Convergence results for the vector penalty-projection and two-step artificial compressibility methods

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Convergence results for the vector penalty-projection and two-step artificial compressibility methods

Philippe Angot, Pierre Fabrie

To cite this version:

Philippe Angot, Pierre Fabrie. Convergence results for the vector penalty-projection and two-step artificial compressibility methods. Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2012, 17 (5), pp. 1383–1405, communicated by Roger Temam.

�10.3934/dcdsb.2012.17.1383�. �hal-00653113v2�

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VolumeX, Number0X, XX200X _pp._X–XX

CONVERGENCE RESULTS FOR THE VECTOR PENALTY-PROJECTION AND TWO-STEP ARTIFICIAL

COMPRESSIBILITY METHODS

Philippe Angot

Aix-Marseille Universit´e

Laboratoire d’Analyse, Topologie, Probabilit´es - CNRS UMR7353 Centre de Math´ematiques et Informatique

13453 Marseille cedex 13 - France

Pierre Fabrie

Universit´e de Bordeaux & IPB

Institut Math´ematiques de Bordeaux - CNRS UMR5251 ENSEIRB-MATMECA, Talence - France

(Communicated by Roger Temam)

Abstract. In this paper, we propose and analyze a new artificial compressibility splitting method which is issued from the recent vector penalty-projection method for the numerical solution of unsteady incompressible viscous flows introduced in [1], [2] and [3]. This method may be viewed as an hybrid two-step prediction-correction method combining an artificial compressibility method and an augmented Lagrangian method without inner iteration. The perturbed system can be viewed as a new approximation to the incompressible Navier- Stokes equations. In the main result, we establish the convergence of solutions to the weak solutions of the Navier-Stokes equations when the penalty parame- ter tends to zero.

1. Introduction and setting of the problem. The artificial compressibility method was introduced by Chorin [6] and Temam [17] for the solution of the unsteady incompressible Stokes or Navier-Stokes equations; see also [20] for the the- oretical analysis. Then, some other numerical schemes to efficiently compute the solutions of Navier-Stokes problems can be viewed as discretizations of perturbed systems of the type of penalization [14] or pseudo-compressibility. This is the case with the famous projection methods from Chorin [7] and Temam [18, 19] and their variants [10], see e.g. [15].

Here, we present a new approximation method for the Navier-Stokes equations modeling incompressible viscous flows in a bounded regular open set Ω endowed with Dirichlet boundary conditions on Γ = ∂Ω (Lipschitz-continuous). With a given source term f, the Navier-Stokes system reads:

2000Mathematics Subject Classification. Primary: 35Q30, 76D05, 76N10, 35A35; Secondary:

65M12, 65N12.

Key words and phrases. Artificial compressibility, Navier-Stokes equations, vector penalty- projection, pseudo-compressibility, penalty method.

1

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 

 

 



∂v

∂t + (v · ∇ )v − 1

R

^e

∆v + ∇ p = f, div v = 0,

v(0) = v

0

, v

_|Γ

= 0, where R

^e

denotes the Reynolds number.

According to the identity, − ∆ϕ = curl curl ϕ −∇ div ϕ, we consider the following approximate method to obtain a solution of the above Navier-Stokes system, with the parameters r ≥ 0, γ > 0 and, ε > 0

 

 



 

 

∂ v e

ε

∂t + (v

ε

· ∇ ) v e

ε

+ 1

2 (div v

ε

) v e

ε

+ 1

R

^e

curl curl v e

ε

− 1

R

^e

∇ div v e

ε

− r ∇ div v e

ε

+ ∇ p

ε

= f

∂ v b

ε

∂t + (v

ε

· ∇ ) v b

ε

+ 1

2 (div v

ε

) v b

ε

+ 1 R

e

curl curl v b

ε

− 1

R

e

∇ div v b

ε

− r ∇ div v b

ε

− 1

ε R

^e

∇ (div v b

ε

+ div v e

ε

) = 0 v

ε

= v e

ε

+ v b

ε

γ ∂p

ε

∂t + γp

ε

+ 1

ε div v

ε

+ r div v e

ε

= 0.

We associate to the previous system the following boundary conditions and initial data,

e

v

ε

(0) = v

0

, v b

ε

(0) = 0, p

ε

(0) = p

0

, e

v

ε

· ν

Γ

= 0, v b

ε

· ν

Γ

= 0,

(curl v e

ε

) ∧ ν

Γ

= 0, (curl v b

ε

) ∧ ν

Γ

= 0, where ν denotes the outward unit normal vector on Γ.

To vanish, at the limit process, the two tangential component of the velocity fields, v e

ε

∧ ν and v b

ε

∧ ν, we use a penalization method which will be detailed below.

This method is close to the artificial compressibility method of Chorin [6] and Temam [17], but presents one important difference. It is a two-step splitting method.

The first equation of the previous system gives a predicted velocity v e

ε

and the second one is the approximate projection of v e

ε

on the free-divergence vector fields. This equation may be seen as an approximate method to solve the well-posed problem (see appendix A) :

div v b

ε

= − div v e

ε

, curl v b

ε

= 0,

b

v

ε

.ν

_|_Γ

= 0.

Remark 1. This approximate method is issued from the Vector Penalty-Projection

(VPP

r,ε

) methods for the numerical solution of unsteady incompressible viscous

flows introduced in [1] and [3]. A fast version of these methods, the so-called

(VPP

ε

) method, is recently proposed also for the numerical solution of the non-

homogeneous Navier-Stokes equations in [2, 4]. It is shown to be very efficient

to compute multiphase flows, i.e. fast, cheap, and robust whatever the density,

viscosity or permeability jumps.

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Even for r = 0, the resulting method which corresponds to a two-step pseudo- compressibility method, is different from the original artificial compressibility method of Chorin [6] and Temam [17, 20].

The new important point is the penalty term 1

ε ( ∇ div v b

ε

+ ∇ div v e

ε

) that appears in the velocity correction step which allows us a direct estimate on the divergence of the velocity. Moreover, this system is quite easy to solve and presents good stability properties, see [1, 2, 3]. The velocity v

ε

and the pressure p

ε

satisfy the equations:

 

 

 

 



∂v

ε

∂t + (v

ε

· ∇ )v

ε

+ 1

2 (div v

ε

) v

ε

+ 1

R

^e

curl curl v

ε

− 1

R

^e

∇ div v

ε

− r ∇ div v

ε

− 1

ε R

^e

∇ div v

ε

+ ∇ p

ε

= f γ ∂p

ε

∂t + γ p

ε

+ 1

ε div v

ε

+ r div v e

ε

= 0, v

ε

(0) = v

0

, p

ε

(0) = p

0

,

v

ε

· ν

Γ

= 0, (curl v

ε

) ∧ ν

Γ

= 0.

