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: bianchin@mat.uniroma2.it (Roberta Bianchini), roberto.natalini@cnr.it (Roberto
Natalini)
1
Dipartimento di Matematica, Università 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
flε (0, x) = Mlε (ρ̄, ερ̄u0 ),
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, ρ̄ > 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 → ρ̂,
uε → û,
and
ρε − ρ̄
→ P̂ ,
ε2
under some consistency conditions of the BGK approximation with respect to the Navier-Stokes
equations, [13], it can be shown that the couple (û, P̂ ) 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 − ρ̄,
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<a< .
4
(1.12)
Finally, we point out that Assumption 1.1 is a necessary condition, also in the case of nonlinear
pressure terms, for the existence of a kinetic entropy for the approximating system, see [8].
1.1. Plan of the paper
In Section 2 we introduce the vector BGK approximation and the general setting of the problem.
Section 3 is dedicated to the discussion on the symmetrizer and the conservative-dissipative form.
In Section 4 we get uniform energy estimates to prove the convergence, in Section 5, of the solutions
to the vector BGK approximation to the solutions to the incompressible Navier-Stokes equations.
Finally, Section 6 is devoted to our conclusions and perspectives.
2. General framework
Let us set
U ε = (f1ε , f2ε , f3ε , f4ε , f5ε ) ∈ R3×5 ,
(2.1)
and let us write the compact formulation of equations (1.7)-(1.4), which reads
∂t U ε + Λ1 ∂x U ε + Λ2 ∂y U ε =
1
(M (U ε ) − U ε ),
τ ε2
(2.2)
with initial data
U0ε = flε (0, x) = Mlε (ρ̄, ερ̄u0 ),
where
Λ1 =
λ
ε Id
0
0
0
0
0
0
0
0
λ
0 − ε Id
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
Λ2 =
l = 1, · · · , 5,
0
0
0
0
0
0
λ
ε Id
0
0
0
0
0
0
0
0
0
λ
0 − ε Id
0
0
Id is the 3 × 3 identity matrix and, for Mlε , l = 1, · · · , 5, in (1.8),
M (U ε ) = (M1ε (wε ), M2ε (wε ), M3ε (wε ), M4ε (wε ), M5ε (wε )).
4
(2.3)
0
0
0
0
0
,
(2.4)
(2.5)
2.1. Conservative variables
We define the following change of variables:
wε =
P5
ε
l=1 fl ,
mε = λε (f1ε − f3ε ), ξ ε = λε (f2ε − f4ε ), kε = f1ε + f3ε , hε = f2ε + f4ε .
This way, the vector BGK model (1.7) reads:
ε
ε
ε
∂t w + ∂x m + ∂y ξ = 0; ε
2
1 A1 (w )
λ
ε
ε
ε
∂t m + ε2 ∂x k = τ ε2 ( εε − m ),
2
)
∂t ξ ε + λε2 ∂y hε = τ 1ε2 ( A2 (w
− ξ ε ),
ε
∂t kε + ∂x mε = τ 1ε2 (2awε − kε ),
∂t hε + ∂y ξ ε = τ 1ε2 (2awε − hε ).
(2.6)
(2.7)
We make a slight modification of system (2.7). Set w̄ = (ρ̄, 0, 0) and
wε ⋆ := wε − w̄ = (w1ε − ρ̄, w2ε , w3ε ),
k ε ⋆ = kε − 2aw̄,
hε ⋆ = hε − 2aw̄.
In the following, we are going to work with the modified variables. System (2.7) reads:
∂t wε ⋆ + ∂x mε + ∂y ξ ε = 0;
λ2
1 A1 (w ε ⋆ +w̄)
ε
ε⋆
− mε ),
∂t m + ε2 ∂x k = τ ε2 (
ε
ε
⋆
2
∂t ξ ε + λε2 ∂y hε ⋆ = τ 1ε2 ( A2 (w ε +w̄) − ξ ε ),
∂t kε ⋆ + ∂x mε = τ 1ε2 (2awε ⋆ − kε ⋆ ),
∂t hε ⋆ + ∂y ξ ε = τ 1ε2 (2awε ⋆ − hε ⋆ ).
(2.8)
(2.9)
Notice from (1.9) that
A1 (wε ) =
(q1ε )2
ρε
q1ε
+ ρε − ρ̄
=
q1ε q2ε
ρε
(w2ε )2
w1ε
w2ε
+ w1ε − ρ̄
=
w2ε w3ε
w1ε
w2ε ⋆
(w2ε ⋆ )2
ε⋆
w1ε ⋆ +ρ̄ + w1
ε
⋆
ε
⋆
w2 w3
w1ε ⋆ +ρ̄
ε⋆
= A1 (w + w̄),
and, similarly,
A2 (w⋆ ) =
q2ε
(q2ε )2
ρε
q1ε q2ε
ρε
+ ρε − ρ̄
=
w3ε
(w3ε )2
w1ε
w2ε w3ε
w1ε
+ w1ε − ρ̄
=
w3ε ⋆
w2ε ⋆ w3ε ⋆
w1ε ⋆ +ρ̄
(w3ε ⋆ )2
w1ε ⋆ +ρ̄
+ w1ε ⋆
ε⋆
= A2 (w + w̄).
From now on, we will omit the apexes ε ⋆ for wε ⋆ , kε ⋆ , hε ⋆ , and the apex ε for mε , ξ ε , when there is
no ambiguity.
5
Let us define the 15 × 15 matrix
C=
Let us set
Id
Id
Id
Id
Id
ελId
0
−ελId
0
0
0
ελId
0
−ελId 0
.
2
2
ε Id
0
ε Id
0
0
2
2
0
ε Id
0
ε Id
0
(2.10)
W = (w, ε2 m, ε2 ξ, ε2 k, ε2 h) := CU − (w̄, 0, 0, 0, 0).
(2.11)
Thus, we can write the translated system (2.9) in the compact form
∂t W + B1 ∂x W + B2 ∂y W =
1
(M̃ (W ) − W ),
τ ε2
(2.12)
with initial conditions
W0 = CU0 − (w̄, 0, 0, 0, 0),
(2.13)
where
B1 = CΛ1
C −1
=
0
0
0
0
0
1
ε2 Id
0
0
Id
0
0
0
0
0
0
0
λ2
ε2
0
0
0
0
0
0
0
0
, B2 = CΛ2 C −1 =
0
0
0
0
0
0
0
0
0
0
1
ε2 Id
0
0
0
Id
0
0
0
0
0
0
0
λ2 ,
ε2
0
0
(2.14)
M̃ (W ) = CM (C −1 W ) = CM (U ).
