Convergence of a vector BGK approximation for the incompressible Navier-Stokes equations
arXiv:1705.04026v1 [math.AP] 11 May 2017
Roberta Bianchini1 , Roberto Natalini2
Abstract We present a rigorous convergence result for the smooth solutions to a singular semilinear hyperbolic approximation, a vector BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof is based on the use of a constant right symmetrizer, weighted with respect to the parameter of the singular pertubation system. This symmetrizer provides a conservative-dissipative form for the system and this allow us to perform uniform energy estimates and to get the convergence by compactness. Keywords: vector BGK schemes, incompressible Navier-Stokes equations, symmetrizer, conservative-dissipative form.
1. Introduction We want to study the convergence of a singular perturbation approximation to the Cauchy problem for the incompressible Navier-Stokes equations on the D dimensional torus TD : ( ∂t uN S + ∇ · (uN S ⊗ uN S ) + ∇P N S = ν∆uN S , (1.1) ∇ · uN S = 0, with (t, x) ∈ [0, +∞) × TD , and initial data uN S (0, x) = u0 (x), with ∇ · u0 = 0.
(1.2)
Here uN S and ∇P N S are respectively the velocity field and the gradient of the pressure term, and ν > 0 is the viscosity coefficient. Email addresses:
[email protected] (Roberta Bianchini),
[email protected] (Roberto Natalini) 1 Dipartimento di Matematica, Universit` a degli Studi di Roma ”Tor Vergata”, via della Ricerca Scientifica 1, I00133 Rome, Italy - Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy. 2 Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy.
Preprint
May 12, 2017
We consider a semilinear hyperbolic approximation, called vector BGK model, [13, 10], to the incompressible Navier-Stokes equations (1.1). The general form of this approximation is as follows: ∂t flε +
λl 1 · ∇x flε = 2 (Ml (ρε , ερε uε ) − flε ), ε τε
(1.3)
with initial data ρ, ε¯ ρu0 ), flε (0, x) = Mlε (¯
u0 in (1.2),
l = 1, · · · , L,
(1.4)
where flε and Mlε take values in RD+1 , with the Maxwellian functions Mlε Lipschitz continuous, λl = (λl1 , · · · , λlD ) are constant velocities, and L ≥ D + 1. Moreover, ρ¯ > 0 is a given constant value, and ε and τ are positive parameters. Denoting by fl εj , Ml εj for j = 0, · · · , D the D + 1 components of flε , Mlε for each l = 1, · · · , L, let us set ε
ρ =
L X
fl ε0 (t, x)
and
qjε
=
ερε uεj
l=1
=
L X
fl εj (t, x).
(1.5)
l=1
In [13, 10], it is numerically studied the convergence of the solutions to the vector BGK model to the solutions to the incompressible Navier-Stokes equations. More precisely, assuming that, in a suitable functional space, ρε → ρˆ,
ˆ, uε → u
and
ρε − ρ¯ → Pˆ , ε2
under some consistency conditions of the BGK approximation with respect to the Navier-Stokes equations, [13], it can be shown that the couple (ˆ u, Pˆ ) is a solution to the incompressible NavierStokes equations. The aim of the present paper is to provide a rigorous proof of this convergence in the Sobolev spaces. Vector BGK models come from the ideas of kinetic approximations for compressible flows. They are inspired by the hydrodynamic limits of the Boltzmann equation: see [3, 4, 14] for the limit to the compressible Euler equations, and see [15, 16] for the incompressible Navier-Stokes equations. In this regard, one of the main directions has been the approximation of hyperbolic systems with discrete velocities BGK models, as in [11, 20, 25, 8, 27]. Similar results have been obtained for convection-diffusion systems under the diffusive scaling [23, 9, 22, 2]. In the framework of the BGK approximations, one of the first important contributions was given in computational physics by the so called Lattice-Boltzmann methods, see for instance [28, 29]. Under some assumptions on the physical parameters, LBMs approximate the incompressible Navier-Stokes equations by scalar velocities models of kinetic equations, and a rigorous mathematical result on the validity of these kinds of approximations was proved in [21]. Other partially hyperbolic approximations of the Navier-Stokes equations were developed in [12, 26, 18, 17]. The vector BGK systems studied in the present paper are a combination of the ideas of discrete velocities BGK approximations and LBMs. They are called vector BGK models since, unlike the LBMs [28, 29], they associate every scalar velocity with one vector of unknowns. Another fruitful 2
property of vector BGK models is their natural compatibility with a mathematical entropy, [8], which provides a nice analytical structure and stability properties. The work of the present paper takes its roots in [13, 10], where vector BGK approximations for the incompressible Navier-Stokes equations were introduced. Here we prove a rigorous local in time convergence result for the smooth solutions to the vector BGK system to the smooth solutions to the Navier-Stokes equations. In this paper we focus on the two dimensional case in space. Following [13], let us set D = 2, L = 5, and wε = (ρε , qε ) = (ρε , q1ε , q2ε ) = (ρε , ερε uε1 , ερε uε2 ) =
5 X
flε ∈ R3 .