The vanishing of the tangential component at the limit process, is fullfilled by a penalization method, which implies that this boundary condition is satisfied at the order ε for the approximate solution.

1.1. Notations. Let Ω be a regular bounded and connected open set of R

^d

, for d = 2 or 3. We note H

^s

(Ω) the classical Sobolev space, and k · k

^H^s

the associated norm. The norm of a function in L

^p

(Ω) is denoted by k · k

^L^p

, and if B is a Banach space, we denote by k . k

^L^p^,B

the norm in L

^p

(]0, T [; B).

L

^p

(Ω) = (L

^p

(Ω))

^d

H

_div

(Ω) = { v ∈ (L

²

(Ω))

^d

, div v ∈ L

²

(Ω) } H = { v ∈ (L

²

(Ω))

^d

, div v = 0, v · ν

Γ

= 0 } H

¹_ν

(Ω) = { v ∈ (H

¹

(Ω))

^d

, v · ν

Γ

= 0 } G = { v ∈ (L

²

(Ω))

^d

, ∃ q ∈ H

¹

(Ω), v = ∇ q } 1.2. Mathematical recalls.

Proposition 1. Under the previous hypothesis, one has the following properties:

L

²

(Ω) = H ⊕ G , Ker (curl ) = G.

Moreover, there exists one constant C > 0 depending only on Ω such that:

k u k

²H¹

= k u k

²L²

+ k∇ u k

²L²

≤ C k u k

²L²

+ k div u k

²L²

+ k curl u k

²L²

, ∀ u ∈ H

¹_ν

(Ω). (1) Besides, if we suppose that the open set Ω is simply-connected, there exist two constants λ

0

and λ

1

depending only on Ω such that:

k u k

²L²

≤ λ

0

k div u k

²L²

+ k curl u k

²L²

, ∀ u ∈ H

¹_ν

(Ω), k u k

²L²

+ k∇ u k

²L²

≤ λ

1

k div u k

²L²

+ k curl u k

²L²

, ∀ u ∈ H

¹_ν

(Ω) and we have:

Ker (curl ) ∩ H = { 0 } .

Proof. All these results may be found in [9] and [8].

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For a Banach space E we introduce the Nikolskii space defined for 1 ≤ q < + ∞ , 0 < σ < 1:

N

_q^σ

(]0, T [; E) =

f ∈ L

^q

(]0, T [; E), sup

0<h<T

k f ( · + h) − f ( · ) k

L^q(]0,T−h[;E)

h

^σ

< + ∞

,

endowed with the following norm:

k f k

^Nq^σ

=

k f k

^qL^q(]0,T[;E)

+ sup

0<h<T

1 h

^σ

k f ( · + h) − f ( · ) k

L^q(]0,T−h[;E)

q

¹_q

.

Let us recall the following property see for example [5] page 105,

Proposition 2. Let H be an Hilbert space and f a function given in L

²

(]0, T [; H) such that, for some 0 < σ < 1,

Z

R

| τ |

^2σ

kF ( ˜ f )(τ) k

²H

dτ ≤ C

²

,

where f ˜ denotes the extension by 0 of the function f outside [0, T ]. Then f ∈ N

₂^σ

(]0, T [; H ) and we have

k f k

^N2^σ

≤ M

σ

(1 + C),

where M

σ

is a constant depending only on σ.

We now recall the important compactness theorem, see for example [16]

Theorem 1.1. Aubin-Lions-Simon

Let B

0

, B

1

, B

2

three Banach spaces with B

0

⊂ B

1

⊂ B

2

with continuous imbedding.

Suppose moreover that the injection of B

0

in B

1

is compact.

Then, for all 1 ≤ q ≤ + ∞ and 0 < σ < 1, the imbedding

L

^q

(]0, T [; B

0

) ∩ N

_q^σ

(]0, T [; B

2

) ֒ → L

^q

(]0, T [; B

1

) is compact.

2. Main result. We associate to the previous approximate system, the variational problem where the tangential components of the velocities v e

ε

and v b

ε

are penalized.

This problem is studied in the next section.

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Find ( v e

ε

, v b

ε

, p

ε

) in

L

^∞

(]0, T [; L

²

(Ω)) ∩ L

²

(]0, T [; H

¹_ν

(Ω))

2

× L

^∞

(]0, T [; L

²₀

(Ω)) satisfying in D

^′

(]0, T [),

 

 



 

  Z

Ω

∂ v e

ε

∂t · ϕ dω + Z

Ω

(v

ε

· ∇ ) v e

ε

+ 1

2 (div v

ε

) v e

ε

· ϕ dω + 1

R

^e

Z

Ω

curl v e

ε

· curl ϕ dω + 1 R

^e

Z

Ω

div v e

ε

div ϕ dω + r

Z

Ω

div v e

ε

div ϕ dω − Z

Ω

p

ε

div ϕ dω + 1

ε Z

Γ

( v e

ε

∧ ν) · (ϕ ∧ ν) dσ = Z

Ω

f · ϕ dx, Z

Ω

∂ v b

ε

∂t · ψ dω + Z

Ω

(v

ε

· ∇ ) v b

ε

+ 1

2 (div v

ε

) v b

ε

· ψ dω + 1

R

^e

Z

Ω

curl v b

ε

· curl ψ dω + 1 R

^e

Z

Ω

div v b

ε

div ψ dω + r

Z

Ω

div v b

ε

div ψ dω + 1 ε R

^e

Z

Ω

div v e

ε

+ div v b

ε

div ψ dω + 1

ε Z

Γ

( v b

ε

∧ ν) · (ψ ∧ ν) dσ = 0,

v

ε

= v e

ε

+ v b

ε

,

γ Z

Ω

∂p

ε

∂t π dω + γ Z

Ω

p

ε

π dω + 1 ε Z

Ω

π div v

ε

dω + r Z

Ω

π div v e

ε

dω = 0,

∀ (ϕ, ψ, π) ∈ (H

¹_ν

(Ω))

²

× L

²₀

(Ω), e

v

ε

(0) = v

0

, v b

ε

(0) = 0, p

ε

(0) = p

0

.

