Large eddy simulation for turbulent magnetohydrodynamic flows

Large eddy simulation for turbulent magnetohydrodynamic flows

J. Math. Anal. Appl. 377 (2011) 516–533 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com...

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J. Math. Anal. Appl. 377 (2011) 516–533

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Large eddy simulation for turbulent magnetohydrodynamic flows A. Labovsky a,∗,1 , C. Trenchea b,2 a b

Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, United States Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 26 April 2010 Available online 16 November 2010 Submitted by J. Guermond Keywords: Large eddy simulation Magnetohydrodynamics Deconvolution

We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ1 , δ2 tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD. © 2010 Elsevier Inc. All rights reserved.

1. Introduction Magnetically conducting fluids arise in important applications including plasma physics, geophysics and astronomy. In many of these, turbulent MHD (magnetohydrodynamics [2]) flows are typical. The difficulties of accurately modeling and simulating flows are magnified many times over in the MHD case. They are evinced by the more complex dynamics of the flow due to the coupling of Navier–Stokes and Maxwell equations via the Lorentz force and Ohm’s law. The flow of an electrically conducting fluid is affected by Lorentz forces, induced by the interaction of electric currents and magnetic fields in the fluid. The Lorentz forces can be used to control the flow and to attain specific engineering design goals such as flow stabilization, suppression or delay of flow separation, reduction of near-wall turbulence and skin friction, drag reduction and thrust generation. There is a large body of literature dedicated to both experimental and theoretical investigations on the influence of electromagnetic force on flows (see e.g. [23,35,36,22,52,16,53,24,46,8]). The MHD equations are related to engineering problems such as plasma confinement, controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, MHD sea water propulsion. The MHD effects arising from the macroscopic interaction of liquid metals with applied currents and magnetic fields are exploited in metallurgical processes to control the flow of metallic melts: the electromagnetic stirring of molten metals [37], electromagnetic turbulence control in induction furnaces [54], electromagnetic damping of buoyancy-driven flow during solidification [41], and the electromagnetic shaping of ingots in continuous casting [43]. The turbulent flow of an electrically and magnetic conducting fluid is more complex than the turbulent flow of a nonconducting fluid and has more parameter regimes. The invariants of 3D MHD are the total energy (velocity and magnetic

* 1 2

Corresponding author. E-mail address: [email protected] (A. Labovsky). Partially supported by the US Air Force Office of Scientific Research under grant number FA9550-08-1-0415. Partially supported by Air Force grant FA9550-09-1-0058.

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.10.070

©

2010 Elsevier Inc. All rights reserved.

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

517

field), the magnetic and cross helicity (see [14,28]). Although the kinetic helicity is a rugged invariant for 3D Euler flows, it is not one for MHD systems, but still an important quantity (see [39]). The magnetic helicity is not conserved when a mean magnetic field is present, see e.g. [34,47,48,7,38]. Note that a strong alignment of the vorticity with the Lorentz force or the velocity and the curl of the Lorentz force is likely to produce a sizabile change in u · (∇ × u ). Also, a flow that is instantaneously nonhelical and/or irrotational will not remain so if ∇ × ( j × B ) has a nonzero projection on the velocity. The mathematical description of the problem proceeds as follows. Assuming the fluid to be viscous and incompressible, the governing equations are the Navier–Stokes and pre-Maxwell equations, coupled via the Lorentz force and Ohm’s law (see e.g. [45]). Let Ω = (0, L )3 be the flow domain, and u (t , x), p (t , x), B (t , x) be the velocity, pressure, and the magnetic field of the flow, drived by the velocity body force f and magnetic field force curl g. Then u , p , B satisfy the MHD equations:





ut + ∇ · uu T − Bt +

1

Rem ∇ · u = 0,

1 Re

S

u +

2

  ∇( B · B ) − S ∇ · B B T + ∇ p = f ,

curl(curl B ) + curl( B × u ) = curl g ,

∇ · B = 0,

(1.1)

in Q = (0, T ) × Ω , with the initial data:

u (0, x) = u 0 (x),

B (0, x) = B 0 (x)

in Ω,

(1.2)

and with periodic boundary conditions (with zero mean):



Φ(t , x + L ei ) = Φ(t , x),

i = 1, 2, 3,

Φ(t , x) dx = 0,

(1.3)

Ω

for Φ = u , u 0 , p , B , B 0 , f , g. Here Re, Rem , and S are nondimensional constants that characterize the flow: the Reynolds number, the magnetic Reynolds number and the coupling number, respectively. For derivation of (1.1), physical interpretation and mathematical analysis, see [12,10,26,44,21] and the references therein. Denote the modified pressure P := 2S | B |2 + p. If whiteaδ1 , whiteaδ2 denote two local, spacing averaging operators

that commute with the differentiation, then averaging (1.1) gives the following non-closed equations for u δ1 , B δ2 , P δ1 in (0, T ) × Ω :





δ

ut 1 + ∇ · uu T δ1 − δ2

Bt +

1 Re m



1 Re

  u δ1 − S ∇ · B B T δ1 + ∇ P δ1 = f δ1 , 









curl curl B δ2 + ∇ · Bu T δ2 − ∇ · u B T δ2 = curl g δ2 ,

∇ · u δ2 = 0,

∇ · B δ2 = 0.

(1.4)

Note that we have replaced the term ∇ × ( B × u ) with its equivalent ∇ · ( Bu ) − ∇ · (u B ), using the continuity equation. The usual closure problem which we study here arises because uu T δ1 = u δ1 u T δ1 , B B T δ1 = B δ1 B T δ1 , u B T δ2 = u δ1 B T δ2 . To isolate the turbulence closure problem from the difficult problem of wall laws for near wall turbulence, we study (1.1) hence (1.4) subject to (1.3). The closure problem is to replace the tensors uu T δ1 , B B T δ1 , u B T δ2 with tensors T (u δ1 , u δ1 ), T ( B δ2 , B δ2 ), T (u δ1 , B δ2 ), respectively, depending only on u δ1 , B δ2 and not u , B. There are many closure models proposed in large eddy simulation reflecting the centrality of closure in turbulence simulation. Calling w , q, W the resulting approximations to u δ1 , P δ1 , B δ2 , we are led to considering the following model T

wt + ∇ · T (w , w ) − Wt +

1

Re m ∇ · w = 0,

1 Re

T

 w − ∇ · S T ( W , W ) + ∇ q = f δ1 ,

curl(curl W ) + ∇ · T ( W , w ) − ∇ · T ( w , W ) = curl g δ2 ,

∇ · W = 0.