(2.15)
Here,
1
1
(M̃ (W ) − W ) =
τ ε2
τ
0
A1 (w+w̄)
ε
−
A2 (w+w̄)
ε
−
ε2 m
ε2
ε2 ξ
ε2
2aw −
ε2 k
ε2
2aw −
ε2 h
ε2
6
1
=
τ
0
1
ε
1
ε
w2
w22
w1 +ρ̄ + w1
w2 w3
w1 +ρ̄
w3
w2 w3
w1 +ρ̄
w32
w1 +ρ̄
ε2 m
ε2
ε2 ξ
ε2
−
−
+ w1
2
2aw − εε2k
2
2aw − εε2h
0
1
=
τ
0
0 1
1
1 0
ε
0 0
0 0
1
0 0
ε
1 0
2aId
2aId
0
0
0
1
0
0
0
0
0
0
0
0
0
− ε12 Id
W + 1
τ
0
− ε12 Id
0
0
1
0
0
− ε2 Id
0
0
0
0
− ε12 Id
=: −LW + N (w + w̄),
1
ε
1
ε
0
w22
w1 +ρ̄
w2 w3
w1 +ρ̄
0
w2 w3
w1 +ρ̄
w32
w1 +ρ̄
0
0
(2.16)
where −L is the linear part of the source term of (2.12), while N is the remaining nonlinear one.
Thus, we can rewrite system (2.12) as follows:
∂t W + B1 ∂x W + B2 ∂y W = −LW + N (w + w̄).
(2.17)
3. The weighted constant right symmetrizer and the conservative-dissipative form
According to the theory of semilinear hyperbolic systems, see for instance [24, 5], we need a
symmetric formulation of system (2.17) in order to get energy estimates. However, we are dealing
with a singular perturbation system, so any symmetrizer for system (2.17) is not enough. In other
words, we look for a symmetrizer which provides a suitable dissipative structure for system (2.17).
In this context, notice that the first equation of system (2.17) reads
∂t w + ∂x m + ∂y ξ = 0,
i.e. the first term of the source vanishes, and w is a conservative variable. We want to take advantage
of this conservative property, in order to simplify the algebraic structure of the linear part of the
source term. To this end, rather than a classical Friedrichs left symmetrizer, see again [24, 5], we
look for a right symmetrizer for (2.17), which provides the conservative-dissipative form introduced
in [7]. More precisely, the right multiplication easily provides the conservative structure in [7], while
the dissipation is proved a posteriori. Besides, the symmetrizer Σ presents constant ε-weighted
entries and this allow us to control the nonlinear part N of the source term (2.16) of system (2.17).
To be complete, we point out that the inverse matrix Σ−1 is a left symmetrizer for system (2.17),
according to the definitions given in [24, 5]. However, the product −Σ−1 L is a full matrix, so the
7
symmetrized version of system (2.17), obtained by the left
the conservative-dissipative form in [7].
Let us explicitly write the symmetrizer
Id
εσ1
εσ2
2
2
εσ1
2λ aε Id
0
2 aε2 Id
Σ=
εσ
0
2λ
2
2aε2 Id
ε3 σ1
0
2
2aε Id
0
ε3 σ2
where
multiplication by Σ−1 , does not provide
2aε2 Id 2aε2 Id
ε3 σ1
0
0
ε3 σ2
2aε4 Id
0
0
2aε4 Id
,
0 0 1
0 1 0
σ1 = 1 0 0 and σ2 = 0 0 0 .
1 0 0
0 0 0
(3.1)
(3.2)
It is easy to check that Σ is a constant right symmetrizer for system (2.17) since, taking B1 , B2 and
L in (2.14) and (2.16) respectively,
0
0
T
T
B1 Σ = ΣB1 ,
B2 Σ = ΣB2 ,
−LΣ =
,
(3.3)
0T −L̃
where 0 is the 3 × 3 null matrix, 0 is the 3 × 12 vector with zero entries, L̃ is a 12 × 12 matrix.
Now, we define the following change of variables:
W = ΣW̃ = Σ(w̃, ε2 m̃, ε2 ξ̃, ε2 k̃, ε2 h̃),
(3.4)
with W in (2.11). System (2.17) reads:
Σ∂t W̃ + B1 Σ∂x W̃ + B2 Σ∂y W̃ = −LΣW̃ + N ((ΣW̃ )1 + w̄),
(3.5)
where (ΣW̃ )1 is the first component of the unknown vector ΣW̃ . Now, we want to show that Σ in
(3.1) is strictly positive definite. Thus,
(ΣW̃ , W̃ )0 = ||w̃||20 + 2λ2 aε6 (||m̃||20 + ||ξ̃||20 ) + 2aε8 (||k̃||20 + ||h̃||20 ) + 2(ε3 σ1 m̃, w̃)0
˜ w̃)0 + 4aε4 (k̃ + h̃, w̃)0 + 2ε7 (σ1 k̃, m̃)0 + 2ε7 (σ2 h̃, ξ)
˜0
+2(ε3 σ2 ξ,
= ||w̃||20 + 2λ2 aε6 (||m̃||20 + ||ξ̃||20 ) + 2aε8 (||k̃||20 + ||h̃||20 ) + I1 + I2 + I3 + I4 + I5 .
Now, taking two positive constants δ, µ and by using the Cauchy inequality, we have:
||w̃2 ||20
||w̃1 ||20
− δε6 ||m̃1 ||20 −
;
δ
δ
||w̃1 ||20
||w̃3 ||20
I2 = 2ε3 [(ξ̃3 , w̃1 )0 + (ξ̃1 , w̃3 )0 ] ≥ −δε6 ||ξ̃3 ||20 −
− δε6 ||ξ̃1 ||20 −
;
δ
δ
I1 = 2ε3 [(m̃2 , w̃1 )0 + (m̃1 , w̃2 )0 ] ≥ −δε6 ||m̃2 ||20 −
8
(3.6)
I3 = 4aε4 [(k̃, w̃)0 + (h̃, w̃)0 ] ≥ −2aµ||w̃||20 −
I4 = 2ε7 [(k̃2 , m̃1 )0 + (k̃1 , m̃2 )0 ] ≥ −
2aε8
2aε8
||k̃||20 − 2aµ||w̃||20 −
||h̃||20 ;
µ
µ
ε8
ε8
||k̃2 ||20 − δε6 ||m̃1 ||20 − ||k̃1 ||20 − δε6 ||m̃2 ||20 ;
δ
δ
ε8
ε8
I5 = 2ε7 [(h̃3 , ξ˜1 )0 + (h̃1 , ξ˜3 )0 ] ≥ − ||h̃3 ||20 − δε6 ||ξ̃1 ||20 − ||h̃1 ||20 − δε6 ||ξ̃3 ||20 .
δ
δ
Thus, putting them all together,
"
"
"
#
#
#
2
1
1
2
2
2
(ΣW̃ , W̃ )0 ≥ ||w̃1 ||0 1 − − 4aµ + ||w̃2 ||0 1 − − 4aµ + ||w̃3 ||0 1 − − 4aµ
δ
δ
δ
+ε6 ||m̃ε1 ||20 [2λ2 a − 2δ] + ε6 ||m̃ε2 ||20 [2λ2 a − 2δ] + ε6 ||m̃ε3 ||20 [2λ2 a]
+ε6 ||ξ̃1ε ||20 [2λ2 a − 2δ] + ε6 ||ξ̃2ε ||20 [2λ2 a] + ε6 ||ξ̃3ε ||20 [2λ2 a − 2δ]
"
"
#
#
#
"
2a
2a
1
1
2a
+ ε8 ||k̃2 ||20 2a −
+ ε8 ||k̃3 ||20 2a −
−
−
+ε8 ||k̃1 ||20 2a −
µ
δ
µ
δ
µ
"
"
"
#
#
#
2a
2a
2a
1
1
+ε8 ||h̃1 ||20 2a −
+ ε8 ||h̃2 ||20 2a −
+ ε8 ||h̃3 ||20 2a −
.