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Then the velocity v

ε

and the pressure p

ε

satisfy in D

^′

(]0, T [),

 

 

 

 

 Z

Ω

∂v

ε

∂t · ϕ dω + Z

Ω

(v

ε

· ∇ )v

ε

+ 1

2 (div v

ε

) v

ε

· ϕ dω + 1

R

^e

Z

Ω

curl v

ε

· curl ϕ dω + 1 R

^e

Z

Ω

div v

ε

div ϕ dω +r

Z

Ω

div v

ε

div ϕ dω + 1 ε R

^e

Z

Ω

div v

ε

div ϕ dω

− Z

Ω

p

ε

div ϕ dω + 1 ε

Z

Γ

(v

ε

∧ ν ) · (ϕ ∧ ν) dσ

= Z

Ω

f · ϕ dω, ∀ ϕ ∈ H

¹_ν

(Ω), v

ε

(0) = v

0

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Remark 2. In order to establish the strong convergence of the sequence (v

ε

)

ε>0

when ε → 0, we use in Section 4 the Leray’s orthogonal decomposition in the

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bounded domain. The curl-free component vanishes with the penalty term introduced by our method, whereas the divergence-free component strongly converges thanks to an estimate of a fractional derivative in time, see [20]. However, this requires to consider velocity fields having only their normal component which is zero on the boundary. Since at the limit process, we aim at solving the Navier- Stokes problem with homogeneous Dirichlet boundary condition, we also penalize the tangential part of the velocity fields.

We prove in section 3 the following results.

Lemma 2.1. Let us suppose that f belongs to L

²

(]0, T [; L

²

(Ω)). Then, there exists at least a solution to the system (2). This solution is unique in two space dimension.

For the dimension d ≤ 3, this solution satisfies the following energy inequality:

1 2

d

dt r ε k v e

ε

k

²L²

+ ε k v b

ε

k

²L²

+ k v

ε

k

²L²

+ γε k p

ε

k

²L²

+ γε k p

ε

k

²L²

+ r ε

2 R

^e

k curl v e

ε

k

²L²

+ ε

2 R

^e

k curl v b

ε

k

²L²

+ 1

2 R

^e

k curl v

ε

k

²L²

+ 1

2 R

^e

k div v

ε

k

²L²

+ r ε

2 R

^e

k div v e

ε

k

²L²

+ ε

2 R

^e

k div v b

ε

k

²L²

+ εr

²

k div v e

ε

k

²L²

+ εr k div v b

ε

k

²L²

+ 1

2ε R

^e

k div v

ε

k

²L²

+ r | ( v e

ε

∧ ν) |

²L²(Γ)

+ | ( v b

ε

∧ ν ) |

²L²(Γ)

+ 1

ε | (v

ε

∧ ν) |

²L²(Γ)

≤ λ R

^e

2 (1 + rε) k f k

²L²

For the dimension d = 2, one has the following energy equality:

1 2

d

dt r ε k v e

ε

k

²L²

+ k v

ε

k

²L²

+ γε k p

ε

k

²L²

+ r ε

R

^e

k curl v e

ε

k

²L²

+ 1

R

^e

k curl v

ε

k

²L²

+ γ ε k p

ε

k

²L²

+ r ε

R

e

k div v e

ε

k

²L²

+ 1

R

^e

k div v

ε

k

²L²

+ εr

²

k div v e

ε

k

²L²

+ 1

ε R

e

k div v

ε

k

²L²

+ | ( v e

ε

∧ ν ) |

²L²Γ

+ 1

ε | (v

ε

∧ ν) |

²L²(Γ)

= r ε Z

Ω

f v e

ε

dω + Z

Ω

f v

ε

dω.

This result is quite classical and we only give the sketch of proof in the section 3.

In fact, we can precise the previous energy inequality if we suppose that the data f belongs to L

^∞

(R

+

; L

²

(Ω)). This shows the absolute stability of the approximate method.

Theorem 2.2. Suppose that the data f satisfies f ∈ L

^∞

(R

+

, L

²

(Ω)),

then, there exists a constant α independent of the data, such that r ε k v e

ε

(t) k

²L²

+ ε k v b

ε

(t) k

²L²

+ k v

ε

(t) k

²L²

+ γε k p

ε

(t) k

²L²

≤ e

⁻^αt

(1 + r ε) k v

0

k

²L²

+ γ ε k p

0

k

²L²

+ λ R

^e

α (1 + r ε) k f k

²L^∞, L²

, ∀ t ∈ R

+

.

The main goal of this paper is to prove the following convergence theorem:

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Theorem 2.3. For d ≤ 3, there exists a subsequence (v

ε_k

, p

ε_k

)

k

solution of (3) that converges to a solution (v, p) to the system of Navier-Stokes equations with homogeneous Dirichlet boundary conditions.

For d = 2, the solution (v, p) is unique, and for all sequences ε

k

, (v

ε_k

, p

ε_k

)

ε_k

converges to (v, p). Moreover, for all sequences ε

k

, (v

εk

)

k

converges strongly to v in L

²

(0, T ; H

¹_ν

(Ω)).

We now give an interpretation of the pressure and precise its convergence. Let us define

q

ε

= p

ε

− 1 + ε ε R

^e

+ r

div v

ε

.

The scalar function q

ε

appears to be the effective approximate pressure, and we have

Theorem 2.4. The function ∇ q

εk

satisfies

• if d = 3, ∇ q

ε_k

converges weakly to ∇ p in H

⁻¹

((0, T ) × Ω)

3

• if d = 2, ∇ q

εk

converges strongly to ∇ p in H

⁻¹

((0, T ) × Ω)

2

These convergence results for both velocity and pressure are proved in Section 4.

3. Energy estimates. We first establish the following existence result.

Proposition 3. For v

ε0

, p

0

given in L

²

(Ω) × L

²₀

(Ω), there exists at least a solution of the system (2) satisfying for d = 3:

e

v

ε

∈ L

^∞

(]0, T [; L

²

(Ω) ∩ L

²

(]0, T [; H

¹_ν

(Ω)), ∂ v e

ε

∂t ∈ L

⁴³

(]0, T [; (H

¹_ν

(Ω))

^′

) b

v

ε

∈ L

^∞

(]0, T [; L

²

(Ω) ∩ L

²

(]0, T [; H

¹_ν

(Ω)), ∂ v b

ε

∂t ∈ L

⁴³

(]0, T [; ( H

¹_ν

(Ω))

^′

) p

ε

∈ L

^∞

(]0, T [; L

²₀

(Ω)), ∂p

ε

∂t ∈ L

²

(]0, T [; L

²₀

(Ω)) e

v

ε

(0) = v

ε o

in (H

¹_ν

(Ω))

^′

, v b

ε

(0) = 0 in (H

¹_ν

(Ω))

^′

, p

ε

(0) = p

0

in L

²₀

(Ω).