(1.5)

With any reasonable averaging operator, the true averages u δ1 , B δ2 , P δ1 are smoother than u , B , P . We consider the simplest, accurate closure model that is exact on constant flows (i.e., u δ1 = u , B δ2 = B) is





uu T δ1 ≈ u δ1 u T δ1 δ1 =: T u δ1 , u δ1 ,









B B T δ1 ≈ B δ2 B T δ2 δ1 =: T B δ2 , B δ2 , u B T δ2 ≈ u δ1 B T δ2 δ2 =: T u δ1 , B δ2 , leading to

(1.6)

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533





w t + ∇ · w w T δ1 − Wt +

1

Rem ∇ · w = 0,

1 Re

   w − S ∇ · W W T δ1 + ∇ q = f δ1 , 





(1.7a)



curl(curl W ) + ∇ · W w T δ2 − ∇ · w W T δ2 = curl g δ2 ,

(1.7b)

∇ · W = 0,

(1.7c)

δ1

δ2

subject to w (x, 0) = u 0 (x), W (x, 0) = B 0 (x) and periodic boundary conditions (with zero means). The first to introduce a regularization of the 3D Navier–Stokes equations was Leray [29], who proved that its solution converges to the weak solution of the 3D NSE. Recently such analysis was done for numerous regularizations in [27]. For the MHD turbulence, Linshiz and Titi [32] studied the NS-α regularization of the momentum equation, with no averaging of the other MHD system’s couple equations. The Lagrangian-averaged magnetohydrodynamics-α model proposed in [19] is also conserving the Alfvén waves. In this report we show that the LES MHD model (1.7) has the mathematical properties (conservation of kinetic energy, magnetic helicity, approximate conservation of the cross helicity, preservation of Alfvén waves) expected of a model derived from the MHD equations by an averaging operation. The model considered can be developed for quite general averaging operators, see e.g. [1,42,25,9,30,31]. The choice of averaging operator in (1.7) is a differential filter due to Germano [17]. Let the δ > 0 denote the averaging radius, related to the finest computationally feasible mesh. (We use different lengthscales for the Navier–Stokes and Maxwell equations, see e.g. [40] for the treatment of large eddy simulation of stratified flows). Given φ ∈ L 20 (Ω), φ δ ∈ H 2 (Ω) ∩ L 20 (Ω) is the unique solution of

A δ φ δ := −δ 2 φ δ + φ δ = φ

in Ω,

(1.8)

subject to periodic boundary conditions. Under periodic boundary conditions, this averaging operator commutes with differentiation, and with this averaging operator, the model (1.6) has consistency O (δ 2 ), i.e.,





uu T δ1 = u δ1 u T δ1 + O δ1 2 ,





B B T δ1 = B δ2 B T δ2 δ1 + O δ2 2 ,





u B T δ2 = u δ1 B T δ2 δ2 + O δ1 2 + δ2 2 , for smooth u , B. We prove that the model (1.7) has a unique, weak solution w , W that converges in the appropriate sense w → u, W → B, as δ1 , δ2 → 0. In Section 2 we address the question of global existence and uniqueness of the solution for the closed MHD model. Section 2.3 treats the limit consistency of the model and verifiability. The conservation of the kinetic energy and helicity for the approximate deconvolution model is presented in Section 3. Section 4 shows that the model preserves the Alfvén waves, with the velocity tending to the velocity of Alfvén waves in the MHD, as the radii δ1 , δ2 tend to zero. Finally, Section 5 presents the computational results: we apply the LES-MHD model to the two-dimensional Chorin’s problem and verify the predicted accuracy of the model. We also compare the conserved quantities: plot the energy of the model vs. the energy of the averaged MHD. 2. Well-posedness of the LES-MHD model 2.1. Notations and preliminaries We shall use the standard notations for function spaces in the space periodic case (see [51]). Let H m p (Ω) denote the space of functions (and their vector-valued counterparts also) that are locally in H m (R3 ), are periodic of period L and have zero mean, i.e. satisfy (1.3). We recall the solenoidal space D(Ω) = {φ ∈ C ∞ (Ω): φ periodic with zero mean, ∇ · φ = 0}, and the closures of D(Ω) in the usual L 2 (Ω) and H 1 (Ω) norms:



H = φ ∈ H 20 (Ω), ∇ · φ = 0 in D(Ω)

2

,



V = φ ∈ H 21 (Ω), ∇ · φ = 0 in D(Ω)

2

.

Definition 2.1. Let (u 0 δ1 , B 0 δ2 ) ∈ H , f δ1 , curl g δ2 ∈ L 2 (0, T ; V ). The measurable functions w , W : [0, T ] × Ω → R3 are the weak solutions of (1.7) if w , W ∈ L 2 (0, T ; V ) ∩ L ∞ (0, T ; H ), and w , W satisfy



t  w (t )φ dx +

1 Re

∇ w (τ )∇φ + w (τ ) · ∇ w (τ )δ1 φ − S W (τ ) · ∇ W (τ )δ1 φ dx dτ

0 Ω

Ω

 =

t  u 0 φ dx + δ1

Ω

f (τ )δ1 φ dx dτ , 0 Ω

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533



t 

1

W (t )ψ dx +

Rem

0 Ω

Ω

 =

519

∇ W (τ )∇ψ + w (τ ) · ∇ W (τ )δ2 ψ − W (τ ) · ∇ w (τ )δ2 ψ dx dτ

t  B 0 δ2 ψ dx +

curl g (τ )δ2 ψ dx dτ ,

(2.1)

0 Ω

Ω

∀t ∈ [0, T ), φ, ψ ∈ D(Ω). Also, it is easy to show that for any u , v ∈ H 1 (Ω) with ∇ · u = ∇ · v = 0, the following identity holds

∇ × (u × v ) = v · ∇ u − u · ∇ v .

(2.2)

2.2. Existence and uniqueness of a solution The first result states that the weak solution of the MHD LES model (1.7) exists globally in time, for large data and general Re, Rem > 0 and that it satisfies an energy equality while initial data and the source terms are smooth enough. Theorem 2.2. Let δ1 , δ2 > 0 be fixed. For any (u 0 δ1 , B 0 δ2 ) ∈ V and ( f δ1 , curl g δ2 ) ∈ L 2 (0, T ; H ), there exists a unique weak solution w , W to (1.7). The weak solution also belongs to L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) and w t , W t ∈ L 2 ((0, T ) × Ω). Moreover, the following energy equality holds for t ∈ [0, T ]:

t E (t ) +

t

ε(τ ) dτ = E (0) + 0

P(τ ) dτ , 0

where

E (t ) =

δ1 2   2

 2 1   2 δ2 2 S    ∇ W (t , ·)2 + S  W (t , ·)2 , ∇ w (t , ·)0 +  w (t , ·)0 + 0 0 2

2

2

2        δ1 2   w (t , ·)2 + 1 ∇ w (t , ·)2 + δ2 S  W (t , ·)2 + S ∇ W (t , ·)2 , ε(t ) = 0 0 0 0 Re Re Rem Rem   δ   δ2 1 P(t ) = f (t ), w (t ) + S curl g (t ), W (t ) .

The proof, using the semigroup approach proposed in [6] for the Navier–Stokes equations, is given in Appendix A, along with a regularity result. Remark 2.1. The modified pressure is recovered from the weak solution via the classical DeRham theorem (see [29]). 2.3. Accuracy of the model We address now the question of consistency, i.e., we show that when δ1 , δ2 go to zero, the solution of the closed model (1.7) converges to a weak solution of the MHD equations (1.1). Theorem 2.3. For any two sequences δ1n , δ2n → 0 as n → ∞, the corresponding solution of (1.7) satisfies

( w δ1n , W δ2n , qδ1n ) → (u , B , P ), 4

where (u , B , P ) ∈ L ∞ (0, T ; H ) ∩ L 2 (0, t ; V ) × L 3 (0, T ; L 2 (Ω)) is a weak solution of the MHD equations (1.1). The sequences 4 3

{ w δ1n }n∈N , { W δ2n }n∈N converge strongly to u , B in L (0, T ; L 2 (Ω)) and weakly in L 2 (0, T ; H 1 (Ω)), respectively, while {qδ1n }n∈N con4

verges weakly to P in L 3 (0, T ; L 2 (Ω)). Proof. The proof follows that of Theorem 3.1 in [27], and is an easy consequence of Theorem 2.4 and Proposition 2.6. Let

2

τu , τ B , τ Bu denote the model’s consistency errors

τu = u δ1 u δ1 ,T − uu T ,

τ B = B δ2 B δ2 ,T − B B T ,

τ Bu = B δ2 u δ1 ,T − Bu T ,

(2.3)

where u , B is a solution of the MHD equations obtained as a limit of a subsequence of the sequence w δ1 , W δ2 . We prove that u δ1 − w L ∞ (0, T ; L 2 ( Q )) , B δ2 − W L ∞ (0, T ; L 2 ( Q )) are bounded by τu L 2 ( Q T ) , τ B L 2 ( Q T ) , τ Bu L 2 ( Q T ) .

