Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method
aa r X i v : . [ m a t h . A P ] J u l STRONG CONVERGENCE OF A VECTOR-BGK MODELTO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONSVIA THE RELATIVE ENTROPY METHOD
ROBERTA BIANCHINI ∗ Abstract.
The aim of this paper is to prove the strong convergence of the solutions to avector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on thetwo-dimensional torus. This result holds in any interval of time [0 , T ], with
T >
0. We also providethe global in time uniform boundedness of the solutions to the approximating system. Our argumentis based on the use of local in time H s -estimates for the model, established in a previous work,combined with the L -relative entropy estimate and the interpolation properties of the Sobolevspaces. Key words.
Vector-BGK models, incompressible Navier-Stokes equations, dissipative entropy,relative entropy, entropy inequality, diffusive relaxation.
1. Introduction.
In this paper we deal with the incompressible Navier-Stokesequations in two space dimensions,(1.1) ( ∂ t u NS + ∇ · ( u NS ⊗ u NS ) + ∇ P NS = ν ∆ u NS , ∇ · u NS = 0 , with ( t, x ) ∈ [0 , + ∞ ) × T , and initial data(1.2) u NS (0 , x ) = u ( x ) , ∇ · u = 0 . In (1.1), u NS and ∇ P NS are respectively the velocity field and the gradient ofthe pressure term, and ν > ∂ t f ε + λε ∂ x f ε = τε ( M ( w ε ) − f ε ) ,∂ t f ε + λε ∂ y f ε = τε ( M ( w ε ) − f ε ) ,∂ t f ε − λε ∂ x f ε = τε ( M ( w ε ) − f ε ) ,∂ t f ε − λε ∂ y f ε = τε ( M ( w ε ) − f ε ) ,∂ t f ε = τε ( M ( w ε ) − f ε ) , where(1.4) w ε = ( ρ ε , ερ ε u ε , ερ ε u ε ) = ( ρ ε , ερ ε u ε ) = ( ρ ε , q ε ) = X i =1 f εi . Its main properties are as follows: ∗ ´Ecole Normale Sup´erieure de Lyon, UMPA, ENS-Lyon, 46, all´ee d’Italie, 69364-Lyon Cedex 07,France and Consiglio Nazionale delle Ricerche, IAC, via dei Taurini 19, I-00185 Rome, Italy. Mailaddress: [email protected] 1 ROBERTA BIANCHINI • f εi , M i ( w ε ) , i = 1 , · · ·
5, are vector-valued functions taking values in R ; • ρ ε ( t, x ) on R + × T is the approximating density, taking values in R + ; • u ε ( t, x ) = ( u ε ( t, x ) , u ε ( t, x )) on R + × T is the approximating vector field,taking values in R ; • the discrete velocities are λ = ( λ, , λ = (0 , λ ) , λ = ( − λ, ,λ = (0 , − λ ) , λ = (0 , , where λ is a positive constant value.Precise compatibility conditions to be satisfied by the constant parameters of themodel and the Maxwellian functions, together with their explicit expressions, will beprovided in details in Section 2.BGK models were introduced by Bhatnagar, Gross and Krook as a modified versionof the Boltzmann equation, characterized by the relaxation of the collision operator.Since they present most of the basic properties of hydrodynamics, they are consideredinteresting models even though they do not contain all of the relevant features of theBoltzmann equation. Essentially, vector-BGK models are inspired by the hydrody-namic limits of the Boltzmann equation [3, 4, 14, 18, 21], but later they have beengeneralized as approximating equations for different kinds of systems. In this regard,one of the main directions has been the approximation of hyperbolic systems withdiscrete velocities BGK models, as in [11, 26, 32, 8, 34]. Similar results have been ob-tained for convection-diffusion systems under the diffusive scaling [29, 10, 27, 2, 25, 22].Originally, they presented continuous velocities, see [34], but later on discrete veloc-ities BGK models inspired by the relaxation method have been introduced, see [31]for a survey. In the spirit of the relaxation approximations, the main advantage ofdiscrete velocities BGK models is to deal with semilinear systems, see [32, 12, 23].Here we spend few words on our main result and we provide a sketch of the strat-egy. We prove the strong convergence in the Sobolev spaces, for any interval of time[0 , T ] , T >
0, of the vector-BGK model presented in (1.3) to the incompressibleNavier-Stokes equations on the two-dimensioanl torus. To achieve this result, thenovelty relies in using local in time H s -estimates from a previous work, see [6], com-bined with the L -relative entropy estimate and the standard interpolation Theorem.More precisely, part of the results of [6] provides uniform (in ε ) estimates of Gronwalltype in the Sobolev spaces, which hold in [0 , T ∗ ], where T ∗ > M > t ∈ [0 , T ∗ ], see Theorem 3.2. Thus, the interpolation Theorem for Sobolevspaces applied to the relative entropy estimate provides a bound for the solutions toour system which is much more precise than the previous pessimistic Gronwall typeestimates. This is the key point in order to close the argument and to prove the strongconvergence for all times of the solutions to (1.3) to (1.1), together with the global intime boundedness of the approximating solution itself, in Theorem 3.1. In particular,Lemma 2.5 plays a crucial role in quantifying the dissipation term coming from theentropy inequality. At the best of our understanding, the expansions in Lemma 2.5are the only way to establish the relative entropy inequality when, as in our case, theexplicit dependency of the kinetic entropy on the singular parameter is not known.We point out that we start from initial data in (2.9) that are small perturbation ofthe Maxwellians and, thanks to the uniform bounds, in the end we prove that every-thing is done in a bounded set of the densities. This local setting perfectly fits theframework described in [8].The relative entropy method, [17, 19], represents an efficient mathematical tool for ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES
The paper is organized as follows. In Section 2 weintroduce the vector-BGK model and provide some preliminary results. Section 3 isdevoted to the relative entropy inequality and the strong convergence of the modelfor all times, in the Sobolev spaces. In the last part of this section we also show theglobal in time boundedness of the solutions to our model.