If d = 2, the unique solution of (2) satisfies the following regularity results:

e

v

ε

∈ L

^∞

(]0, T [; L

²

(Ω) ∩ L

²

(]0, T [; H

¹_ν

(Ω)),

∂ v e

ε

∂t ∈ L

²

(]0, T [; (H

¹_ν

(Ω))

^′

)) + L

⁴³

(]0, T [; L

⁴³

(Ω)) b

v

ε

∈ L

^∞

(]0, T [; L

²

(Ω) ∩ L

²

(]0, T [; H

¹_ν

(Ω)),

∂ v b

ε

∂t ∈ L

²

(]0, T [; (H

¹_ν

(Ω))

^′

)) + L

⁴³

(]0, T [; L

⁴³

(Ω)) p

ε

∈ L

^∞

(]0, T [; L

²₀

(Ω)), ∂p

ε

∂t ∈ L

²

(]0, T [; L

²₀

(Ω)) e

v

ε

(0) = v

ε o

in L

²

(Ω), v b

ε

(0) = 0 in L

²

(Ω), p

ε

(0) = p

0

in L

²₀

(Ω).

Remark 3. In the three-dimensional case, the equalities e

v

ε

(0) = v

ε o

in (H

¹ν

(Ω))

^′

, v b

ε

(0) = 0 in (H

¹ν

(Ω))

^′

,

are valid in the trace sense.

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Proof. For fixed parameters ε > 0, r ≥ 0 and γ > 0, we build approximate solutions by a classical Galerkin process.

Let us introduce the self-adjoint operator A = curl curl − ∇ div defined on the domain H

¹_ν

(Ω) ∩ (H

²

(Ω))

^d

. Then, for the approximation of the two fields of velocity, we use as special basis the eigenfunctions of this operator associated with the following boundary conditions:

u · ν

Γ

= 0, (curl u) ∧ ν

Γ

= 0.

For the pressure, one can use as special basis the eigenfunctions of the self-adjoint operator A = − ∆ with domain H

²

(Ω) associated to the Neumann boundary conditions.

This approximate finite dimensional system is then a classical ordinary differential equation which has a unique solution. Next, to perform the limit we use the same strategy as for the classical Navier-Stokes equations i.e. a priori estimates and compactness results using an estimate on the temporal derivative, see for example [13], [20],[5].

Now, we will focus our attention on the estimates on the time derivative according to the dimension d. Let us begin with the three-dimensional case. We have to estimate the two nonlinear terms v

ε

· ∇ w, w div v

ε

, with either w = v e

ε

or w = v b

ε

and the pressure p

ε

.

Suppose first that d = 3. By Sobolev embedding, the two nonlinear terms of the form v

ε

· ∇ w and w div v

ε

belong to L

⁴³

(0, T ; H

¹_ν

(Ω)

^′

) since we have, for example for all ϕ ∈ H

¹_ν

(Ω) :

Z

Ω

(v

ε

· ∇ )w · ϕ dω + 1

2 Z

Ω

(div v

ε

) w · ϕ dω

≤ C k v k

L³

k w k

H_ν¹

k ϕ k

L⁶

+ k w k

L³

k v

ε

k

H_ν¹

k ϕ k

L⁶

≤ C k v

ε

k

1 2

L²

k v

ε

k

1 2

H_ν¹

k w k

H¹_ν

+ k w k

1 2

L²

k w k

1 2

H_ν¹

k v

ε

k

^H¹_ν

k ϕ k

^H_ν¹

. The bounds of the linear terms are straightforward and we have

Z

Ω

curl v e

ε

· curl ϕ dω +

Z

Ω

div v e

ε

div ϕ dω + r

Z

Ω

div v e

ε

div ϕ dω

≤ C k v e

ε

k

^Hν¹

k ϕ k

^Hν¹

Z

Ω

p

ε

div ϕ dω

≤ C k p

ε

k

L²

k ϕ k

H_ν¹

Z

Ω

div v e

ε

+ div v b

ε

div ϕ dω

≤ C k v e

ε

k

H_ν¹

+ k v b

ε

k

H_ν¹

k ϕ k

H_ν¹

. and, with standard trace theorems

Z

Γ

(v

ε

∧ ν) · (ϕ ∧ ν) dω

≤ C k v

ε

k

H_ν¹

k ϕ k

H_ν¹

. Thus it follows, from equation (2) that

∂ v e

ε

∂t ∈ L

⁴³

(]0, T [; ( H

¹_ν

(Ω))

^′

), ∂ v b

ε

∂t ∈ L

⁴³

(]0, T [; ( H

¹_ν

(Ω))

^′

)

∂p

ε

∂t ∈ L

²

(]0, T [; L

²₀

(Ω))

(10)

These estimates show that the velocities ( v e

ε

, v b

ε

) are equal almost everywhere to continuous functions with values in (H

¹_ν

(Ω))

^′

. Besides, the pressure p

ε

is equal almost everywhere to a continuous function with value in L

²₀

(Ω).

For the two-dimensional case, the situation is quite different. We observe first that the velocity fields v b

ε

and v e

ε

belong to L

⁴

(]0, T [; L

⁴

(Ω)), so that:

v

ε

∇ ) v e

ε

+ 1 2 div v

ε

v e

ε

∈ L

⁴³

(]0, T [; L

⁴³

(Ω)), v

ε

∇ ) v b

ε

+ 1

2 div v

ε

v b

ε

∈ L

⁴³

(]0, T [; L

⁴³

(Ω)).

So it follows from the equation (2) that

∂ v e

ε

∂t ∈ L

²

(]0, T [; (H

¹_ν

(Ω))

^′

) + L

⁴³

(]0, T [; L

⁴³

(Ω))

∂ v b

ε

∂t ∈ L

²

(]0, T [; ( H

¹_ν

(Ω))

^′

) + L

⁴³

(]0, T [; L

⁴³

(Ω))

∂p

ε

∂t ∈ L

²

(]0, T [; L

²₀

(Ω))

We now observe that the two velocity fields v e

ε

and v b

ε

belong to L

⁴

(]0, T [; L

⁴

(Ω)) ∩ L

²

(]0, T [; (H

¹_ν

(Ω))

^′

) which is the dual space of

L

²

(]0, T [; ( H

¹_ν

(Ω))

^′

) + L

⁴³

(]0, T [; L

⁴³

(Ω)).

Thus the functions ( v e

ε

, v b

ε

) are equal almost everywhere to continuous functions with values in L

²

(Ω).

This ends the proof of proposition 3.

3.1. Stability. In the case of three-dimensional vector spaces, we do not have an equality for the conservation of the energy, we have only an inequality. Nevertheless for two-dimensional vector spaces, the weak solutions satisfy the energy equality.