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

Theorem 2.4. Under the assumption (u , B ) ∈ L 4 (0, T ; V ), the errors e = u δ1 − w, E = B δ2 − W satisfy

    e (t )2 + S  E (t )2 + 0

t 

0

   ∇ e (s)2 + S curl E (s)2 ds 0 0

1  Re

Rem

0

t  C Φ(t )



0

where Φ(t ) = exp{Re3





2

2 

Reτu (s) + S τ B (s)0 + Rem τ Bu (s) − τ Bu T (s)0 ds,

t 0

∇ u 40 ds, Re3m

t

∇ u 40 ds + Rem Re2

0

t 0

(2.4)

∇ B 40 }.

Proof. The errors e , E satisfy the following momentum equation





et + ∇ · u δ1 u δ1 , T − w w T δ1 − Et +

1 Re m

curl curl E + ∇ ·



1 Re

      e + S ∇ · B δ2 B δ2 ,T − W W T δ1 + ∇ P δ1 − q = ∇ · τ δu1 + S τ δB1 ,

B δ2 u δ1 , T

    2  2 T , − W w T δ2 − ∇ · u δ1 B δ2 ,T − w W T δ2 = ∇ · τ δBu − τ δBu

along with the corresponding conservation of mass equation and homogeneous boundary conditions. Taking the inner product with A δ1 e , S A δ2 E, respectively, we obtain after some calculation that

d

 δ2 δ2 S 1 S e 20 + S E 20 + δ12 ∇ e 20 + S δ22 curl E 20 + ∇ e 20 + curl E 20 + 1 e 20 + 2 curl curl E 20 dt Re Rem Re Rem   2       −e · ∇ u δ1 e − S ∇ · E B δ2 e − S ∇ · Eu δ1 E + Se · ∇ B δ2 E dx + Re τu + S τ B 20 + Rem τ Bu − τ Bu T 0  Ω

     3/2 1/2  1/2 1/2  1/2 3/2   C ∇ e 0 e 0 ∇ u δ1 0 + 2S E 0 ∇ E 0 ∇ B δ2 0 ∇ e 0 + S E 0 ∇ E 0 ∇ u δ1 0  2 + Re τu + S τ B 20 + Rem τ Bu − τ Bu T 0 . Using Young’s and Gronwall’s inequality we deduce

    e (t )2 + S  E (t )2 + 0

t 

0

   ∇ e (s)2 + S curl E (s)2 ds 0 0

1  Re

Rem

0

t  C Ψ (t )







2

2 

Reτu (s) + S τ B (s)0 + Rem τ Bu (s) − τ Bu T (s)0 ds,

0

where

Ψ (t ) = exp Re

t 3

 δ 4 ∇ u 1  ds, Re3

t

m

0

0

 δ 4 ∇ u 1  ds + Rem Re2

t

0

0

 δ 4 ∇ B 2  ds . 0

0

The use of the stability bounds ∇ u δ1 0  ∇ u 0 , ∇ B δ2 0  ∇ B 0 concludes the proof.

2

Finally we give bounds on the consistency errors (2.3) as δ1 , δ2 → 0 in L 1 ((0, T ) × Ω) and L 2 ((0, T ) × Ω). Proposition 2.5. Assuming ( f , curl g ) ∈ L 2 (0, T ; V ), then

τu L 1 (0,T ; L 1 (Ω))  23/2 δ1 T 1/2 Re1/2 C ( T ), 1/2

τ B L 1 (0,T ; L 1 (Ω))  23/2 δ2 T 1/2 τ Bu L 1 (0,T ; L 1 (Ω))  21/2 T 1/2 where

Rem

1 S

S

C ( T ),

1/2  δ1 Re1/2 +δ2 Rem C ( T ),

 Rem 2 2 2 2 C ( T ) = u 0 0 + S B 0 0 + Re f L 2 (0,T ; H −1 (Ω)) + curl g L 2 (0,T ; H −1 (Ω)) . S

(2.5)

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

521

Proof. Using (1.8) and the stability bounds we obtain

 δ       u 1 − u 2  δ 2 ∇ u δ1  ∇ u δ1 − u   2δ 2 ∇ u 2 , 0 0 0 hence by (2.3) and Cauchy–Schwarz inequality we have

    τu L 1 (0,T ; L 1 (Ω))  u + u δ1  L 2 (0,T ; L 2 (Ω)) u δ1 − u  L 2 (0,T ; L 2 (Ω)) √  2 u L 2 (0,T ; L 2 (Ω)) 2δ1 ∇ u L 2 (0,T ; L 2 (Ω)) . Similarly

    τ B L 1 (0,T ; L 1 (Ω))   B + B δ2 L 2 (0,T ; L 2 (Ω))  B δ2 − B  L 2 (0,T ; L 2 (Ω)) √  2 B L 2 (0,T ; L 2 (Ω)) 2δ2 ∇ B L 2 (0,T ; L 2 (Ω)) ,       τ Bu L 1 (0,T ; L 1 (Ω))   B δ2 − B L 2 ( Q ) u δ1  L 2 ( Q ) + B L 2 ( Q ) u δ1 − u L 2 ( Q ) √ √  2δ2 ∇ B L 2 ( Q ) u L 2 ( Q ) + 2δ1 ∇ u L 2 ( Q ) B L 2 ( Q ) . The classical energy estimates for the MHD system (1.1) (the a priori estimates can be found, e.g., in [44]) yield (2.5).

2

Assuming more regularity on (u , B ) leads to the sharper bounds on the consistency errors. Remark 2.2. Let (u , B ) ∈ L 2 (0, T ; H 2 (Ω)). Then

τu L 1 (0,T ; L 1 (Ω))  C δ12 , τ B L 1 (0,T ; L 1 (Ω))  C δ22 ,   τ Bu L 1 (0,T ; L 1 (Ω))  C δ12 + δ22 , where C = C ( T , Re, Rem , (u , B ) L 2 (0, T ; L 2 (Ω)) , (u , B ) L 2 (0, T ; H 2 (Ω)) ). Proof. The result is obtained as in the proof of Proposition 2.5, using the bounds

 δ  u 1 − u  2  δ12 u L 2 (0,T ; L 2 (Ω)) , L (0, T ; L 2 (Ω))   δ B 2 − B 2  δ 2  B 2 , 2 2 L (0, T ; L (Ω))

which follow from (1.8).

L (0, T ; L (Ω))

2

2

Next we estimate the L 2 -norms of the consistency errors filtering errors e , E.

