Stability and asymptotic analysis of a fluid-particle interaction model

STABILITY AND ASYMPTOTIC ANALYSIS OF A FLUID-PARTICLE INTERACTION MODEL JOS´ E A. CARRILLO, THIERRY GOUDON Abstract. We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models. 1. Introduction Fluid-particle interaction is of primarily importance in sedimentation analysis of disperse suspensions of particles in fluids, one of the issues being the separation of the solid grains from the fluid by external forces: gravity settling processes or centrifugal forces. These procedures find their applications in biotechnology, medicine, waste-water recycling and mineral processing [11]. On the other hand, aerosols and sprays can be also modelled by fluid-particle type interactions in which bubbles of suspended substances are seen as solid particles [5, 6]. Eventually, such problems also arises in combustion theory, when modelling Diesel engines or rocket propulsors [45, 46]. In what follows, we describe a single specie of disperse particles by a density function f(t, x, ξ): f(t, x, ξ) dξdxgives the number of particles enclosed at time t≥0 in the infinitesimal domain of the phase space centered on (x, ξ)∈R3×R3, with volume dξdx. A macroscopic description of the dispersed phase is obtained by looking at averages with respect to the ξvariable like the macroscopic density RR3fdξ, the macroscopic momentum RR3ξf dξand so on. The surrounding fluid is described by its density n(t, x)≥0 and its velocity field u(t, x)∈R3. In this work, we will consider the fluid as compressible and we will describe it by the compressible Euler 2000 Mathematics Subject Classification. 35Q99 35B25. Key words and phrases. Fluid-Particles Interaction. Vlasov-Euler system. Stability. Hydrodynamic Limit. 1 2 JOS´ E A. CARRILLO, THIERRY GOUDON equations. The coupled system of the kinetic transport equation for the particles and the compressible Euler equation for the fluid will be referred as the Vlasov-Euler system. Throughout the paper, particles are assumed spherically shaped with constant radius a > 0, so that the mass of a particle is given by mP= 4 3ρPπa3,ρPbeing the particle mass per unit volume. The evolution of the two phases is coupled through friction force terms. Indeed, the surrounding fluid produces a friction force on the particle, which is proportional to the relative velocity F(t, x, ξ) = 6πµa ¡u(t, x)−ξ¢, with µ > 0 being the (dynamic) viscosity of the fluid. Accordingly, the force exerted by the particles on the fluid is given by the sum 6πµa ZR3¡ξ−u(t, x)¢fdξ. Furthermore, both phases are affected by the external forces, which are supposed to derive from a time independent potential Φ(x). Precisely, we denote by −mP∇xΦ the force exerted on the particles, and αρF∇xΦ the force exerted per unit volume on the fluid, where ρFis the fluid mass per unit volume and α∈Ris a dimensionless parameter (which measures the ratio of the strength of the external force on each phase). External forces can be produced by a great deal of relevant physical mechanisms: gravity, centrifugal, electric or magnetic forces on charged particles/fluid... Throughout the paper, we assume that the potential Φ does not depend on the unknowns f, n, u. It is worthwhile to have in mind the simplest example of settling gravity processes. Here, the force exerted on each particle is the sum of the weight and the buoyancy force: −4 3ρPπa3g³1−ρF ρP´e3, where gis the gravity acceleration and e3the upwards unit vector. Therefore, in such a case we have Φ(x) = (1 −ρF/ρP)gx3α=1 1−ρF/ρP . Note in particular that the sign