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Large Eddy Simulation for turbulent flows with critical regularization Hani Ali Département de mathématiques, Université Paris-Sud, Bât. 425, 91405 Orsay Cedex, France

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Article history: Received 4 October 2011 Available online 30 April 2012 Submitted by Pierre Lemarie-Rieusset Keywords: Turbulence simulation and modeling Large-eddy simulations Partial differential equations

abstract In this paper, we establish the existence of a unique ‘‘regular’’ weak solution to the Large Eddy Simulation (LES) models of turbulence with critical regularization. We first consider the critical LES for the Navier–Stokes equations and we show that its solution converges to a solution of the Navier–Stokes equations as the averaging radii converge to zero. Then we extend the study to the critical LES for magnetohydrodynamic equations. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Let us consider the Navier–Stokes equations in a three dimensional torus T3 , div v = 0,

(1.1)

v,t + div (v ⊗ v ) − ν ∆v + ∇ p = f ,

(1.2)

subject to v (x, 0) = v0 (x). Here, v is the fluid velocity field, p is the pressure, f is the external body force, ν stands for the viscosity. Eqs. (1.1) and (1.2) are known to be the idealized physical model to compute Newtonian fluid flows. They are also known to be unstable in numerical simulations when the Reynolds number is high, thus when the flow is turbulent. Therefore, numerical turbulent models are needed for real simulations of turbulent flows. In many practical applications, knowing the mean characteristics of the flow by averaging techniques is sufficient. However, averaging the nonlinear term in NSE leads to the well-known closure problem. To be more precise, if v denotes the filtered/averaged velocity field then the Reynolds averaged Navier-Stokes (RANS) equations v ,t + div (v ⊗ v ) − ν ∆v + ∇ p + div R(v , v ) = f ,

(1.3)

where R(v , v ) = v ⊗ v − v ⊗ v, the Reynolds stress tensor, is not closed because we cannot write it in terms of v alone. The main essence of turbulence modeling is to derive simplified, reliable and computationally realizable closure models. In [1,2] Layton and Lewandowski suggested an approximation of the Reynolds stress tensor given by

R(v , v ) = v ⊗ v − v ⊗ v .

(1.4)

This is an equivalent form to the approximation div (v ⊗ v ) ≈ div (v ⊗ v ).

E-mail addresses: [email protected], [email protected] 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.04.066

(1.5)

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H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

Hence Layton and Lewandowski studied the following Large Scale Model considered as a Large Eddy Simulation (LES) model: div w = 0,

(1.6)

w,t + div (w ⊗ w ) − ν ∆w + ∇ q = f ,

(1.7)

considered in (0, T ) × T3 and subject to w (x, 0) = w0 (x) = v0 and periodic boundary conditions with mean value equal to zero. Where they denoted (w , q) as the approximation of (v , p). The averaging operator chosen in (1.7) is a differential filter, [3–5,1,6,7], that commutes with differentiation under periodic boundary conditions and is defined as follows. Let α > 0, given a periodic function ϕ ∈ L2 (T3 ), define its average ϕ to be the unique solution of

− α 2 ∆ϕ + ϕ = ϕ,

(1.8)

The main goal in using such a model is to filter eddies of scale less than the numerical grid size α in numerical simulations. The Laplacian in the above expression has a smoothing effect and this allow us to prove existence and uniqueness of the solution. In some cases, the use of the smoothing effect of the Laplacian can be unnecessary. However, we may use other filters, as the top-hat filter [8,9] which are not smoothing, or as differential filter with fractional order Laplace operator [10–12]. Moreover, Layton and Neda [13] observed by using the classical dimensional analysis arguments of Kolmogorov coupled with precise mathematical knowledge of the model’s kinetic energy balance that the energy spectra of the LES model (1.6) and (1.7) should scale as 2

5

E (k ) ∼ = ϵα3 |k |− 3 , 2

E (k ) ∼ = ϵα3 α −2 |k |−

for α ≤ 11 3

,

1

|k |

,

for α ≥

(1.9) 1

|k |

,

(1.10)

where ϵα is the time averaged energy dissipation rate of the model’s solution given by

ϵα =

ν 2 2 2 ∥ w ∥ + α ∥ w ∥ . 2 1,2 3 L

(1.11)

Here, ∥w ∥21,2 represents the W 1,2 (T3 )3 semi-norm which is given by

∥w ∥21,2 =

|k |2 | w (k )|2 .

k ∈Z3

Thus the application of a smooth filter strongly affects the shape of the energy spectra. In particular, we observe that there −5

are two different power laws for the energy cascade. For wave numbers k such that |k | ≤ α1 we obtain the usual |k | 3 Kolmogorov power law. This implies that the large scale statistics of the flow of size greater than the length scale α are consistent with the Kolmogorov theory for 3D turbulent flows. On the other hand, for |k | ≥ α1 we obtain a steeper power law. This implies a faster decay of energy in comparison to direct numerical simulation (DNS), which suggests, in terms of numerical simulation, a smaller resolution requirement in computing turbulent flows. For a general overview of LES models, the readers are referred to Berselli et al. [8] and references cited therein. Notice that the Layton–Lewandowski model (1.6) and (1.7) differs from the one introduced by Bardina et al. [14] where the following approximation of the Reynolds stress tensor is used:

R(v , v ) = v ⊗ v − v ⊗ v .

(1.12)

In [1,2] Layton and Lewandowski have proved that (1.6) and (1.7) have a unique regular solution. They have also shown that there exists at least a sequence αj which converges to zero and such that the sequence (wαj , qαj ) converges to a distributional solution (v , p) of the Navier–Stokes equations. We remark that many of these results established in the above cited papers have been extended to the following three dimensional magnetohydrodynamic equations (MHD): div B = div v = 0 ∂t v − ν1 ∆v + div (v ⊗ v ) − div (B ⊗ B ) + ∇ p = 0, ∂t B − ν2 ∆B + div (v ⊗ B ) − div (B ⊗ v ) = 0, B dx = v dx = 0 T3

B (0) = B0 ,

(1.13) (1.14) (1.15) (1.16)

T3

v (0) = v0 .