(1.6)
l=1
Fix λ, τ > 0 and let ε > 0 be a small parameter, which is going to zero in the singular perturbation limit. Thus, we get a five velocities model (15 scalar equations): ∂t f1ε + λε ∂x f1ε = τ 1ε2 (M1 (wε ) − f1ε ), λ 1 ε ε ε ε ∂t f2 + ε ∂y f2 = τ ε2 (M2 (w ) − f2 ), (1.7) ∂t f3ε − λε ∂x f3ε = τ 1ε2 (M3 (wε ) − f3ε ), λ 1 ε ε ε ε ∂t f4 − ε ∂y f4 = τ ε2 (M4 (w ) − f4 ), ∂t f5ε = τ 1ε2 (M5 (wε ) − f5ε ).
Here the Maxwellian functions Mj ∈ R3 have the following expressions: M1,3 (wε ) = awε ± where
A1 (wε ) , 2λ
A1 (wε ) =
(q1ε )2 ρε
M2,4 (wε ) = awε ± q1ε
+ P (ρε ) ,
q1ε q2ε ρε
A2 (wε ) , 2λ
A2 (wε ) =
P (ρε ) = ρε − ρ¯,
and
M5 (wε ) = (1 − 4a)wε , q2ε
(q2ε )2 ρε
q1ε q2ε ρε
+
P (ρε )
,
(1.8)
(1.9)
(1.10)
ν , (1.11) 2λ2 τ where ν is the viscosity coefficient in (1.1). In the following, our main goal is to obtain uniform energy estimates for the solutions to the vector BGK model (1.7) in the Sobolev spaces and to get the convergence by compactness. In [13, 10], an L2 estimate was obtained by using the entropy function associated with the vector BGK model, whose existence is proved in [8]. However, there is no explicit expression for the kinetic entropy, so we do not know the weights, with respect to the singular parameter, of the terms of the classical symmetrizer derived by the entropy, see [19] for the one dimensional case and [7, 21] for the general case. For this reason, the existence of an entropy is not enough to control the higher order estimates. Moreover, our pressure term is given by (1.10) a=
3
and it is linear with respect to ρε , so the estimates in [13, 10] no more hold. To solve this problem, we use a constant right symmetrizer, whose entries are weighted in terms of the singular parameter in a suitable way. Besides, the symmetrization obtained by the right multiplication provides the conservative-dissipative form introduced in [7]. The dissipative property of the symmetrized system holds under the following hypothesis. Assumption 1.1 (Dissipation condition). We assume the following structural condition: 1 0 0; # 4a ΘΣ + (1 − 1/β)ΓΣ 4a2 − 2aζ − η > 0; # " 4a 2 ΘΣ + (1 − β)ΓΣ 4a − 2aζ − η > 0.