Proof. Through classical computations one obtains, with equations (2) and (3):

1 2

d

dt k v e

ε

k

²L²

+ 1

R

^e

k curl v e

ε

k

²L²

+ 1

R

^e

k div v e

ε

k

²L²

+ r k div v e

ε

k

²L²

+ 1

ε | ( v e

ε

∧ ν) |

²L²(Γ)

− Z

Ω

p

ε

div v e

ε

dω = Z

Ω

f · v e

ε

dω,

(4)

1 2

d

dt k v b

ε

k

²L²

+ 1

R

^e

k curl v b

ε

k

²L²

+ 1

R

^e

k div v b

ε

k

²L²

+ r k div v b

ε

k

²L²

+ 1

ε | ( v b

ε

∧ ν) |

²L²(Γ)

+ 1 ε R

^e

Z

0

div v

ε

div v b

ε

dω = 0,

(5)

1 2

d

dt k v

ε

k

²L²

+ 1

R

^e

k curl v

ε

k

²L²

+ 1

R

^e

k div v

ε

k

²L²

+ 1

ε R

^e

k div v

ε

k

²L²

+ r k div v

ε

k

²L²

+ 1

ε | (v

ε

∧ ν) |

²L²(Γ)

− Z

Ω

p

ε

div v

ε

dω = Z

Ω

f · v

ε

dω,

(6)

γ ε 2

d

dt k p

ε

k

²L²

+ γ ε k p

ε

k

²L²

+ Z

Ω

p

ε

div v

ε

dω + r ε Z

Ω

p

ε

div v e

ε

dω = 0. (7)

(11)

Multiplying (4) by r ε and (5) by ε and summing with (6) and (7), one obtains:

1 2

d

dt r ε k v e

ε

k

²L²

+ ε k v b

ε

k

²L²

+ k v

ε

k

²L²

+ γε k p

ε

k

²L²

+ r ε

R

^e

k curl v e

ε

k

²L²

+ ε

R

^e

k curl v b

ε

k

²L²

+ 1

R

^e

k curl v

ε

k

²L²

+ γ ε k p

ε

k

²L²

+ r ε

R

^e

k div v e

ε

k

²L²

+ ε

R

^e

k div v b

ε

k

²L²

+ 1

R

^e

k div v

ε

k

²L²

+ r | ( v e

ε

∧ ν) |

²L²(Γ)

+ | ( v b

ε

∧ ν) |

²L²(Γ)

+ 1

ε | (v

ε

∧ ν ) |

²L²(Γ)

+ εr

²

k div v e

ε

k

²L²

+ rε k div v b

ε

k

²L²

+ 1

ε R

^e

k div v

ε

k

²L²

= − 1 R

^e

Z

Ω

div v

ε

div v b

ε

dω + r ε Z

Ω

f · v e

ε

dω + Z

Ω

f · v

ε

dω,

Let us now give some bounds of the right-hand side terms.

The term 1

R

^e

(div v

ε

, div v b

ε

) is bounded by 1

2 ε R

^e

k div v

ε

k

²L²

+ ε

2 R

^e

k div v b

ε

k

²L²

. According to the estimate of the L

²

norm in H

¹_ν

(Ω) given by equation (1), we bound the source terms in the following way:

Z

Ω

f · v d ω

≤ k f k

L²

k v k

L²

≤ √

λ k f k

L²

k div v k

²L²

+ k curl v k

²L²

¹2

≤ 1

2 R

e

k div v k

²L²

+ k curl v k

²L²

+ R

^e

λ 2 k f k

²L²

.

Using these bounds, we get from the previous equation the following fundamental estimate:

1 2

d

dt r ε k v e

ε

k

²L²

+ ε k v b

ε

k

²L²

+ k v

ε

k

²L²

+ γε k p

ε

k

²L²

+ γε k p

ε

k

²L²

+ r ε

2 R

e

k curl v e

ε

k

²L²

+ ε

R

e

k curl v b

ε

k

²L²

+ 1

2 R

e

k curl v

ε

k

²L²

+ 1

2 R

^e

k div v

ε

k

²L²

+ r ε

2 R

^e

k div v e

ε

k

²L²

+ ε

2 R

^e

k div v b

ε

k

²L²

+ εr

²

k div v e

ε

k

²L²

+ εr k div v b

ε

k

²L²

+ 1

2ε R

^e

k div v

ε

k

²L²

+ r | ( v e

ε

∧ ν) |

²L²(Γ)

+ | ( v b

ε

∧ ν) |

²L²(Γ)

+ 1

ε | (v

ε

∧ ν) |

²L²(Γ)

≤ λ R

^e

2 (1 + ε r) k f k

²L²

.

(8)

After integration in time, we deduce from the previous estimate that there exists a

continuous function g defined on [0, T ] such that: for all t > 0,

(12)

1 2

r ε k v e

ε

(t) k

²

+ ε k v b

ε

(t) k

²

+ k v

ε

(t) k

²L²

+ γε k p

ε

(t) k

²L²

+ γ ε

Z

t

0

k p

ε

(s) k

²L²

ds + rε 2 R

e

Z

t

0

k curl v e

ε

(s) k

²L²

ds + ε R

e

Z

t

0

k curl v b

ε

(s) k

²L²

ds

+ 1

2 R

^e

Z

t

0

k curl v

ε

(s) k

²L²

ds + rε 2 R

^e

Z

t

0

k div v e

ε

(s) k

²L²

ds

+ ε

2 R

^e

Z

t

0

k div v b

ε

(s) k

²L²

ds + 1 2 R

^e

Z

t

0

k div v

ε

(s) k

²L²

ds + r

Z

t

0

| ( v e

ε

(s) ∧ ν) |

²L²(Γ)

ds + Z

t

0

| ( v b

ε

(s) ∧ ν ) |

²L²(Γ)

ds + 1

ε Z

t

0

| (v

ε

(s) ∧ ν ) |

²L²(Γ)

ds + εr

²

Z

t

0

k div v e

ε

(s) k

²L²

ds + εr

Z

t

0

k div v b

ε

(s) k

²L²

ds + 1 2ε R

^e

Z

t

0

k div v

ε

(s) k

²L²

ds ≤ g(t),

(9)

with

g(t) = r ε k v e

ε

(0) k

²

+ε k v b

ε

(0) k

²

+ k v

ε

(0) k

²L²

+γε k p

ε

(0) k

²L²

+ λ R

^e

2 (1+rε)

Z

t

0

k f (s) k

²L²

ds.

This inequality is the key point to establish the convergence result.

To improve the convergence result in the two-dimensional case, we use the following energy equality derived as above without using (5).