−
−
µ
δ
µ
µ
δ
(3.7)
Now, we can prove the following lemma.
Lemma 3.1. If Assumption 1.1 is satisfied and λ is big enough, then Σ is strictly positive definite.
Proof. From (3.7), we take
1
;
1 < µ < 4a
1
2
, 2a(1−
δ > max{ 1−4aµ
1 };
)
µ
q
λ > aδ .
(3.8)
Notice that we can choose the constant velocity λ as big as we need, therefore the third inequality
is automatically verified.
Now, we consider the linear part −LΣ of the source term of (3.5). Explicitly
0
0
0
0
0
0 −2λ2 aId + σ12
σ1 σ2
(2a − 1)εσ1
2aεσ1
2 aId + σ 2
0
σ
σ
−2λ
2aεσ
(2a
− 1)εσ2
−LΣ =
1 2
2
2
2
0 (2a − 1)εσ1
2aεσ2
2a(2a − 1)ε Id
4a2 ε2 Id
2
2
0
2aεσ1
(2a − 1)εσ2
4a ε Id
2a(2a − 1)ε2 Id
9
.
(3.9)
Thus,
(−LΣW̃ , W̃ )0 = −2λ2 aε4 (||m̃||20 + ||ξ̃||20 ) + 2a(2a − 1)ε6 (||k̃||20 + ||h̃||20 ) + ε4 ||m̃1 ||20 + ε4 ||m̃2 ||20
˜0
+ε4 ||ξ̃1 ||20 + ε4 ||ξ̃3 ||20 + 2ε4 (σ1 σ2 ξ̃, m̃)0 + 2(2a − 1)ε5 (σ1 k̃, m̃)0 + 4aε5 (σ1 h̃, m̃)0 + 4aε5 (σ2 k̃, ξ)
˜ 0 + 8a2 ε6 (h̃, k̃)0
+2(2a − 1)ε5 (σ2 h̃, ξ)
= (−2λ2 a+1)ε4 (||m̃1 ||20 +||m̃2 ||20 +||ξ̃1 ||20 +||ξ̃3 ||20 )−2λ2 aε4 (||m̃3 ||20 +||ξ̃2 ||20 )+2a(2a−1)ε6 (||k̃||20 +||h̃||20 )
+J1 + J2 + J3 + J4 + J5 + J6 .
Now, taking a positive constant ω and by using the Cauchy inequality, we have
J1 = 2ε4 (ξ̃3 , m̃2 )0 ≤ ε4 (||ξ̃3 ||20 + ||m̃2 ||20 );
(
)
6
6
ε
ε
J2 = 2(2a − 1)ε5 [(k̃2 , m̃1 )0 + (k̃1 , m̃2 )0 ] ≤ (1 − 2a)
||k̃2 ||20 + ε4 ω||m̃1 ||20 + ||k̃1 ||20 + ε4 ω||m̃2 ||20 ;
ω
ω
(
)
6
6
ε
ε
||h̃2 ||20 + ε4 ω||m̃1 ||20 + ||h̃1 ||20 + ε4 ω||m̃2 ||20 ;
J3 = 4aε5 [(h̃2 , m̃1 )0 + (h̃1 , m̃2 )0 ] ≤ 2a
ω
ω
(
)
6
6
ε
ε
||k̃3 ||20 + ε4 ω||ξ̃1 ||20 + ||k̃1 ||20 + ε4 ω||ξ̃3 ||20 ;
J4 = 4aε5 [(k̃3 , ξ˜1 )0 + (k̃1 , ξ̃3 )0 ] ≤ 2a
ω
ω
(
)
6
6
ε
ε
||h̃3 ||20 + ε4 ω||ξ̃1 ||20 + ||h̃1 ||20 + ε4 ω||ξ̃3 ||20 ;
J5 = 2(2a − 1)ε5 [(h̃3 , ξ̃1 )0 + (h̃1 , ξ˜3 )0 ] ≤ (1 − 2a)
ω
ω
J6 = 8a2 ε6 (h̃, k̃)0 ≤ 4a2 ε6 {||h̃||20 + ||k̃||20 }.
Putting them all together, we have
(−LΣW̃ , W̃ )0 ≤ ε4 ||m̃1 ||20 [−2λ2 a + 1 + ω] + ε4 ||m̃2 ||20 [−2λ2 a + 2 + ω] − 2λ2 aε4 ||m̃3 ||20
"
#
1
2a(4a − 1) +
+ε
+ 1 + ω] − 2λ aε
+ε
+ 2 + ω] + ε
ω
"
"
"
#
#
#
(1 − 2a)
2a
1
6
2
6
2
6
2
+ε ||k̃2 ||0 2a(4a − 1) +
+ ε ||k̃3 ||0 2a(4a − 1) +
+ ε ||h̃1 ||0 2a(4a − 1) +
ω
ω
ω
"
#
#
"
(1
−
2a)
2a
+ ε6 ||h̃3 ||20 2a(4a − 1) +
.
(3.10)
+ε6 ||h̃2 ||20 2a(4a − 1) +
ω
ω
4
||ξ̃1 ||20 [−2λ2 a
2
4
||ξ̃2 ||20
4
||ξ̃3 ||20 [−2λ2 a
This way, we obtain the following Lemma.
10
6
||k̃1 ||20
Lemma 3.2. If Assumption 1.1 is satisfied and λ is big enough, then the symmetrized linear part
of the source term −LΣ given by (3.9) is negative definite.
Proof. We need ω and λ satisfying:
ω >
Recalling (3.8), we take
λ > max
Then, we take ω >
1
2a(1−4a)
q
λ > 2+ω .
2a
1
2a(1−4a) ,
(r
δ
,
a
s
;
(3.11)
)
4a(1 − 4a) + 1
.
4a2 (1 − 4a)
(3.12)
which ends the proof.
4. Energy estimates
Here we provide ε-weighted energy estimates for the solution W ε to (2.17). Let us introduce T ε the
maximal time of existence of the unique solution W̃ ε for fixed ε to system (3.5), see [24]. In the
following, we consider the time interval [0, T ], with T ∈ [0, T ε ). Our setting is represented by the
Sobolev spaces H s (T2 ), with s > 3.
4.1. Zero order estimate
We assume the following condition.
Assumption 4.1. Let λ satisfies (3.12) and
s
4+
λ>
1
τ
+
1
a(1−4a)
4a
.
