17)

By the same way we obtain

     HN (B )∇ u|∇|B dx ≤ C (α, N )∥B ∥ 1 ∥u∥V ∥∇ u∥ 1 .  V2 V2

(8.18)

T3

Therefore,

    ∥u∥2 1 + ∥B ∥2 1 + min(ν1 , ν2 ) ∥∇ u∥2 1 + ∥∇ B ∥2 1

1 d 2 dt

V2

V2

≤ ∥HN (u)∥V ∥u∥

1 2 1

V2

∥∇ u∥

V2

+ C (α, N )∥B ∥

1

V2

3 2 1

V2

+ ∥HN (u)∥V ∥B ∥

∥B ∥V ∥∇ u∥

1

V2

1 2 1

V2

∥∇ B ∥

V2

3 2 1

V2

+ C (α, N )∥B ∥

1

V2

∥u∥V ∥∇ u∥

1

V2

.

(8.19)

By the Young inequality,



d dt

∥u ∥2 1 + ∥ B ∥2 1 V2

c

≤

min(ν1 , ν2 )

+

V2



  + min(ν1 , ν2 ) ∥∇ u∥2 1 + ∥∇ B ∥2 1 V2

∥HN (u)∥ ∥u∥

C (α, N ) min(ν1 , ν2 )

4

2 1

+

V2

∥B ∥2 1 ∥B ∥2V + V2

c min(ν1 , ν2 ) C (α, N )

min(ν1 , ν2 )

V2

∥HN (u)∥ ∥B ∥2 1 4 V

V2

∥B ∥2 1 ∥u∥2V . V2

Therefore,

   ∥u∥2 1 + ∥B ∥2 1 + min(ν1 , ν2 ) ∥∇ u∥2 1 + ∥∇ B ∥2 1 dt V2 V2 V2 V2   c C (α, N ) C (α, N ) ≤ ∥B ∥2 1 ∥HN (u)∥4V + ∥B ∥2V + ∥u∥2V min(ν1 , ν2 ) min(ν1 , ν2 ) min(ν1 , ν2 ) V2 d



(8.20)

H. Ali / Nonlinear Analysis: Real World Applications 14 (2013) 1563–1584

+

c min(ν1 , ν2 )

∥HN (u)∥4V ∥u∥2 1 V2



≤ C (t ) ∥u∥2 1 + ∥B ∥2 1 V2

where C (t ) =

 3

c min(ν1 ,ν2 )

1583

 (8.21)

V2

 1 C (α,N ) C (α,N ) ∥HN (u)∥4V + min ∥B ∥2V + min ∥u∥2V ∈ L1 ([0, T ]). We conclude that u, B ∈ L∞ ((0, T ], V 2 ) (ν1 ,ν2 ) (ν1 ,ν2 )

∩ L2 ([0, T ], V 2 ) thanks to Grönwall’s lemma.



The pressure p is reconstructed from u, HN (u), B , HN (B ) (as we are working with periodic boundary conditions) and its 1

regularity results from the fact that (HN (u) · ∇)u, (HN (B ) · ∇)B ∈ L2 ([0, T ], H − 2 ). Next, we will deduce that the deconvolution regularization gives rise to a suitable weak solution to the MHD equations. Theorem 8.3. Let (uα,N , Bα,N , pα,N ) be the solution of (8.1) with θ ≥ 41 , Then when α → 0 and/or N → ∞, (uα,N , Bα,N , pα,N ) tends to a weak solution (u, B , p) to the MHD equations. The convergence to u and the convergence to B are weak in L2 ([0, T ]; V ) ∩ L

10 3

10

([0, T ]; L 3 (T3 )3 ) and strong in Lq ([0, T ]; Lq (T3 )3 ) for all 2 ≤ q <

L ([0, T ]; L (T3 )) for all 1 < q < q

q

2ν

 0

T



5 3

5 3

5 3

10 3

and the convergence to p is strong in

and weak in L ([0, T ]; L (T3 )). Furthermore the solution verifies in addition

  |∇ u|2 + |∇ B |2 φ dxdt ≤

T





0

T3

|u|2 (φt + ν1 ∆φ) + |B |2 (φt + ν2 ∆φ) dxdt T3

T





 2   |u| + |B |2 u + 2pu · ∇φ dxdt

+ 0

T3 T





(uB ) B · ∇φ dxdt

−4 0

(8.22)

T3

for all T ∈ (0, +∞] and for all non negative functions φ ∈ C ∞ and supp φ b T3 × (0, T ). Proof. As in Lemma 7.2 we can show that HN (uα,N ) → u HN (Bα,N ) → B

strongly in Lp (0, T ; Lp (T3 )3 ) for all 2 ≤ p <

10

strongly in Lp (0, T ; Lp (T3 )3 ) for all 2 ≤ p <

.

3 10 3

(8.23)

.

(8.24)

The above Lp convergence combined with the fact that uα,N and Bα,N belong to the energy space of the solutions of the Navier–Stokes equations and the Aubin–Lions compactness lemma allows us to take the limit α → 0 or/and N → ∞ in (8.2). The rest can be done in the exact way as in [6], so we omit the details.  References [1] E. Olson, E.S. Titi, Viscosity versus vorticity stretching: global well-posdness for a family of Navier–Stokes-alpha-like models, Nonlinear Analysis 66 (2007) 2427–2458. [2] M. Holst, E. Lunsain, G. Tsogtgerel, Analysis of a general family of regularized Navier–Stokes and MHD models, Journal of Nonlinear Science 20 (2010) 523–567. [3] H. Ali, Z. Ammari, On the Hausdorff dimension of singular sets for the Leray-α Navier–Stokes equations with fractional regularization, Dynamics of Partial Differential Equations 9 (2012) 261–271. [4] T. Kato, H. Fujita, On the Navier Stokes initial value problem, I. Archive for Rational Mechanics and Analysis 16 (1964) 269–315. [5] C. Foias, D. Holm, E.S. Titi, The three dimensional viscous Camassa–Holm equations and their relation to the Navier–Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations 14 (2002) 1–35. [6] W. Layton, R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior, Analysis and Applications 6 (2008) 23–49. [7] V. Vyalov, Partial regularity of solutions to the mangnetohydrodynamic equations, Journal of Mathematical Sciences 150 (2008) 1771–1786. [8] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Communications on Pure and Applied Mathematics 35 (1982) 771–831. [9] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific Journal of Mathematics 66 (1976) 535–552. [10] H. Ali, Etude mathématiques de quelques modèles de turbulence, Ph.D. Thesis of the University of Rennes 1, 2011. [11] P. Constantin, C. Foias, Navier Stokes Equations, The University of Chicago Press, 1988. [12] C. Doering, J. Gibbon, Applied Analysis of the Navier–Stokes Equations, Cambridge University Press, 1995. [13] C. Foias, O. Manley, R. Rosa, R. Temam, Navier–Stokes Equations and Turbulence, Cambridge University Press, 2001. [14] R. Lewandowski, Approximations to the Navier–Stokes Equations, In progress, 2012. [15] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, in: CBMS Regional Conference Series, SIAM, Philadelphia, 1983. [16] L. Evans, Partial Differential Equations, American Mathematical Society, 1998. [17] R. Adams, J.J. Fournier, Sobolev spaces, Pure and Apllied Mathematics (2003). [18] J. Bergh, J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin, 1976. [19] J.L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires, Gauthiers-Villard, 1969. [20] J. Simon, Compact sets in the spaces Lp (0, t ; b), Annali di Matematica Pura ed Applicata 146 (1987) 65–96.

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