Convergence of a vector BGK approximation for the incompressible Navier-Stokes equations

# Convergence of a vector BGK approximation for the incompressible Navier-Stokes equations

[    [
###### 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.

Bianchini]Roberta Bianchini

Natalini]Roberto Natalini

11footnotetext: Dipartimento di Matematica, Università degli Studi di Roma ”Tor Vergata”, via della Ricerca Scientifica 1, I-00133 Rome, Italy - Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy. 22footnotetext: Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy.

Convergence of a vector BGK approximation for the incompressible Navier-Stokes equations

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 dimensional torus :

 {∂tuNS+∇⋅(uNS⊗uNS)+∇PNS=νΔuNS,∇⋅uNS=0, (1.1)

with and initial data

 uNS(0,x)=u0(x),with∇⋅u0=0. (1.2)

Here and are respectively the velocity field and the gradient of the pressure term, and is the viscosity coefficient.

We consider a semilinear hyperbolic approximation, called vector BGK model, CN (); VBouchut (), to the incompressible Navier-Stokes equations (1.1). The general form of this approximation is as follows:

 ∂tfεl+λlε⋅∇xfεl=1τε2(Ml(ρε,ερεuε)−fεl), (1.3)

with initial data

 fεl(0,x)=Mεl(¯ρ,ε¯ρu0),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak u0in(???),l=1,⋯,L, (1.4)

where and take values in with the Maxwellian functions Lipschitz continuous, are constant velocities, and . Moreover, is a given constant value, and and are positive parameters. Denoting by for the components of for each , let us set

 ρε=L∑l=1flε0(t,x)%andqεj=ερεuεj=L∑l=1flεj(t,x). (1.5)

In CN (); VBouchut (), 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→^ρ,%uε→^u,andρε−¯ρε2→^P,

under some consistency conditions of the BGK approximation with respect to the Navier-Stokes equations, CN (), it can be shown that the couple is a solution to the incompressible Navier-Stokes 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 Golse1 (); Golse2 (); Cercignani () for the limit to the compressible Euler equations, and see Esposito (); Laure () 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 Brenier (); XinJin (); Natalini (); Bouchut (); Perthame1 (). Similar results have been obtained for convection-diffusion systems under the diffusive scaling Toscani (); BGN (); Lattanzio (); Aregba2 (). 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 Succi (); Wolf (). 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 Yong1 (). Other partially hyperbolic approximations of the Navier-Stokes equations were developed in BNP (); Raugel (); Imene (); Imene1 ().

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 Succi (); Wolf (), they associate every scalar velocity with one vector of unknowns. Another fruitful property of vector BGK models is their natural compatibility with a mathematical entropy, Bouchut (), which provides a nice analytical structure and stability properties. The work of the present paper takes its roots in CN (); VBouchut (), 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 CN (), let us set and

 wε=(ρε,qε)=(ρε,qε1,qε2)=(ρε,ερεuε1,ερεuε2)=5∑l=1fεl∈R3. (1.6)

Fix and let be a small parameter, which is going to zero in the singular perturbation limit. Thus, we get a five velocities model (15 scalar equations):

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂tfε1+λε∂xfε1=1τε2(M1(wε)−fε1),∂tfε2+λε∂yfε2=1τε2(M2(wε)−fε2),∂tfε3−λε∂xfε3=1τε2(M3(wε)−fε3),∂tfε4−λε∂yfε4=1τε2(M4(wε)−fε4),∂tfε5=1τε2(M5(wε)−fε5). (1.7)

Here the Maxwellian functions have the following expressions:

 M1,3(wε)=awε±A1(wε)2λ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak M2,4(wε)=awε±A2(wε)2λ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak M5(wε)=(1−4a)wε, (1.8)

where

 (1.9)
 P(ρε)=ρε−¯ρ, (1.10)

and

 a=ν2λ2τ, (1.11)

where 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 CN (); VBouchut (), an estimate was obtained by using the entropy function associated with the vector BGK model, whose existence is proved in Bouchut (). 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 HN () for the one dimensional case and Bianchini (); Yong1 () 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) and it is linear with respect to so the estimates in (CN, ; VBouchut, ) 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 Bianchini (). The dissipative property of the symmetrized system holds under the following hypothesis.

###### Assumption 1.1 (Dissipation condition).

We assume the following structural condition:

 0

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 Bouchut ().