(4.1)
Lemma 4.1. If Assumptions 1.1 and 4.1 are satisfied, then the following zero order energy estimate
holds:
||w̃(T )||20 + ε6 (||m̃(T )||20 + ||ξ̃(T )||20 ) + ε8 (||k̃(T )||20 + ||h̃(T )||20 )
Z T
ε4 (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε6 (||k̃(t)||20 + ||h̃(t)||20 ) dt
+
0
2
≤ cε
||u0 ||20
+ c(||u||L∞ ([0,T ]×T2 ) )
Z
T
0
||w̃(t)||20 + ε6 (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε8 (||k̃(t)||20 + ||h̃(t)||20 ) dt.
(4.2)
11
Proof. We consider the symmetrized compact system (3.5) and we multiply W̃ through the L2 -scalar
product. Thus, we have:
1d
(ΣW̃ , W̃ )0 + (LΣW̃ , W̃ )0 = (N ((ΣW̃ )1 + w̄), W̃ )0 .
2 dt
Integrating in time, we get:
1
(ΣW̃ (T ), W̃ (T ))0 +
2
+
Z
Z
T
0
1
(LΣW̃ (t), W̃ (t))0 dt ≤ (ΣW̃ (0), W̃ (0))0
2
T
|(N ((ΣW̃ (t))1 + w̄), W̃ (t))0 | dt.
(4.3)
0
Consider (3.7) and let us introduce the following positive constants:
ΓΣ := 1 − 4aµ − 2δ , ∆Σ := 2(λ2 a − δ) ΘΣ := 2a(1 − µ1 ) − 1δ .
(4.4)
Similarly, from (3.10), we define:
∆LΣ := 2(λ2 a − 1) − ω, ΘLΣ := 2a(1 − 4a) − ω1 .
(4.5)
Thus, from (4.3), we get:
ΓΣ ||w̃(T )||20 + ε6 ∆Σ (||m̃(T )||20 + ||ξ̃(T )||20 ) + ε8 ΘΣ (||k̃(T )||20 + ||h̃(T )||20 )
+2
Z
T
0
ε4 ∆LΣ (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε6 ΘLΣ (||k̃(t)||20 + ||h̃(t)||20 ) dt
≤ (ΣW̃0 , W̃0 )0 + 2
Notice that, from (3.4),
Z
T
|(N ((ΣW̃ (t))1 + w̄), W̃ (t))0 | dt.
(4.6)
0
(ΣW̃0 , W̃0 )0 = (ΣΣ−1 W0 , Σ−1 W0 )0 = (Σ−1 W0 , W0 )0 ,
where W0 = W (0, x) = (w(0, x), ε2 m(0, x), ε2 ξ(0, x), ε2 k(0, x), ε2 h(0, x)), and, from (2.6), (2.8) and
the initial conditions (1.4),
w(0, x) = w0 − w̄ = (0, ερ̄u01 , ερ̄u02 );
0)
= (ρ̄u01 , ερ̄u01 2 , ερ̄u01 u02 );
m(0, x) = λε (f1 0 − f3 0 ) = A1 (w
ε
0)
= (ρ̄u02 , ερ̄u01 u02 , ερ̄u02 2 );
ξ(0, x) = λε (f2 0 − f4 0 ) = A2 (w
ε
k(0, x) = f1 0 + f3 0 − 2aw̄ = 2aw0 − 2aw̄ = 2a(0, ερ̄u01 , ερ̄u02 );
h(0, x) = f2 0 + f4 0 − 2aw̄ = 2aw0 − 2aw̄ = 2a(0, ερ̄u01 , ερ̄u02 ).
12
(4.7)
Besides, the explicit expression of the constant symmetric matrix Σ−1 is given by
1
−1
−1
0
0
Id
Id
1−4a Id
ε2 (1−4a)
ε2 (1−4a)
1
0
H1
0
σ
0
ε3 (1−4λ2 a2 ) 1
1
−1
σ
0
0
H2
0
Σ =
ε3 (1−4λ2 a2 ) 2
1
1
−1
0
H3
ε2 (1−4a) Id ε3 (1−4λ2 a2 ) σ1
ε4 (1−4a) Id
1
1
−1
Id
0
σ
Id
H4
ε2 (1−4a)
ε3 (1−4λ2 a2 ) 2
ε4 (1−4a)
where
H1 =
2a
0
0
0
2a
ε2 (4λ2 a2 −1)
ε2 (4λ2 a2 −1)
0
0
1
2λ2 aε2
0
H3 =
4λ2 a2 −2λ2 a+1
ε4 (4a−1)(4λ2 a2 −1)
0
0
;
2a
0
0
0
1
2λ2 aε2
ε2 (4λ2 a2 −1)
H2 =
0
0
0
4λ2 a2 −2λ2 a+1
ε4 (4a−1)(4λ2 a2 −1)
0
0
4λ2 a2 −2λ2 a+1
ε4 (4a−1)(4λ2 a2 −1)
H4 =
0
0
2a−1
2aε4 (4a−1)
0
0
2a−1
2aε4 (4a−1)
0
0
4λ2 a2 −2λ2 a+1
ε4 (4a−1)(4λ2 a2 −1)
It is easy to check that
(Σ−1 W0 , W0 )0 = ρ̄2 ε2 ||u0 ||20 +
,
0
0
2a
ε2 (4λ2 a2 −1)
(4.8)
;
;
.
ρ̄2 ε4
2aρ̄2 ε4
2 2
2 2
(||u
||u01 u02 ||20 ≤ cε2 ||u0 ||20 ,
||
+
||u
||
)
+
0
0
1
0
2
0
4λ2 a2 − 1
λ2 a
(4.9)
and so, from (4.6) we get the following inequality:
ΓΣ ||w̃(T )||20 + ε6 ∆Σ (||m̃(T )||20 + ||ξ̃(T )||20 ) + ε8 ΘΣ (||k̃(T )||20 + ||h̃(T )||20 )
Z
+2
T
0
ε4 ∆LΣ (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε6 ΘLΣ (||k̃(t)||20 + ||h̃(t)||20 ) dt
≤
cε2 ||u0 ||20
Z
+2
T
|(N ((ΣW̃ (t))1 + w̄), W̃ (t))0 | dt.
0
13
(4.10)
It remains to deal with the last term of (4.10). Recall that w = (ρ − ρ̄, ερu1 , ερu2 ). From (2.16),
0
0
u w
1 2
u1 w3
1
(4.11)
N ((ΣW̃ )1 + w̄) = N (w + w̄) =
.
0
τ
u2 w2
u2 w3
0
0
Thus,
(N (w + w̄), W̃ )0 =
≤
1
{(u1 w2 , ε2 m̃2 )0 + (u1 w3 , ε2 m̃3 )0 + (u2 w2 , ε2 ξ˜2 )0 + (u2 w3 , ε2 ξ̃3 )0 }
τ
1
{||u1 w2 ||20 + ε4 ||m̃2 ||20 + ||u1 w3 ||20 + ε4 ||m̃3 ||20 + ||u2 w2 ||20 + ε4 ||ξ̃2 ||20 + ||u2 w3 ||20 + ε4 ||ξ̃3 ||20 }
2τ
≤ c(||u||∞ )||w||20 +
ε4
(||m̃||20 + ||ξ̃||20 ).