of Φ depends on the ratio of the densities . Another example is given by considering a centrifugal force: Φ(x) = R∞ |x|ϕ(z) dz≥0, for some function ϕ:R+→R+which tends to 0 at infinity. Finally, the particles are also subject to Brownian motion, which translates into diffusion in the variable ξ, the diffusion coefficient being defined FLUID-PARTICLE INTERACTION MODELS 3 by the Einstein formula [25] kθ0 mP 6πµa mP =kθ0 mP 9µ 2a2ρP . Here kstands for the Boltzmann constant, and θ0>0 denotes the temperature of the surrounding fluid, assumed to be constant. Therefore, we arrive at the following PDEs system ∂tf+ξ·∇xf−∇xΦ·∇ξf=9µ 2a2ρP divξ³(ξ−u)f+kθ0 mP∇ξf´,(1.1) ∂tn+ divx(nu) = 0,(1.2) (1.3) ρF³∂t(nu)+Divx(nu ⊗u)+αn∇xΦ´+∇xp(n)=6πµa ZR3 (ξ−u)fdξ. In (1.3), p(n) is a general pressure law, for instance p(n) = Cγnγ,γ≥1, Cγ>0. The system is completed by the initial condition f0≥0, n0≥0, u0∈R3. Besides, if fluid and particles evolve in a domain Ω ⊂R3, we should also add boundary conditions. Boundary conditions will be considered in such a way that the total number of particles and mass of the fluid remain constant on the evolution. The precise boundary conditions are discussed in Section 3. Although the fluid has been described by the compressible Euler equations, the fluid viscosity plays a major role in the definition of the friction forces. This is completely coherent since in a lot of practical situations viscosity effects are negligible on the fluid evolution. For instance, such a Vlasov-Euler description was introduced for reacting flows of sprays in [45, 46] and it is used in Los Alamos National Laboratory by O’Rourke and collaborators [41, 1, 2] to derive the code KIVA-II, as well as in the French Atomic Commission, [5]. This remark will be completely clarified once we choose relevant dimensionless parameters. Coupled kinetic-fluid model have been introduced in [13]. Derivation of such coupled micro/macro systems are discussed with various viewpoints in [26, 33, 36]. Mathematical analysis of a system coupling the Vlasov equation (1.1) to the constant density incompressible Navier-Stokes equation can be found in [31], while hydrodynamic limits for this model are considered in [28, 29], following the preliminary work [32]. Similar singular perturbations problems are dealt with in [10, 22, 23, 24, 27, 35]. Existence of a smooth solution to the Cauchy problem (1.1), (1.2), (1.3) on a small enough interval of time, is discussed in [5]. 4 JOS´ E A. CARRILLO, THIERRY GOUDON We wish to address some stability and asymptotic limit questions for the problem (1.1)-(1.3). To this end, it is more convenient to work on the dimensionless version of (1.1)-(1.3). This is discussed in Section 2. The cornerstone of our analysis relies on a dissipation property of the system (1.1)-(1.3), which is presented in Section 3. In turn, this leads to the definition of (non trivial) equilibrium states which can be shown to be nonlinearly stable. Then, we can also guess on formal grounds the asymptotic behavior of the system with respect to the evolution of the physical parameters of the problem. In the spirit of hydrodynamic limits, we obtain coupled two-phase macroscopic equations. This is explained in Section 4. However a complete and rigorous proof of convergence for this problem is far beyond the scope of this paper. Instead, we refer to the progresses obtained in [40]. In Section 5 we explore the limiting two-phase systems and we show the dissipation of a quantity that implies again the nonlinear stability of non trivial equilibrium states. In the gravity settling case these equilibrium states correspond to sedimentation profiles. Despite of the lack of a rigorous proof of convergence in the studied asymptotic limits, we bring out the relevancy of our analysis by studying rigorously the linearized version of (1.1)-(1.3) in the Section 6. 