τu , τ B , τ Bu , which were used in Theorem 2.4 to estimate the

Proposition 2.6. If the solution u , B of (1.1) satisfies

    (u , B ) ∈ L 4 (0, T ) × Ω ∩ L 2 0, T ; H 2 (Ω) ,

then the model consistency errors satisfy the following bound

τu L 2 ( Q )  C δ1 ,

τ B L 2 ( Q )  C δ2 ,

τ Bu L 2 ( Q )  C (δ1 + δ2 ),

where C = C ( (u , B ) L 4 ((0, T )×Ω) , (u , B ) L 2 (0, T ; H 2 (Ω)) ). Proof. As in the proof of Proposition 2.5, using the stability bounds we have

  τu L 2 ( Q )  2 u L 4 ( Q ) u δ1 − u  L 4 ( Q )

T  δ  3/2 u 1 − u   2 u 4 L (Q )

  δ  ∇ u 1 − u 32 dt L (Ω) L 2 (Ω)

0

T 2

3/2

1/4 4δ14 ∇ u L 2 (Ω) u 3L 2 (Ω) dt

u L 4 ( Q ) 0

 4δ1 u L 4 ( Q ) u L 2 (0,T ; H 1 (Ω)) u L 2 (0,T ; H 2 (Ω)) .

1/4

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

Similarly we deduce

τ B L 2 ( Q )  4δ2 B L 4 ( Q ) B L 2 (0,T ; H 1 (Ω)) B L 2 (0,T ; H 2 (Ω)) , and

    τ Bu L 2 ( Q )  u L 4 ( Q )  B δ2 − B L 4 ( Q ) + B L 4 ( Q ) u δ2 − u L 4 ( Q )  2δ2 u L 4 ( Q ) B L 2 (0,T ; H 1 (Ω)) B L 2 (0,T ; H 2 (Ω)) + 2δ1 B L 4 ( Q ) u L 2 (0,T ; H 1 (Ω)) u L 2 (0,T ; H 2 (Ω)) .

2

As in Remark 2.2, assuming extra regularity on (u , B ) leads to the sharper bounds. Remark 2.3. Let

    (u , B ) ∈ L 4 (0, T ) × Ω ∩ L 4 0, T ; H 2 (Ω) .

Then

τu L 2 ( Q )  C δ12 ,

τ B L 2 ( Q )  C δ22 ,

  τ Bu L 2 ( Q )  C δ12 + δ22 ,

where C = C ( (u , B ) L 4 ((0, T )×Ω) , (u , B ) L 4 (0, T ; H 2 (Ω)) ). 3. Conservation laws It is well known that kinetic energy and helicity are critical in the organization of the flow. We prove now that the model (1.7) inherits some of the original properties of the 3D MHD equations (1.1): it conserves the kinetic energy, magnetic helicity and approximates the cross helicity. E=

Let recall that, in the absence of the forcing, with zero1 kinematic viscosity and magnetic diffusivity, the energy

1 (u (x) · u (x) + S B (x) · B (x)) dx, the cross helicity H C = 2 Ω (u (x) · B (x)) dx and the magnetic helicity H M = 12 Ω (A(x) · 2 Ω B (x)) dx (where A is the vector potential, B = ∇ × A) are the three invariants of the MHD equations (1.1) (see e.g. [14]). We introduce the following characteristic quantities of the model

E LES =

1 2

H C ,LES =

 ( A δ1 w , w ) + S ( A δ2 W , W ) ,

1

( A δ1 w , A δ2 W ), 2 1 δ2  δ2 1 H M ,LES = A δ2 W , A , where A = A − δ2 A. 2 The next result is devoted to proving that these quantities are conserved by (1.7) with the periodic boundary conditions and 1 1 f = g = 0, Re = Re = 0. Also, note that m

E LES → E ,

H C ,LES → H C ,

H M ,LES → H M ,

as δ1,2 → 0.

Theorem 3.1. The following conservation laws hold, ∀ T > 0

E LES ( T ) = E LES (0),

(3.1a)

H C ,LES ( T )  H C ,LES (0) +

C ( T ) max δi2 , i =1,2

H M ,LES ( T ) = H M ,LES (0). Proof. Consider (1.7) with





1 Re

=

(3.1c) 1 Re m

= 0. Multiplying (1.7a), (1.7b) by A δ1 w and S A δ2 W , respectively, and using the identity

(∇ × v ) × u , w = (u · ∇ v , w ) − ( w · ∇ v , u )

we obtain

1 d 2 dt

 ( A δ1 w , w ) + S ( A δ2 W , W ) = S ( W · ∇ W , w ) − S ( w · ∇ W , W ) + S ( W · ∇ w , W ),

which by (1.7c) yields (3.1a):

1 d 2 dt

(3.1b)

 ( A δ1 w , w ) + S ( A δ2 W , W ) = 0.

(3.2)

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523

To prove (3.1b), multiply (1.7a), (1.7b) by A δ1 W and A δ2 w, respectively, and use the identity (u · ∇ v , w ) = −(u · ∇ w , v ) to get



∂ A δ1 w ,W ∂t





+

∂ A δ2 W , w = 0. ∂t

(3.3)

Recall that from (1.8) we have

w = A δ1 w + δ12  w , Then (3.3) gives



∂ A δ1 w , A δ2 W ∂t



W = A δ2 W + δ22  W .



  ∂ A δ2 W ∂ A δ1 w 2 ∂ A δ2 W 2 , A δ1 w = , δ2  W + , δ1  w . + ∂t ∂t ∂t

Hence



d dt

(3.4)

( A δ1 w , A δ2 W ) = δ22

∂ A δ1 w , W ∂t



 + δ12

∂ A δ2 W , w , ∂t

(3.5)

(3.6)

which proves (3.1b). δ2

Next, we multiply (1.7b) by A δ2 A , and integrating over Ω

1 d 2 dt

δ

∇ × A δ2 A 2 , A

δ2 

  δ  δ  + w · ∇ W , A 2 − W · ∇ w , A 2 = 0.

(3.7)

Since the cross-product of two vectors is orthogonal to each of them,



∇ ×A

δ2 

δ2 

× w, ∇ × A

= 0,

(3.8)

it follows from (3.8) and (3.2) that



δ2 

δ2

w · ∇A , ∇ × A

=



δ2 

∇ ×A

 δ · ∇A 2 , w .

(3.9)

δ2

Since W = ∇ × A , from (3.7) and (3.9) we obtain (3.1c).

2

4. Alfvén waves In this section we prove that our model possesses a very important property of the MHD, namely the ability of the magnetic field to transmit transverse inertial waves – Alfvén waves. We follow the argument typically used to prove the existence of Alfvén waves in MHD, see, e.g., [13]. Using the density ρ and permeability μ, we write the equations of the model (1.7) in the form





w t + ∇ · w w T δ1 + ∇ p δ1 =

1

ρμ

(∇ × W ) × W

δ1

− ν ∇ × (∇ × w ),

∂W δ = ∇ × ( w × W ) 2 − η∇ × (∇ × W ), ∂t ∇ · w = 0, ∇ · W = 0, 1 , Re

(4.1a) (4.1b) (4.1c)

1 . Rem

where ν = η= Assume a uniform, steady magnetic field W 0 , perturbed by a small velocity field w. We denote the perturbations in current density and magnetic field by j model and W p , with

∇ × W p = μ j model .

(4.2)

Also, the vorticity of the model is

ωmodel = ∇ × w .