1 2

d

dt r ε k v e

ε

k

²L²

+ k v

ε

k

²L²

+ γε k p

ε

k

²L²

+ r ε

R

e

k curl v e

ε

k

²L²

+ 1

R

^e

k curl v

ε

k

²L²

+ γ ε k p

ε

k

²L²

+ r ε

R

^e

k div v e

ε

k

²L²

+ 1

R

^e

k div v

ε

k

²L²

+ εr

²

k div v e

ε

k

²L²

+ 1

ε R

^e

k div v

ε

k

²L²

+ r | ( v e

ε

∧ ν) |

²L²(Γ)

+ 1

ε | (v

ε

∧ ν) |

²L²(Γ)

= r ε Z

Ω

f · v e

ε

dω + Z

Ω

f · v

ε

dω.

(10)

This concludes the proof of lemma 2.1.

3.2. Uniform stability for the approximate solution. In this section, we deal with the stability of the proposed approximation method. We notice that this property is valid for all solutions satisfying the energy inequality (8), as it is the case when they are built by a finite dimensional approximation method such as the Galerkin method for example.

Proof. Let us write

χ

ε

(t) = ε r k v e

ε

(t) k

²L²

+ ε k v b

ε

(t) k

²L²

+ k v

ε

(t) k

²L²

+ γε k p

ε

(t) k

²L²

.

We note λ > 0 the smallest eigenvalue of the self-adjoint operator A = curl curl −

∇ div with the domain

D = H

¹_ν

(Ω) ∩ (H

²

(Ω))

^d

,

(13)

and we introduce α = min

λ Re

, 1

. Classically, the inequality (8) leads to the differential inequality

d

dt χ

ε

(t) + αχ

ε

(t) ≤ λ R

^e

(1 + r ε) k f (t) k

²L²

, which implies the following uniform bound

χ

ε

(t) ≤ e

⁻^αt

χ

ε

(0) + λ R

e

α (1 + r ε) k f k

^L^∞^,L²

. This concludes the proof of theorem 2.2.

4. Convergence analysis and compactness results.

4.1. Compactness results for the velocity. Let us introduce the Leray projection w

ε

of a velocity field v

ε

(t) ∈ H

¹_ν

(Ω) defined as follows

v

ε

= w

ε

+ ∇ q

ε

, div w

ε

= 0,

w

ε

· ν

_|Γ

= 0, ∇ q

ε

· ν

_|Γ

= 0, Z

Ω

q

ε

dω = 0.

By the estimate (9), we see that the irrotational part of v

ε

goes to zero with ε.

Thus it remains to bring to the fore the behavior of the free divergence part w

ε

and to obtain an estimate on a fractional time derivative of this term. We detail the different steps of this strategy.

From the regularity of the Leray projector (see R. Temam [20] page 18), one has:

k w

ε

k

^L^∞^,L²

≤ c k v

ε

k

^L^∞^,L²

,

k w

ε

k

L²,H¹

≤ c k v

ε

k

L²,H¹

. (11) Moreover, we can easily prove the following lemma.

Lemma 4.1. There exists two constants depending only on T and Ω such that:

k∇ q

ε

k

L²,H¹

≤ c √ ε,

k∇ q

ε

k

L^∞,L²

≤ c (12) Proof: The function q

ε

belongs to H

²

(Ω) and satisfies

− ∆q

ε

(t) = − div v

ε

(t),

∇ q

ε

(t) · ν

_|Γ

= 0.

This implies using the estimate (9)

k ∆q

ε

(t) k

L²,L²

= k div v

ε

k

L²,L²

≤ C √ ε.

Besides, we have Z

Ω

∇ q

ε

· ∇ q

ε

dω = − Z

Ω

∆q

ε

q

ε

dω, so that, with Poincar´e-Neumann inequality, we get

k∇ q

ε

(t) k

²L²,L²

≤ C k ∆q

ε

(t) k

^L²^,L²

k q

ε

k

^L²^,L²

,

≤ C √

ε k∇ q

ε

(t) k

L²,L²

.

(14)

The regularity properties of the Neumann problem give

k∇ q

ε

k

L²,H¹

≤ C k q

ε

k

L²,H²

≤ C k ∆q

ε

k

L²,L²

≤ C √

ε. (13)

Moreover, by orthogonality of the Leray projector in L

²

, one has

k∇ q

ε

k

^L^∞^,L²

≤ k v

ε

k

^L^∞^,L²

≤ C. (14) This concludes the proof of the lemma 4.1.

So by interpolation and using estimates (13)-(14), we have proved the result below.

Corollary 1. The function q

ε

satisfies:

∇ q

ε

strongly converges to 0 in L

^p

(]0, T [; L

²

(Ω))

, ∀ p, 1 ≤ p < + ∞ .

Now we have to write the equation satisfied by w

ε

. As the Leray projection is orthogonal in L

²

(Ω), this equation reads

 

 



 

 

∀ ϕ ∈ H

¹_ν

(Ω), div ϕ = 0, Z

Ω

∂w

ε

∂t · ϕ dω + Z

Ω

(v

ε

· ∇ )v

ε

+ 1

2 (div v

ε

)v

ε

· ϕ dω + 1

R

^e

Z

Ω

curl v

ε

· curl ϕ dω + 1 ε

Z

Γ

(v

ε

∧ ν) · (ϕ ∧ ν) dγ

= Z

Ω

f · ϕ dω in L

¹

(0, T ).

(15)

Now we introduce the extension by 0 of w

ε

(resp. v

ε

) outside [0, T ] denoted, only in this part, by w f

ε

(resp. v e

ε

) and we take the Fourier transform in time of the equation (15) to obtain

 

 



 

 

∀ ϕ ∈ H

¹_ν

(Ω), div ϕ = 0, iτ

Z

Ω

F ( w f

ε

)(τ ) · ϕ dω + Z

Ω

F

( v e

ε

· ∇ ) v e

ε

+ 1

2 (div v e

ε

) v e

ε

(τ) · ϕ dω

+ 1 R

e

Z

Ω

curl F ( v e

ε

)(τ) · curl ϕ dω + 1 ε

Z

Γ

F ( v e

ε

∧ ν)(τ) · (ϕ ∧ ν) dγ

= Z

Ω

F ( f e )(τ) · ϕ dω + 1

√ 2π Z

Ω

v

ε

(0) · ϕ dω − e

⁻^{iτ T}

√ 2π Z

Ω

v

ε

(T ) · ϕ dω.

Following Boyer-Fabrie [5, page 253], we take ϕ = F ( w f

ε

)(τ) as test function in the previous equation to obtain for all τ ∈ R:

iτ Z

Ω

|F ( w f

ε

)(τ) |

²

dω = − Z

Ω

F ( v e

ε

· ∇ ) v e

ε

(τ) · F ( w f

ε

)(τ ) dω

− 1 2

Z

Ω

F (div v e

ε

) v e

ε

(τ ) · F ( w f

ε

)(τ) dω

− 1 R

^e

Z

Ω

curl F ( v e

ε

)(τ) · curl F ( w f

ε

)(τ) dω

− 1 ε

Z

Γ

F ( v e

ε

∧ ν)(τ) · ( F ( w f

ε

)(τ) ∧ ν) dγ +

Z

Ω

F ( f e )(τ) · F ( w f

ε

)(τ ) dω + 1

√ 2π Z

Ω

v

ε

(0) · F ( w f

ε

)(τ) dω − e

⁻^{iτ T}

√ 2π Z

Ω

v

ε

(T) · F ( w f

ε

)(τ) dω.