(1.17)

Here v is the fluid velocity field, p is the fluid pressure, B is the magnetic field, and v0 and B0 are the corresponding initial data. The interested readers are referred to [15,16] and references cited therein.

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

293

This paper has two main correlated goals. The first one is to study the Large Eddy Simulation for the Navier–Stokes equations (LES for NSE) with a general filter −θ : div w = 0,

(1.18) θ

w,t + div (w ⊗ w ) − ν ∆w + ∇ q = f , 2θ

θ

θ

(1.19)

θ

α (−∆) ϕ + ϕ = ϕ,

(1.20) θ

where the nonlocal operator (−∆) is defined through the Fourier transform

(−∆ )θ ϕ(k ) = |k |2θ ϕ (k ).

(1.21)

Our task is to show that for θ ≥ (see Theorem 2.1), we get global in time existence of a unique weak solution (w , q) to Eqs. (1.18)–(1.20) such that: (w , q) are spatially periodic with period L, 1 6

w (t , x)dx = 0

q(t , x)dx = 0

and

for t ∈ [0, T ),

(1.22)

T3

T3

and w (0, x) = w0 (x) = v0

θ

in T3 .

(1.23)

We note that the value θ = is optimal in order to get existence and uniqueness of weak solutions for all times. The LES model studied by Layton–Lewandowski in [2] corresponds to the case where θ = 1. The LES for NSE with θ = 1 can be also addressed as the zeroth order Approximate Deconvolution Model referring to the family of models in [17]. The value θ = 16 is consistent with the critical regularization value needed to get existence and uniqueness to the simplified Bardina model studied in [18]. We note also that fractional order Laplace operator has been used in others α models of turbulence in [10–12]. A simple modification of the scaling arguments in [13] yield that the energy spectrum of w should scale as 1 6

2

5

Eθ ( k ) ∼ = ϵθ3 |k |− 3 , 2

for α ≤

1

|k |

5

∼ ϵ 3 α −2θ |k |− 3 +(−2θ) , Eθ ( k ) = θ

,

for α ≥

(1.24) 1

|k |

,

(1.25)

where ϵθ is the average total energy dissipation rate given by

ϵθ =

ν ∥w ∥22 + α 2θ ∥w ∥21+2θ,2 . 3

(1.26)

L

Here, ∥w ∥21+2θ,2 represents the W 1+2θ,2 (T3 )3 semi-norm which is given by

∥w ∥21+2θ,2 =

|k |2+4θ | w (k )|2 .

k ∈Z3 5

Therefore, we would get the corresponding energy spectra which has a faster decaying power law |k |− 3 +(−2θ ) than the 5

usual Kolmogorov |k |− 3 power law, in the subrange |k | ≥ α1 . This signifies that the LES model with fractional order Laplace operator is also a good candidate subgrid scale model of turbulence. The second goal of this paper is to study the LES model for magnetohydrodynamic (LES for MHD) equation with a general filter −θ . Hence, we consider the following LES for MHD problem div w = div W = 0

(1.27) θ

θ

∂t w − ν1 ∆w + div (w ⊗ w ) − div (W ⊗ W ) + ∇ q = 0, θ

θ

∂t W − ν2 ∆W + div (w ⊗ W ) − div (W ⊗ w ) = 0, W dx = w dx = 0, T3

W (0) = W0 ,

(1.28) (1.29) (1.30)

T3

w (0) = w0 ,

(1.31)

where the boundary conditions are taken to be periodic, and as before we take the same filter −θ and denote the approximation of (v , B , p) solution of the averaged MHD equations by (w , W , q). The case when θ = 1 is studied in [15], there the authors gave a mathematical description of the problem, performed the numerical analysis of the model and verified their physical fidelity. The method carried out in [15] relies on using the semigroup approach in order to show the existence of the solution. In this paper, we show by using the Galerkin method [19] that for θ ≥ 16 (see Theorem 3.1), we get global in time existence of a unique weak solution (w , W , q) to Eqs. (1.27)–(1.31). Let us mention that the idea to consider the LES for MHD with critical regularization is a new feature for this work.

294

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

The Approximate Deconvolution Model for Navier–Stokes equations with θ > 34 is studied in [20] and the Approximate Deconvolution Model for magnetohydrodynamic equations with θ = 1 is studied in [21]. As mentioned in [20] the value θ > 34 is not optimal in order to prove the existence and the uniqueness of the solutions in the deconvolution case. ‘‘The exponent ‘‘3/4’’ looks like a ‘‘critical exponent’’. We conjecture that we can get an existence and uniqueness result for lower exponents, but concerning the convergence towards the mean Navier–Stokes equations, we think that it is the best exponent, but this question remains an open one.’’ Based on this work, we will study in a forthcoming paper the Approximate Deconvolution Model for both Navier–Stokes equations and magnetohydrodynamic equations with critical regularizations. Finally, one may ask questions about the relation between the regularization parameter θ and the model consistency errors. These questions are addressed in [22] where θ = 1. Therefore, the issue is to find the relation between the model consistency errors and the regularization parameter θ . This paper is organized as follows. In Section 2 we prove the global existence and uniqueness of the solution for the LES for NSE with critical regularization. We also prove that the solution (w , q) of the LES for NSE converges in some sense to a solution of the Navier–Stokes equations when α goes to zero. Section 3 treats the questions of global existence, uniqueness and convergence for the LES for MHD with critical regularization. 2. The critical LES for NSE Before formulating the main results of this paper, we fix notation of function spaces that we shall employ. We denote by Lp (T3 ) and W s,p (T3 ), s ≥ −1, 1 ≤ p ≤ ∞ the usual Lebesgue and Sobolev spaces over T3 , and the Bochner spaces C (0, T ; X ), Lp (0, T ; X ) are defined in the standard way. In addition we introduce s,p Wdiv

s,p

= w ∈ W (T3 ) ; 3

w = 0; div w = 0 , T3

endowed with the norms

∥w ∥2s,2 =

|k |2s | w (k )|2 .