(4.29)
Recalling definition (4.4), ∆Σ = 2(λ2 a−δ), and so the second inequality is satisfied for λ big enough. Precisely, we take λ as in Assumption 4.1 and r ηΓΣ (1 + 2a) δ + . (4.30) λ> a 2a Moreover, the first condition of (4.29) is verified if η>
2ζ , ζ > 4a . ζ − 4a
(4.31)
Since ΘΣ and ΓΣ are positive, taking 1 − β < 0, i.e. β > 1, the last inequality is verified if 2aζ + From (4.31), 2aζ +
4a − 4a2 > 0. η
4a 4a 4a − 4a2 > 8a2 + − 4a2 = 4a2 + > 0, η η η
then the last inequality in (4.29) holds under (4.31). Now, the third condition in (4.29) is satisfied if ΘΣ ζ< + 2(a − 1/η). 2aΓΣ (1 − 1/β) Thus, if η >
1 , we can take a 4a < ζ
1 such that: 4a
1, we need 1
0, 8a2 − ΘΣ
which is already satisfied thanks to Lemma 3.1. This way, from (4.28), (4.24) and (4.29), we get some positive constants Γ1Σ , ∆1Σ , Θ1Σ such that 2 2 8 1 2 2 2 2 c(||u||L∞ ([0,t],H s (T2 )) )t ˜ ˜ ˜ Γ1Σ ||w(t)||2s + ε6 ∆1Σ (||m(t)|| ˜ , s + ||ξ(t)||s ) + ε ΘΣ (||k(t)||s + ||h(t)||s ) ≤ cε ||u0 ||s e (4.35) and, in particular, c(||u||L∞ ([0,(t)],H s (T2 )) )t ||w(t)||2s ≤ cε2 ||u0 ||2s e , (4.36)
i.e.
||ρ(t) − ρ¯||2s c(||u||L∞ ([0,t],H s (T2 )) )t + ||ρu(t)||2s ≤ c||u0 ||2s e . ε2
(4.37)
Thus, we are able to prove that the time T ε of existence of the solutions to the vector BGK scheme is bounded form below by a positive time T ⋆ , which is independent of ε. Proposition 4.2. There exist ε0 and T ⋆ fixed such that T ⋆ < T ε for all ε ≤ ε0 . This also yields, for ε ≤ ε0 , the uniform bounds: ||u(t)||s ≤ M,
t ∈ [0, T ⋆ ],
||ρ(t) − ρ¯||s ≤ εM, i.e. ||ρ(t)||s ≤ ρ¯|T2 | + εM,
(4.38) t ∈ [0, T ⋆ ],
(4.39)
and ||ρu(t)||s ≤ M (¯ ρ|T2 | + εM ),
t ∈ [0, T ⋆ ].
(4.40)
Proof. Let u0 ∈ H s (T2 ) and, from (1.4), recall that ρ0 = ρ¯. Then, there exists a positive constant M0 such that ||u0 ||s ≤ M0 , and ˜ 0. ||ρ0 u0 ||s = ρ¯||u0 ||s ≤ ρ¯M0 =: M 19
(4.41)
˜ 0 be any fixed constant, and Let M > M ( ) ||ρ(t) − ρ¯||2 s T0ε := sup t ∈ [0, T ε ] + ||ρu(t)||2s ≤ M 2 , ∀ε ≤ ε0 . ε2
(4.42)
Notice that, from (4.42),
||ρ − ρ¯||∞ ≤ cS ||ρ − ρ¯||s ≤ cS M ε, t ∈ [0, T0ε ], where cS is the Sobolev embedding constant, i.e. ρ¯ − cS M ε ≤ ρ ≤ ρ¯ + cS M ε, t ∈ [0, T0ε ]. Taking ε0 such that ρ¯ − cS M ε0 > ρ2¯ , i.e. ρ¯ > 2cS M ε0 , we have ρ> Now, since s > 3 =
D 2
ρ¯ , 2
t ∈ [0, T0ε ].
(4.43)
+ 2, ||u||s ≤ ||ρu||s ||1/ρ||s .
Moreover, |T2 | ||ρ||s ||1/ρ||s ≤ c + ρ¯ c(¯ ρ)
!
≤ c1 + c2 ||ρ||s .
From (4.42), ρ + M ε), ||ρ||s ≤ c(|T2 |¯ so ||1/ρ||s ≤ c1 + c2 M ε, and ||u||s ≤ cM (c1 + c2 M ε). From (4.37), ||ρ(t) − ρ¯||2s + ||ρu(t)||2s ≤ cM02 ec(M (c1 +c2 M ε))t , ε2 We take T ⋆ ≤ T0ε such that ⋆ cM02 ec(M (c1 +c2 M ε0 ))T ≤ M 2 ,
t ∈ [0, T0ε ].
i.e. T⋆ ≤
1 log(M 2 /(cM02 )) ∀ε ≤ ε0 . c(M (c1 + c2 M ε0 ))
(4.44)
This way, ||u(t)||s ≤ cM (c1 + c2 M ε),
t ∈ [0, T ⋆ ] and ||ρu||s ≤ M ∀ε ≤ ε0 .