### 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

 Uε=(fε1,fε2,fε3,fε4,fε5)∈R3×5, (2.1)

and let us write the compact formulation of equations (1.7)-(1.4), which reads

 ∂tUε+Λ1∂xUε+Λ2∂yUε=1τε2(M(Uε)−Uε), (2.2)

with initial data

 Uε0=fεl(0,x)=Mεl(¯ρ,ε¯ρu0),l=1,⋯,5, (2.3)

where

 Λ1=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λεId00000000000−λεId000000000000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak Λ2=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000000λεId00000000000−λεId000000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (2.4)

is the identity matrix and, for , in (1.8),

 M(Uε)=(Mε1(wε),Mε2(wε),Mε3(wε),Mε4(wε),Mε5(wε)). (2.5)

### 2.1 Conservative variables

We define the following change of variables:

 wε=∑5l=1fεl,\leavevmode\nobreak \leavevmode\nobreak mε=λε(fε1−fε3),\leavevmode\nobreak \leavevmode\nobreak ξε=λε(fε2−fε4),\leavevmode\nobreak \leavevmode\nobreak kε=fε1+fε3,\leavevmode\nobreak \leavevmode\nobreak hε=fε2+fε4. (2.6)

This way, the vector BGK model (1.7) reads:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂twε+∂xmε+∂yξε=0;∂tmε+λ2ε2∂xkε=1τε2(A1(wε)ε−mε),∂tξε+λ2ε2∂yhε=1τε2(A2(wε)ε−ξε),∂tkε+∂xmε=1τε2(2awε−kε),∂thε+∂yξε=1τε2(2awε−hε). (2.7)

We make a slight modification of system (2.7). Set and

 wε⋆:=wε−¯w=(wε1−¯ρ,wε2,wε3),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak kε⋆=kε−2a¯w,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak hε⋆=hε−2a¯w. (2.8)

In the following, we are going to work with the modified variables. System (2.7) reads:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂twε⋆+∂xmε+∂yξε=0;∂tmε+λ2ε2∂xkε⋆=1τε2(A1(wε⋆+¯w)ε−mε),∂tξε+λ2ε2∂yhε⋆=1τε2(A2(wε⋆+¯w)ε−ξε),∂tkε⋆+∂xmε=1τε2(2awε⋆−kε⋆),∂thε⋆+∂yξε=1τε2(2awε⋆−hε⋆). (2.9)

Notice from (1.9) that

 A1(wε)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝qε1(qε1)2ρε+ρε−¯ρqε1qε2ρε⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝wε2(wε2)2wε1+wε1−¯ρwε2wε3wε1⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝wε⋆2(wε⋆2)2wε⋆1+¯ρ+wε⋆1wε⋆2wε⋆3wε⋆1+¯ρ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=A1(wε⋆+¯w),

and, similarly,

 A2(w⋆)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝qε2qε1qε2ρε(qε2)2ρε+ρε−¯ρ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝wε3wε2wε3wε1(wε3)2wε1+wε1−¯ρ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝wε⋆3wε⋆2wε⋆3wε⋆1+¯ρ(wε⋆3)2wε⋆1+¯ρ+wε⋆1⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=A2(wε⋆+¯w).

From now on, we will omit the apexes for , and the apex for when there is no ambiguity.

Let us define the matrix

 C=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝IdIdIdIdIdελId0−ελId000ελId0−ελId0ε2Id0ε2Id000ε2Id0ε2Id0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (2.10)

Let us set

 W=(w,ε2m,ε2ξ,ε2k,ε2h):=CU−(¯w,0,0,0,0). (2.11)

Thus, we can write the translated system (2.9) in the compact form

 ∂tW+B1∂xW+B2∂yW=1τε2(~M(W)−W), (2.12)

with initial conditions

 W0=CU0−(¯w,0,0,0,0), (2.13)

where

 B1=CΛ1C−1=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝01ε2Id000000λ2ε20000000Id00000000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,\parB2=CΛ2C−1=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝001ε2Id00000000000λ2ε20000000Id00⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (2.14)
 ~M(W)=CM(C−1W)=CM(U). (2.15)

Here,

 =1τ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000001ε⎛⎜⎝010100000⎞⎟⎠−1ε2Id0001ε⎛⎜⎝001000100⎞⎟⎠0−1ε2Id002aId00−1ε2Id02aId000−1ε2Id⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠W+1τ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝01ε⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0w22w1+¯ρw2w3w1+¯ρ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠1ε⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0w2w3w1+¯ρw23w1+¯ρ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠00⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠
 =:−LW+N(w+¯w), (2.16)

where is the linear part of the source term of (2.12), while is the remaining nonlinear one. Thus, we can rewrite system (2.12) as follows:

 ∂tW+B1∂xW+B2∂yW=−LW+N(w+¯w). (2.17)

## 3 The weighted constant right symmetrizer and the conservative-dissipative form

According to the theory of semilinear hyperbolic systems, see for instance Majda (); Benzoni (), 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

 ∂tw+∂xm+∂yξ=0,

i.e. the first term of the source vanishes, and 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 Majda (); Benzoni (), we look for a right symmetrizer for (2.17), which provides the conservative-dissipative form introduced in Bianchini (). More precisely, the right multiplication easily provides the conservative structure in Bianchini (), while the dissipation is proved a posteriori. Besides, the symmetrizer presents constant -weighted entries and this allow us to control the nonlinear part of the source term (2.16) of system (2.17). To be complete, we point out that the inverse matrix is a left symmetrizer for system (2.17), according to the definitions given in Majda (); Benzoni (). However, the product is a full matrix, so the symmetrized version of system (2.17), obtained by the left multiplication by does not provide the conservative-dissipative form in Bianchini ().