2. Modelling Issues 2.1. Dimensionless System. Let us first remark that TS=mP 6πµa =2ρPa2 9µ is the natural relaxation time for the kinetic equation (1.1), usually referred in these applications as the Stokes settling time. We consider as usual Vth =skθ0 mP as the measure of the fluctuation of particles velocity, called their thermal speed. Let us introduce time and length units T and L respectively, which define the time and length units of the observation. We associate a velocity unit U = L/T. Then, we can define dimensionless variables. Adopting the convention that primed quantities are dimensionless, we set t= T t0, x = L x0, ξ =Vthξ0 n(Tt0,Lx0) = n0(t0, x0)u(Tt0,Lx0) = U u0(t0, x0) p(Tt0,Lx0) = Pp0(t0, x0), where Pis a pressure unit, and f0(t0, x0, ξ0) = 4 3πa3V3 th f(Tt0,Lx0,Vthξ0). FLUID-PARTICLE INTERACTION MODELS 5 Note that ρFndxgives the mass of fluid enclosed in a domain with volume dx, so that nis already dimensionless. We eventually associate to the external potential a velocity VSby setting Φ(Tt0,Lx0) = VSL TS Phi0(t0, x0). Therefore, (1.1) can be recast as 1 T∂t0f0+Vth Lξ0·∇x0f0−VS TSVth ∇x0Φ0·∇ξ0f0= =1 TSVth divξ0¡¡Vthξ0−Uu0¢f0+Vth∇ξ0f0¢, while (1.2) and (1.3) become 1 T∂t0n0+U Ldivx0(n0u0)=0, U T∂t0(n0u0) + U2 LDivx0(n0u0⊗u0) + P ρFL∇x0p0(n0) + αVS TS n0∇x0Φ0 =1 TS ρP ρFZR3¡Vthξ0−Uu0¢f0dξ0. Let us define the following dimensionless quantities (2.1)          β=T LVth =Vth U,1 ²=T TS , η=VST VthTS , χ =PT ρFLU =fracPρFU2. Hence, dropping the primes we are led to (2.2)                  ∂tf+βξ ·∇xf−η∇xΦ·∇ξf=1 ²divξ³¡ξ−1 βu¢f+∇ξf´, ∂tn+ divx(nu) = 0, ∂t(nu) + Divx(nu ⊗u) + χ∇xp(n) + αβη n∇xΦ = 1 ² ρP ρF (J−ρu), where we use the notation ρ(t, x) = ZR3 f(t, x, ξ) dξ, J(t, x) = βZR3 ξ f(t, x, ξ) dξ. For further purposes, it is also convenient to define the kinetic pressure P(t, x) = ZR3 ξ⊗ξ f(t, x, ξ) dξ. 6 JOS´ E A. CARRILLO, THIERRY GOUDON Remark 1. Let us go back to the example of the gravity to explain in this particular case the meaning of VS. Consider a particle with radius aand mass per unit volume ρP. The particle is dropped in a fluid at rest, having viscosity µand mass per unit volume ρF. Hence, the motion of the particle is simply described by the ODE 4 3πρPa3d2 dt2X(t) = −6πµa d dtX(t)−4 3πρPa3g¡1−ρF ρP¢. Then, as time becomes large, the velocity of the particle d dtX(t)has a limit, the modulus of which is VS=TSg¯¯1−ρF ρP¯¯. This is referred as the Stokes settling velocity. Remark 2. Still for the example of gravity, the dimensionless coefficient αβη is nothing but the inverse of the Froude number of the flow. Accordingly, 1/η can be seen as the (reduced) Froude number of the dispersed phase. Remark 3. We have mentioned in the Introduction that it could be strange to neglect the viscosity term µ∆xuin the fluid equation, while viscosity plays a crucial role in the definition of the friction force. However, when writing the equation in dimensionless form, the corresponding diffusion coefficient reads 2 9³a L´2T TS ρP ρF . Hence, we realize that this term is negligible under the natural modelling assumption a¿L, whatever will be our scaling assumptions on ρP/ρFand T/TS. 2.2. Boundary Conditions. From now on, we consider the problem (2.2) with the space variable x∈Ω⊂R3, with either Ω = R3itself or a bounded domain with smooth