(15)

As we look for an estimate independent of ε, we have to pay a special attention to the imaginary part of the penalty term:

A

ε

= − 1 ε

Z

Γ

F ( v e

ε

∧ ν)(τ) · ( F ( w f

ε

)(τ) ∧ ν) dγ. (16) By writing w

ε

= v

ε

− ∇ q

ε

, we have:

1 ε

Z

Γ

F ( v e

ε

∧ ν )(τ ) · ( F ( w f

ε

)(τ) ∧ ν) dγ = 1 ε Z

Γ

|F ( v e

ε

)(τ) ∧ ν |

²

dγ

− 1 ε Z

Γ

F ( ∇ q e

ε

)(τ ) ∧ ν

· F ( v e

ε

)(τ) ∧ ν) dγ.

So, the imaginary part of A

ε

is bounded as follows:

1 ε Z

Γ

F ( ∇ q e

ε

)(τ) ∧ ν

· F ( v e

ε

)(τ ) ∧ ν) dγ

≤ 1

ε |F ( ∇ q e

ε

)(τ) |

L²(Γ)

|F ( v e

ε

)(τ) ∧ ν |

L²(Γ)

,

≤ C 1

ε kF ( ∇ q e

ε

)(τ ) k

H¹

|F ( v e

ε

)(τ ) ∧ ν |

L²(Γ)

.

(17)

From estimates (12) and (9), we have

k∇ q

ε

k

L2,H¹

≤ C √ ε,

| v

ε

∧ ν |

^L²^,L²(Γ)

≤ C √ ε.

So there exists a function f

ε⁴

(τ) ∈ L

¹

(R) bounded independently on ε such that:

1 ε Z

Γ

F ( ∇ q e

ε

)(τ) ∧ ν

· F ( v e

ε

)(τ) ∧ ν ) dγ

≤ f

ε⁴

(τ) (18) Now we can derive the estimate of | τ |

Z

Ω

|F ( w f

ε

)(τ) |

²

dω, and we have

| τ | Z

Ω

|F ( w f

ε

)(τ) |

²

dω ≤ Z

Ω

F ( v e

ε

· ∇ ) v e

ε

(τ) · F ( w f

ε

)(τ)dω + 1

2 Z

Ω

F (div v e

ε

) v e

ε

(τ) · F ( w f

ε

)(τ)dω + 1

R

^e

Z

Ω

curl F ( v e

ε

)(τ) · curl F ( w f

ε

)(τ)dω + 1

ε Z

Γ

F ( ∇ q e

ε

)(τ) ∧ ν

· F ( v e

ε

)(τ ) ∧ ν) dγ

+

Z

Ω

F (f )(τ ) · F ( w f

ε

)(τ) dω + 1

√ 2π Z

Ω

v

ε

(0) · F ( w f

ε

)(τ)dω + 1

√ 2π Z

Ω

v

ε

(T ) · F ( w f

ε

)(τ)dω

≤ f

_ε¹

(τ) + f

_ε²

(τ) + f

_ε³

(τ ) + f

_ε⁴

(τ) + f

_ε⁵

(τ) + f

_ε⁶

(τ) + f

_ε⁷

(τ ).

(19)

We now estimate each term of the right-hand side of the previous inequality for d ≤ 3.

Term f

ε¹

= Z

Ω

F ( v e

ε

· ∇ ) v e

ε

(τ) · F ( w f

ε

)(τ)dω

(16)

According to the energy estimate (9), the function v

ε

is bounded in L

²

(0, T ; H

¹_ν

(Ω)) and hence, by Sobolev injection, it is bounded in L

⁶⁵

(0, T ; L

⁶

(Ω)). So by Hausdorff- Young theorem,

( F ( w f

ε

))

ε

is bounded in L

⁶

(R; L

⁶

(Ω)). (20) We also have the inequality:

k v e

ε

· ∇ v e

ε

k

_L⁶5

≤ k∇ v

ε

k

L²

k v

ε

k

L³

≤ C k∇ v

ε

k

3 2

L²

k v

ε

k

1 2

L²

,

which implies, according to (9), that v

ε

· ∇ v

ε

is bounded in L

⁴³

(0, T ; L

⁶⁵

(Ω)), and necessarily in L

⁶⁵

(0, T ; L

⁶⁵

(Ω)). So, by Hausdorff-Young theorem, the family of functions F ( v e

ε

· ∇ ) v e

ε

is bounded in L

⁶

(R; L

⁶⁵

(Ω)). Then with H¨ older inequality, (f

ε¹

)

ε

is bounded in L

³

(R). (21) Term f

_ε²

= 1

2 Z

Ω

F (div v e

ε

) v e

ε

(τ) · F ( w f

ε

)(τ)dω

The same arguments show that

(f

ε²

)

ε

is bounded in L

³

(R). (22) Term f

_ε³

= 1

R

^e

Z

Ω

curl F ( v e

ε

)(τ ) · curl F ( w f

ε

)(τ)dω

According to the regularity of the Leray projection recalled above and estimate (9), one has:

(f

_ε³

)

ε

is bounded in L

¹

(R). (23) Terms f

_ε⁴

= 1

ε Z

Γ

F ( ∇ q e

ε

)(τ) ∧ ν

· F ( v e

ε

)(τ) ∧ ν) dγ

As we have seen by the estimate (18),

(f

_ε⁴

)

ε

is bounded in L

¹

(R). (24) Term f

_ε⁵

=

Z

Ω

F ( ˜ f )(τ ) · F ( w f

ε

)(τ) dω

By hypothesis, ˜ f is a given function in L

²

(R; L

²

(Ω)) and from estimate (9), we get that v e

ε

is bounded in L

²

(R; L

²

(Ω)), so as w f

ε

is the Leray projection of v e

ε

, the function w f

ε

is also bounded in L

²

(R; L

²

(Ω)). Finally, we obtain

(f

_ε⁵

)

ε

is bounded in L

¹

(R). (25) Terms f

ε⁶

= 1

√ 2π Z

Ω

v

ε

(0) · F ( w f

ε

)(τ)dω

and f

ε⁷

=

^√¹_2π

R

Ω

v

ε

(T ) · F ( w f

ε

)(τ )dω These two terms come from the Dirac measure when we derive discontinuous functions. Let us consider f

_ε⁷

.

f

_ε⁷

(τ) ≤ 1

√ 2π k v

ε

(T) k

L²

kF ( w f

ε

)(τ) k

L²

.