2. Presentation of the model, formal limit, and intermediate results.
First, we aim at providing a relative entropy inequality for a vector-BGK modelapproximating the two-dimensional incompressible Navier-Stokes equations. After,this inequality will allow us to extend for long times the local convergence for smoothsolutions achieved in [6]. Let us introduce the setting that will be taken into accounthereafter.Our approximating vector-BGK model has been presented in (1.3), together witha list of the main properties. We point out that, in order to get consistency withthe incompressible Navier-Stokes equations, the Maxwellian functions M i ( w ε ) , i =1 , · · · , , need to satisfy the following compatibility conditions: • P i =1 M i ( w ε ) = w ε ; • P i =1 λ ij M i ( w ε ) = A j ( w ε ) , j = 1 ,
2, with A j in (2.2).We provide here the explicit expressions of the Maxwellian functions(2.1) M , ( w ε ) = aw ε ± A ( w ε )2 λ , M , ( w ε ) = aw ε ± A ( w ε )2 λ , M ( w ε ) = (1 − a ) w ε , ROBERTA BIANCHINI (2.2) A ( w ε ) = q ε q ε ) ρ ε + P ( ρ ε ) q ε q ε ρ ε , A ( w ε ) = q ε q ε q ε ρ ε ( q ε ) ρ ε + P ( ρ ε ) , (2.3) P ( ρ ε ) = (( ρ ε ) − ¯ ρ )2 ¯ ρ , where ¯ ρ > Assumptions a = ν λ τ , < a < , where ν is the viscosity coefficient in (1.1). Moreover, we also take the parameter λ > • guarantee the positivity of the symmetrizer in [6]; • satisfy the sub-characteristic condition, i.e. the positivity of the spectrum ofthe Jacobian matrices of the Maxwellians, see [8, 33].The change of variables introduced in [6],(2.5) w ε = X i =1 f εi , m ε = λε ( f ε − f ε ) , ξ ε = λε ( f ε − f ε ) ,k ε = f ε + f ε , h ε = f ε + f ε . allows us to recover the consistency with respect to (1.1) in a simple way at the formallevel. This way, the vector-BGK model (1.3) reads:(2.6) ∂ t w ε + ∂ x m ε + ∂ y ξ ε = 0 ,∂ t m ε + λ ε ∂ x k ε = τε ( A ( w ε ) ε − m ε ) ,∂ t ξ ε + λ ε ∂ y h ε = τε ( A ( w ε ) ε − ξ ε ) ,∂ t k ε + ∂ x m ε = τε (2 aw ε − k ε ) ,∂ t h ε + ∂ y ξ ε = τε (2 aw ε − h ε ) . Hereafter, we will drop the apex ε where there is no ambiguity. Moreover, we denoteby M i ( w ) := f i the solutions to system (1.3) after taking the limit under the diffusivescaling. The relaxation formulation (2.6) of the system gives(2.7) m = λε ( f − f ) := λε ( M ( w ) − M ( w )) = A ( w ) ε − τ λ ∂ x k + O ( ε ) ,ξ = λε ( f − f ) := λε ( M ( w ) − M ( w )) = A ( w ) ε − τ λ ∂ y h + O ( ε ) ,k = f + f = M ( ¯ w ) + M ( w ) = 2 aw + O ( ε ) ,h = f + f = M ( w ) + M ( w ) = 2 aw + O ( ε ) . Recalling that, from Assumptions 2.1 ν = 2 aτ λ , formally we get ∂ t w + ∂ x A ( w ) ε + ∂ y A ( w ) ε = ν ∆ w + O ( ε ) . ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES w, A ( w ) , A ( w ) in (1.4)-(2.2), ∂ t ρ − ¯ ρερu ερu + ∂ x ρu ερu + ρ − ¯ ρ ρε ερu u + ∂ y ρu ερu u ερu + ρ − ¯ ρ ρε = ν ∆ ρ − ¯ ρερu ερu + O ( ε ) . Dividing the last two lines by ε , this yields ( ∂ t ( ρ − ¯ ρ ) + ∇ · u = ν ∆( ρ − ¯ ρ ) + O ( ε ) ,∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) + ∇ ( ρ − ¯ ρ )2¯ ρε = ν ∆( ρ u ) + O ( ε ) , which is the compressible approximation to the incompressible Navier-Stokesequations provided by the scaled isentropic Euler equations.Now we find an expression of the formal limit in terms of the original kinetic vari-ables (1.3). The limit solution is obtained by solving the linear system (2.7) in theunknowns M i ( w ) , i = 1 , · · · ,
5, so providing(2.8) M ( w ) = M ( w ) − aελτ ∂ x w, M ( w ) = M ( w ) − aελτ ∂ y w, M ( w ) = M ( w ) + aελτ ∂ x w, M ( w ) = M ( w ) + aελτ ∂ y w, M ( w ) = M ( w ) . In order to avoid further complications due to the initial layer, in our convergenceproof the two-dimensional vector-BGK model is endowed with the following initialdata:(2.9) f εi (0 , x ) = M i (¯ ρ, ε ¯ ρ ¯ u ) , i = 1 , · · · , , where u is in (1.2) and ¯ ρ is a positive constant value. According to the theorydeveloped by Bouchut [8], the existence of a kinetic entropy for system (1.3) is sub-jected to the existence of a convex entropy for the limit solution to (1.3) under thehyperbolic scaling. The hyperbolic parameter of the vector-BGK approximation (1.3)is represented by τ and the limit equations approximated by (1.3) in the vanishingparameter of the hyperbolic scaling τ are the isentropic Euler equations. The con-vergence of the hyperbolic-scaled system is guaranteed by the structural propertiesof our vector-BGK model listed before, see [13], while a rigorous proof is provided in[37]. A convex entropy for the limit equation in hyperbolic scaling, i.e, the isentropicEuler equations, is given by(2.10) η ( w ε ) = 12 | q ε | ρ ε + k ( ρ ε ) . Here we collect some preliminary results, which holdfor local times, essentially due to our previous work [6]. Let us start with the followingremark.
ROBERTA BIANCHINI
Remark • In [6], the compressible pressure P ( ρ ε ) in (2.3) is linear. More precisely, from[[6], (10)], ˜ P ( ρ ε ) = ρ ε − ¯ ρ. In this paper, we consider the case of a quadratic pressure P ( ρ ε ) in (2.3). Asimple remark shows that, from (2.3), P ( ρ ε ) = ( ρ ε ) − ¯ ρ ρ = 2 ¯ ρ ( ρ ε − ¯ ρ ) + ( ρ ε − ¯ ρ ) ρ = ( ρ ε − ¯ ρ ) + ( ρ ε − ¯ ρ ) ρ . Thus, the estimates in [6] still hold here: the quadratic pressure only providesan additional quadratic term in the fifth and the ninth line of the nonlinearvector N ( w + ¯ w ) in [[6], (26)]. These supplementary quadratic terms can behandled exactly as the other ones in the energy estimates in [6]. However, wepoint out that the same argument holds exactly in the same way for a generalcompressible pressure P ( ρ ε ) = ( kγ − [( ρ ε ) γ − ¯ ρ γ ] , γ > ,k [ ρ ε log ( ρ ε ) − ¯ ρlog (¯ ρ )] , γ = 1 , where k is a positive constant value. • In [[6], (18)-(19)], we consider a translated version of the relaxation system(2.6). Of course this is an equivalent formulation of the approximating model,and since the translation vector (¯ ρ, ,
0) in [[6], (18)] is constant in t and x ,most of the energy estimates in [6] can be used here. • A further change of variables, involving the dissipative constant right sym-metrizer Σ in [[6], (28)] is defined in [[6], (30)]. However, here the energyestimates from [6] are expressed in terms of the original relaxation variables(2.5) to avoid further complications. The explicit change of variables is writ-ten in [[6], (78)].Taking into account Remark 2.2, we state some results that will be applied below.Hereafter, we denote by T ε the maximum time of existence of the solution to thesemilinear vector-BGK approximation (1.3) with initial data (2.9), see [30]. Of course T ε could depend on ε . In the following, we recall and adapt some results from [6],showing that there exist ε and a fixed and positive time T ∗ , independent of ε anddepending on the Sobolev norm of the initial data, such that, for ε ≤ ε , some localin time H s -estimates on the solutions to the approximating system hold uniformlywith respect to ε . In this context, we consider the constant vector (¯ ρ, ,
0) and thetranslated variables(2.11) w ∗ ( t, x ) = w ( t, x ) − (¯ ρ, , ,k ∗ ( t, x ) = k ( t.x ) − a (¯ ρ, , ,h ∗ ( t, x ) = h ( t.x ) − a (¯ ρ, , . ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES Lemma
Consider the vector-BGK system (1.3) with initial data (2.9), and u in (1.2) beloging to H s ( T ) , for s > . Then, the following estimates hold true. (2.12) k w ∗ ( t ) k s + ε ( k m ( t ) k s + k ξ ( t ) k s ) + k k ∗ ( t ) k s + k h ∗ ( t ) k s + Z T ε k w ∗ ( θ ) k s + k m ( θ ) k s + k ξ ( θ ) k s + 1 ε ( k k ∗ ( θ ) k s + k h ∗ ( θ ) k s ) dθ ≤ cε ( k u k s + k∇ u k s )+ c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) Z T k w ∗ ( θ ) k s + ε ( k m ( θ ) k s + k ξ ( θ ) k s ) dθ + c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) Z T k k ∗ ( θ ) k s + k h ∗ ( θ ) k s dθ, t < T ε . (2.13) k w ∗ ( t ) k s + ε ( k m ( t ) k s + k ξ ( t ) k s ) + k k ∗ ( t ) k s + k h ∗ ( t ) k s ≤ cε ( k u k s + k∇ u k s ) e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t , t < T ε . (2.14) k ∂ t w ∗ ( t ) k s − + ε ( k ∂ t m ( t ) k s − + k ∂ t ξ ( t ) k s − ) + k ∂ t k ∗ ( t ) k s − + k ∂ t h ∗ ( t ) k s − ≤ cε ( k u k s − + k∇ u k s − + k∇ u k s − ) e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t , t < T ε . Moreover, there exists ε , M and T ∗ < T ε fixed such that, for ε ≤ ε , (2.15) | ρ u ( t ) | ∞ ≤ M, | ρ ( t ) − ¯ ρ | ∞ ≤ εM, t ∈ [0 , T ∗ ] , (2.16) | ρ ( t ) | ∞ ≤ ¯ ρ + εM, | u ( t ) | ∞ ≤ M ¯ ρ + εM , t ∈ [0 , T ∗ ] . (2.17) Z T | ρ ( t ) − ¯ ρ | ∞ dt ≤ c ( M ) ε , T ∈ [0 , T ∗ ] . Proof.