k ∈Z3 \{0}

We present our main results, restricting ourselves to the critical case θ =

1 , 6

and for simplicity we drop some indices of θ

so sometimes we will write ‘‘ϕ ’’ instead of ‘‘ϕ θ ’’, expecting that no confusion will occur. 2.1. Existence and uniqueness results for the LES for NSE Theorem 2.1. Assume that θ =

1 . 6

1

5

,2

6 Let f ∈ L2 0, T ; W − 6 ,2 be a divergence free function and w0 ∈ Wdiv . Then there exist

(w , q) a unique ‘‘regular’’ weak solution to (1.18)–(1.23) such that 1 ,2 1+ 1 ,2 6 w ∈ C 0, T ; Wdiv ∩ L2 0, T ; Wdiv 6 ,

5

(2.1)

w,t ∈ L2 0, T ; W − 6 ,2 , q ∈ L2

1 0, T ; W 6 ,2 (T3 ) ,

(2.2)

q = 0,

(2.3)

T3

fulfill T

⟨w,t , ϕ⟩ − (w ⊗ w , ∇ϕ) + ν(∇ w , ∇ϕ) dt = 0

T

⟨f , ϕ⟩ dt for all ϕ ∈ L

2

5 ,2 6

0, T ; Wdiv

.

(2.4)

0

Moreover, w (0) = w0 .

(2.5)

Note that the first and the last term in (2.4) are defined in the sense of duality pairing on W

(w ⊗ w , ∇ϕ) and ν(∇ w , ∇ϕ) are defined in the sense of duality pairing on W

−1 ,2 6

×W

1 ,2 6

−5 ,2 6

5

× W 6 ,2 . The terms

.

Remark 2.1. The notion of ‘‘regular weak solution’’ is introduced in [20]. Here, we use the name ‘‘regular’’ for the weak solution since the weak solution is unique and the velocity part of the solution w does not develop a finite time singularity.

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

295

Proof of Theorem 2.1. The proof of Theorem 2.1 follows the classical scheme. We start by constructing approximated solutions (w N , qN ) via Galerkin method. Then we seek for a priori estimates that are uniform with respect to N. Next, we pass to the limit in the equations after having used compactness properties. Finally we show that the solution we constructed is unique thanks to Gronwall’s lemma [19]. 2 1,2 Step 1 (Galerkin approximation) Consider a sequence {ϕr }∞ -orthogonal r =1 consisting of L -orthonormal and W eigenvectors of the Stokes problem subjected to the space periodic conditions. We note that this sequence forms a Hilbertian basis of L2 . We set

w N (t , x) =

N

crN (t )ϕr (x),

and qN (t , x) =

r =1

N

qNk (t )eik ·x .

(2.6)

|k |=1

We look for (w N (t , x), qN (t , x)) that are determined through the system of equations w,Nt , ϕr − (w N ⊗ w N , ∇ϕr ) + ν(∇ w N , ∇ϕr ) = ⟨f , ϕr ⟩,

r = 1, 2, . . . , N ,

(2.7)

and

∆qN = −div div

Π N (w N ⊗ w N ) ,

(2.8)

where the projector Π N assign to any Fourier series

k ∈Z3 \{0}

gk eik ·x the following series

k ∈Z3 \{0},|k |≤N

gk eik ·x .

N

Moreover we require that w satisfies the following initial condition w N (0, ·) = w0N =

N

c0N ϕr (x),

(2.9)

r =1

and 1

strongly in W 6 ,2 (T3 )3

w0N → w0

when N → ∞.

(2.10)

The classical Caratheodory theory [23] then implies the short-time existence of solutions to (2.7) and (2.8). Next we derive estimates on c N that is uniform w.r.t. N. These estimates then imply that the solution of (2.7) and (2.8) constructed on a short time interval [0, T N [ exists for all t ∈ [0, T ].

Step 2 (A priori estimates) Multiplying the rth equation in (2.7) with α 2θ |k |2θ crN (t ) + crN (t ), summing over r = 1, 2, . . . , N, integrating over time from 0 to t and using the following identities

1

1

w,Nt , w N + α 3 (−∆) 6 w N

1 d

1

1 d

∥w N ∥22 + α 3 ∥w N ∥21 ,2 , = 2 dt 2 dt 6 1 1 1 N N N N 2 N 2 −∆w , w + α 3 (−∆) 6 w = ∥w ∥1,2 + α 3 ∥w ∥1+ 1 ,2 ,

(2.11) (2.12)

6

1

1

⟨f , w N + α 3 (−∆) 6 w N ⟩ = ⟨f , w N ⟩,

(2.13)

and

1

1

w N ⊗ w N , ∇(w N + α 3 (−∆) 6 w N )

= wN ⊗ wN , ∇wN |w N |2 = − div w N , =0

(2.14)

2

leads to the a priori estimates 1 2

∥

1 3

∥ + α ∥ w ∥ 1 ,2 + ν

w N 22 t

= 0

N 2

6

t

1 ∥w N ∥21,2 + α 3 ∥w N ∥21+ 1 ,2 ds 6

0

1 1 ⟨f , w N ⟩ ds + ∥w0 ∥22 + α 3 ∥w0 ∥21 ,2 . 2

(2.15)

6

Using the duality norm combined with Young’s inequality we conclude from Eq. (2.15) that sup ∥ t ∈[0,T N [

∥ +α

w N 22

1 3

t 1 sup ∥w ∥ 1 + ν ∥ w N ∥21,2 + α 3 ∥w N ∥2 1 ds ≤ C ,2 1 + ,2 N 2

t ∈[0,T N [

6

0

6

that immediately implies that the existence time is independent of N and it is possible to take T = T N .