According to (9), this term is bounded in L

^∞

(R). Moreover, the set of functions F ( w f

ε

)(τ)

ε

is bounded in L

²

(R; L

²

(Ω)) so, f

ε⁷

ε

is bounded in L

²

(R).

(17)

We treat in the same way the term f

_ε⁶

, and thus:

(f

_ε⁶

)

ε

and (f

_ε⁷

)

ε

are bounded in L

²

(R) (26) We are now able to show that the set of functions ( w f

ε

)

ε

is bounded in an appropriate Nikolskii space.

For all γ < 1, there exists a constant d such that:

| τ |

¹⁻^γ

≤ d

1 + | τ | 1 + | τ |

^γ

, so that,

| τ |

¹⁻^γ

kF ( w f

ε

)(τ) k

²L²

≤ d

kF ( w f

ε

)(τ) k

²L²

+ | τ |

1 + | τ |

^γ

kF ( w f

ε

)(τ) k

²L²

Let us denote f

_ε⁰

(τ) = kF ( w f

ε

)(τ) k

²L²

, which belongs to L

¹

(R), the previous inequality reads with (19):

| τ |

¹⁻^γ

kF ( w f

ε

)(τ) k

²L²

≤ d f

_ε⁰

(τ) + d 1 + | τ |

^γ

f

_ε¹

(τ) + f

_ε²

(τ) + f

_ε³

(τ) + f

_ε⁴

(τ ) + f

_ε⁵

(τ) + f

_ε⁶

(τ) + f

_ε⁷

(τ)

≤ h(τ)

If we suppose that the function τ 7→ 1

1 + | τ |

^γ

belongs to L

^∞

(R) ∩ L

²

(R), then the function τ 7→ h(τ) belongs to L

¹

(R). This condition is satisfied for γ ∈ ]

²₃

, 1[. So we have proved :

Lemma 4.2. Let us suppose that σ ∈ ]0,

¹₆

[, then there exists a constant C such that Z

R

| τ |

^2σ

kF (w

ε

)(τ) k

²L²

dτ ≤ C. (27) Then, from lemma 4.1 and 4.2 we deduce the following key result:

Theorem 4.3. There exists a sequence (ε

k

)

k

which converges to zero and a function v ∈ L

²

(]0, T [; L

²

(Ω)) satisfying div v = 0 such that:

(v

εk

)

k

→ v in L

²

(]0, T [; L

²

(Ω)) strongly.

Proof. The function v

ε

is the sum of two terms ∇ q

ε

and w

ε

. From corollary 1 the first term converges strongly to 0 in L

²

(]0, T [; L

²

(Ω)). Now, from Aubin-Lions- Simon Theorem, it follows from lemma 4.2, that there exists a sequence (ε

k

)

k

such that:

(w

ε_k

)

k

−→ v in L

²

(]0, T [; L

²

(Ω)) strongly . Moreover, since div w

ε_k

= 0, we have div v = 0.

4.2. Convergence of the method. We first give a general convergence theorem

for a subsequence solution of the approximate scheme (3), to a weak solution of

the initial Navier-Stokes problem, in the case d ≤ 3. For the two-dimensional case,

since the weak solution of the Navier-Stokes equation is unique, the whole sequence

of approximate solution v

ε

converges to v. Moreover, in this case, we prove that

the convergence is strong.

(18)

4.2.1. The general case d ≤ 3. Let θ an element of C

^∞

(0, T ), satisfying θ(T ) = 0, and ϕ a free-divergence vector field in H

₀¹

(Ω)

d

∩ H

²

(Ω))

^d

. An integration by parts gives from the equation (3)

− Z

T

0

Z

Ω

v

ε

· ϕ dω θ

^′

(τ) dτ + Z

T

0

Z

Ω

(v

ε

· ∇ )v

ε

+ 1

2 (div v

ε

) v

ε

· ϕ dω θ(τ) dτ + 1

R

^e

Z

T

0

Z

Ω

curl v

ε

· curl ϕ dω θ(τ ) dτ

= Z

T

0

Z

Ω

f ϕ dω θ(τ ) dτ + Z

Ω

v

0

ϕ dω θ(0).

(28)

According to estimate (9), there exists a sequence ε

k

such that v

ε_k

→ v in L

²

(0, T ; L

²

(Ω)) strongly,

div v

ε_k

→ div v = 0 in L

²

(0, T ; L

²

(Ω)) strongly, curl v

ε_k

⇀ curl v in L

²

(0, T ; L

²

(Ω)) weakly.

Since k u k

²L²

+ k div u k

²L²

+ k curl u k

²L²

¹2

is a norm equivalent to the H

¹

-norm on H

¹_ν

(Ω), we have

∇ v

ε_k

⇀ ∇ v in L

²

(0, T ; L

²

(Ω)) weakly.

and so, we can take the limit on the term (v

ε

· ∇ )v

ε

as v

ε_k

· ∇ v

ε_k

→ v · ∇ v in L

¹

(0, T ; L

¹

(Ω)) div v

ε_k

v

ε_k

→ 0 in L

¹

(0, T ; L

¹

(Ω)).

From estimate (9), we have for the tangential traces Z

t

0

| (v

ε

∧ ν )(τ) |

²L²(Γ)

dτ ≤ ε g(t), and since for any function v in H

¹

(Ω)

d

,

| v |

L²(Γ)

≤ C k v k

1 2

L²

k v k

1 2

H¹

, we obtain that

v

ε_k

→ v in L

²

]0, T [; (L

²

(Γ))

^d

strongly.

This implies (v ∧ ν )

_|Γ

= 0, and so, since by construction (v · ν )

_|Γ

= 0, v belongs to H

₀¹

(Ω)

d

Finally, at the limit process we obtain

 

 

 

 



− Z

T

0

Z

Ω

v · ϕ dω θ

^′

(τ) dτ + Z

T

0

Z

Ω

(v · ∇ )v · ϕ dω θ(τ) dτ + 1

R

^e

Z

T

0

Z

Ω

curl v · curl ϕ dω θ(τ ) dτ = Z

T

0

Z

Ω

f · ϕ dω θ(τ) dτ + Z

Ω

v

0

ϕ dω θ(0), div v = 0,

v

_|Γ

= 0.

From the identity Z

Ω

∇ v : ∇ ϕ dω = Z

Ω

curl v · curl ϕ dω + Z

Ω

div v · div ϕ dω,