We discuss each result separately. • Estimate (2.12) follows from [[6], Lemma 4.2], the change of variables [[6],(30)] and the Sobolev embedding theorem. Notice that the dependency of c ( | ρ | L ∞ t L ∞ x , · ) on | ρ | L ∞ t L ∞ x is a consequence of the quadratic pressure in (2.3),see Remark 2.2, and the estimates of the nonlinear term in [[6], Lemma 4.2]. • By applying Gronwall’s inequality to (2.12), one gets (2.13). • Estimate (2.14) follows from [[6], Proposition 3 and (30)]. • For a fixed constant
M > M := ¯ ρ k u k s +1 , let us define(2.18) T ∗ := sup t ∈ [0 ,T ε ) ( | ρ ( t ) − ¯ ρ | ∞ ε + | ρ u ( t ) | ∞ ≤ M ) . The Sobolev embedding theorem applied to (2.13) yields, thanks to the defi-nition of w ∗ in (2.11),(2.19) | ρ ( t ) − ¯ ρ | ∞ ε + | ρ u ( t ) | ∞ ≤ cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t , t ≤ T ∗ . The uniform bounds (2.15)-(2.16) are due to the Sobolev embedding theoremapplied to (2.13) and the definition of T ∗ , which depends on M , M . ROBERTA BIANCHINI • The last uniform bound is a consequence of the Sobolev embedding theoremapplied to (2.12), the previous bounds in (2.15)-(2.16), and the definition of w ∗ in (2.11). Here we recall the defini-tion and the conditions that assure the existence of a kinetic entropy for a discretevelocities BGK model, see [8] for a detailed discussion.Let E be a non-empty set of convex entropies for a given limit system. Assume alsothat E is separable. A general BGK model under the diffusive scaling reads as follows(2.20) ∂ t f i + λ i ε · ∇ x f i = 1 ε ( M i ( u ε ) − f i ) , i = 1 , · · · , L, where L ≥ d , for i = 1 , · · · , L,f i ( t, x ) = ( f i , · · · , f Ni ) : R × R d → R N ,λ i = ( λ i , · · · , λ di ) ,M i ( u ε ) = ( M i , · · · , M Ni ) : R N → R N . and u ε = P Li =1 f i is the approximating vector field, converging to the solution to thelimit system, which is established under some consistency conditions, see [8, 13, 1,2, 10] for a detailed discussion. An important feature of these approximations is theexistence, under some reasonable conditions, of a kinetic entropy. Set D i := { M i ( u ) : u ∈ U} . Definition
A kinetic entropy for system (2.20) is a convex function H ( f ) = P Li =1 H i ( f i ) , with H i : D i → R , such that, for η ( u ) ∈ E , • (E1) H ( M ( u )) = η ( u ) for every u ∈ U , • (E2) H ( M ( u f )) ≤ H ( f ) , where u f := P Li =1 f i ∈ U , f i ∈ D i . Such a property provides an energy inequality which gives robustness for the scheme,and this is the main advantage of these models with respect to another class of dis-crete velocities BGK models used in computational physiscs, the Lattice Boltzmannschemes, see [38, 42]. Indeed, it is easy to see that, multiplying the BGK system(2.20) by ∇ f H ( f ), the minimality (E2) together with the convexity property, providethe following entropy inequality(2.21) ∂ t H ( f ) + Λ · ∇ x H ( f ) = 1 ε ∇ f H ( f ) · ( M ( u ) − f ) ≤ , which means that, according with the definition given in [24], the kinetic entropy H ( f )is dissipative. More precisely, properties (E1)-(E2) under the hypothesis of [[8], Thm.2.1] assure that, for any η ( u ) ∈ E , defining the projector P such that(2.22) P f = L X i =1 = u , then(2.23) η ( u ) = min P f = u H ( f ) = H ( M ( u )) . In this context, the Gibbs principle for relaxation and, in particular, [[40], Prop. 2.1],imply that(2.24) ∇ f H ( M ( u )) ⊥ Ker ( P ) . ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES f − M ( u ) ∈ Ker ( P ), the convexity property of H ( f ) together with condition(2.24) allow us to get the following inequality:(2.25) ∇ f H ( f ) · ( f − M ( u )) ≤ − c | f − M ( u ) | , c = c ( | f | ∞ ) , meaning that the kinetic entropy H ( M ( u )) is strictly dissipative, according to thedefinition given in [24]. This dissipative property is the main ingredient to apply therelative entropy method, which provides a uniform bound for the relative entropy.Roughly speaking, the relative entropy can be seen as a perturbation of the kineticentropy near to the equilibrium represented by the solution to the limit system. Aprecise definition in the context of hyperbolic relaxation is provided in [40]. Fordiffusive relaxation, we will use the following(2.26) ˜ H ( f | ¯ f ) = H ( f ) − H ( M ( ¯ w )) − ∇ f H ( M ( ¯ w )) · ( f − M ( ¯ w ))= X i H i ( f i ) − H i ( M i ( ¯ w )) − ∇ f i H i ( M i ( ¯ w )) · ( f i − M i ( ¯ w )) , where H ( f ) is in Definition 2.4, and M ( ¯ w ) = ( M i ( ¯ w )) i =1 , ··· , are the perturbedMaxwellians in (2.8) evaluated in the solution ¯ w = (¯ ρ, ε ¯ ρ ¯ u ) to the incompressibleNavier-Stokes equations (1.1). The aim of this part is to characterize andto quantify the dissipative terms resulting from the relative entropy estimate. Here-after, we will drop the apex ε when there is no ambiguity. We start with two prelim-inary lemmas. Lemma
Let η ( w ) be defined in (2.10). Let H ( f ) = P i =1 H i ( f i ) be a kineticentropy associated with the vector-BGK model in (1.3), such that H ( M ( w )) = η ( w ) .Then the following entropy expansion is satisfied: ε Z T Z Z X i =1 ∇ f i H i ( f i ) · ( M i − f i ) dt dx dy = − Z T Z Z " ∇ w η ( w )2 aλ τ · ( m − A ( w ) ε ) · ( m − A ( w ) ε ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aλ τ · ( ξ − A ( w ) ε ) · ( ξ − A ( w ) ε ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aε τ · ( k − aw ) · ( k − aw ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aε τ · ( h − aw ) · ( h − aw ) dt dx dy − Z T Z Z " ∇ w η ( w )(1 − a ) τ ε · (4 aw − ( k + h )) · (4 aw − ( k + h )) dt dx dy + O ( ε ) . Proof.