(2.16)

296

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

We deduce from (2.16) that w N ∈ L∞

1

,2

6 0, T ; Wdiv

1 ∩ L2 0, T ; W 1+ 6 ,2 (T3 )3 .

(2.17)

From (2.17) and by using Hölder’s inequality combined with Sobolev injection we get

w N ⊗ w N ∈ L2 0, T ; W

−1 ,2 6

(T3 )3×3 .

(2.18)

From (2.18) and by using (1.20) it follows that

1

w N ⊗ w N ∈ L2 0, T ; W 6 ,2 (T3 )3×3 .

(2.19)

Consequently from the elliptic theory Eq. (2.8) implies that T

∥pN ∥21 ,2 dt < K .

(2.20)

6

0

From Eqs. (2.7), (2.17), (2.19) and (2.20) we also obtain that T

0

∥w,Nt ∥2− 5 ,2 dt < K .

(2.21)

6

Step 3 (Limit N → ∞) It follows from the estimates (2.17)–(2.21) and the Aubin–Lions compactness lemma (see [24] for example) that there are a not relabeled subsequence of (w N , qN ) and a couple (w , q) such that w N ⇀∗ w wN ⇀ w w,Nt ⇀ w,t

1

weakly ∗ in L∞ 0, T ; W 6 ,2 ,

1

5

(2.22)

weakly in L2 0, T ; W 1+ 6 ,2 ,

(2.23)

weakly in L2 0, T ; W − 6 ,2 ,

(2.24)

1

qN ⇀ q weakly in L2 0, T ; W 6 ,2 (T3 ) , wN → w

(2.25)

1

strongly in L2 0, T ; W 6 ,2 (T3 )3 .

(2.26) 1

1

Moreover, due to the a priori estimate of w N in L2 0, T ; W 1+ 6 ,2 (T3 )3 and in L∞ 0, T ; W 6 ,2 (T3 )3 , we have wN → w for all s < 1 +

1 6

wN → w

strongly in L2 0, T ; W s,2 (T3 )3 ,

(2.27)

and

1

strongly in Lq 0, T ; W 6 ,2 (T3 )3 ,

(2.28)

for all q < ∞. From (2.27), (2.28) and (2.19) and by using (1.20) it follows that

1

weakly in L2 0, T ; W 6 ,2 (T3 )3×3 ,

(2.29)

w N ⊗ w N → w ⊗ w strongly in L 0, T ; W r ,2 (T3 )3×3 ,

(2.30)

wN ⊗ wN ⇀ w ⊗ w

q

for all q < 2 and all r < 16 . The above established convergences are clearly sufficient for taking the limit in (2.7) and for concluding that the velocity part w satisfy (2.4). Moreover, from (2.23) and (2.24) one we can deduce by a classical argument (see in [11]) that

1

w ∈ C 0, T ; W 6 ,2 .

(2.31) 1

Furthermore, from the strong continuity of w with respect to the time with value in W 6 ,2 we deduce that w (0) = w0 . 1 3

1 6

Let us mention also that w + α (−∆) w is a possible test function in the weak formulation (2.4). Thus w verifies for all t ∈ [0, T ] the following equality

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

1

∥w (t )∥22 + α 3 ∥w (t )∥21 ,2 + 2ν 6

t

=2 0

t 0

297

1

∥w ∥21,2 + α 3 ∥w ∥21+ 1 ,2 ds 6

1 ⟨f , w ⟩ds + ∥w0 ∥22 + α 3 ∥w0 ∥21 ,2 .

(2.32)

6

Step 4 (Uniqueness) Since the pressure part of the solution is uniquely determined by the velocity part it remain to show the uniqueness to the velocity. Next, we will show the continuous dependence of the solutions on the initial data and in particular the uniqueness. Let (w1 , p1 ) and (w2 , p2 ) any two solutions of (1.18)–(1.20) on the interval [0, T ], with initial values w1 (0) and w2 (0). Let us denote by δ w = w2 − w1 . We subtract the equation for w1 from the equation for w2 and test it with δ w. We get using successively the relation (1.20), the fact that the averaging operator commutes with differentiation under periodic boundary conditions, the norm duality, Young’s inequality and Sobolev embedding theorem: 1 d

1

1

d

1

∥δ w ∥22 + α 3 ∥δ w ∥21 ,2 + ν∥∇δ w ∥22 + να 3 ∥∇δ w ∥21 ,2 2 dt 6 6 1 1 ≤ w2 ⊗ w2 − w1 ⊗ w1 , ∇ δ w + α 3 (−∆) 6 δ w

2 dt

≤ (w2 ⊗ w2 − w1 ⊗ w1 , ∇δ w ) ≤ ≤

4 1

να 3 4 1

να 3

∥δ w ⊗ w1 ∥2− 1 ,2 6

∥δ w ∥21 ,2 ∥w1 ∥21+ 1 . 6

(2.33)

6

Using Gronwall’s inequality we conclude the continuous dependence of the solutions on the initial data in the 1

L∞ [0, T ], W 6 ,2

norm. In particular, if δ w 0 = 0 then δ w = 0 and the solutions are unique for all t ∈ [0, T ]. Since

T > 0 is arbitrary this solution may be uniquely extended for all time. This finishes the proof of Theorem 2.1. 2.2. Limit consistency for the critical LES for NSE In this section, we take the limit α → 0 in order to show the following result: Theorem 2.2. Let (wα , qα ) be the solution of (1.18)–(1.20) for a fixed α . There is a subsequence αj such that (wαj , qαj ) → (v , p) 5 1 ,2 as j → ∞ where (v , p) ∈ L∞ ([0, T ]; L2 (T3 )3 ) ∩ L2 ([0, T ]; Wdiv ) × L 3

5 [0, T ]; L 3 (T3 ) is a weak solution of the Navier–Stokes

equations with periodic boundary conditions and zero mean value constraint. The sequence wαj converges strongly to v in the space Lp ([0, T ]; Lp (T3 )3 ) for all 2 ≤ p < 6r

10 3

, and weakly in Lr 0, T ;

L 3r −4 (T3 )3 for all 2 ≤ r < ∞, while the sequence qαj converges strongly to p in the space Lp ([0, T ]; Lp (T3 )) for all r

and weakly in the space L 2

0, T ; L

3r 3r −4

4 3

≤ p < 53 ,

(T3 ) for all 2 ≤ r < ∞.