2τ
(4.12)
By definition (3.4), explicitly we have:
w = (ΣW̃ ε )1 = w̃ + ε3 σ1 m̃ + ε3 σ2 ξ˜ + 2aε4 (k̃ + h̃),
(4.13)
|(N (w+ w̄), W̃ )0 | ≤ c(||u||∞ ){||w̃||20 +ε6 (||m̃||20 +||ξ̃||20 )+ε8 (||k̃||20 +||h̃||20 )}+
ε4
(||m̃||20 +||ξ̃||20 ). (4.14)
2τ
and so,
Putting them all together, (4.10) yields:
ΓΣ ||w̃(T )||20 + ε6 ∆Σ (||m̃(T )||20 + ||ξ̃(T )||20 ) + ε8 ΘΣ (||k̃(T )||20 + ||h̃(T )||20 )
Z
+2
T
0
ε4 ∆LΣ (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε6 ΘLΣ (||k̃(t)||20 + ||h̃(t)||20 ) dt
≤
Z
+c(||u||L∞ ([0,T ]×T2 ) )
cε2 ||u
T
0
2
0 ||0
+
Z
T 4
ε
0
τ
(4.15)
(||m̃(t)||20
+
||ξ̃(t)||20 )
dt
||w̃(t)||20 + ε6 (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε8 (||k̃(t)||20 + ||h̃(t)||20 ) dt.
14
This gives:
ΓΣ ||w̃(T )||20 + ε6 ∆Σ (||m̃(T )||20 + ||ξ̃(T )||20 ) + ε8 ΘΣ (||k̃(T )||20 + ||h̃(T )||20 )
+
Z
T
ε4 (2∆LΣ − 1/τ )(||m̃(t)||20 + ||ξ̃(t)||20 ) + 2ε6 ΘLΣ (||k̃(t)||20 + ||h̃(t)||20 ) dt
0
≤
cε2 ||u
2
0 ||0
Z
+ c(||u||L∞ ([0,T ]×T2 ) )
T
0
||w̃(t)||20 + ε6 (||m̃(t)||20 + ||ξ̃(t)||20 ) + ε8 (||k̃(t)||20 + ||h̃(t)||20 ) dt,
(4.16)
where, by definition (4.5), 2∆LΣ − 1 = 4λ2 a − 4 − 2ω − τ1 is positive thanks to condition (4.1). This
gives estimate (4.2).
4.2. Higher order estimates
Lemma 4.2. If Assumptions 1.1 and 4.1 are satisfied, then the following H s energy estimate holds:
||w̃(T )||2s + ε6 (||m̃(T )||2s + ||ξ̃(T )||2s ) + ε8 (||k̃(T )||2s + ||h̃(T )||2s )
+
Z
T
0
ε4 (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε6 (||k̃(t)||2s + ||h̃(t)||2s ) dt
Z
≤ cε2 ||u0 ||2s + c(||u ||L∞ ([0,T ],H s (T2 )) )
T
0
||w̃(t)||2s + ε6 (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε8 (||k̃(t)||2s + ||h̃(t)||2s ) dt.
(4.17)
Proof. We take the |α|-derivative, 0 < |α| ≤ s, of the semilinear system given by (2.17). As done
previously, we get:
˜ )||2 ) + ε8 ΘΣ (||D α k̃(T )||2 + ||D α h̃(T )||2 )
ΓΣ ||D α w̃(T )||20 + ε6 ∆Σ (||D α m̃(T )||20 + ||D α ξ(T
0
0
0
+2
Z
T
0
ε4 ∆LΣ (||D α m̃(t)||20 + ||D α ξ̃(t)||20 ) + ε6 ΘLΣ (||D α k̃(t)||20 + ||D α h̃(t)||20 ) dt
2
≤ cε ||D
α
u0 ||20
+2
Now, from (4.11),
|(D α N (w + w̄), D α W̃ )0 | ≤
Z
T
|D α (N ((ΣW̃ (t))1 + w̄), D α W̃ (t))0 | dt.
(4.18)
0
1
{|(D α (u1 w2 ), D α ε2 m̃2 )0 | + |D α (u1 w3 ), D α ε2 m̃3 )0 |
τ
+|(D α (u2 w2 ), D α ε2 ξ˜2 )0 | + |D α (u2 w3 ), D α ε2 ξ̃3 )0 |}
≤
1
˜ 2 )}
{||(D α (u1 w2 )||20 + ||D α (u1 w3 )||20 + ||D α (u2 w2 )||20 + ||D α (u2 w3 )||20 ) + ε4 (||D α m̃||20 + ||D α ξ||
0
2τ
15
≤ c(||u||s )||w||2s +
ε4
(||m̃||2s + ||ξ̃||2s ).
2τ
(4.19)
By using (4.13) we have:
|(D α N (w + w̄), D α W̃ )0 | ≤ c(||u||s )(||w̃||2s + ε6 (||m̃||2s + ||ξ̃||2s ) + ε8 (||k̃||2s + ||h̃||2s )) +
ε4
(||m̃||2s + ||ξ̃||2s ).
2τ
(4.20)
Thus, from (4.18),
ΓΣ ||w̃(T )||2s + ε6 ∆Σ (||m̃(T )||2s + ||ξ̃(T )||2s ) + ε8 ΘΣ (||k̃(T )||2s + ||h̃(T )||2s )
Z T
ε4 (2∆LΣ − 1/τ )(||m̃(t)||2s + ||ξ̃(t)||2s ) + 2ε6 ΘLΣ (||k̃(t)||2s + ||h̃(t)||2s ) dt
+
0
≤ cε2 ||u0 ||2s + c(||u||L∞ ([0,T ],H s (T2 )) )
Z
T
0
||w̃(t)||2s + ε6 (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε8 (||k̃(t)||2s + ||h̃(t)||2s ) dt.
(4.21)
Remark 4.1. In the case s > 3 is not an integer, by using the pseudodifferential operator λs (ξ) =
(1 + |ξ|2 )s/2 in the Fourier space we get the same estimates in a standard way.
Now, we need a bound in the H s -norm for the original variable w = (ρ − ρ̄, ερu), which is the first
component of the unknown vector W in (2.11). By using estimate (4.17) and definition (3.4), we
can prove the following proposition.
Proposition 4.1. If Assumptions 1.1 and 4.1 are satisfied, then the following estimate holds:
||w(t)||2s + ε6 (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε8 (||k̃(t)||2s + ||h̃(t)||2s ) ≤ cε2 ||u0 ||2s ec(||u||L∞ ([0,t],H s (T2 )) )t , (4.22)
and
||ρ(t) − ρ̄||2s
+ ||ρu(t)||2s ≤ c||u0 ||2s ec(||u||L∞ ([0,t],H s (T2 )) )t ,
2
ε
(4.23)
for t ∈ [0, T ε ).
Proof. The Gronwall inequality applied to (4.17) yields:
ΓΣ ||w̃(t)||2s + ε6 ∆Σ (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε8 ΘΣ (||k̃(t)||2s + ||h̃(t)||2s ) ≤ cε2 ||u0 ||2s ec(||u||L∞ ([0,t],H s (T2 )) )t .