boundary. In such a case we denote by ν(x) the outward unit vector at x∈∂Ω. Then, the problem has to be completed with boundary conditions. For the fluid, it is natural to require (2.3) u·ν(x) = 0,for x∈∂Ω for solid boundaries. For the particles, we consider a general boundary condition which describes how particles are reflected by the boundary. Let us denote by f±(t, x, ξ) the trace of fon the set Σ±={(t, x, ξ)∈R+×∂Ω×R3,±ξ·ν(x)≥0}. Details on the theory of traces for transport equations can be found in e.g. to [8, 16, 17, 19]. The boundary condition relates the incoming trace to the FLUID-PARTICLE INTERACTION MODELS 7 outgoing one as follows |ξ·ν(x)|f−(t, x, ξ) = Zξ0·ν(x)>0 K(x, ξ, ξ0)f+(t, x, ξ0)ξ0·ν(x) dξ0 for (t, x, ξ)∈Σ− where the kernel Kis required to satisfy the following properties •Non negativeness: (2.4) K(x, ξ, ξ0)≥0. •Normalization: (2.5) Zξ·ν(x)<0 K(x, ξ, ξ0) dξ= 1. •Preservation of equilibrium: (2.6) |ξ·ν(x)|e−|ξ|2/2=Zξ0·ν(x)>0 K(x, ξ, ξ0)e−|ξ0|2/2ξ0·ν(x) dξ0. The first condition preserves non negativeness of the solution; the second condition implies the mass conservation: boundaries are neither a source nor a sink of particles and we always have d dt ZΩZR3 fdξdx= 0. The simplest example is given by the total accomodation reflexion law |ξ·ν(x)|f−(t, x, ξ) = |ξ·ν(x)|e−|ξ|2/21 2πZξ0·ν(x)>0 f+(t, x, ξ0)ξ0·ν(x) dξ0, which means that particles are re-emitted by the wall according to a Gaussian law with a temperature coinciding with the temperature of the surrounding fluid (here 1 in dimensionless variables.) In some particular cases, periodic boundary conditions are more physically reasonable in certain variables. For instance, in the gravity settling case we might consider the simplest situation of a box Ω = [0, L]2×[0, L3] in which periodic boundary conditions are considered for both x1, x2-directions and no-flux boundary conditions of the type described above are imposed in the boundaries on the vertical direction x3since our given potential depends only on the x3-variable. Again, with these boundary conditions, particles and fluid total masses are preserved and global boundary fluxes of interest are zero over the variables in which periodic boundary conditions are imposed. 8 JOS´ E A. CARRILLO, THIERRY GOUDON 3. Entropy Dissipation and Stability As mentioned in the Introduction, the system enjoys a remarkable dissipation property. This needs a crucial relation between the scaling parameters, and in turn, it leads to stability properties. 3.1. H-Theorem. Let us assume the following hypotheses on the pressure function p: (HP1) p:R+→R+is continuous, strictly increasing, satisfies p(0) = 0 and is of class C2on (0,+∞). (HP2) The enthalpy function h, defined by h(n) := Zn 1 p0(s) sds, n ∈(0,∞), belongs to L1 loc(0,∞). These hypotheses allow us to introduce the internal energy function Π associated to the pressure function pby Π : [0,∞)→R,Π(n) = Zn 0 h(s)ds. It is well-defined due to (HP2), with Π0(n) = h(n) and nΠ00(n) = p0(n) for all n∈R+and with Π(0) = Π0(1) = 0. Remark 4. •Considering a power pressure law p(n) = nγ,γ > 0, we have Π(n) = (nγ−γn)/(γ−1) for γ6= 1 and Π(n) = nln(n)−nfor γ= 1. •Standard physical assumptions give pressure-laws functions with exponents γ > 1, and therefore they give rise to degenerate pressure functions at zero in the sense that p0(0) = 0. Nevertheless, we include here the most general case allowing atypical pressure functions. Let us mention that compressible Euler equations with a pressure function like p(n) = nγ,0< γ < 1, were considered in hydrodynamic limits of granular media [9]. Proposition 