In order to prove Theorem 2.2 we need to reconstruct a uniform estimates for wα . Since in the proof of Theorem 2.1 the dependence of constants on α is not addressed we cannot use it. Instead we show in the proof of Theorem 2.2 that wα 1 ,2 belongs to L∞ ([0, T ]; L2 (T3 )3 ) ∩ L2 ([0, T ]; Wdiv ) uniformly with respect to α . Before proving Theorem 2.2, we first record the following three lemmas. s,2

s+β,2

Lemma 2.1. Let θ ∈ R+ , 0 ≤ β ≤ 2θ , s ∈ R and assume that ϕ ∈ Wdiv . Then ϕ ∈ Wdiv 1

∥ϕ∥s+β,2 ≤

∥ϕ∥s,2 ,

αβ

such that (2.34)

and

∥ϕ∥s,2 ≤ ∥ϕ∥s,2 . Proof. See in [11].

(2.35)

1 ,2

Lemma 2.2. Assume wα belongs to L∞ ([0, T ]; L2 (T3 )3 ) ∩ L2 ([0, T ]; Wdiv ), uniformly with respect to α , then their exists a constant C which is independent from α such that

∥wα ⊗ wα ∥

r

L2

8−3r ,2 0,T ;W 2r

≤ C < ∞ for any

8 3

≤ r < ∞.

(2.36)

298

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

Proof. We have by interpolation that

6r

wα ∈ Lr 0, T ; L 3r −4 (T3 )3

(2.37)

for any r ≥ 2, thus we deduce by using Hölder’s inequality that r

wα ⊗ wα ∈ L 2

3r

0, T ; L 3r −4 (T3 )3×3 .

(2.38)

3r

Thus, by using the Sobolev embedding L 3r −4 (T3 ) ↩→ W

∥wα ⊗ wα ∥

r

L2

8−3r ,2 0,T ;W 2r

=

sup r 3r −8 ,2 ϕ∈L r −2 0,T ;W 2r

8−3r ,2 2r

T 0

(T3 ) we deduce that (wα ⊗ wα , ϕ) dt

∥ϕ∥≤1

T

r

∥wα ⊗ wα ∥ 28−3r ,2 dt

≤

2r

0 T

r

∥wα ⊗ wα ∥ 2 3r dt

≤

3r −4

0 T

∥wα ∥r 6r dt ≤ C < ∞

≤ for any r ≥

8 3

.

(2.39)

3r −4

0

3r Remark 2.2. We note that 8− ≤ 0 for r ≥ 2r

8 . 3 1 ,2

Lemma 2.3. Assume wα belongs to L∞ ([0, T ]; L2 (T3 )3 ) ∩ L2 ([0, T ]; Wdiv ), uniformly with respect to α , then for all p ≥ 1 and q ≥ 34 such that 2 q

+

3 p

> 3,

(2.40)

we have T

∥wα ⊗ wα − wα ⊗ wα ∥qp dt ≤ C α

3q+2p−3pq p

.

(2.41)

0

Proof. We take r = 2q, from the Sobolev injection W T

r

∥wα ⊗ wα − wα ⊗ wα ∥ 23p−6 dt ≤ C α 2p

0

,2

3p−6 ,2 2p

3r +4p−3pr 2p

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

.

(2.42)

From the relation between wα ⊗ wα and wα ⊗ wα we have r

r

∥wα ⊗ wα − wα ⊗ wα ∥ 23p−6 2p

,2

≤ α θ r ∥wα ⊗ wα ∥ 23p−6 2p

+2θ ,2

T

(2.43)

Lemma 2.1 implies that T

∥wα ⊗ wα − wα ⊗ wα ∥ 0

r 2 3p−6 ,2 2p

dt ≤ α

3r +4p−3pr 2p

0

r

∥wα ⊗ wα ∥ 28−3r ,2 dt .

(2.44)

2r

Recall that T

0

r

∥wα ⊗ wα ∥ 28−3r ,2 dt < ∞ for any 2r

8 3

This yields the desired result for any p ≥ 1, q ≥

≤ r < ∞. 4 3

such that

(2.45) 2 q

+

3 p

> 3.

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

Remark 2.3. When

2 q

+

3 p

299

= 3 we have

T

∥wα ⊗ wα − wα ⊗ wα ∥qp dt ≤ C ,

(2.46)

0

where the constant C is independent from α and can be arbitrary large. Proof of Theorem 2.2. The proof of Theorem 2.2 follows the lines of the proof of the Theorem 4 in [2]. The only difference is the strong convergence of the pressure term qα to the pressure term p of the Navier–Stokes equations. We will use Layton–Lewandowski [2] as a reference and only point out the differences between their proof of convergence to a weak solution of the Navier–Stokes equations and the proof of convergence in our study. First, we need to find estimates that are independent from α . From (2.32) we have that the solution of (1.18)–(1.20) satisfies

t

1 ∥wα ∥21,2 + α 3 ∥wα ∥21+ 1 ,2 ds ∥wα (t )∥ + α ∥wα (t )∥ 1 ,2 + 2ν 6 6 0 t 1 = 2 ⟨f , wα ⟩ds + ∥w0 ∥22 + α 3 ∥w0 ∥21 ,2 , 1 3