20
(4.45)
4.3. Time derivative estimate In order to use the compactness tools, we need a uniform bound for the time derivative of the unknown vector field. Proposition 4.3. If Assumptions 1.1 and 4.1 hold, for M0 in (4.41) and M in (4.38), we have: ˜ 2 ) ˜ 2 + ||∂t h|| ˜ 2 ) + ε8 (||∂t k|| ˜ 2s−1 + ||∂t ξ|| ||∂t w||2s−1 + ε6 (||∂t m|| s−1 s−1 s−1 (4.46) ≤ ε2 c(||u0 ||s )ec(M )t ≤ ε2 c(M0 , M ) in [0, T ⋆ ], with T ⋆ in (4.44). This also yields the uniform bound: ||∂t (ρ − ρ¯)||2s−1 + ||∂t (ρu)||2s−1 ≤ c(||u0 ||s ) ≤ M 2 in [0, T ⋆ ]. ε2
(4.47)
˜ ε , from (2.16) we get: Proof. Let us take the time derivative of system (3.5). Defining V˜ = ∂t W ˜ 1 Σ∂x V˜ + Λ ˜ 2 Σ∂y V˜ = −LΣV˜ + ∂t N ((ΣW ˜ )1 + w) ∂t ΣV˜ + Λ ¯ = −LΣV˜ + ∂t N (w + w), ¯ where
1 ∂t N (w + w) ¯ = τ
0 0 2u1 ∂t w2 − εu21 ∂t w1 u2 ∂t w2 + u1 ∂t w3 − εu1 u2 ∂t w1 0 u2 ∂t w2 + u1 ∂t w3 − εu1 u2 ∂t w1 2u2 ∂t w3 − εu22 ∂t w1 0 0
Taking the scalar product with V˜ , we have:
(4.48)
.
1 d (ΣV˜ , V˜ )0 + (LΣV˜ , V˜ )0 ≤ |(∂t N (w + w), ¯ V )0 |. 2 dt
(4.49)
(4.50)
Here, |(∂t N (w+w), ¯ V˜ )0 | =
1 |(2u1 ∂t w2 −εu21 ∂t w1 , ε2 ∂t m ˜ 2 )0 +(u2 ∂t w2 +u1 ∂t w3 −εu1 u2 ∂t w1 , ε2 ∂t m ˜ 3 +ε2 ∂t ξ˜2 )0 τ +(2u2 ∂t w3 − εu22 ∂t w1 , ε2 ∂t ξ˜3 )0 | ≤ c(||u||∞ )||∂t w||20 +
ε4 ˜ 2 ). (||∂t m|| ˜ 20 + ||∂t ξ|| 0 2τ
Similarly to (4.16), we get: ˜ 2 ) + ε8 ΘΣ (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2) ΓΣ ||∂t w|| ˜ 20 + ε6 ∆Σ (||∂t m|| ˜ 20 + |||∂t ξ|| 0 0 0 21
+
Z
T
0
2
≤ cε
||∂t u|t=0 ||20
˜ 2 ) + 2ε6 ΘLΣ (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 ) dt (2∆LΣ − 1/τ )ε4 (||∂t m|| ˜ 20 + ||∂t ξ|| 0 0 0
+ c(||u||L∞ ([0,T ]×T2 ) )
Z
T 0
˜ 2 ) + ε8 (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 ) dt. ||∂t w|| ˜ 20 + ε6 (||∂t m|| ˜ 20 + ||∂t ξ|| 0 0 0 (4.51)
Now, from the first equation given by (2.9), ∂t w|t=0 = −∂x m|t=0 − ∂y ξ|t=0 ,
(4.52)
where, from (2.6), (1.4), and (1.9), m|t=0
ξ|t=0
u01 λ A1 (w0 ) = (f1 |t=0 − f3 |t=0 ) = = ρ¯ εu0 21 , ε ε εu01 u02
u02 λ A2 (w0 ) = (f2 |t=0 − f4 |t=0 ) = = ρ¯ εu01 u02 . ε ε εu0 22
By definition of w in (1.6), ∂t w|t=0 = (∂t ρ|t=0 , ε∂t (ρu)|t=0 ). This implies that ∂t ρ|t=0 = −¯ ρ(∇ · u0 ) = 0, since u0 is divergence free. This way, ∂t u|t=0 = −∂x
u0 21 u01 u02
− ∂y
u01 u02 u0 22
.