1. Assuming the scaling (3.1) ρP ρF =1 β2, η =β, let us define the free energies associated respectively to the particles and the fluid as follows: FP(f) = ZΩZR3³fln(f) + ξ2 2f+ Φf´dξdx, FF(n, u) = ZΩ³n|u|2 2+χΠ(n) + αβηΦn´dx. FLUID-PARTICLE INTERACTION MODELS 9 Then the total free energy functional, F(f, n, u) = FP(f) + FF(n, u), is dissipated along the flow, precisely (3.2) d dt³F(f(t), n(t), u(t))´+1 εZΩZR3¯¯(v−β−1u)pf+ 2∇ξpf¯¯2dξdx≤0. Proof. In the proof below, we perform integration by parts reasoning in the whole space. The slight modifications when dealing with boundaries are detailed at the end. We compute successively the following time derivatives: •Entropy and kinetic energy of the particles d dt ZR3 ZR3³fln(f) + ξ2 2f´dxi dx =−1 εZR3 ZR3³(ξ−β−1u)f+∇ξf´·³∇ξf f+ξ´dξdx −ηZR3 ZR3∇xΦ·ξf dξdx. •Potential energy of the particles d dt ZR3 ZR3 Φfdξdx=βZR3 ZR3∇xΦ·ξf dξdx. •Kinetic energy of the fluid d dt ZR3 n|u|2 2dx=−χZR3 up0(n)·∇xndx−αβη ZR3 nu ·∇xΦ dx +β ε ρP ρFZR3 ZR3 (ξ−β−1u)f·udξdx. •Entropy of the fluid d dt ZR3 Π(n) dx=−ZR3 Π0(n)divx(nu) dx=ZR3 Π00(n)∇xn·nu dx. •Potential energy of the fluid d dt ZR3 nΦ dx=ZR3 nu ·∇xΦ dx. Now, we sum these relations. Taking into account (3.1) and using the fact that RR3u·∇ξfdξ= 0, we arrive at d dt³F(f(t), n(t), u(t))´= =−1 εZR3 ZR3³(ξ−β−1u)2f+|∇ξf|2 f +(ξ−β−1u)f·∇ξf f+ξ·∇ξf−β−1u·∇ξf´dξdx, 16 JOS´ E A. CARRILLO, THIERRY GOUDON already available in the literature [14, Section 4] for the nonlinear case. As a consequence, we obtain directly the following non-linear stability result for the equilibrium solution (fS, nS, uS). Theorem 2. Given (f(t), n(t), u(t)) a solution to the Cauchy problem for the system (2.2) and (fS, nS, uS= 0) with fSand nSgiven respectively by (3.3) and (3.6), such that ZΩZR3 f0dξdx=ZΩZR3 fSdξdxand ZΩ n0dx=ZΩ nSdx, then for any ² > 0, there exists δ > 0such that if RE((f0, n0, u0)|(fS, nS, uS)) ≤δ, we conclude the solution satisfies kf(t)−fSkL1(Ω×R3)≤², kn(t)−nSkL1(Ω) ≤²and ZΩ n(t)|u(t)|2 2dx≤² for all t≥0. 4. Asymptotic Analysis 4.1. A priori Estimates and Moments. We are interested in hydrodynamic limits for the problem (2.2), which means that the relaxation time TSis small compared to the observation time scale T, i.e., we deal with the singular perturbation problem ε→0. We expect that the limit can be described by macroscopic equations, with unknowns that do not depend on the variable ξ. Indeed, coming back to the entropy dissipation (3.2), we realize that the system then relaxes to (4.1) f(t, x, ξ)≃ρ(t, x) (2π)3/2exp ³−|ξ−β−1u(t, x)|2 2´. Hence, of course, the limiting behavior highly depends on the behavior of the other scaling parameters. Since we are interested in hydrodynamic limits, it is convenient to write the macroscopic equations satisfied by the moments of f. Integrating the kinetic equation in (2.2) with respect to ξleads to (4.2) (∂tρ+ divxJ= 0, ∂tJ+β2DivxP+ηβρ∇xΦ = −1 ε(J−ρu). Before we specify the behavior of the parameters with respect to ε, let us emphasize the main features of the problem. First, the entropy dissipation can be used to establish useful a priori estimates. FLUID-PARTICLE INTERACTION MODELS 17 Proposition 2. Assume that (3.1) holds and that the conditions (HP1)- (HP3) on the pressure and the potential and the confinement conditions (HC1)-(HC7) are satisfied. Moreover, we assume further confinement conditions: (HC8) Φis a bounded below function on Ω, i.e., there exists a constant C∈Rsuch that Φ(x)≥Ca.e. x∈Ω. (HC9) (1 + Φ) exp(−νΦ(x)) ∈L1(Ω), for some positive constants C > 0, and 0< ν < 1. (HC10) If h(0+) = −∞ we assume there