First of all, the uniform bounds (2.16) and [[8], Theorem 2.1] provide theexistence of a kinetic entropy for (1.3), such that H ( M ( w )) = η ( w ) in (2 . , ∇ f i H i ( M i ( w )) = ∇ w η ( w ) , i = 1 , · · · , . ROBERTA BIANCHINI
We point out that the spectrum of the Jacobian matrices of the Maxwellians in (2.1)is positive provided that the parameter a in the expressions (2.1) is positive and λ > f ε remain in a bounded set, close enough to the hyperbolic equilibrium.Now we consider the following expansion(2.27)1 ε X i =1 ∇ f i H i ( f i ) · ( M i − f i )= 1 ε X i =1 ∇ f i H i ( M i ) · ( f i − M i ) + 1 ε X i =1 ∇ f i H i ( M i ) · ( f i − M i ) · ( f i − M i )+ O ( | f i − M i | ε )= − ε X i =1 ∇ f i H i ( M i ) · ( f i − M i ) · ( f i − M i )+ O ( | f i − M i | ε ) . where the first term vanishes thanks to the orthogonality property [[40], Proposition2.1]. For i = 1 , · · · ,
4, the first term of the last equality reads − ε Z T Z Z ∇ f i H i ( M i ) · ( f i − M i ) · ( f i − M i ) dt dx dy = − ε Z T Z Z ∇ H i ( aw ± A i ( w )2 λ ) · ( f i − M i ) · ( f i − M i ) dt dx dy. Note that, from (1.4)-(2.2) and Lemma 2.3, w = ρερu ερu = O (1) O ( ε ) O ( ε ) ,A ( w ) = ερu ε ρu + ρ − ¯ ρ ρ ε ρu u = O ( ε ) O ( ε ) O ( ε ) ,A ( w ) = ερu ε ρu u ε ρu + ρ − ¯ ρ ρ = O ( ε ) O ( ε ) O ( ε ) . This way, ∇ f i H ( M i ( w )) = ∇ f i H aw ± A i ( w )2 λ ! = ∇ f i H ( aw ) + O ( ε ) . ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES ∇ f i H ( M i ( w )) = ∇ w η ( w ) . Differentiating again the previous equivalent expressions,(2.28) ∇ f i H i ( aw ) = 1 a ∇ w η ( w ) + O ( ε ) . Thus, the last equality yields − ε Z T Z Z ∇ f i H i ( aw ± A i ( w )2 λ ) · ( f i − M i ) · ( f i − M i ) dt dx dy ≤ − ε Z T Z Z a ∇ w η ( w ) · ( f i − M i ) · ( f i − M i ) dt dx dy + c ( | w | L ∞ t L ∞ x ) ε | f i − M i | L ∞ t L ∞ x . Now, from (1.3)-(2.5),(2.29) M − f ε = ∂ t f + λε ∂ x f = 12 ( ∂ t k + ∂ x m ) + 12 λε ( ε ∂ t m + λ ∂ x k ) ,M − f ε = ∂ t f − λε ∂ x f = 12 ( ∂ t k + ∂ x m ) − λε ( ε ∂ t m + λ ∂ x k ) ,M − f ε = ∂ t f + λε ∂ y f = 12 ( ∂ t h + ∂ y ξ ) + 12 λε ( ε ∂ t ξ + λ ∂ y h ) ,M − f ε = ∂ t f − λε ∂ y f = 12 ( ∂ t h + ∂ y ξ ) − λε ( ε ∂ t ξ + λ ∂ y h ) . Lemma 2.3 and the previous equalities imply that c ( | w | L ∞ t L ∞ x ) ε | f i − M i | L ∞ t L ∞ x = O ( ε ) , and so, by using the change of variables (2.5), − ε Z T Z Z X i =1 ∇ f i H i ( f i ) · ( M i − f i ) dt dx dy = − ε Z T Z Z X i =1 a ∇ w η ( w ) · ( M i − f i ) · ( M i − f i ) dt dx dy + O ( ε )= Z T Z Z " ∇ w η ( w )2 aλ τ · ( m − A ( w ) ε ) · ( m − A ( w ) ε ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aλ τ · ( ξ − A ( w ) ε ) · ( ξ − A ( w ) ε ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aε τ · ( k − aw ) · ( k − aw ) dt dx dy − Z T Z Z " ∇ w η ( w )2 aε τ · ( h − aw ) · ( h − aw ) dt dx dy + O ( ε ) . ROBERTA BIANCHINI
The expansion − ε Z T Z Z H i ( f ) · ( M − f ) dt dx dy = − Z T Z Z " ∇ w η ( w )(1 − a ) τ ε · (4 aw − ( k + h )) · (4 aw − ( k + h )) dt dx dy + O ( ε )is obtained in analogous way. Lemma
Consider the limit solution M i , for i = 1 , · · · , , in (2.8). Then (2.30) ∇ f i H i ( M i ) = ∇ f i H i ( M i ) ∓ aελτ ∇ f i H i ( M i ) ∂ x j ¯ w + O ( ε )= ∇ w η ( ¯ w ) ∓ λετ ∇ w η ( ¯ w ) ∂ x j ¯ w + O ( ε ) , j = 1 , . Proof.
The proof follows by Taylor expansions and (2.28), in the spirit of Lemma2.3.
3. Relative entropy estimate for the vector-BGK model.
Our main resultis stated here.
Theorem
Consider the vector-BGK model in (1.3) for the two-dimensionalincompressible Navier-Stokes equations in (1.1) on [0 , + ∞ ) × T , endowed with akinetic entropy H ( f ε ) , whose existence and properties are given by Lemma 2.5. Let ¯ u = (¯ u , ¯ u ) , ∇ ¯ P be a smooth velocity field and pressure satisfying the incompressibleNavier-Stokes equations (1.1) on [0 , + ∞ ) × T and { f ε } be a family of smooth solutionsto (1.3) and emanating from smooth initial data u in (1.2) and f = ( f i (0 , x )) i =1 , ··· , in (2.9). Then, defining w ε = P i f εi = ( ρ ε , ερ ε u ε ) , the following estimate holds forany T > and for ε ≤ ε , where ε is fixed and it depends on M = ¯ ρ k u k s +1 , sup t ∈ [0 ,T ] k ρ ( t ) − ¯ ρ k s ′ ε + k u ( t ) − ¯ u ( t ) k s ′ ≤ cε − δ , with s > , < s ′ < s and δ := s − s ′ s . Moreover, for ε ≤ ε , the solutions ( ρ ε , u ε ) tothe approximating system (1.3) are globally bounded in time, and for ε → , ∇ (( ρ ε ) − ¯ ρ ) ε ⇀ ⋆ ∇ ¯ P in L ∞ t H s − x . The global in time convergence proof is based on the use of the relative entropyinequality, which is stated here.
Theorem
Under the hypothesis of Theorem 3.1, let T ∗ be defined in (2.18).Then the relative entropy method provides the following estimate: sup t ∈ [0 ,T ∗ ] k ρ ( t ) − ¯ ρ k ε + k ρ u ( t ) − ¯ ρ ¯ u ( t ) k ≤ c √ ε. Proof.