(4.3)

Since w · ∇ w is quadratic in the small quantity w, it can be neglected in the Navier–Stokes equation (4.1a), and therefore

∂w 1 δ + ∇ p δ1 = (∇ × W p ) × W 0 1 − ν ∇ × (∇ × w ). ∂t ρμ

(4.4)

The leading order terms in the induction equation (4.1b) are

∂Wp δ = ∇ × ( w × W 0 ) 2 − η∇ × (∇ × W p ). ∂t

(4.5)

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

Using (4.2), we rewrite (4.4) as

∂w 1 δ1 + ∇ p δ1 = j model × W 0 + ν  w . ∂t ρ

(4.6)

Taking the curl of (4.6), using the identity (2.2) and ∇ W 0 = 0, we obtain from (4.3) that

∂ ωmodel 1 δ1 = W 0 · ∇ j model + ν ωmodel . ∂t ρ

(4.7)

Similarly, taking curl of (4.5) and using (4.2), (4.3) yields

μ

∂ j model δ = W 0 · ∇ ωmodel 2 + ημ j model . ∂t

(4.8)

We now eliminate j model from (4.7) by taking the time derivative of (4.7) and substituting for

∂ j model using (4.8). This yields ∂t

 δ1 ∂ ωmodel ∂ 2 ωmodel 1 1 δ2 = W0 · ∇ W 0 · ∇ ωmodel + η j model + ν . ρ μ ∂t ∂t2 1 The linearity of A − δ1 implies

δ  ∂ 2 ωmodel 1 η ∂ ωmodel δ1 δ2  1 = W · ∇ W · ∇ ω + W 0 · ∇( j model ) + ν  . 0 0 model 2 ρμ ρ ∂t ∂t

(4.9)

To eliminate the term containing  j model from (4.9), we take the Laplacian of (4.7):



1 ∂ ωmodel δ1 = W 0 · ∇( j model ) + ν 2 ωmodel . ∂t ρ

(4.10)

Then from (4.9)–(4.10) we obtain δ  ∂ ωmodel ∂ 2 ωmodel 1 δ2  1 = W · ∇ W · ∇ ω + (η + ν ) − ην 2 ωmodel . 0 0 model ρμ ∂t ∂t2

(4.11)

Next we look for plane-wave solutions of the form

ωmodel ∼ ω0 e i(k·x−θ t ) ,

(4.12)

where k is the wavenumber. It follows from (4.12) that

∂ 2 ωmodel = −θ 2 ωmodel , ∂t2

∂ ωmodel = −i θ ωmodel , ∂t ∂ ωmodel = i θ k2 ωmodel ,  ∂t

2 (ωmodel ) = k4 ωmodel .

The substitution of (4.12) into the wave equation (4.11) gives

−θ 2 ωmodel =

1

ρμ



W 0 · ∇ W 0 · ∇ ωmodel

δ2 

δ1

+ (η + ν )i θ k2 ωmodel − ην k4 ωmodel .

(4.13)

Note that by (1.8) we have

  = W 0 · ∇ ωmodel + O δ22 ,      δ2  W 0 · ∇ W 0 · ∇ ωmodel δ1 = ( W 0 · ∇)2 ωmodel + O δ12 + O δ22 , W 0 · ∇ ωmodel

δ2

therefore

−θ 2 ωmodel =

1

ρμ

  ( W 0 · ∇)2 ωmodel + (η + ν )i θ k2 ωmodel − ην k4 ωmodel + O δ12 + δ22 .

(4.14)

It follows from (4.12) that

( W 0 · ∇)2 ωmodel = − W 02k2 ωmodel ,

(4.15)

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

525

where k is the component of k parallel to W 0 , which by (4.14) implies

−θ 2 = −

W 02 k2

ρμ

  + (η + ν )i θ k2 − ην k4 + O δ12 + δ22 .

Solving this quadratic equation for θ gives the dispersion relationship

θ =−

(η + ν )k2 2

  i±

W 02 k2



ρμ

(ν − η)2k4 4

  + O δ12 + δ22 .

Hence, for a perfect fluid (ν = η = 0) we obtain

θ = ± v˜ a k ,





v˜ a = v a + O δ12 + δ22 ,



where v a is the Alfvén velocity W 0 / ρμ. When ν = 0 and η is small (i.e. for high Rem ) we have

θ = ± v˜ a k −

ηk2 2

i,

which represents a transverse wave with a group velocity equal to ± v a + O (δ12 + δ22 ). In conclusion, model (1.7) preserves the Alfvén waves and the group velocity of the waves v˜ a tends to the true Alfvén velocity v a as the radii tend to zero. 5. Computational results In this section we present the computational results for two LES-MHD models: the two-dimensional Chorin’s model (circular motion in a square) of electrically conducting fluid, and the model of wave propagation. We compare the solution obtained by the LES-MHD model to the average of the known true solution and present the rates of convergence. We also compare the energy of the model to the energy of the averaged MHD. We employ the Backward Euler time discretization along with the finite element discretization in spacial variables, using the Taylor–Hood polynomials (piecewise quadratics for velocity and magnetic field, and piecewise linear for the pressure). We take the filtering widths δ1 = δ2 = h, a typical choice of filtering widths in real life applications, and verify the claimed second order accuracy of the model (the time step is take small enough t = h2 ). First, consider the MHD flow in Ω = (0.5, 1.5) × (0.5, 1.5), with the Reynolds number and magnetic Reynolds number Re = 105 , Rem = 105 , the final time T = 1/4. If we take

f =

1 2 1 2

π sin(2π x)e−4π

2

t /Re

− xe 2t



,

π sin(2π y )e−4π t /Re − ye2t  t 2 e (x − (cos π x sin π y + π x sin π x sin π y + π y cos π x cos π y )e −2π t /Re ) , ∇×g= t 2 e (− y − (sin π x cos π y + π x cos π x cos π y + π y sin π x sin π y )e −2π t /Re ) 2

the solution is



u=

− cos(π x) sin(π y )e −2π

2

t /Re

,

sin(π x) cos(π y )e −2π t /Re  1 2 p = − cos(2π x) + cos(2π y ) e −4π t /Re , 2  xet B= . − yet 2

Although the theoretical results were obtained only for the periodic boundary conditions, we apply the LES-MHD model to the problem with Dirichlet boundary conditions. The results presented were obtained using the software FreeFEM++. The velocity and magnetic field are sought in the finite element space of piecewise quadratic polynomials, and the pressure in the space of piecewise linears. In order to draw conclusions about the convergence rate, we take the time step t = h2 and compare the model’s solution ( w , W ) to the average (u , B ) of the true solution. According to Theorem 2.4 and Remark 2.3, the second order accuracy is expected. The computational results in Table 5.1 verify the claimed accuracy of the model.

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

Table 5.1 Approximating the average solution, Re = 105 , Rem = 105 . h

w − u L 2 (0, T ;L 2 (Ω))

1/4 1/8 1/16 1/32 1/64

0.0247837 0.0245241 0.0131042 0.00434599 0.00120907

rate

W − B L 2 (0, T ;L 2 (Ω))

rate

0.0152 0.9042 1.5923 1.8458

0.0253257 0.0268628 0.0132399 0.00412013 0.001116

−0.085 1.0207 1.6841 1.8844

Table 5.2 Wave propagation test problem, Re = 104 , Rem = 104 . h

w − u L 2 (0, T ;L 2 (Ω))

1/4 1/8 1/16 1/32 1/64

0.0128497 0.00860029 0.00390914 0.0012649 0.000346

rate

W − B L 2 (0, T ;L 2 (Ω))

0.58 1.14 1.63 1.87

0.0114325 0.0051792 0.00187599 0.00055774 0.00014841

rate

w − u L 2 (0, T ; H 1 (Ω))

rate

1.14 1.47 1.75 1.91

0.120428 0.0866042 0.0490398 0.018325 0.005388

0.48 0.82 1.42 1.77

Fig. 5.1. LES-MHD energy vs. averaged MHD.