exists 0< s1<1such that sup ½Π(n) nh(n),0< n < s1¾<+∞. We suppose that the initial data (f0, n0, u0)satisfies f0≥0,n0≥0and that the quantities ZΩ ZR3 f0¡1 + |ln(f0)|+ξ2 2+|Φ|¢dξdx ZΩ¡n0+n0|u0|2+|Π(n0)|+n0βη|αΦ|¢dx are finite and bounded uniformly with respect to all the parameters ε, β, η, α, ρP/ρF. Then, we have (i) f(1 + ξ2+|Φ|+|ln(f)|)is bounded in L∞(R+; L1(Ω ×R3)). (ii) n,|Π(n)|and βη|αΦ|nare bounded in L∞(R+; L1(Ω)). (iii) √n u is bounded in L∞(R+; L2(Ω)). (iv) 1 √ε¡(ξ−β−1u)√f+ 2∇ξ√f¢=D √εis bounded in L2(R+×Ω×R3). In this statement “bounded” means “bounded uniformly with respect to all the parameters ε, β, η, α,ρP/ρF”. Deducing these estimates from the relative entropy is somehow classical. We will give some details of the proof. To this end, we need the following claim. Lemma 1. Let Ωbe a subset of RD, possibly RDitself. Let U: Ω →R+ such that (1+U)e−νU ∈L1(Ω) for some 0< ν < 1. Let g: Ω →R+. Then, we have 0≤ZΩ gln−(g) dy≤νZUg dy+ZΩ (1 + νU)e−νU dy. 18 JOS´ E A. CARRILLO, THIERRY GOUDON Proof. We follow arguments in [20]. We split Z0≤g≤1 g(−ln(g)) dy= =Z0≤g≤e−νU g(−ln(g)) dy+Ze−νU ≤g≤1 g(−ln(g)) dy ≤eZ0≤g≤e−νU g e³−ln ¡g e¢´dy−Z0≤g≤e−νU gln(e) dy +νZνU≥−ln(g)≥0 Ug dy. Then, since s7→ s(−ln(s)) is non decreasing on (0,1/e), we can dominate eZ0≤g≤e−νU g e³−ln ¡g e¢´dy≤Z0≤g≤e−νU e−νU ¡−ln ¡e−νU ¢+ ln(e)¢dy. Hence, since g≥0, we obtain Z0≤g≤1 g(−ln(g)) dy≤ZΩ e−νU ¡νU + 1¢dy+νZΩ Ug dy. Proof of Proposition 2. Integration of (3.2) yields FP(f(t)) + FF(n(t), u(t)) + Zt 0ZΩZR3 D2 εdξdxds≤C0, where C0depends on the initial data. In order to obtain all the conclusions of the claim it suffices to control the negative terms in the free energy functional by the positive terms and prove that all positive terms are uniformly bounded. By hypotheses (HC3) and (HC8) we can replace Φ (resp. αβηΦ) by Φ + C1(resp. αβηΦ + C2) and thus, we can assume that the contribution of the potential energies is non negative. To control the negative term coming from the particles free energy functional, we apply Lemma 1 using (HC9) with y= (x, v), g(y) = f(x, v) and U(y) = ξ2/2 + Φ(x). We obtain ZΩZR3 fln+(f) dξdx+ (1 −ν)ZΩZR3 f(ξ2/2 + Φ) dξdx+FF(n(t), u(t)) ≤C0+ZΩZR3¡νξ2/2 + νΦ(x)+1¢e−ν(ξ2/2+Φ(x)) dξdx. FLUID-PARTICLE INTERACTION MODELS 19 Now, we need to control from below the negative term coming from the fluid free energy functional, i.e., −ZΩ Π−(n) dx. This control is quite similar to what it has already been done for the particles free energy. Let us distinguish three cases: •If h(0+)>−∞, then Π(n) = e Π(n)−h(0+)nwith e Π(n)≥0 and thus, ZΩ Π−(n) dx≤h(0+)MF. •If p(n) = n, one can use exactly the same argument as for the particles. In fact, Lemma 1 using (HC9) with y=x,g(y) = n(x) and U(y) = Φ(x) gives ZΩ nln+(n) dx+ (1 −ν)ZΩ nΦ dx+ZΩ n|u|2 2dx ≤ FF(n(t), u(t)) + ZΩ¡νΦ(x)+1¢e−νΦ(x)dx. •In the general case when h(0+) = −∞, one proceeds in a similar way to the above arguments using assumptions (HC7) and (HC10). These arguments correspond exactly to [14, Theorem 18, Step 7] and we refer to it for details. As a conclusion, assumptions (HC7) and (HC10) gives us again an estimate of the type ZΩ Π−(n) dx≤B+νZΩ nΦ dx with B∈R. As a consequence of the estimates in each case, one is always able to prove that ZΩ Π+(n) dx+ (1 −ν)ZΩ nΦ dx+ZΩ n|u|2 2dx≤ FF(n(t), u(t)) + C1 with C1∈R. Putting together the estimates for the particle and the fluid free energies, one deduces that the positive integrals ZΩZR3 fln+(f) dξdx, ZΩZR3 f(ξ2/2 + Φ) dξdx, Zt 0ZΩZR3 D2 εdξdxds ZΩ n|u|2 2dx, ZΩ Π+(n) dxand ZΩ nΦ dx 20 JOS´ E A. CARRILLO, THIERRY GOUDON are uniformly bounded. Coming