(4.53)
Thus, ˜ 2 ) + ε8 ΘΣ (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2) ΓΣ ||∂t w|| ˜ 20 + ε6 ∆Σ (||∂t m|| ˜ 20 + |||∂t ξ|| 0 0 0 +
Z
0
≤
T
˜ 2 ) + 2ε6 ΘLΣ (||∂t k|| ˜ 2 + ||∂t h||2 ) dt (2∆LΣ − 1/τ )ε4 (||∂t m|| ˜ 20 + ||∂t ξ|| 0 0 0
cε2 ||∇u0 ||20
Z + c(M )
T 0
(4.54)
˜ 2 ) + ε8 (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 ) dt, ||∂t w|| ˜ 20 + ε6 (||∂t m|| ˜ 20 + ||∂t ξ|| 0 0 0
where the last inequality follows form the Sobolev embedding theorem and from (4.38). Similarly, taking the |α|-derivative, for |α| ≤ s − 1, of (4.48) and multiplying by D α V˜ through the scalar product, we get: 1 d (ΣD α V˜ , D α V˜ )0 + (LΣD α V˜ , D α V˜ )0 ≤ |(D α ∂t N (w + w), ¯ D α V )0 |, 2 dt 22
(4.55)
where
1 |(D α (2u1 ∂t w2 − εu21 ∂t w1 ), ε2 ∂t D α m ˜ 2 )0 τ +(D α (u2 ∂t w2 + u1 ∂t w3 − εu1 u2 ∂t w1 ), ε2 ∂t D α m ˜ 3 + ε2 ∂t D α ξ˜2 )0
|(D α ∂t N (w + w), ¯ D α V˜ )0 | =
+(D α (2u2 ∂t w3 − εu22 ∂t w1 ), ε2 ∂t D α ξ˜3 )0 | 4 ε4 ˜ 2 ) ≤ c(M )||∂t w||2 + ε (||∂t m|| ˜ 2 ), (||∂t m|| ˜ 2s−1 + ||∂t ξ|| ˜ 2s−1 + ||∂t ξ|| s−1 s−1 s−1 2τ 2τ where the last inequality follows from (4.38). Finally, we obtain:
≤ c(||u||s−1 )||∂t w||2s−1 +
˜ 2 ) ˜ 2 + ||∂t h|| ˜ 2 ) + ε8 ΘΣ (||∂t k|| ˜ 2s−1 + ||∂t ξ|| ΓΣ ||∂t w|| ˜ 2s−1 + ε6 ∆Σ (||∂t m|| s−1 s−1 s−1 +
Z
0
≤
cε2 ||∇u
T
˜ 2 ) + ε6 ΘLΣ (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 ) dt (2∆LΣ − 1/τ )ε4 (||∂t m|| ˜ 2s−1 + ||∂t ξ|| s−1 s−1 s−1
2 0 ||s−1
Z + c(M )
T 0
˜ 2 ) + ε8 (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 ) dt. ||∂t w|| ˜ 2s−1 + ε6 (||∂t m|| ˜ 2s−1 + ||∂t ξ|| s−1 s−1 s−1 (4.56)
Lemma 4.3. If Assumption 1.1 and 4.1 hold, then there exists a positive constant c such that: ˜ 2 ) + ε8 (||∂t k|| ˜ 2 + ||∂t h|| ˜ 2 )). ||∂t w||2s−1 ≤ c(||∂t w|| ˜ 2s−1 + ε6 (||∂t m|| ˜ 2s−1 + ||∂t ξ|| s−1 s−1 s−1
(4.57)
Proof. The proof of Proposition 4.1 can be adapted here with slight modifications. We end the proof by applying the Gronwall inequality to (4.56) and using Lemma 4.3. 5. Convergence to the Navier-Stokes equations Now we state our main result. Theorem 5.1. Let s > 3. If Assumptions 1.1 and 4.1 hold, there exists a subsequence W ε = (wε ⋆ , ε2 mε , ε2 ξ ε , ε2 kε ⋆ , ε2 hε ⋆ ), with wε ⋆ = (ρε − ρ¯, ερε uε ) and ρ¯ > 0, of the solutions to the vector BGK model (2.12) with initial data (2.13) and u0 ∈ H s (T2 ) in (1.2), such that (ρε , uε ) → (¯ ρ, uN S ) in C([0, T ⋆ ], H s′ (T2 )), with T ⋆ in (4.44), s − 1 < s′ < s, and where uN S is the unique solution to the Navier-Stokes equations in (1.1), with initial data u0 above and P N S the incompressible pressure. Moreover, ∇(ρε − ρ¯) ⋆ ⇀ ∇P N S in L∞ ([0, T ⋆ ], H s−3 (T2 )). ε2