(4.24)
Recalling (4.13),
w̃ = w − ε3 σ1 m̃ − ε3 σ2 ξ˜ − 2aε4 (k̃ + h̃).
(4.25)
Thus,
||w̃||2s = ||w||2s + ε6 (||m̃1 ||2s + ||m̃2 ||2s + ||ξ̃1 ||2s + ||ξ̃3 ||2s ) + 4a2 ε8 ||k̃ + h̃||2s
16
−2ε3 (w, σ1 m̃)s − 2ε3 (w, σ2 ξ̃)s − 4aε4 (w, k̃ + h̃)s + 2ε6 (σ1 m̃, σ2 ξ̃)s
+4aε7 (σ1 m̃, k̃ + h̃)s + 4aε7 (σ2 ξ̃, k̃ + h̃)s
= ||w||2s + ε6 (||m̃1 ||2s + ||m̃2 ||2s + ||ξ̃1 ||2s + ||ξ̃3 ||2s ) + 4a2 ε8 ||k̃ + h̃||2s + Y1 + Y2 + Y3 + Y4 + Y5 + Y6 . (4.26)
Now, taking two positive constants η, ζ and using the Cauchy inequality, from (4.26) we have:
2
Y1 = −2ε3 (w, σ1 m̃)s = −2ε3 [(w1 , m̃2 )s + (w2 , m̃1 )s ] ≥ − ||wη1||s − ε6 η||m̃2 ||2s −
2
Y2 = −2ε3 (w, σ2 ξ̃)s = −2ε3 [(w1 , ξ˜3 )s + (w3 , ξ˜1 )s ] ≥ − ||wη1||s − ε6 η||ξ̃3 ||2s −
−2a
2
ζ ||w||s
Y3 = −4aε4 (w, k̃ + h̃)s ≥
||w2 ||2s
η
||w3 ||2s
η
− ε6 η||m̃1 ||2s ;
− ε6 η||ξ̃1 ||2s ;
− 2aζε8 ||k̃ + h̃||2s ;
Y4 = 2ε6 (m̃2 , ξ˜3 )s ≥ −ε6 (||m̃ε2 ||2s + ||ξ̃3ε ||2s );
2aε8
η ||k̃1
Y5 = 4aε7 [(m̃2 , k̃1 + h̃1 )s + (m̃1 , k̃2 + h̃2 )s ] ≥ −2aε6 η||m̃2 ||2s −
8
2
− 2aε
η ||k̃2 + h̃2 ||s ;
Y6 = 4aε7 [(ξ̃3 , k̃1 + h̃1 )s + (ξ̃1 , k̃3 + h̃3 )s ] ≥ −2aε6 η||ξ̃3 ||2s −
8
2
− 2aε
η ||k̃3 + h̃3 ||s .
2aε8
η ||k̃1
+ h̃1 ||2s − 2aε6 η||m̃1 ||2s
+ h̃1 ||2s − 2aε6 η||ξ̃1 ||2s
The left hand side of (4.24) and the previous calculations yield the following inequality:
ΓΣ ||w̃||2s + ε6 ∆Σ (||m̃||2s + ||ξ̃||2s ) + ε8 ΘΣ (||k̃||2s + ||h̃||2s )
"
#
#
"
1
2a
2 2a
||w1 ||2s + ΓΣ 1 − −
(||w2 ||2s + ||w3 ||2s )
≥ ΓΣ 1 − −
η
ζ
η
ζ
+ε6 (||m̃1 ||2s +||ξ̃1 ||2s )[∆Σ +ΓΣ (1−η−2aη)]+ε(||m̃2 ||2s +||ξ̃3 ||2s )[∆Σ +ΓΣ (−η−2aη)]+ε6 (||m̃3 ||2s +||ξ̃2 ||2s )∆Σ
"
#
"
#
4a
2a
+ε8 ||k̃1 +h̃1 ||2s ΓΣ 4a2 −2aζ−
+ε8 ΓΣ (||k̃2 +h̃2 ||2s +||k̃3 +h̃3 ||2s ) 4a2 −2aζ−
+ε8 θΣ (||k̃||2s +||h̃||2s ).
η
η
(4.27)
Fixed β > 1, the Cauchy inequality yields ||k̃ + h̃||2s ≥ (1 − β1 )||k̃||2s + (1 − β)||h̃||2s , then the last term
of (4.27) is bounded from below by the following expression:
8
ε
||k̃1 ||2s
"
ΘΣ +(1−1/β)ΓΣ
"
4a
4a −2aζ −
η
2
##
8
+ε
"
(||k̃2 ||2s +||k̃3 ||2s )
17
ΘΣ +(1−1/β)ΓΣ
"
2a
4a −2aζ −
η
2
##
"
+ε8 ||h̃1 ||2s ΘΣ + (1− β)ΓΣ
"
4a
4a2 − 2aζ −
η
##
"
+ ε8 (||h̃2 ||2s + ||h̃3 ||2s ) ΘΣ + (1− β)ΓΣ
"
##
2a
4a2 − 2aζ −
.
η
(4.28)
Thus, in order to get estimate (4.22), we require:
1 − η2 − 4a
ζ > 0;
∆Σ − ηΓΣ (1 + 2a)"> 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 <
ΘΣ
,
2aΓΣ (1 − 1/β)
18
(4.32)
with η and ζ satisfying (4.31). In particular, we show that there exists β > 1 such that:
4a <
ΘΣ
,
2aΓΣ (1 − 1/β)
2
δ
From (4.4), ΓΣ = 1 − 4aµ −
we require:
i.e. 8a2 ΓΣ (1 − 1/β) < ΘΣ .
(4.33)
and, from Lemma 3.1, 0 < ΓΣ < 1. Thus, in order to verify (4.33),
8a2 (1 − 1/β) < ΘΣ ,
(4.34)
which is automatically verified if 8a2 ≤ ΘΣ . Otherwise, it yields β <
Finally, since β > 1, we need
1<
8a2
.
8a2 − ΘΣ
8a2
, i.e. ΘΣ > 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
Γ1Σ ||w(t)||2s + ε6 ∆1Σ (||m̃(t)||2s + ||ξ̃(t)||2s ) + ε8 Θ1Σ (||k̃(t)||2s + ||h̃(t)||2s ) ≤ cε2 ||u0 ||2s ec(||u||L∞ ([0,t],H s (T2 )) )t ,
(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) − ρ̄||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) − ρ̄||s ≤ εM, i.e. ||ρ(t)||s ≤ ρ̄|T2 | + εM,
(4.38)
t ∈ [0, T ⋆ ],
(4.39)
and
||ρu(t)||s ≤ M (ρ̄|T2 | + εM ),
t ∈ [0, T ⋆ ].
(4.40)
Proof. Let u0 ∈ H s (T2 ) and, from (1.4), recall that ρ0 = ρ̄. Then, there exists a positive constant
M0 such that ||u0 ||s ≤ M0 , and
||ρ0 u0 ||s = ρ̄||u0 ||s ≤ ρ̄M0 =: M̃0 .