We start by recalling the definition of the relative entropy in (2.26),˜ H ( f | ¯ f ) = H ( f ) − H ( M ) − ∇ f H ( M ) · ( f − M )= X i H i ( f i ) − H i ( M i ) − ∇ f i H i ( M i ) · ( f i − M i ) , ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES M i = M i (¯ ρ, ε ¯ ρ ¯ u ) , i = 1 , · · · , , are in (2.8), ¯ ρ is a constantdensity, ¯ u is the smooth solution to (1.1), and the associated entropy-flux is given by˜ Q ( f | ¯ f ) = λε H ( f ) − H ( f ) − ( H ( M ) − H ( M )) H ( f ) − H ( f ) − ( H ( M ) − H ( M )) − λε ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M ) ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M ) . Hereafter, we adopt the following notation, H i := H i ( M i ). Now we proceed to getthe desired inequality. Z T Z Z ∂ t ˜ H ( f | ¯ f ) + ∇ x · ˜ Q ( f | ¯ f ) dt dx dy = Z T Z Z ∂ t H ( f ) + λε ∂ x ( H ( f ) − H ( f )) + λε ∂ y ( H ( f ) − H ( f )) dt dx dy − Z T Z Z ∂ t H ( M ) + λε ∂ x ( H ( M ) − H ( M )) dt dx dy − Z T Z Z λε ∂ y ( H ( M ) − H ( M )) dt dx dy − Z T Z Z ∂ t ( ∇ f H ( M )( f − M ) + ∇ f H ( M )( f − M )+ ∇ f H ( M )( f − M ) + ∇ f H ( M )( f − M )+ ∇ f H ( M )( f − M )) dt dx dy − Z T Z Z λε ∂ x ( ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M )) dt dx dy − Z T Z Z λε ∂ y ( ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M )) dt dx dy = I + I + I + I . First of all, I is already estimated in Lemma 2.3. Now, let us consider I .4 ROBERTA BIANCHINI
The following expansions are based on Lemma 2.6. − Z T Z Z ∂ t ( H + H + H + H + H ) dt dx dy − Z T Z Z λε ∂ x ( H − H ) + λε ∂ y ( H − H ) dt dx dy = − Z T Z Z ( ∇ w η ( ¯ w ) − aελτ ∇ f H ∂ x ¯ w )( ∂ t M + λε ∂ x M ) dt dx dy − Z T Z Z ( ∇ w η ( ¯ w ) + aελτ ∇ f H ∂ x ¯ w )( ∂ t M − λε ∂ x M ) dt dx dy − Z T Z Z ( ∇ w η ( ¯ w ) − aελτ ∇ f H ∂ y ¯ w )( ∂ t M + λε ∂ y M ) dt dx dy − Z T Z Z ( ∇ w η ( ¯ w ) + aελτ ∇ f H ∂ y ¯ w )( ∂ t M − λε M ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ t M dt dx dy − Z T Z Z ∇ f H ( ∂ t M + λε ∂ x M ) − ∇ f H ( ∂ t M − λε ∂ x M ) dt dx dy − Z T Z Z ∇ f H ( ∂ t M + λε ∂ y M ) − ∇ f H ( ∂ t M − λε ∂ y M ) dt dx dy + O ( ε )= − Z T Z Z ∇ w η ( ¯ w ) · ∂ t ( M + M + M + M + M ) dt dx dy − Z T Z Z ∇ w η ( ¯ w )[ λε ∂ x ( M − M ) + λε ∂ y ( M − M )] dt dx dy + 2 aτ λ Z T Z Z ( ∇ w η ( ¯ w ) · ∂ x ¯ w ) ∂ x ¯ w + ( ∇ w η ( ¯ w ) · ∂ y ¯ w ) ∂ y ¯ w dt dx dy + O ( ε )= − Z T Z Z ∇ w η ( ¯ w )[ ∂ t ¯ w + ∂ x A ( ¯ w ) ε + ∂ y A ( ¯ w ) ε − ν ( ∂ xx ¯ w + ∂ yy ¯ w )] dt dx dy + Z T Z Z τ " ∇ w η ( ¯ w )2 aλ · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) dt dx dy + Z T Z Z τ " ∇ w η ( ¯ w )2 aλ · ( ε ∂ t ¯ ξ + λ ∂ y ¯ h ) · ( ε ∂ t ¯ ξ + λ ∂ y ¯ h ) dt dx dy + O ( ε ) . Remark
ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES ε ∂ t ¯ m ) · ∂ x ¯ k, ( ε ∂ t ¯ ξ ) · ∂ y ¯ h, | ε ∂ t ¯ m | , | ε ∂ t ¯ ξ | , where ∂ x ¯ k = 2 a∂ x ¯ w = 2 a∂ x ¯ ρε ¯ ρ ¯ u ε ¯ ρ ¯ u = O ( ε ) ,∂ y ¯ k = 2 a∂ y ¯ w = O ( ε ) ,ε ∂ t ¯ m = ε ∂ t [ A ( ¯ w ) ε − ν∂ x ¯ w ] = ε ∂ t " ¯ ρ ¯ u ε ¯ ρ ¯ u + ε ¯ Pε ¯ ρ ¯ u ¯ u − ν∂ x ¯ w = O ( ε ) ,ε ∂ t ¯ ξ = ε ∂ t [ A ( ¯ w ) ε − ν∂ y ¯ w ] = ε ∂ t " ¯ ρ ¯ u ε ¯ ρ ¯ u ¯ u ε ¯ ρ ¯ u + ε ¯ P − ν∂ y ¯ w = O ( ε ) . This way, every remainder term is O ( ε ).Next, we consider I . I = − Z T Z Z ∂ t [ ∇ f H ( M )( f − M ) + ∇ f H ( M )( f − M )] dt dx dy − Z T Z Z ∂ t [ ∇ f H ( M )( f − M ) + ∇ f H ( M )( f − M )] dt dx dy − Z T Z Z ∂ t [ ∇ f H ( M )( f − M )] dt dx dy = − Z T Z Z ∂ t [ ∇ w η ( ¯ w ) · ( w − ¯ w )] dt dx dy + ελτ Z T Z Z ∂ t [ ∇ w η ( ¯ w ) · ∂ x ¯ w · ( f − f − ( M − M ))] dt dx dy + ελτ Z T Z Z ∂ t [ ∇ η ( ¯ w ) · ∂ y ¯ w · ( f − f − ( M − M ))] dt dx dy + O ( ε )= − Z T Z Z ∂ t [ ∇ w η ( ¯ w ) · ( w − ¯ w )] dt dx dy + ε τ Z T Z Z ∂ t [ ∇ w η ( ¯ w ) · ∂ x ¯ w · ( m − A ( ¯ w ) ε + 2 aλ τ ∂ x ¯ w )] dt dx dy + ε τ Z T Z Z ∂ t [ ∇ w η ( ¯ w ) · ∂ y ¯ w · ( ξ − A ( ¯ w ) ε + 2 aλ τ ∂ y ¯ w )] dt dx dy + O ( ε )6 ROBERTA BIANCHINI = Z T Z Z " ∇ w η ( ¯ w ) · [ ∂ x A ( ¯ w ) ε + ∂ y A ( ¯ w ) ε − aλ τ ∂ xx ¯ w − aλ τ ∂ yy ¯ w ] · ( w − ¯ w )+ ∇ w η ( ¯ w ) · [ ∂ x m + ∂ y ξ − ∂ x A ( ¯ w ) ε − ∂ y A ( ¯ w ) ε + 2 aλ τ ∂ xx ¯ w + 2 aλ ∂ yy ¯ w ]+ τ ∇ w η ( ¯ w ) · ∂ x ¯ w · ( ε ∂ t m + λ ∂ x k ) − τ λ ∇ w η ( ¯ w ) · ∂ x ¯ w · ∂ x k + τ ∇ w η ( ¯ w ) · ∂ y ¯ w · ( ε ∂ t ξ + λ ∂ y h ) − τ λ ∇ w η ( ¯ w ) · ∂ y ¯ w · ∂ y h dt dx dy + O ( ε ) . It remains to deal with the last term. I = − Z T Z Z λε ∂ x [ ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M )] dt dx dy − Z T Z Z λε ∂ y [ ∇ f H ( M )( f − M ) − ∇ f H ( M )( f − M )] dt dx dy = − Z T Z Z λε ∂ x " ( ∇ w η ( ¯ w ) − ελτ ∇ w η ( ¯ w ) ∂ x ¯ w )( f − M ) − ( ∇ w η ( ¯ w ) + ελτ ∇ w η ( ¯ w ) ∂ x ¯ w )( f − M ) dt dx dy − Z T Z Z λε ∂ y " ( ∇ w η ( ¯ w ) − ελτ ∇ w η ( ¯ w ) ∂ y ¯ w )( f − M ) − ( ∇ w η ( ¯ w ) + ελτ ∇ w η ( ¯ w ) ∂ y ¯ w )( f − M ) dt dx dy + O ( ε )= − Z T Z Z λε ∂ x [ ∇ w η ( ¯ w ) · (( f − f ) − ( M − M ))] dt dx dy − Z T Z Z λε ∂ y [ ∇ w η ( ¯ w ) · (( f − f ) − ( M − M ))] dt dx dy + λ τ Z T Z Z ∂ x [ ∇ w η ( ¯ w ) · ∂ x ¯ w · ( f + f − ( M + M ))] dt dx dy + λ τ Z T Z Z ∂ y [ ∇ w η ( ¯ w ) · ∂ y ¯ w · ( f + f − ( M + M ))] dt dx dy + O ( ε )= − Z T Z Z ∂ x [ ∇ w η ( ¯ w ) · ( m − A ( ¯ w ) ε + 2 aτ λ ∂ x ¯ w )] dt dx dy − Z T Z Z ∂ y [ ∇ w η ( ¯ w ) · ( ξ − A ( ¯ w ) ε + 2 aτ λ ∂ y ¯ w )] dt dx dy + λ τ Z T Z Z ∂ x [ ∇ w η ( ¯ w ) · ∂ x ¯ w · ( k − a ¯ w )] dt dx dy + λ τ Z T Z Z ∂ y [ ∇ w η ( ¯ w ) · ∂ y ¯ w · ( h − a ¯ w )] dt dx dy + O ( ε ) ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES − Z Z Z T ∇ w η ( ¯ w ) · [ ∂ x m + ∂ y ξ − A ( ¯ w ) ε − ∂ y A ( ¯ w ) ε + 2 aτ λ ∂ xx ¯ w + 2 aτ λ ∂ yy ¯ w ] dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( m − A ( w ) ε ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( A ( w ) ε − A ( ¯ w ) ε ) dt dx dy − aτ λ Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ∂ x ¯ w dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( ξ − A ( w ) ε ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( A ( w ) ε − A ( ¯ w ) ε ) dt dx dy − aτ λ Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ∂ y ¯ w dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( ∂ x k − a∂ x ¯ w ) + ∇ w η ( ¯ w ) · ∂ xx ¯ w · ( k − a ¯ w ) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( ∂ y h − a∂ y ¯ w ) + ∇ w η ( ¯ w ) · ∂ yy ¯ w · ( h − a ¯ w ) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w )( ∂ x ¯ w ) ( k − a ¯ w ) + ∇ w η ( ¯ w )( ∂ y ¯ w ) ( h − a ¯ w ) dt dx dy + O ( ε )= − Z T Z Z ∇ w η ( ¯ w ) · [ ∂ x m + ∂ y ξ − A ( ¯ w ) ε − ∂ y A ( ¯ w ) ε ] dt dx dy − aτ λ Z T Z Z ∂ xx ¯ w + ∂ yy ¯ w dt dx dy + Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · τ ( ε ∂ t m + λ ∂ x k ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( A ( w ) ε − A ( ¯ w ) ε ) dt dx dy − Z T Z Z aτ λ ∇ w η ( ¯ w ) · ∂ x ¯ w · ∂ x ¯ w dt dx dy + Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · τ ( ε ∂ t ξ + λ ∂ y h ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( A ( w ) ε − A ( ¯ w ) ε ) dt dx dy − Z T Z Z aτ λ ∇ w η ( ¯ w ) · ∂ y ¯ w · ∂ y ¯ w dt dx dy ROBERTA BIANCHINI + λ τ Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ∂ x k + ∇ w η ( ¯ w ) · ∂ y ¯ w · ∂ y h dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w ) · ∂ xx ¯ w · ( k − aw ) + 2 a ∇ w η ( ¯ w ) · ∂ xx ¯ w · ( w − ¯ w ) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w ) · ∂ yy ¯ w · ( h − aw ) + 2 a ∇ w η ( ¯ w ) · ∂ yy ¯ w · ( w − ¯ w ) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w )[( ∂ x ¯ w ) ( k − a ¯ w ) + ( ∂ y ¯ w ) ( h − a ¯ w )] dt dx dy + O ( ε ) . As an intermediate step, let us look at the sum I + I = 2 τ Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( ε ∂ t m + λ ∂ x k ) dt dx dy + 2 τ Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( ε ∂ t ξ + λ ∂ y h ) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ( A ( w ) ε − A ( ¯ w ) ε − A ′ ( ¯ w ) ε ( w − ¯ w )) dt dx dy − Z T Z Z ∇ w η ( ¯ w ) · ∂ y ¯ w · ( A ( w ) ε − A ( ¯ w ) ε − A ′ ( ¯ w ) ε ( w − ¯ w )) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w ) · [ ∂ xx ¯ w · ( k − aw ) + ∂ yy ¯ w · ( h − aw )] dt dx dy − aτ λ Z T Z Z ∇ w η ( ¯ w ) · ∂ x ¯ w · ∂ x ¯ w + ∇ w η ( ¯ w ) · ∂ y ¯ w · ∂ y ¯ w dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w )( ∂ x ¯ w ) ( k − a ¯ w ) + ∇ w η ( ¯ w )( ∂ y ¯ w ) ( h − a ¯ w ) dt dx dy + O ( ε ) . We analyse each line separately. • The first one can be written as τ aλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) · ( ε ∂ t m + λ ∂ x k ) dt dx dy + O ( ε ) . • Similarly for the second line. • The third/fourth lines are equivalent to |∇ w η ( ¯ w ) | L ∞ t L ∞ x Z T Z Z | w − ¯ w | dt dx dy + O ( ε ) . • The fifth line can estimated by c ( |∇ w η ( ¯ w ) | L ∞ t L ∞ x ) Z T Z Z ε | ∂ xx ¯ w | + ε | ∂ yy ¯ w | dt dx dy + c ( |∇ w η ( ¯ w ) | L ∞ t L ∞ x ) Z T Z Z | k − aw | ε + | h − aw | ε dt dx dy, where the first term is O ( ε ), while the second one is absorbed by the dissi-pation in I . ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES • The sixth term can be written as − τaλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) dt dx dy − τaλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t ¯ ξ + λ ∂ y ¯ h ) · ( ε ∂ t ¯ ξ + λ ∂ y ¯ h ) dt dx dy + O ( ε ) . • The last term presents the following form λ τ Z T Z Z ∇ w η ( ¯ w )( ∂ x ¯ w ) ( k − a ¯ w ) + ∇ w η ( ¯ w )( ∂ y ¯ w ) ( h − a ¯ w ) dt dx dy = λ τ Z T Z Z ∇ w η ( ¯ w )( ∂ x ¯ w ) ( k − aw ) − a ∇ w η ( ¯ w )( ∂ x ¯ w ) ( w − ¯ w ) dt dx dy + λ τ Z T Z Z ∇ w η ( ¯ w )( ∂ y ¯ w ) ( h − aw ) − a ∇ w η ( ¯ w )( ∂ y ¯ w ) ( w − ¯ w ) dt dx dy ≤ c ( |∇ w η ( ¯ w ) | L ∞ t L ∞ x ) Z T Z Z | w − ¯ w | + | k − aw | ε + | h − aw | ε dt dx dy + O ( ε ) , where the right-hand side is controlled by using the dissipation coming from I . Remark µ i ( ∇ w η ( w )) , µ i ( ∇ w η ( ¯ w )) the eigenvalues of ∇ w η ( w ) , ∇ w η ( ¯ w ) respectively, by simple calculations one gets that Z T | µ i ( ∇ w η ( w ( t ))) − µ i ( ∇ w η ( ¯ w ( t ))) | ∞ dt ≤ c Z T | ρ ( t ) − ¯ ρ | ∞ dt = O ( ε ) , where the last equality follows from Lemma 2.3. Thus, we can write12 aλ Z T Z Z ( ∇ w η ( w ) · ( ε ∂ t m + λ ∂ x k )) · ( ε ∂ t m + λ ∂ x k ) dt dx dy + 12 aλ Z T Z Z ( ∇ w η ( w ) · ( ε ∂ t ξ + λ ∂ y h )) · ( ε ∂ t ξ + λ ∂ y h ) dt dx dy = 12 aλ Z T Z Z ( ∇ w η ( ¯ w ) · ( ε ∂ t m + λ ∂ x k )) · ( ε ∂ t m + λ ∂ x k ) dt dx dy + 12 aλ Z T Z Z ( ∇ w η ( ¯ w ) · ( ε ∂ t ξ + λ ∂ y h )) · ( ε ∂ t ξ + λ ∂ y h ) dt dx dy + O ( ε ) . ROBERTA BIANCHINI