Since the flow is not ideal (nonzero power input, nonzero viscosity/magnetic diffusivity, non-periodic boundary conditions), the energy is not conserved. But we expect the energy of the model to approximate the energy of the averaged MHD. Indeed, Fig. 5.1 shows that the graph of the model’s energy is hardly distinguishable from that of the averaged MHD. Finally, we introduce another test problem – the two-dimensional wave propagation with the nonlinear magnetic field increasing in time. Consider the MHD flow in Ω = (0, 1) × (0, 1), with the Reynolds number and magnetic Reynolds number Re = 104 , Rem = 104 (see Table 5.2), the final time T = 1/8. We construct the solution as

 u=

2 0.75 + 0.25 cos(2π (x − t )) sin(2π ( y − t ))e −8π t ν 2 0.75 − 0.25 sin(2π (x − t )) cos(2π ( y − t ))e −8π t ν

p=−



B=

1 

,

    2 cos 4π (x − t ) + cos 4π ( y − t ) e −16π t ν ,

64 y 3 et x3 et

,

and compute the right-hand sides accordingly. As before, we compare the model solution to the average of the known true solution of the problem. The following table verifies the claimed convergence rates in the L 2 (0, T ; L 2 (Ω)) and L 2 (0, T ; H 1 (Ω)) norms.

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

527

Appendix A We use the semigroup approach, based on the machinery of nonlinear differential equations of accretive type in Banach spaces. We define the operator A ∈ L ( V , V ) by setting



 A ( w 1 , W 1 ), ( w 2 , W 2 ) =

 

1 Re

∇ w1 · ∇ w2 +

S Rem



curl W 1 · curl W 2 dx,

(A.1)

Ω

for all ( w i , W i ) ∈ V . The operator A is an unbounded operator on H = {φ ∈ H 20 (Ω), ∇ · φ = 0 in D(Ω) }2 , with the domain D (A ) = {( w , W ) ∈ V ; ( w ,  W ) ∈ H } and we denote again by A its restriction to H . We define also a continuous tri-linear form B0 on V × V × V by setting

  B0 ( w 1 , W 1 ), ( w 2 , W 2 ), ( w 3 , W 3 ) =





    ∇ · w 2 w 1T δ1 w 3 − S ∇ · W 2 W 1T δ1 w 3

Ω

     + ∇ · W 2 w 1T δ2 W 3 − ∇ · w 2 W 1T δ2 W 3 dx

(A.2)

and a continuous bilinear operator B(·) : V → V with



   B( w 1 , W 1 ), ( w 2 , W 2 ) = B0 ( w 1 , W 1 ), ( w 1 , W 1 ), ( w 2 , W 2 )

for all ( w i , W i ) ∈ V . The following properties of the trilinear form B0 hold (see [33,44,20,15])

  B0 ( w 1 , W 1 ), ( w 2 , W 2 ), ( A δ1 w 2 , S A δ2 W 2 ) = 0,     B0 ( w 1 , W 1 ), ( w 2 , W 2 ), ( A δ1 w 3 , S A δ2 W 3 ) = −B0 ( w 1 , W 1 ), ( w 3 , W 3 ), ( A δ1 w 2 , S A δ2 W 2 ) , for all ( w i , W i ) ∈ V . Also

       B0 ( w 1 , W 1 ), ( w 2 , W 2 ), ( w 3 , W 3 )   C ( w 1 , W 1 ) ( w 2 , W 2 ) m m 1

2 +1

 δ   w 3 1 , W 3 δ2 

m3

(A.3)

(A.4)

for all ( w 1 , W 1 ) ∈ H m1 (Ω), ( w 2 , W 2 ) ∈ H m2 +1 (Ω), ( w 3 , W 3 ) ∈ H m3 (Ω) and

m1 + m2 + m3  m1 + m2 + m3 >

d 2 d 2

,

if mi =

,

if mi =

d

for all i = 1, . . . , d,

2 d

for any of i = 1, . . . , d.

2

In terms of V , H , A , B(·) we can rewrite (1.7) as

    ( w , W ) + A ( w , W )(t ) + B ( w , W )(t ) = f δ1 , curl g δ2 ,   ( w , W )(0) = u δ01 , B δ02 , d

t ∈ (0, T ),

dt

(A.5)

where ( f , curl g ) = P ( f , curl g ), and P : L (Ω) → H is the Hodge projection. Let us define the modified nonlinearity B N (·) : V → V by setting 2



BN ( w , W ) =

B( w , W )  2 N ( w , W ) 1

if ( w , W ) 1  N ,

(A.6)

B( w , W ) if ( w , W ) 1 > N .

By (A.4) we have for the case of ( w 1 , W 1 ) 1 , ( w 2 , W 2 ) 1  N

   B N ( w 1 , W 1 ) − B N ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 )     = B0 ( w 1 − w 2 , W 1 − W 2 ), ( w 1 , W 1 ), ( w 1 − w 2 , W 1 − W 2 )     + B0 ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 ), ( w 1 − w 2 , W 1 − W 2 )         C ( w 1 − w 2 , W 1 − W 2 ) ( w 1 , W 1 )  w 1 − w 2 δ1 , W 1 − W 2 δ2  

ν  2

1/2

1

 2 2 ( w 1 − w 2 , W 1 − W 2 ) + C N ( w 1 − w 2 , W 1 − W 2 ) , 1

where ν = inf{1/ Re, S / Rem }. In the case of ( w i , W i ) 1 > N we have

0

1

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   B N ( w 1 , W 1 ) − B N ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 )    B0 ( w 1 − w 2 , W 1 − W 2 ), ( w 1 , W 1 ), ( w 1 − w 2 , W 1 − W 2 ) ( w 1 , W 1 ) 21    N2 N2 + − B0 ( w 2 , W 2 ), ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 ) 2 2 ( w 1 , W 1 ) 1 ( w 2 , W 2 ) 1  3/2  1/2  C N ( w 1 − w 2 , W 1 − W 2 )1 ( w 1 − w 2 , W 1 − W 2 )0  2 + C N ( w 1 − w 2 , W 1 − W 2 )1  2 2 ν  ( w 1 − w 2 , W 1 − W 2 )1 + C N ( w 1 − w 2 , W 1 − W 2 )0 . N2

=

2

For the case of ( w 1 , W 1 ) 1 > N, ( w 2 , W 2 ) 1  N (similar estimates are obtained when ( w 1 , W 1 ) 1  N, ( w 2 , W 2 ) 1 > N) we have

   B N ( w 1 , W 1 ) − B N ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 )  =

N2

( w 1 , W 1 ) 21

 − 1−

  B0 ( w 1 − w 2 , W 1 − W 2 ), ( w 1 , W 1 ), ( w 1 − w 2 , W 1 − W 2 ) N2

( w 1 , W 1 ) 21



  B0 ( w 2 , W 2 ), ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 )

 3/2  1/2  C N ( w 1 − w 2 , W 1 − W 2 )1 ( w 1 − w 2 , W 1 − W 2 )0     + C N ( w 1 − w 2 , W 1 − W 2 ) ( w 1 − w 2 , W 1 − W 2 ) 

ν  2

1

1/2

 2 2 ( w 1 − w 2 , W 1 − W 2 )1 + C N ( w 1 − w 2 , W 1 − W 2 )0 .

Combining all the cases above we conclude that

   B N ( w 1 , W 1 ) − B N ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 )   2 2 ν  ( w 1 − w 2 , W 1 − W 2 )1 + C N ( w 1 − w 2 , W 1 − W 2 )0 .

