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. 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