Let us explicitly write the symmetrizer

 Σ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Idεσ1εσ22aε2Id2aε2Idεσ12λ2aε2Id0ε3σ10εσ202λ2aε2Id0ε3σ22aε2Idε3σ102aε4Id02aε2Id0ε3σ202aε4Id⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (3.1)

where

 σ1=⎛⎜⎝010100000⎞⎟⎠\leavevmode\nobreak \leavevmode\nobreak and\leavevmode\nobreak \leavevmode\nobreak σ2=⎛⎜⎝001000100⎞⎟⎠. (3.2)

It is easy to check that is a constant right symmetrizer for system (2.17) since, taking and in (2.14) and (2.16) respectively,

 B1Σ=ΣBT1,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak B2Σ=ΣBT2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak −LΣ=(000T−~L), (3.3)

where is the null matrix, 0 is the vector with zero entries, is a matrix.

Now, we define the following change of variables:

 W=Σ~W=Σ(~w,ε2~m,ε2~ξ,ε2~k,ε2~h), (3.4)

with in (2.11). System (2.17) reads:

 Σ∂t~W+B1Σ∂x~W+B2Σ∂y~W=−LΣ~W+N((Σ~W)1+¯w), (3.5)

where is the first component of the unknown vector . Now, we want to show that in (3.1) is strictly positive definite. Thus,

 (Σ~W,~W)0=||~w||20+2λ2aε6(||~m||20+||~ξ||20)+2aε8(||~k||20+||~h||20)+2(ε3σ1~m,~w)0
 +2(ε3σ2~ξ,~w)0+4aε4(~k+~h,~w)0+2ε7(σ1~k,~m)0+2ε7(σ2~h,~ξ)0
 =||~w||20+2λ2aε6(||~m||20+||~ξ||20)+2aε8(||~k||20+||~h||20)+I1+I2+I3+I4+I5. (3.6)

Now, taking two positive constants and by using the Cauchy inequality, we have:

 I1=2ε3[(~m2,~w1)0+(~m1,~w2)0]≥−δε6||~m2||20−||~w1||20δ−δε6||~m1||20−||~w2||20δ;
 I2=2ε3[(~ξ3,~w1)0+(~ξ1,~w3)0]≥−δε6||~ξ3||20−||~w1||20δ−δε6||~ξ1||20−||~w3||20δ;
 I3=4aε4[(~k,~w)0+(~h,~w)0]≥−2aμ||~w||20−2aε8μ||~k||20−2aμ||~w||20−2aε8μ||~h||20;
 I4=2ε7[(~k2,~m1)0+(~k1,~m2)0]≥−ε8δ||~k2||20−δε6||~m1||20−ε8δ||~k1||20−δε6||~m2||20;
 I5=2ε7[(~h3,~ξ1)0+(~h1,~ξ3)0]≥−ε8δ||~h3||20−δε6||~ξ1||20−ε8δ||~h1||20−δε6||~ξ3||20.

Thus, putting them all together,

 (Σ~W,~W)0≥||~w1||20[1−2δ−4aμ]+||~w2||20[1−1δ−4aμ]+||~w3||20[1−1δ−4aμ]
 +ε6||~mε1||20[2λ2a−2δ]+ε6||~mε2||20[2λ2a−2δ]+ε6||~mε3||20[2λ2a]
 +ε6||~ξε1||20[2λ2a−2δ]+ε6||~ξε2||20[2λ2a]+ε6||~ξε3||20[2λ2a−2δ]
 +ε8||~k1||20[2a−2aμ−1δ]+ε8||~k2||20[2a−2aμ−1δ]+ε8||~k3||20[2a−2aμ]
 +ε8||~h1||20[2a−2aμ−1δ]+ε8||~h2||20[2a−2aμ]+ε8||~h3||20[2a−2aμ−1δ]. (3.7)

Now, we can prove the following lemma.

###### Lemma 3.1.

If Assumption 1.1 is satisfied and is big enough, then is strictly positive definite.

###### Proof.

From (3.7), we take

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩1<μ<14a;δ>max{21−4aμ,12a(1−1μ)};λ>√δa. (3.8)

Notice that we can choose the constant velocity as big as we need, therefore the third inequality is automatically verified. ∎

Now, we consider the linear part of the source term of (3.5). Explicitly

 −LΣ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000000−2λ2aId+σ21σ1σ2(2a−1)εσ12aεσ10σ1σ2−2λ2aId+σ222aεσ2(2a−1)εσ20(2a−1)εσ12aεσ22a(2a−1)ε2Id4a2ε2Id02aεσ1(2a−1)εσ24a2ε2Id2a(2a−1)ε2Id⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.9)

Thus,

 (−LΣ~W,~W)0=−2λ2aε4(||~m||20+||~ξ||20)+2a(2a−1)ε6(||~k||20+||~h||20)+ε4||~m1||20+ε4||~m2||20
 +ε4||~ξ1||20+ε4||~ξ3||20+2ε4(σ1σ2~ξ,~m)0+2(2a−1)ε5(σ1~k,~m)0+4aε5(σ1~h,