19
(4.41)
Let M > M̃0 be any fixed constant, and
(
)
2
||ρ(t)
−
ρ̄||
s
T0ε := sup t ∈ [0, T ε ]
+ ||ρu(t)||2s ≤ M 2 , ∀ε ≤ ε0 .
ε2
(4.42)
Notice that, from (4.42),
||ρ − ρ̄||∞ ≤ cS ||ρ − ρ̄||s ≤ cS M ε, t ∈ [0, T0ε ],
where cS is the Sobolev embedding constant, i.e.
ρ̄ − cS M ε ≤ ρ ≤ ρ̄ + cS M ε, t ∈ [0, T0ε ].
Taking ε0 such that ρ̄ − cS M ε0 > ρ̄2 , i.e. ρ̄ > 2cS M ε0 , we have
ρ>
Now, since s > 3 =
D
2
ρ̄
,
2
t ∈ [0, T0ε ].
(4.43)
+ 2,
||u||s ≤ ||ρu||s ||1/ρ||s .
Moreover,
|T2 | ||ρ||s
||1/ρ||s ≤ c
+
ρ̄
c(ρ̄)
!
≤ c1 + c2 ||ρ||s .
From (4.42),
||ρ||s ≤ c(|T2 |ρ̄ + M ε),
so
||1/ρ||s ≤ c1 + c2 M ε,
and
||u||s ≤ cM (c1 + c2 M ε).
From (4.37),
||ρ(t) − ρ̄||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:
||∂t w||2s−1 + ε6 (||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ) + ε8 (||∂t k̃||2s−1 + ||∂t h̃||2s−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 − ρ̄)||2s−1
+ ||∂t (ρu)||2s−1 ≤ c(||u0 ||s ) ≤ M 2 in [0, T ⋆ ].
ε2
(4.47)
Proof. Let us take the time derivative of system (3.5). Defining Ṽ = ∂t W̃ ε , from (2.16) we get:
∂t ΣṼ + Λ̃1 Σ∂x Ṽ + Λ̃2 Σ∂y Ṽ = −LΣṼ + ∂t N ((ΣW̃ )1 + w̄) = −LΣṼ + ∂t N (w + w̄),
where
1
∂t N (w + w̄) =
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 Ṽ , we have:
(4.48)
.
1 d
(ΣṼ , Ṽ )0 + (LΣṼ , Ṽ )0 ≤ |(∂t N (w + w̄), V )0 |.
2 dt
(4.49)
(4.50)
Here,
|(∂t N (w+w̄), Ṽ )0 | =
1
|(2u1 ∂t w2 −εu21 ∂t w1 , ε2 ∂t m̃2 )0 +(u2 ∂t w2 +u1 ∂t w3 −εu1 u2 ∂t w1 , ε2 ∂t m̃3 +ε2 ∂t ξ̃2 )0
τ
+(2u2 ∂t w3 − εu22 ∂t w1 , ε2 ∂t ξ̃3 )0 |
≤ c(||u||∞ )||∂t w||20 +
ε4
˜ 2 ).
(||∂t m̃||20 + ||∂t ξ||
0
2τ
Similarly to (4.16), we get:
˜ 2 ) + ε8 ΘΣ (||∂t k̃||2 + ||∂t h̃||2 )
ΓΣ ||∂t w̃||20 + ε6 ∆Σ (||∂t m̃||20 + |||∂t ξ||
0
0
0
21
+
Z
T
0
2
≤ cε
||∂t u|t=0 ||20
˜ 2 ) + 2ε6 ΘLΣ (||∂t k̃||2 + ||∂t h̃||2 ) dt
(2∆LΣ − 1/τ )ε4 (||∂t m̃||20 + ||∂t ξ||
0
0
0
+ c(||u||L∞ ([0,T ]×T2 ) )
Z
T
0
˜ 2 ) + ε8 (||∂t k̃||2 + ||∂t h̃||2 ) dt.
||∂t w̃||20 + ε6 (||∂t m̃||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 ) =
= ρ̄ εu0 21 ,
ε
ε
εu01 u02
u02
λ
A2 (w0 )
= (f2 |t=0 − f4 |t=0 ) =
= ρ̄ 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 = −ρ̄(∇ · 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 k̃||2 + ||∂t h̃||2 )
ΓΣ ||∂t w̃||20 + ε6 ∆Σ (||∂t m̃||20 + |||∂t ξ||
0
0
0
+
Z
0
≤
T
(2∆LΣ − 1/τ )ε4 (||∂t m̃||20 + ||∂t ξ̃||20 ) + 2ε6 ΘLΣ (||∂t k̃||20 + ||∂t h||20 ) dt
cε2 ||∇u0 ||20
Z
+ c(M )
T
0
(4.54)
˜ 2 ) + ε8 (||∂t k̃||2 + ||∂t h̃||2 ) dt,
||∂t w̃||20 + ε6 (||∂t m̃||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 Ṽ through the
scalar product, we get:
1 d
(ΣD α Ṽ , D α Ṽ )0 + (LΣD α Ṽ , D α Ṽ )0 ≤ |(D α ∂t N (w + w̄), D α V )0 |,
2 dt
22
(4.55)
where
1
|(D α (2u1 ∂t w2 − εu21 ∂t w1 ), ε2 ∂t D α m̃2 )0
τ
+(D α (u2 ∂t w2 + u1 ∂t w3 − εu1 u2 ∂t w1 ), ε2 ∂t D α m̃3 + ε2 ∂t D α ξ˜2 )0
|(D α ∂t N (w + w̄), D α Ṽ )0 | =
+(D α (2u2 ∂t w3 − εu22 ∂t w1 ), ε2 ∂t D α ξ˜3 )0 |
ε4
ε4
(||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ) ≤ c(M )||∂t w||2s−1 + (||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ),
2τ
2τ
where the last inequality follows from (4.38). Finally, we obtain:
≤ c(||u||s−1 )||∂t w||2s−1 +
ΓΣ ||∂t w̃||2s−1 + ε6 ∆Σ (||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ) + ε8 ΘΣ (||∂t k̃||2s−1 + ||∂t h̃||2s−1 )
+
Z
0
≤
cε2 ||∇u
T
(2∆LΣ − 1/τ )ε4 (||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ) + ε6 ΘLΣ (||∂t k̃||2s−1 + ||∂t h̃||2s−1 ) dt
2
0 ||s−1
Z
+ c(M )
T
0
˜ 2 ) + ε8 (||∂t k̃||2 + ||∂t h̃||2 ) dt.
||∂t w̃||2s−1 + ε6 (||∂t m̃||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:
||∂t w||2s−1 ≤ c(||∂t w̃||2s−1 + ε6 (||∂t m̃||2s−1 + ||∂t ξ̃||2s−1 ) + ε8 (||∂t k̃||2s−1 + ||∂t h̃||2s−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 − ρ̄, ερε uε ) and ρ̄ > 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 ) → (ρ̄, 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 − ρ̄) ⋆
⇀ ∇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 − ρ̄||s
≤ M,
ε
t∈[0,T ⋆ ]
sup
sup ||ρε uε ||s ≤ N,
t∈[0,T ⋆ ]
sup
||∂t (ρε − ρ̄)||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 → ρ̄ 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/ρ̄ strongly in C([0, T ⋆ ], H s′ (T2 )),
since we can take ρ̄ such that ρε >
uε =
ρ̄
2
as in (4.43). Then
mε
m⋆
=: u⋆ strongly in C([0, T ⋆ ], H s′ (T2 )).