Now we consider the total sum, given by I + I + I + I ≤ |∇ w η ( ¯ w ) | L ∞ t L ∞ x Z T Z Z | w − ¯ w | dt dx dy − τ aλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t m + λ ∂ x k ) · ( ε ∂ t m + λ ∂ x k ) dt dx dy − τ aλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t ξ + λ ∂ y h ) · ( ε ∂ t ξ + λ ∂ y h ) dt dx dy + τaλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t m + λ ∂ x k ) · ( ε ∂ t ¯ m + λ ∂ x ¯ k ) dt dx dy + τaλ Z T Z Z ∇ w η ( ¯ w ) · ( ε ∂ t ξ + λ ∂ y h ) · ( ε ∂ t ¯ ξ + λ ∂ y ¯ h ) dt dx dy − c ( |∇ w η ( w ) | L ∞ t L ∞ x (1 − δ ))2 aτ ε Z T Z Z | k − aw | + | h − aw | dt dx dy − c ( |∇ w η ( w ) | L ∞ t L ∞ x )(1 − a ) τ ε Z T Z Z | aw − ( k + h ) | dt dx dy + O ( ε ) . The Gronwall inequality, together with the definition of w in (1.4), yields thefollowing estimate(3.1) sup t ∈ [0 ,T ∗ ] k ρ ( t ) − ¯ ρ k ε + k ρ u ( t ) − ¯ ρ ¯ u ( t ) k ≤ c √ ε, where the local time T ∗ is defined in (2.18).Now we prove Theorem 3.1. Proof.
We start by using the interpolation properties of Sobolev spaces, see [39],for 0 < s ′ < s and t ∈ [0 , T ∗ ], which gives(3.2) k ρ u ( t ) − ¯ ρ ¯ u ( t ) k s ′ ≤ k ρ u ( t ) − ¯ ρ ¯ u ( t ) k − s ′ /s k ρ u ( t ) − ¯ ρ ¯ u ( t ) k s ′ /ss ≤ cε s − s ′ s ( M + cM e t | u | L ∞ t L ∞ x ) s ′ /s , where the last inequality follows by • the H s - bound of the solution to the incompressible Navier-Stokes equationson the two-dimensional torus, i.e. k ¯ ρ ¯ u ( t ) k s ≤ k ¯ ρ u k s ≤ M ; • the Gronwall inequality applied to estimate (2.12), k ρ u ( t ) k s ≤ cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t . Taking s ′ big enough, the Sobolev embedding theorem yields(3.3) | ρ u ( t ) − ¯ ρ ¯ u ( t ) | ∞ ≤ c S k ρ u ( t ) − ¯ ρ ¯ u ( t ) k s ′ ≤ cε s − s ′ s ( M + cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s , ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES | ρ u ( t ) | ∞ ≤ M + cε s − s ′ s ( M + cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s . Similarly,(3.4) | ρ ( t ) − ¯ ρ | ∞ ≤ c S k ρ ( t ) − ¯ ρ k s ′ ≤ c S k ρ ( t ) − ¯ ρ k − s ′ /s k ρ ( t ) − ¯ ρ k s ′ /ss ≤ cε s − s ′ )2 s ( cεM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s , i.e.(3.5) | ρ ( t ) | ∞ ≤ ¯ ρ + cε s − s ′ )2 s ( cεM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s . Now, since u − ¯ u = 1 ρ ( ρ u − ¯ ρ ¯ u ) + ¯ u ρ (¯ ρ − ρ ) , then from (3.4)-(3.3)-(3.4),(3.6) | u ( t ) − ¯ u ( t ) | ∞ ≤ ρ cε s − s ′ s ( M + cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s + cε s − s ′ )2 s ( cεM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s ! , i.e.,(3.7) | u ( t ) | ∞ ≤ M + 1¯ ρ cε s − s ′ s ( M + cM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s + cε s − s ′ )2 s ( cεM e c ( | ρ | L ∞ t L ∞ x , | u | L ∞ t L ∞ x ) t ) s ′ /s ! . Recalling the definition of T ∗ in (2.18) and taking M = 4 M , estimate (3.7)implies that there exists ε fixed such that, for ε ≤ ε and t ≤ T ∗ , | u ( t ) | ∞ ≤ M + cε − δ < M = M , for 0 < δ = s ′ s < . Similarly, for t ≤ T ∗ ,(3.8) | ρ ( t ) − ¯ ρ | ∞ ε + | ρ u ( t ) | ∞ ≤ M + cε − δ < M = M . Now let us assume T ∗ < T ε . Then, by definition (2.18), | ρ ( T ∗ ) − ¯ ρ | ∞ ε + | ρ u ( T ∗ ) | ∞ = 4 M = M. ROBERTA BIANCHINI
On the other hand, estimate (3.8) implies that there exists a fixed ε , depending on M and small enough such that, for ε ≤ ε , | ρ ( T ∗ ) − ¯ ρ | ∞ ε + | ρ u ( T ∗ ) | ∞ ≤ M + cε − δ < M . Now, by contradiction one gets that T ∗ = T ε for ε ≤ ε . As a consequence, for ε ≤ ε the solutions ( ρ ε , u ε ) to the approximating system evaluated in T ε are bounded.This way, the Continuation Principle, see [30], implies that they are globally boundedin time. Moreover, since the uniform bounds in Lemma 2.3 are based on the L ∞ t L ∞ x boundedness of ( ρ ε , u ε ), it turns out that they hold globally in time for ε ≤ ε . Inthe end, we proved: • the global in time existence and uniform boundedness of ( ρ ε , u ε ) in H s ( T )for a fixed ε ≤ ε depending on M ; • the strong convergence in [0 , T ], for any T >
0, of the solutions ( ρ ε , u ε ) tothe approximating system (1.3) to the solutions (¯ ρ, ¯ u ) to the incompressibleNavier-Stokes equations in H s ′ ( T ), for 0 < s ′ < s and s > • the rate of this strong convergence.Finally, the convergence to the gradient of the limit incompressible pressure ∇ P NS in (1.1) is discussed in details in [6]. Acknowledgements.