(A.7)

2

The operator B N is continuous from V to V . Indeed, as above we have (using (A.4) with m1 = 1, m2 = 0, m3 = 1)

      B N ( w 1 , W 1 ) − B N ( w 2 , W 2 ), ( w 3 , W 3 )   B0 ( w 1 − w 2 , W 1 − W 2 ), ( w 1 , W 1 ), ( w 3 , W 3 )     + B0 ( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 ), ( w 3 , W 3 )       C N ( w 1 − w 2 , W 1 − W 2 ) ( w 3 , W 3 ) . 1

1

(A.8)

Now consider the operator Γ N : D (Γ N ) → H defined by

ΓN = A + BN ,

D (Γ N ) = D (A ).

Here we used (A.4) with m1 = 1, m2 = 1/2, m3 = 0 and interpolation results (see e.g. [18,50,15]) to show that

        B N ( w , W )  C ( w , W )3/2 A ( w , W )1/2  C N A ( w , W )1/2 . 0 1 0 0

(A.9)

Lemma A.1. There exists α N > 0 such that Γ N + α N I is m-accretive (maximal monotone) in H × H . Proof. By (A.7) we have that



 (Γ N + λ)( w 1 , W 1 ) − (Γ N + λ)( w 2 , W 2 ), ( w 1 − w 2 , W 1 − W 2 ) 2 ν  ( w 1 − w 2 , W 1 − W 2 )1 , for all ( w i , W i ) ∈ D (Γ N ), 2

for λ  C N . Next we consider the operator

F N ( w , W ) = A ( w , W ) + B N ( w , W ) + α N ( w , W ), with

for all ( w , W ) ∈ D (F N ),

(A.10)

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533



529



D (F N ) = ( w , W ) ∈ V ; A ( w , W ) + B N ( w , W ) ∈ H . By (A.8) and (A.10) we see that F N is monotone, coercive and continuous from V to V . We infer that F N is maximal monotone from V to V and the restriction to H is maximal monotone on H with the domain D (F N ) ⊇ D (A ) (see e.g. [11,4]). Moreover, we have D (F N ) = D (A ). For this we use the perturbation theorem for nonlinear m-accretive operators and split F N into a continuous and an ω -m-accretive operator on H

 ε F N1 = 1 − A,





D F N1 = D (A ),

2



ε

F N2 = A + B N (·) + α N I ,







D F N2 = ( w , W ) ∈ V , F N2 ( w , W ) ∈ H .

2

As seen above by (A.9) we have

 2        F ( w , W )  ε A ( w , W ) + B N ( w , W ) + α N ( w , W ) N 0 0 0 0 2

    C2  ε A ( w , W )0 + α N ( w , W )0 + N , 2ε





for all ( w , W ) ∈ D F N1 = D (A ),

where 0 < ε < 1. Since F N1 + F N2 = Γ N + α N I we infer that Γ N + α N I with domain D (A ) is m-accretive in H as claimed.

2

Proof of Theorem 2.2. As a consequence of Lemma A.1 (see, e.g., [4,5]) we have that for (u 0 δ1 , B 0 δ2 ) ∈ D (A ) and ( f δ1 , curl g δ2 ) ∈ W 1,1 ([0, T ], H ) the equation

d dt

    ( w , W ) + A ( w , W )(t ) + B N ( w , W )(t ) = f δ1 , curl g δ2 ,

t ∈ (0, T ),

( w , W )(0) = (u 0 δ1 , B 0 δ2 ),

(A.11) L ∞ (0, T ; D (A )).

has a unique strong solution ( w N , W N ) ∈ W ([0, T ]; H ) ∩ By a density argument (see, e.g., [5,33]) it can be shown that if (u 0 δ1 , B 0 δ2 ) ∈ H and ( f δ1 , curl g δ2 ) ∈ L 2 (0, T , V ) then there exist absolute continuous functions ( w N , W N ) : [0, T ] → V that satisfy ( w N , W N ) ∈ C ([0, T ]; H ) ∩ L 2 (0, T : V ) ∩ W 1,2 ([0, T ], V ) and (A.11) a.e. in (0, T ), where d/dt is considered in the strong topology of V . First, we show that D (A ) is dense in H . Indeed, if ( w , W ) ∈ H we set ( w ε , W ε ) = ( I + ε Γ N )−1 ( w , W ), where I is the unity operator in H . Multiplying the equation 1,∞

( w ε , W ε ) + εΓN ( w ε , W ε ) = ( w , W ) by ( w ε , W ε ) it follows by (A.3), (A.7) that

      ( w ε , W ε )2 + 2εν ( w ε , W ε )2  ( w , W )2 0

1

0

and by (A.6)

  ( w ε − w , W ε − W )

−1

   1/2  1/2 = ε Γε ( w ε , W ε )−1  ε N ( w ε , W ε )0 ( w ε , W ε )1 .

Hence, {( w ε , W ε )} is bounded in H and ( w ε , W ε ) → ( w , W ) in V as ε → 0. Therefore, ( w ε , W ε )  ( w , W ) in H as ε → 0, which implies that D (Γ N ) is dense in H . δ δ Secondly, let (u 0 δ1 , B 0 δ2 ) ∈ H and ( f δ1 , curl g δ2 ) ∈ L 2 (0, T , V ). Then there are sequences {(u 0 n1 , B 0 n2 )} ⊂ D (Γ N ), δ1 δ2 1, 1 {( f n , curl g n )} ⊂ W ([0, T ]; H ) such that

 

δ

δ





u 0 n1 , B 0 n2 → u 0 δ1 , B 0 δ2 δ1

δ2 

f n , curl g n





→ f , curl g δ1

in H , δ2







in L 2 0, T ; V , δ

δ

n as n → ∞. Let ( w nN , W N ) ∈ W 1,∞ ([0, T ]; H ) be the solution to problem (A.11) where ( w , W )(0) = (u 0 n1 , B 0 n2 ) and

( f δ1 , curl g δ2 ) = ( f nδ1 , curl g nδ2 ). By (A.10) we have d  dt

    w n − w m , W n − W m 2 + ν  w n − w m , W n − W m 2 N N N N N N N N 0 1 2

   2  δ1 δ2  n m 2 2 ,  2C N  w nN − w m +  f nδ1 − f m , curl g nδ2 − g m N, WN − WN −1 0

ν

for a.e. t ∈ (0, T ). By the Gronwall inequality we obtain

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A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

 n    δ1 δ2  2  w − w m , W n − W m (t )2  e 2C N t  u 0 nδ1 − u 0 m , B 0 nδ2 − B 0 m N N N N 0 0 +

2e 2C N t

t

ν

 δ1 2  δ1 δ2   fn − fm (τ )−1 dτ . , curl g nδ2 − g m

0

Hence









n w N (t ), W N (t ) = lim w nN (t ), W N (t ) n→∞

exists in H uniformly in t on [0, T ]. Similarly we obtain

 n 2  n 2  w (t ) +  W (t ) + N

N

0

t 

0

     ∇ w n (s)2 + S curl W n (s)2 ds N N 0 0

1  Re

Rem

0

  t  2   δ1 2  δ2 2  δ1 2 δ2          C N u0n 0 + B 0n 0 + f n (s) −1 + curl g n (s) −1 ds , 0

and

   T  t  2  d n  δ1 2  δ2 2  2  δ1 2 δ2 n           f n (s) −1 + curl g n (s) −1 ds .  dt w N , W N (t ) dt  C N u 0 n 0 + B 0 n 0 + −1