→
ρε
ρ̄
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 +w̄) − mε ),
ε
ε
2
ε∂t ξ ε + λε ∂y hε = τ1 ( A2 (wε2 +w̄) − 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 +w̄) − ν∂x wε + ε2 λ2 τ 2 (∂tx kε + ∂xx mε ) − ε2 τ ∂t mε ;
ε
ξ = A2 (wε +w̄) − ν∂y wε + ε2 λ2 τ 2 (∂ty hε + ∂yy ξ ε ) − ε2 τ ∂t ξ 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 +w̄)
ε
+
∂y A2 (w ε +w̄)
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)
Recall that W ε = ΣW̃ ε by definition (3.4), with W ε , W̃ ε in (2.11) and (3.4) respectively. This
yields:
wε = w̃ε + ε3 σ1 m̃ε + ε3 σ2 ξ̃ ε + 2aε4 k̃ε + 2aε4 h̃ε ;
2 ε
ε
2 4 ε
5
ε
ε m = εσ1 w̃ + 2aλ ε m̃ + ε σ1 k̃ ;
(5.6)
ε2 ξ ε = εσ2 w̃ε + 2aλ2 ε4 ξ̃ ε + ε5 σ2 h̃ε ;
2
ε
2
ε
5
ε
6
ε
ε k = 2aε w̃ + ε σ1 m̃ + 2aε k̃ ;
2 ε
ε h = 2aε2 w̃ε + ε5 σ2 ξ̃ ε + 2aε6 h̃ε .
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
∂t wε +
∂x A1 (wε + w̄) ∂y A2 (wε + w̄)
+
− ν∆wε
ε
ε
= O(ε2 ).
(5.8)
s−3
The last two equations and the previous bounds (5.1) and (5.2) yield:
∂t (ρε uε ) + ∇ · (ρε uε ⊗ uε ) +
∇(ρε − ρ̄)
− ν∆(ρuε )
ε2
= O(ε),
(5.9)
s−3
and, in particular,
||∇(ρε − ρ̄)||s−3
≤ c,
ε2
i.e. there exists ∇P ⋆ ∈ L∞ ([0, T ⋆ ], H s−3 (T2 )) such that
∇(ρε − ρ̄) ⋆
⇀ ∇P ⋆ in L∞ ([0, T ⋆ ], H s−3 (T2 )).
ε2
(5.10)
(5.11)
Moreover, since ρε → ρ̄ 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 ) ⇀⋆ ρ̄∂t u⋆ in L∞ ([0, T ⋆ ], H s−3 (T2 )),
(5.12)
while
∇ · (ρε uε ⊗ uε ) ⇀⋆ ρ̄∇ · (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 − ρ̄)
ε ε
⋆
⋆
⋆
⋆
⋆
−ν∆(ρ u ) ⇀ ρ̄ ∂t u +∇·(u ⊗u )+
−ν∆u . (5.14)
∂t (ρ u )+∇·(ρ u ⊗u )+
ε2
ρ̄
ε ε
ε ε
ε
On the other hand, the first equation of (5.8) yields
∂t (ρε − ρ̄) + ∇ · (ρε uε ) − ν∆(ρε − ρ̄) = O(ε2 ).
25
(5.15)
Notice that ||∂t (ρε − ρ̄)||s−1 = O(ε) and ||∆(ρε − ρ̄)||s−2 = O(ε) thanks to (5.1), while ρε → ρ̄
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.
26
[8] F. Bouchut, Construction of BGK Models with a Family of Kinetic Entropies for a Given
System of Conservation Laws, J. of Stat. Phys. 95 (2003).
[9] F. Bouchut, F. Guarguaglini, R. Natalini, Diffusive BGK Approximations for Nonlinear
Multidimensional Parabolic Equations, Indiana Univ. Math. J. 49 (2000), 723-749.
[10] F. Bouchut, Y. Jobic, R. Natalini, R. Occelli, V. Pavan, Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations, preprint June 2016,
submitted.
[11] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer.
Anal. 21 (1984), 1013-1037.
[12] Y. Brenier, R. Natalini, M. Puel, On a relaxation approximation of the incompressible
Navier-Stokes equations, Proc. Amer. Math. Soc. 132 4 (2003), 1021-1028.
[13] M. Carfora, R. Natalini, A discrete kinetic approximation for the incompressible NavierStokes equations, ESAIM: Math. Modelling Numer. Anal. 42 (2008), 93-112.
[14] C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases,
Springer-Verlag, New York (1994).
[15] A. DeMasi, R. Esposito, J. Lebowitz, Incompressible Navier-Stokes and Euler Limits of
the Boltzmann equation, Comm. Pure Appl. Math. 42 (1990), 1189-1214.
[16] F. Golse, L. Saint-Raymond, The Navier–Stokes limit of the Boltzmann equation for
bounded collision kernels, Invent. math. 155 81 (2004).
[17] I. Hachicha, Approximations hyperboliques des équations de Navier-Stokes, Ph. D. Thesis,
Université d’Évry-Val d’Essone (2013).
[18] I. Hachicha, Global existence for a damped wave equation and convergence towards a solution
of the Navier-Stokes problem. Nonlinear Anal. 96 (2014), 68-86.
[19] B. Hanouzet, R. Natalini, Global Existence of Smooth Solutions for Partially Dissipative
Hyperbolic Systems with a Convex Entropy, Arch. Rational Mech. Anal. 169 (2003), 89-117.
[20] Jin, Z. Xin, The relaxation schemes for system of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235-277.
[21] M. Junk, W.-A- Yong, Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation,
Asymptotic Anal. 35 165 (2003).
[22] C. Lattanzio, R. Natalini, Convergence of diffusive BGK approximations for nonlinear
strongly parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) n. 2, 341-358.
27
[23] P. L. Lions, G. Toscani, Diffusive limits for finite velocity Boltzmann kinetic models, Revista
Mat. Iberoamer. 13 (1997), 473-513.
[24] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space
Variables, Springer-Verlag, New York (1984).
[25] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar
conservation laws, J. Diff. Eq. 148 (1998), 292-317.
[26] M. Paicu and G. Raugel, A hyperbolic perturbation of the Navier-Stokes equations, (Une
perturbation hyperbolique des équations de Navier-Stokes.). ESAIM, Proc. 21 (2007), 65-87.
[27] B. Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics
and its Applications 21, Oxford University Press (2000).
[28] S. Succi, The lattice Boltzmann equation for fluid dynamics and beyond, Numerical Mathematics and Scientific Computation, Oxford Science Publications, the Clarendon Press, Oxford
University Press, New York (2001).
[29] D. A. Wolf-Gladrow, Lattice-gas cellular automata and Lattice Boltzmann models. An
introduction, Lecture Notes in Mathematics, Springer-Verlag, Berlin (2000).
28