2 2

2

6

0

notice that since α → 0 , we can assume that, 0 < α ≤ ℓ, where ℓ is a single length scale; consequently we can bound the right hand side by a constant C which is independent from α . Therefore wα belong to the energy space 1,2 L∞ ([0, T ]; L2 (T3 )3 ) ∩ L2 ([0, T ]; Wdiv ), uniformly with respect to α . From the Aubin–Lions compactness Lemma (the same arguments as in Section 2.1) we can find a subsequence (wαj , qαj ) and (v , p) such that when αj tends to zero we have: +

wαj ⇀∗ v

weakly∗ in L∞ ([0, T ]; L2 (T3 )3 ),

wαj ⇀ v

weakly in L ([0, T ]; W

3

wαj ⇀ v

weakly in Lr

wαj → v

strongly in Lp ([0, T ]; Lp (T3 )3 ),

for all r ≥ 2 and all 2 ≤ p <

10 3

≤ p < . Thus from (2.38) we obtain by identification of weak limits that 3r r wαj ⊗ wαj ⇀ v ⊗ v weakly in L 2 0, T ; L 3r −4 (T3 )3×3 , 4 3

(2.47) (2.48) (2.49) (2.50)

and

strongly in Lp ([0, T ]; Lp (T3 )3×3 ),

wαj ⊗ wαj → v ⊗ v for all

1 ,2

(T3 ) ), 6r 0, T ; L 3r −4 (T3 )3 ,

2

(2.51)

5 3

(2.52)

for all r ≥ 2. Having (2.41) and (2.51) at hand we deduce that strongly in Lp (0, T ; Lp (T3 )3×3 )

wαj ⊗ wαj → v ⊗ v for all

≤ p < 53 . Then, from (2.38) and (2.53) we deduce that 3r r wαj ⊗ wαj ⇀ v ⊗ v weakly in L 2 0, T ; L 3r −4 (T3 )3×3

(2.53)

4 3

(2.54)

for all r ≥ 2. Further, we have

q(t ) = R

k,l

wk

αj

αj (t )

wl

,

(2.55)

where the linear map R is defined by R : Ls (T3 )9 −→ Ls (T3 )

(2.56)

(u )k,l=1,2,3 −→ (−∆) ∂k ∂l (u ) −1

kl

kl

(2.57)

By the theory of Riesz transforms, R is a continuous map for any s ∈]1, ∞[. Consequently, from (2.54) we have T

r

∥qαj ∥ 2 3r dt < ∞, 0

3r −4

for all r ≥ 2.

(2.58)

300

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

From (2.53) we deduce that for almost all t > 0, wαj ⊗ wαj (t ) → v ⊗ v (t ) for all

for all

4 3

strongly in Lp (T3 )3×3

(2.59)

≤ p < 53 . Using the dominate convergence theorem and the continuity of the operator R, we conclude that

qαj → p strongly in Lp ([0, T ]; Lp (T3 ))

(2.60)

≤ p < 53 . Finally we deduce from (2.60) and (2.58) that 3r r qαj ⇀ p weakly in L 2 0, T ; L 3r −4 (T3 ) ,

(2.61)

4 3

for all r ≥ 2. These convergence results allow us to prove in the same way as in [2] that (v , p) is a weak solution to the Navier–Stokes equations, so we will not repeat it. 3. Application to the LES for magnetohydrodynamic (LES for MHD) equations In this section, we consider the critical LES regularization for magnetohydrodynamic (LES for MHD) equations, given by div w = div W = 0

(3.1)

1

∂t w − ν1 ∆w + div w ⊗ w 6 − div ∂t W − ν2 ∆W + div

1 w⊗W6

W ⊗W

− div

1 6

+ ∇ q = 0,

1 W ⊗w6

(3.2)

= 0,

(3.3)

w dx = 0,

W dx = T3

(3.4)

T3

W (0) = W0 ,

w (0) = w0 ,

(3.5)

Here, the unknowns are the averaging fluid velocity field w (t , x), the averaging fluid pressure q(t , x), and the averaging magnetic field W (t , x). Note that when α = 0, we formally retrieve the MHD equations [25,26]. They showed in both papers [25,26] that the classical properties of the Navier–Stokes equations can be extended to the MHD system but as the Navier–Stokes equations the problem of the global existence and uniqueness of the solutions of the three-dimensional MHD equations are open. The aim in this section is to extend the results of existence uniqueness and convergence established above for the LES for NSE to the LES for MHD. We know, thanks to the work [15], that for θ = 1 these results hold true. Further, when θ = 16 , we proved in the above section the existence of a unique ‘‘regular’’ weak solution to the LES for NSE. Therefore, it is intersecting to find the critical value of regularization needed to establish global in time existence of a unique ‘‘regular’’ weak solution to LES for MHD. Existence and uniqueness of solutions of other modifications of the MHD equations have been studied in [27–29]. We divide this section into two subsections. One is devoted to prove the existence of a unique ‘‘regular’’ weak solution to the LES for MHD with θ = 61 . The second one is devoted to prove that this solution converges to a weak solution to the MHD equations when α tends to zero. 3.1. Existence and uniqueness results for the LES for MHD First, we establish the global existence and uniqueness of solutions for the LES for MHD equations with θ = We have the following theorem: Theorem 3.1. Assume that θ = solution to (3.1)–(3.5) such that w, W ∈ C

1

,2

6 0, T ; Wdiv

5

1 . 6

1

q ∈ L2

1 0, T ; W 6 ,2 (T3 ) ,

,2

6 Assume w0 and W0 are both in Wdiv . Then there exist (w , W , q) a unique ‘‘regular’’ weak

1 + 1 ,2 ∩ L2 0, T ; Wdiv 6 ,

w,t , W,t ∈ L2 0, T ; W − 6 ,2 ,

1 . 6

(3.7) q = 0,

T3

(3.6)

(3.8)

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

301

fulfill T

⟨w,t , ϕ⟩ − (w ⊗ w , ∇ϕ) + (W ⊗ W , ∇ϕ) + ν1 (∇ w , ∇ϕ) dt = 0 0 T

⟨W,t , ϕ⟩ − (w ⊗ W , ∇ϕ) + (W ⊗ w , ∇ϕ) + ν2 (∇ w , ∇ϕ) dt = 0 5 ,2 6 . for all ϕ ∈ L2 0, T ; Wdiv

0

(3.9)

Moreover, w (0) = w0

and

W (0) = W0 .