0

0

Hence on a sequence we have





n w nN , W N → (w N , W N )

d dt



n → w nN , W N

d dt

weakly in L 2 (0, T ; V ),

  ( w N , W N ) weakly in L 2 0, T ; V ,

where d( w N , W N )/dt is considered in the sense of V -valued distributions on (0, T ). We proved that ( w N , W N ) ∈ C ([0, T ]; H ) ∩ L 2 (0, T ; V ) ∩ W 1,2 ([0, T ]; V ). It remains to prove that ( w N , W N ) satisfies Eq. (A.11) a.e. on (0, T ). Let ( w , W ) ∈ V be arbitrary but fixed. We multiply the equation

d dt









δ

δ



n w nN , W N + Γ N w nN , W Nn = f n1 , curl g n2 ,

a.e. t ∈ (0, T ),

n by ( w nN − w , W N − W ), integrate on (s, t ) and get

       w n (t ), W n (t ) − ( w , W )2 −  w n (s), W n (s) − ( w , W )2 N N N N 0 0

1  2

t 



δ

δ









n f n1 (τ ), curl g n2 (τ ) − Γ N ( w , W ), w nN (τ ), W N (τ ) − ( w , W ) dτ .

s

After we let n → ∞ we get



  ( w N (t ), W N (t )) − ( w N (s), W N (s))  , w N (s), W N (s) − ( w , W ) t−s 

1 t−s

t









s

Let t 0 denote a point at which ( w N , W N ) is differentiable and



 1 f (t 0 ), curl g (t 0 ) = lim δ1



f δ1 (τ ), curl g δ2 (τ ) − Γ N ( w , W ), w N (τ ), W N (τ ) − ( w , W ) dτ .

t 0 +h



δ2

h→0

h

t0



f δ1 (h), curl g δ2 (h) dh.

(A.12)

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

Then by (A.12) we have



d( w N , W N ) dt



(t 0 ) − f , curl g δ1

δ2



531

 (t 0 ) + Γ N ( w , W ), ( w N , W N )(t 0 ) − ( w , W )  0.

Since ( w , W ) is arbitrary in V and Γ N is maximal monotone in V × V we conclude that

d( w N , W N ) dt

  (t 0 ) + Γ N ( w N , W N )(t 0 ) = f δ1 , curl g δ2 (t 0 ).

If we multiply (A.11) by ( A δ1 w N , S A δ2 W N ), use (A.3) and integrate in time we obtain 2 2         w N (t )2 + S  W N (t )2 + δ1 ∇ w N (t )2 + δ2 S curl W N (t )2

1  2

0

t  +

0

0

2

         ∇ w N (s)2 + δ1 2  w N (s)2 + S curl W N (s)2 + δ2 2 curl curl W N (s)2 ds 0 0 0 0

1  Re

Rem

0

=

0

2

2 2        u 0 δ1 2 + S  B 0 δ2 2 + δ1 ∇ u 0 δ1 2 + δ2 S curl B 0 δ2 2

1 

0

2

t +

  δ  f 1 (s)

0

−1

0

2

0

2

        w N (s) + S curl g δ2 (s)  W N (s) ds. 1 −1 1

0

Using the Cauchy–Schwarz and Gronwall inequalities implies

  ( w N , W N )(t )  C δ ,δ 1 2 1

for all t ∈ (0, T ),

where C δ1 ,δ2 is independent of N. In particular, for N sufficiently large it follows from (A.6) that B N = B and ( w N , W N ) = ( w , W ) is a solution to (1.7). In the following we prove the uniqueness of the weak solution. Let ( w 1 , W 1 ) and ( w 2 , W 2 ) be two solutions of system (A.5) and set ϕ = w 1 − w 2 , Φ = B 1 − B 2 . Thus (ϕ , Φ) is a solution to the problem

d dt

    (ϕ , Φ) + A (ϕ , Φ)(t ) = −B ( w 1 , W 1 )(t ) + B ( w 2 , W 2 )(t ) ,

t ∈ (0, T ),

(ϕ , Φ)(0) = (0, 0). We take ( A δ1 ϕ , S A δ2 Φ) as test function, integrate in space, use the incompressibility condition (A.3) and the estimate (A.4) to get

1 d

   1  S  ϕ 20 + δ1 2 ∇ ϕ 20 + S Φ 20 + S δ2 2 ∇Φ 20 + ∇ ϕ 20 + δ12 ϕ 20 + ∇Φ 20 + δ22 Φ 20 Re Rem   = B0 (ϕ , Φ), ( w 1 , W 1 ), ( A δ1 ϕ , S A δ2 Φ)    1/2  3/2  C ( w 1 , W 1 )0 (ϕ , Φ)0 (∇ ϕ , ∇Φ)0      C δ ,δ ( w 1 , W 1 ) ϕ 2 + δ1 2 ∇ ϕ 2 + S Φ 2 + S δ2 2 ∇Φ 2 .

2 dt

1

0

2

0

0

0

0

Applying the Gronwall’s lemma we deduce that (ϕ , Φ) vanishes for all t ∈ [0, T ], and hence the uniqueness of the solution. 2 A.1. Regularity Theorem A.2. Let m ∈ N, (u 0 , B 0 ) ∈ V ∩ H m−1 (Ω) and ( f , curl g ) ∈ L 2 (0, T ; H m−1 (Ω)). Then there exists a unique solution w , W , q to Eq. (1.7) such that

    ( w , W ) ∈ L ∞ 0, T ; H m+1 (Ω) ∩ L 2 0, T ; H m+2 (Ω) ,





q ∈ L 2 0, T ; H m (Ω) .

Proof. The result is already proved when m = 0 in Theorem 2.2. For any m ∈ N∗ , we assume that

    ( w , W ) ∈ L ∞ 0, T ; H m (Ω) ∩ L 2 0, T ; H m+1 (Ω)

(A.13)

so it remains to prove













D m w , D m W ∈ L ∞ 0, T ; H 1 (Ω) ∩ L 2 0, T ; H 2 (Ω) ,

532

A. Labovsky, C. Trenchea / J. Math. Anal. Appl. 377 (2011) 516–533

where D m denotes any partial derivative of total order m. We take the mth derivative of (1.7) and have

 

Dm w

 t





1

   D m w + D m ( w · ∇ w )δ1 − S D m ( W · ∇ W )δ1 = D m f δ1 ,

Re 1

  + ∇ × ∇ × D m W + D m ( w · ∇ W )δ2 − D m ( W · ∇ w )δ2 = ∇ × D m g δ2 , Rem     ∇ · D m w = 0, ∇ · D m W = 0, Dm W

t

D m w (0, ·) = D m u 0 δ1 , D m W (0, ·) = D m B 0 δ2 , with periodic boundary conditions and zero mean, and the initial conditions with zero divergence and mean. Taking A δ1 D m w , A δ1 D m W as test functions we obtain

         D m w 2 + δ1 2 ∇ D m w 2 + S  D m W 2 + S δ2 2 ∇ D m W 2

1 d  2 dt

0

0

0

0

         1  ∇ D m w 2 + δ 2  D m w 2 + 1 ∇ D m W 2 + δ 2  D m W 2 + 1 2 0 0 0 0 Re Rem   m m  = D f D w + ∇ × g D m W dx − X ,

(A.14)

Ω

where

 X =









D m ( w · ∇ w ) − S D m ( W · ∇ W ) D m w + D m ( w · ∇ W ) − D m ( W · ∇ w ) D m W dx.

Ω

Now we apply (A.4) and use the induction assumption (A.13)

X =

3   m 

|α |m

α

D α w i D m −α D i w j D m w j − S D α W i D m −α D i W j D m w j

i , j =1 Ω

− D α w i D m −α D i W j D m W j − D α W i D m −α D i w j D m W j 3/2

1/2

3/2

1/2

 w m+1 w m+2 w m + W m+1 W m+2 w m 1/2

1/2

3/2

1/2

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