back to the estimate given by Lemma 1 yields the bound on fln−(f).Analogously for the estimate on Π−(n) coming back to the estimate listed in each of the above cases. Let us turn to macroscopic quantities. Corollary 1. We have the following expansions J=ρu +β√εK, P=ρII +1 β2J⊗u+√εK where the components of the vector Kand of the matrix Kare bounded in L2(R+; L1(Ω)). Proof. We write Jand Pby introducing the entropy dissipation as follows J=ZR3 uf dξ+β√εZR3 D √εpfdξ, and P=√εZR3 ξpf⊗D √εdξ+1 β2ZR3 βξpf⊗upfdξ−ZR3 ξpf⊗2∇ξpfdξ. Then, the conclusion follows from an application of the Cauchy-Schwarz inequality, and remarking that RR3ξ√f⊗2∇ξ√fdξ=−RR3fII dξ, by integrating by parts. In what follows, we distinguish two different scalings. 4.2. Flowing Regime. We suppose ρP ρF =1 β2, η =βa fixed positive constant, (i.e., βdoes not depend on the small parameter ε; actually this can be slightly relaxed, assuming that βtends to some 0 < β∞<∞). Coming back to the physical quantities, this scaling assumption means VS¿U = Vth, while ρPand ρFhave the same order. We suppose that αis also a fixed quantity. Note that the ratio ρP/ρFas well as αcan take any fixed value; we only suppose that they remain large compared to ε→0. Therefore, we are concerned with the behavior as ε→0 of (4.3)          ∂tfε+β³ξ·∇xfε−∇xΦ·∇ξfε´=1 εdivξ³¡ξ−1 βuε¢f+∇ξfε´, ∂t(nε) + divx(nεuε) = 0, ∂t(nεuε)+Divx(nεuε⊗uε)+χ∇xp(nε)+αβ2nε∇xΦ= 1 εβ2(Jε−ρεuε), FLUID-PARTICLE INTERACTION MODELS 21 with   ρε Jε Pε (t, x) = ZR3  1 βξ ξ⊗ξ fε(t, x, xi) dξ. The macroscopic quantities satisfy the following moment system (4.4)      ∂tρε+ divxJε= 0, 1 β2∂tJε+ DivxPε+ρε∇xΦ = −1 εβ2(Jε−ρεuε). Using the fluid momentum equation, we are led to (4.5) ∂t(nεuε+β−2Jε)+Divx(nεuε⊗uε+Pε)+χ∇xp(nε)+(αβ2nε+ρε)∇xΦ = 0. Now, coming back to (4.1), it is tempting to infer Jε≃ρεuε,Pε≃ρεII +β−2ρεuε⊗uε, which is made more precise in Corollary 1. Theorem 3. Let fε, nε, uεbe a family of solutions of (4.3). Let us suppose that, as ε→0, ρε→ρ, nε→n, uε→u in the sense of distributions. Suppose also that the non linear terms pass to the limit ρεuε→ρu, ρεuε⊗uε→ρu ⊗u, nεuε⊗uε→nu ⊗u, p(nε)→p(n) in D0(R+×Ω). Then, (ρ, n, u)satisfy the following system (4.6)                ∂tρ+ divx(ρu) = 0, ∂tn+ divx(nu) = 0, ∂t¡(n+β−2ρ)u¢+ Divx¡(n+β−2ρ)u⊗u¢ +∇x¡ρ+χp(n)¢+ (αβ2n+ρ)∇xΦ=0. 4.3. Bubbling Regime. In this case, we assume ρP ρF =1 β2, η =β, β =1 √ε, and thus, according to (3.1), we also have ρP ρF =ε, η =1 √ε. 22 JOS´ E A. CARRILLO, THIERRY GOUDON (This can be slightly modified to β−β∞/√ε→0 with 0 < β∞<∞.) Coming back to the physical quantities, it means that VS= U ¿ Vth. Finally, we assume that α= sign(α)εwith sign(α) = ±1. Therefore, we are concerned with the behavior as ε→0 of (4.7)              ∂tfε+1 √ε³ξ·∇xfε−∇xΦ·∇ξfε´=1 εdivξ³¡ξ−√εuε¢f+∇ξfε´, ∂tnε+ divx(nεuε) = 0, ∂t(nεuε)+Divx(nεuε⊗uε)+χ∇xp(nε)+sign(α)nε∇xΦ=(Jε−ρεuε), with ρε(t, x) = ZR3 fε(t, x, ξ) dξ, Jε(t, x) = ZR3 1 √εξ fε(t, x, ξ) dξ. We still denote Pε(t, x) = ZR3 ξ⊗ξ fε(t, x, ξ) dξ. Then, we obtain the following moment equations (4.8)    ∂tρε+ divxJε= 0, ε∂tJε+ DivxPε+ρε∇xΦ = −(Jε−ρεuε). Using the formal ansatz (4.1), and Corollary 1, the distribution function can be approximated as fε≃ρε (2π)3/2e−ξ2/2, and Pε≃ρεII, but it remains to describe the behavior of Jε. Letting εgo to 0 in the first order moment equation yields ∇xρ+ρ∇xΦ = −J+ρu. Inserting this result in the continuity equation, and passing to the limit in the fluid equation, we are led to the following claim. Theorem 4. Let fε, nε, uεbe a family of solutions of (4.7). Let us suppose that ρε→ρ, Jε→J, nε→n, uε→u