(3.10)

Note that the time derivative terms in (3.9) are defined in the sense of duality pairing on W in (3.9) are defined in the sense of duality pairing on W

−1 ,2 6

×W

1 ,2 6

−5 ,2 6

5

× W 6 ,2 . The other terms

.

Proof of Theorem 3.1. We only sketch the proof since is similar to the Navier–Stokes equations case. The proof is obtained 1

1

1

1

by taking the inner product of (3.2) with α 3 (−∆) 6 w +w, (3.3) with α 3 (−∆) 6 W +W and then adding them, the existence of a solution to the critical LES for MHD can be derived thanks to the Galerkin method. Notice that (w , W ) satisfy the following estimates

1 1 ∥w (t )∥22 + α 3 ∥w (t )∥21 ,2 + ∥W (t )∥22 + α 3 ∥W (t )∥21 ,2 6 6 t t 1 1 2 2 + 2ν1 ∥w ∥1,2 + α 3 ∥w ∥1+ 1 ,2 ds + 2ν2 ∥W ∥21,2 + α 3 ∥W ∥21+ 1 ,2 ds 6

0

6

0

1 1 = ∥w0 ∥22 + α 3 ∥w0 ∥21 ,2 + ∥W0 ∥22 + α 3 ∥W0 ∥21 ,2 . 6

(3.11)

6

The averaging pressure q is reconstructed from w and W (as we work with periodic boundary conditions) and its regularity 1

results from the fact that w ⊗ w and W ⊗ W ∈ L2 [0, T ]; W 6 ,2 (T3 )3×3 .

It remains to prove the uniqueness. Let (w1 , W1 , q1 ) and (w2 , W2 , q2 ), be two solutions, δ w = w2 − w1 , δ W = W2 − W1 ,

δ q = q2 − q1 . Then one has

∂t δ w − ν1 ∆δ w + div (w2 ⊗ w2 ) − div (w1 ⊗ w1 ) − div (W2 ⊗ W2 ) + div (W1 ⊗ W1 ) + ∇δ q = 0, ∂t δ W − ν2 ∆δ W + div (w2 ⊗ W2 ) − div (w1 ⊗ W1 ) − div (W2 ⊗ w2 ) + div (W1 ⊗ w1 ) = 0, 1

(3.12)

1

and δ w = 0, δ W = 0 at initial time. One can take α 3 (−∆) 6 δ w + δ w as test in the first equation of (3.12) and 1 3

1 6

α (−∆) δ W + δ W as test in the second equations of (3.12). Since w1 is divergence-free we have T

0

w1 ⊗ δ w : ∇δ w = −

T

0

T3

(w1 · ∇)δ w · δ w = 0,

(3.13)

T3

Thus we obtain by using the fact that the averaging operator commutes with differentiation under periodic boundary conditions T

0

1

1

div (w2 ⊗ w2 ) − div (w1 ⊗ w1 ) · α 3 (−∆) 6 δ w + δ w

T3 T

(div (w2 ⊗ w2 ) − div (w1 ⊗ w1 )) · δ w

= 0

T3 T

δ w ⊗ w2 : ∇δ w .

=− 0

(3.14)

T3

Similarly, because w1 is divergence-free we have T

0

w1 ⊗ δ W : ∇δ W = − T3

T

0

(w1 · ∇)δ W · δ W = 0, T3

(3.15)

302

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

and thus we have the following identity T

0

1

1

div (w2 ⊗ W2 ) − div (w1 ⊗ W1 ) · α 3 (−∆) 6 δ W + δ W

T3 T

(div (w2 ⊗ W2 ) − div (w1 ⊗ W1 )) · δ W

= 0

T3 T

δ w ⊗ W2 : ∇δ W .

=− 0

(3.16)

T3

Concerning the remaining terms we get by integrations by parts and by using the fact that the averaging operator commutes with differentiation under periodic boundary conditions T

0

1 1 −div (W2 ⊗ W2 ) + div (W1 ⊗ W1 ) · α 3 (−∆) 6 δ w + δ w T3 T

(−div (W2 ⊗ W2 ) + div (W1 ⊗ W1 )) · δ w

= 0

T3 T

W1 ⊗ δ W : ∇δ w + δ W ⊗ W2 : ∇δ w

= 0

(3.17)

T3

and similarly T

0

1 1 −div (W2 ⊗ w2 ) + div (W1 ⊗ w1 ) · α 3 (−∆) 6 δ W + δ W T3 T

(div (W2 ⊗ w2 ) − div (W1 ⊗ w1 )) · δ W

= 0

T3 T

−(W1 · ∇)δ w · δ W + δ W ⊗ w2 : ∇δ W .