The author is grateful to Roberto Natalini, for his con-stant support, and to Laure Saint-Raymond, for useful discussions.
REFERENCES[1]
D. Aregba-Driollet, R. Natalini,
Discrete Kinetic Schemes for Multidimensional Conserva-tion Laws,
SIAM J. Num. Anal. (2000), 1973-2004.[2] D. Aregba-Driollet, R. Natalini, S.Q. Tang,
Diffusive kinetic explicit schemes for nonlineardegenerate parabolic systems,
Math. Comp. (2004), 63-94.[3] C. Bardos, F. Golse, C. D. Levermore,
Fluid dynamic limits of hyperbolic equations. I.Formal derivations,
J. Stat. Phys. (1991), 323-344.[4] C. Bardos, F. Golse, C. D. Levermore,
Fluid dynamic limits of kinetic equations - II Con-vergence proofs for the Boltzmann-equation,
Comm. Pure Appl. Math. (1993), 667-753.[5] F. Berthelin, A. E. Tzavaras, A. Vasseur
From Discrete Velocity Boltzmann Equations toGas Dynamics Before Shocks,
J. Stat. Phys. (2009), 153-173.[6]
R. Bianchini, R. Natalini,
Convergence of a vector-BGK approximation for the incompressibleNavier-Stokes equations, to appear on
Kinetic and Related Models (2019).[7]
R. Bianchini,
Uniform asymptotic and convergence estimates for the Jin-Xin model under thediffusion scaling,
SIAM J. Math. Anal. (2) (2018), 1877-1899.[8] F. Bouchut,
Construction of BGK Models with a Family of Kinetic Entropies for a GivenSystem of Conservation Laws,
J. Stat. Phys. (2003).[9] F. Bouchut, Y. Jobic, R. Natalini, R. Occelli, V. Pavan,
Second-order entropy satisfyingBGK-FVS schemes for incompressible Navier-Stokes equations, SMAI Journal of computa-tional mathematics, (2018), 1-56.[10] F. Bouchut, F. Guarguaglini, R. Natalini,
Diffusive BGK Approximations for NonlinearMultidimensional Parabolic Equations,
Indiana Univ. Math. J. (2000), 723-749.[11] Y. Brenier,
Averaged multivalued solutions for scalar conservation laws,
SIAM J. Numer.Anal. (1984), 1013-1037.[12] Y. Brenier, R. Natalini, M. Puel,
On a relaxation approximation of the incompressibleNavier-Stokes equations,
Proc. Amer. Math. Soc.
M. Carfora, R. Natalini,
A discrete kinetic approximation for the incompressible Navier-Stokes equations,
ESAIM: Math. Modelling Numer. Anal. (2008), 93-112.[14] C. Cercignani, R. Illner, M. Pulvirenti,
The Mathematical Theory of Dilute Gases,
Springer-Verlag, New York (1994).[15]
I.L. Chern,
Long-time effect of relaxation for hyperbolic conservation laws,
Comm. Math.Phys. (1995), 39-55.ELATIVE ENTROPY FOR A BGK MODEL FOR 2D NAVIER-STOKES [16] J.-F. Coulombel, T. Goudon,
The strong relaxation limit of the multidimensional isothermalEuler equations,
Trans. Amer. Math. Soc. (2) (2007), 637-648.[17]
C. M. Dafermos,
Stability of motions of thermoelastic fluids,
J. Thermal Stresses (1979),127-134.[18] A. DeMasi, R. Esposito, J. Lebowitz,
Incompressible Navier-Stokes and Euler Limits of theBoltzmann equation,
Comm. Pure Appl. Math. (1989), 1189-1214.[19] R. J. DiPerna,
Uniqueness of solutions to hyperbolic conservation laws,
Indiana Univ. Math.J. (1979), 137-188.[20] F. Golse, C. D. Levermore, L. Saint-Raymond,
La m´ethode de l’entropie relative pour leslimites hydrodynamiques de mod`eles cin´etiques,
S´eminaire ´Equations aux d´eriv´ees partielles
Vol. 1999-2000 (2000), 1-21.[21]
F. Golse, L. Saint-Raymond,
The Navier-Stokes limit of the Boltzmann equation for boundedcollision kernels,
Invent. math.
81 (2004).[22]
L. Gosse, G. Toscani,
Space localization and well-balanced scheme for discrete kinetic modelsin diffusive regimes,
SIAM J. Numer. Anal. I. Hachicha,
Approximations hyperboliques des quations de Navier-Stokes, Ph. D. Thesis,Universit´e d’´Evry-Val d’Essone (2013).[24]
B. Hanouzet, R. Natalini,
Global Existence of Smooth Solutions for Partially DissipativeHyperbolic Systems with a Convex Entropy,
Arch. Rational Mech. Anal. (2003), 89-117.[25]
S. Jin, L. Pareschi, G. Toscani,
Diffusive relaxation schemes for multiscale discrete-velocitykinetic equations,
SIAM J. Numer. Anal. (1998), 2405-2439.[26] S. Jin, Z. Xin,
The relaxation schemes for system of conservation laws in arbitrary spacedimensions,
Comm. Pure Appl. Math. (1995), 235-277.[27] C. Lattanzio, A. Tzavaras,
Relative entropy in diffusive relaxation,
SIAM J. Math. Anal. (3) (2013), 1563-1584.[28] C. Lin, J.-F. Coulombel,
The strong relaxation limit of the multidimensional Euler equations,Nonlinear Diff. Eq. and Appl., (3) (2013), 447- 461.[29] P. L. Lions, G. Toscani,
Diffusive limits for finite velocity Boltzmann kinetic models,
RevistaMat. Iberoamer. (1997), 473-513.[30] A. Majda,
Compressible Fluid Flow and Systems of Conservation Laws in Several SpaceVariables,
Springer-Verlag, New York (1984).[31]
C. Mascia,
Twenty-eight years with Hyperbolic Conservation Laws with Relaxation,
ActaMath. Scientia (4) (2015), 807-831.[32] R. Natalini,
A discrete kinetic approximation of entropy solutions to multidimensional scalarconservation laws,
J. Diff. Eqs. (1998), 292-317.[33]
R. Natalini,
Convergence to equilibrium for the relaxation approximations of conservationlaws,
Comm. Pure Appl. Math. (8) (1996), 795-823.[34] B. Perthame,
Kinetic Formulation of Conservation Laws,
Oxford Lecture Series in Mathemat-ics and its Applications , Oxford University Press (2002).[35]
L. Saint-Raymond,
From the BGK model to the NavierStokes equations,
Annales scientifiquesde l’ ´Ecole Normale Sup´erieure (2) (2003), 271-317.[36] L. Saint-Raymond,
Hydrodynamic limits: some improvements of the relative entropy method,
Annales de l’Institut Henri Poincar´e (C) Non Linear Analysis (3) (2009), 705-744.[37] A. Sepe,
Convergence of a BGK approximation of the isentropic Euler equations,
J. Hyper.Differential Equations (2) (2011), 233-255.[38] S. Succi,
The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,
Numerical Math-ematics and Scientific Computation, Oxford Science Publications, the Clarendon Press,Oxford University Press, New York (2001).[39]
M. Taylor,
Partial differential equations III,
Applied Mathematical Sciences 117, Springer (1996).[40]
A. E. Tzavaras,
Relative Entropy in Hyperbolic Relaxation,
Comm. Math. Sci. (2) (2005),119-132.[41] W.-A. Yong,
Entropy and Global Existence for Hyperbolic Balance Laws,
Arch. RationalMech. Anal. (2) (2004), 247-266.[42]
D. A. Wolf-Gladrow,
Lattice-gas cellular automata and Lattice Boltzmann models. Anintroduction, Lecture Notes in Mathematics,