23
Proof. First of all, consider the previous bounds in (4.38), (4.39), (4.40) and (4.47): ||ρε − ρ¯||s ≤ M, ε t∈[0,T ⋆ ] sup
sup ||ρε uε ||s ≤ N,
t∈[0,T ⋆ ]
sup
||∂t (ρε − ρ¯)||s−1 ≤ M1 , ε t∈[0,T ⋆ ]
(5.1)
sup ||∂t (ρε uε )||s−1 ≤ N1 ,
(5.2)
t∈[0,T ⋆ ]
where M, M1 , N, N1 are positive constants. The Lions-Aubin Lemma in [6] implies that, for s − 1 < s′ < s, ρε → ρ¯ strongly in C([0, T ⋆ ], H s′ (T2 )), and there exists m⋆ such that mε = ρε uε → m⋆ strongly in C([0, T ⋆ ], H s′ (T2 )). Notice also that uε =
mε , where ρε 1/ρε → 1/¯ ρ strongly in C([0, T ⋆ ], H s′ (T2 )),
since we can take ρ¯ such that ρε > uε =
ρ¯ 2
as in (4.43). Then
mε m⋆ =: u⋆ strongly in C([0, T ⋆ ], H s′ (T2 )). → ρε ρ¯
Now, consider system (2.9) in the following formulation: ∂t wε + ∂x mε + ∂y ξ ε = 0; ε ¯ 2 ε ε∂t mε + λε ∂x kε = τ1 ( A1 (wε2 +w) − mε ), ε ¯ ε 2 ε∂t ξ ε + λε ∂y hε = τ1 ( A2 (wε2 +w) − ξε ), . (2aw ε −k ε ) ε ε , ε∂t k + ε∂x m = τε (2aw ε −hε ) ε ε ε∂t h + ε∂y ξ = , τε
From (5.3) and 2aλ2 τ = ν as in (1.11), it follows that ( ε ¯ mε = A1 (wε +w) − ν∂x wε + ε2 λ2 τ 2 (∂tx kε + ∂xx mε ) − ε2 τ ∂t mε ; A2 (w ε +w) ¯ ξ= − ν∂y wε + ε2 λ2 τ 2 (∂ty hε + ∂yy ξ ε ) − ε2 τ ∂t ξ ε . ε
(5.3)
(5.4)
Substituting the expansions above in the first equation of (5.3), we get the following equation: ∂t wε +
∂x A1 (w ε +w) ¯ ε
+
∂y A2 (w ε +w) ¯ ε
− ν∆wε
= ε2 τ ∂tx mε + ε2 τ ∂ty ξ ε − ε2 λ2 τ 2 (∂txx kε + ∂xxx mε + ∂tyy hε + ∂yyy ξ ε ). 24
(5.5)
˜ ε by definition (3.4), with W ε , W ˜ ε in (2.11) and (3.4) respectively. This Recall that W ε = ΣW yields: wε = w ˜ ε + ε3 σ1 m ˜ ε + ε3 σ2 ξ˜ε + 2aε4 k˜ε + 2aε4 ˜hε ; 2 ε ˜ε + 2aλ2 ε4 m ˜ ε + ε5 σ1 k˜ε ; ε m = εσ1 w ˜ε; (5.6) ε2 ξ ε = εσ2 w ˜ ε + 2aλ2 ε4 ξ˜ε + ε5 σ2 h 2 ε 2 ε 5 ε 6 ε ˜ ε k = 2aε w ˜ + ε σ1 m ˜ + 2aε k ; 2 ε 2 ε 5 ˜ε. ε h = 2aε w ˜ + ε σ2 ξ˜ε + 2aε6 h
From (4.46), (4.22)-(4.45) and (5.6) it follows that, for a fixed constant value c > 0,
τ ε2 ||∂tx mε + ∂ty ξ ε − λ2 τ (∂txx kε + ∂xxx mε + ∂tyy hε + ∂yyy ξ ε )||s−3 = O(ε2 ),
(5.7)
then
ε + w) ε + w) ∂ A (w ¯ ∂ A (w ¯
x 1 y 2 + − ν∆wε
∂t wε +
ε ε
= O(ε2 ).