= 0

(3.18)

T3

Therefore by adding (3.14)–(3.18) and using the fact that the averaging operator commutes with differentiation under periodic boundary conditions we obtain d

d 1 1 1 1 ∥δ w ∥22 + α 3 ∥∇ 6 δ w ∥22 + ∥δ W ∥22 + α 3 ∥∇ 6 δ W ∥22 2dt 2dt 1 1 1 2 1+ 16 3 + ν1 ∥∇δ u∥2 + α ∥∇ δ u∥22 + ν2 ∥∇δ B ∥22 + α 3 ∥∇ 1+ 6 δ B ∥22 = δ w ⊗ w2 : ∇δ w + δ w ⊗ W2 : ∇δ W − δ W ⊗ W2 : ∇δ w − δ W ⊗ w2 : ∇δ W . T3

T3

T3

(3.19)

T3

By the norm duality

≤ ∥δ w ⊗ w2 ∥ 1 ∥∇δ w ∥ 1 , δ w ⊗ w : ∇δ w 2 ,2 − 6 ,2 6 T3 δ w ⊗ W2 : ∇δ W ≤ ∥δ w ⊗ W2 ∥− 1 ,2 ∥∇δ W ∥ 1 ,2 , 6 6 T3 δ W ⊗ W2 : ∇δ w ≤ ∥δ W ⊗ W2 ∥− 1 ,2 ∥∇δ w ∥ 1 ,2 ,

(3.21)

δ W ⊗ w2 : ∇δ W ≤ ∥δ W ⊗ w2 ∥− 1 ,2 ∥∇δ W ∥ 1 ,2 . 6 6

(3.23)

6

T3

(3.20)

(3.22)

6

T3

By Young’s inequality,

≤ δ w ⊗ w : ∇δ w 2 T3

δ w ⊗ W2 : ∇δ W ≤ T3

1 1

α 3 ν1

1

∥δ w ⊗ w2 ∥2− 1 ,2 +

1 1

α 3 ν2

α 3 ν1

6

4

∥∇δ w ∥21 ,2 ,

(3.24)

6

1

2

∥δ w ⊗ W2 ∥− 1 ,2 + 6

α 3 ν2 4

∥∇δ W ∥21 ,2 , 6

(3.25)

H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

≤ δ W ⊗ W : ∇δ w 2 T3

δ W ⊗ w2 : ∇δ W ≤ T3

303

1

1

α 3 ν1

∥δ W ⊗ W2 ∥2− 1 ,2 +

1 3

α ν1

4

6

∥∇δ w ∥21 ,2 ,

(3.26)

∥∇δ W ∥21 ,2 .

(3.27)

6

1

1

2

1

α 3 ν2

∥δ W ⊗ w2 ∥− 1 ,2 +

α 3 ν2 4

6

6

By Hölder’s inequality combined with Sobolev injection 1 1 3

α ν1 1 1 3

α ν2 1 1 3

α ν1 1 1 3

α ν2

∥δ w ⊗ w2 ∥2− 1 ,2 ≤ 6

∥δ w ⊗ W2 ∥2− 1 ,2 ≤ 6

∥δ W ⊗ W2 ∥2− 1 ,2 ≤ 6

∥δ W ⊗ w2 ∥2− 1 ,2 ≤ 6

1 1 3

α ν1

∥δ w ∥21 ,2 ∥w2 ∥21+ 1 ,2

(3.28)

∥δ w ∥21 ,2 ∥W2 ∥21+ 1 ,2

(3.29)

∥δ W ∥21 ,2 ∥W2 ∥21+ 1 ,2

(3.30)

∥δ W ∥21 ,2 ∥w2 ∥21+ 1 ,2

(3.31)

1 1 3

α ν2

6

1 1 3

α ν1 1 1 3

α ν2

6

6

6

6

6

6

6

Hence, d

1 d ∥δ w ∥22 + ∥δ W ∥22 + α 3 ∥δ w ∥21 ,2 + ∥δ W ∥21 ,2 2dt 2dt 6 6 1 + min(ν1 , ν2 ) ∥δ w ∥21,2 + ∥δ W ∥21,2 + α 3 min(ν1 , ν2 ) ∥δ w ∥21+ 1 ,2 + ∥δ W ∥21+ 1 ,2 6

≤

1

1

α 3 min(ν1 , ν2 )

6

∥δ w ∥21 ,2 + ∥δ W ∥21 ,2 ∥w2 ∥21+ 1 ,2 + ∥W2 ∥21+ 1 ,2 6

6

6

We conclude that δ u = δ B = 0 thanks to Grönwall’s Lemma.

(3.32)

6

3.2. Limit consistency for the critical LES for MHD Next, we will deduce that the LES for MHD with critical regularization gives rise to a weak solution to the MHD equations. Theorem 3.2. Let (wα , Wα , qα ) be the solution of (3.1)–(3.5) for a fixed α . There is a subsequence αj such that the triplet (wαj , Wαj , qαj ) converges to (v , B , p) as j → ∞. This convergence holds in the same sense of Theorem 2.2, where (v , W , p) ∈ 5 5 1,2 2 [L∞ ([0, T ]; L2 (T3 )3 )∩ L2 ([0, T ]; Wdiv )] × L 3 [0, T ]; L 3 (T3 ) is a weak solution of the MHD equations with periodic boundary conditions and zero mean value constraint. Proof of Theorem 3.2. As in the proof of Theorem 2.2 we can show that for all

4 3

≤p<

5 3

we have

wα ⊗ wα → w ⊗ w

strongly in Lp (0, T ; Lp (T3 )3×3 ),

(3.33)

wα ⊗ Wα → w ⊗ B

strongly in Lp (0, T ; Lp (T3 )3×3 ),

(3.34)

Wα ⊗ w α → B ⊗ w

strongly in L (0, T ; L (T3 )

),

(3.35)

Wα ⊗ Wα → B ⊗ B

strongly in Lp (0, T ; Lp (T3 )3×3 ).

(3.36)

p

p

3×3

p p

The above L L convergences combined with the fact that wα and Wα belong to the energy space of the solutions of the Navier–Stokes equations and the Aubin–Lions compactness Lemma allow us to take the limit α → 0 in order to deduce that (wα , Wα , qα ) converge to (v , B , p) a weak solution to the MHD equations. The rest can be done in exactly way as in [2], so we omit the details. Acknowledgments The author thanks Prof. R. Lewandowski and Prof. W. Layton for their interesting discussions about this paper as well as the anonymous referee for his helpful suggestions and insightful comments.

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H. Ali / J. Math. Anal. Appl. 394 (2012) 291–304

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