(5.8)
s−3
The last two equations and the previous bounds (5.1) and (5.2) yield:
ε−ρ ∇(ρ ¯ )
ε − ν∆(ρu )
∂t (ρε uε ) + ∇ · (ρε uε ⊗ uε ) +
ε2
= O(ε),
(5.9)
s−3
and, in particular,
||∇(ρε − ρ¯)||s−3 ≤ c, ε2 i.e. there exists ∇P ⋆ ∈ L∞ ([0, T ⋆ ], H s−3 (T2 )) such that
∇(ρε − ρ¯) ⋆ ⇀ ∇P ⋆ in L∞ ([0, T ⋆ ], H s−3 (T2 )). ε2
(5.10)
(5.11)
Moreover, since ρε → ρ¯ and uε → u⋆ in C([0, T ⋆ ], H s′ (T2 )), from ||∂t (ρε uε )||s−1 ≤ N1 as in (5.2), it follows also that ∂t (ρε uε ) ⇀⋆ ρ¯∂t u⋆ in L∞ ([0, T ⋆ ], H s−3 (T2 )), (5.12) while ∇ · (ρε uε ⊗ uε ) ⇀⋆ ρ¯∇ · (u⋆ ⊗ u⋆ ) in L∞ ([0, T ⋆ ], H s−3 (T2 )).
(5.13)
Thus, from (5.9) we have the weak⋆ convergence in L∞ ([0, T ⋆ ], H s−3 (T2 )), i.e. ! ∇P ⋆ ∇(ρε − ρ¯) ε ε ⋆ ⋆ ⋆ ⋆ ⋆ −ν∆(ρ u ) ⇀ ρ¯ ∂t u +∇·(u ⊗u )+ −ν∆u . (5.14) ∂t (ρ u )+∇·(ρ u ⊗u )+ ε2 ρ¯ ε ε
ε ε
ε
On the other hand, the first equation of (5.8) yields ∂t (ρε − ρ¯) + ∇ · (ρε uε ) − ν∆(ρε − ρ¯) = O(ε2 ). 25
(5.15)
Notice that ||∂t (ρε − ρ¯)||s−1 = O(ε) and ||∆(ρε − ρ¯)||s−2 = O(ε) thanks to (5.1), while ρε → ρ¯ and uε → u⋆ in C([0, T ⋆ ], H s′ (T2 )). This way, from (5.15) we finally recover the divergence free condition ∇ · u⋆ = 0. (5.16)
6. Conclusions and perspectives We proved the convergence of the solutions to the vector BGK model to the solutions to the incompressible Navier-Stokes equations on the two dimensional torus T2 . It could be worth extending these results to the whole space and to a general bounded domain with suitable boundary conditions, but new ideas are needed to approach these cases. Rather than the more classical kinetic entropy approach, in this paper our main tool was the use of a constant right symmetrizer, which provides the conservative-dissipative form introduced in [7], and allow us to get higher order energy estimates. Nevertheless, we do not have an estimate for the rate of convergence, in terms of the difference ||uε − uN S ||s , with uε , uN S the velocity fields associated with the BGK system in (1.5) and the Navier-Stokes equations in (1.1) respectively. References [1] D. Aregba-Driollet, R. Natalini, Discrete Kinetic Schemes for Multidimensional Conservation Laws, SIAM J. Num. Anal. 37 (2000), 1973-2004. [2] D. Aregba-Driollet, R. Natalini, S.Q. Tang, Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems, Math. Comp. 73 (2004), 63-94. [3] C. Bardos, F. Golse, C. D. Levermore, Fluid dynamic limits of hyperbolic equations. I. Formal derivations, J. Stat. Phys. 63 (1991), 323-344. [4] C. Bardos, F. Golse, C. D. Levermore, Fluid dynamic limits of kinetic equations - II Convergence proofs for the Boltzmann-equation, Comm. Pure Appl. Math. 46 (1993), 667-753. [5] S. Benzoni-Gavage, D. Serre, Multidimensional Hyperbolic Partial Differential Equations, Oxford University Press (2007). [6] A. Bertozzi, A. Majda, Vorticity and Incompressible Flow, Cambridge University Press (2002). [7] S. Bianchini, B. Hanouzet, R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), 1559-1622.
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