in the sense of distributions. Suppose also that the non linear terms pass to the limit ρεuε→ρu, √ε Jε⊗uε→0, nεuε⊗uε→nu⊗u, p(nε)→p(n) FLUID-PARTICLE INTERACTION MODELS 23 in D0(R+×Ω). Then, J=ρ(u− ∇xΦ) − ∇xρ, and (ρ, n, u)satisfy the following system (4.9)            ∂tρ+ divx¡ρ(u−∇xΦ) −∇xρ¢= 0, ∂tn+ divx(nu) = 0, ∂t(nu) + Divx(nu ⊗u) + ∇x(χp(n) + ρ) + (sign(α)n+ρ)∇xΦ = 0. Remark 7. The continuity equation for the particle density ρin (4.9) is often referred as the Smoluchowski equation. Note that in the limit equations (4.9), the behavior of the fluid remains dependent on the evolution of the particles, even in absence of external forces, through the ∇xρ. It is worth pointing out the difference with incompressible models, as studied in [28], where this term disappears due to the incompressibility constraint. Remark 8. For both the flowing and the bubbling regimes, the scaling assumption on the parameter α, which measures the ratio of the effect of the external force on the fluid and on the particles, is particularly clear when looking at the example of gravity force. Indeed, we have seen that αis a simple function of ρP/ρFwhich then imposes the scaling behavior. Remark 9. For both regimes, the available bounds, provided by the entropy dissipation, leads to some weak compactness properties on the sequences fε, ρε,nεand √nεuε. However, we do not have any estimate on the product ρεuε, and thus on Jε. Therefore, this is certainly a great mathematical difficulty when dealing with a rigorous proof of convergence. When Navier- Stokes equation is considered, one obtains an additional H1estimate. Then, in dimension two, this combines with the L ln L estimate on ρ, so that ρεuεis a bounded sequence of integrable functions which passes to the limit, see [28]. We refer to [40] for an attempt dealing with a relative entropy approach. Remark 10. When considering the problem in a bounded domain, boundary conditions have also to be discussed for the limit equations. For the flowing regime the physical boundary condition u·ν(x) = 0 on ∂Ω, preserved by the asymptotic procedure, is enough since ρand nsatisfy the same transport equation with velocity u. For the bubbling regime, the limit equation for the particles density is a convection-diffusion equation. While the boundary condition for the equation in nis still the physical boundary condition u· ν(x) = 0 on ∂Ω, the boundary condition for the equation in ρ, which comes from the asymptotic analysis, is the no-flux boundary condition ρ(∇xΦ−u)·ν(x) + ∂νρ=ρ∇xΦ·ν(x) + ∂νρ= 0 24 JOS´ E A. CARRILLO, THIERRY GOUDON on ∂Ω. We refer to [39, 7] for similar considerations and rigorous proofs in a different physical context. 5. Properties of the Limit Systems We will show that both limit systems (4.6) and (4.9) have first order terms with a hyperbolic structure while keeping the main qualitative property of the original Vlasov-Euler system, that is, the nonlinear stability of steady states. 5.1. Hyperbolicity. Consider the equivalent flowing regime system to (4.6) given by (5.1)            ∂tρ+ divx(ρu) = 0, ∂tr+ divx(ru) = 0, ∂t¡ru¢+ Divx¡ru ⊗u¢+∇x¡ρ+χp(n)¢+αβ2r∇xΦ = 0, with r=n+β−2ρ. Therefore, the first order term has a flux function Ffr :R5−→ M5×3(R) given by: Ffr(U) = µjρ r, j, j⊗j r+ (χp(r−β−2ρ) + ρ)II¶ with U= (ρ, r, j) and j=ru. Taking the component in the x1-direction given by F1 fr(U) = µj1ρ r, j1,j2 1 r+χp(r−β−2ρ) + ρ, j1j2 r,j1j3 r¶, it is easy to check that its jacobian matrix has real eigenvalues given by j1 r(triple) and two simple eigenvalues j1 r±pρ r+χp0(n)n r. Therefore, the system (5.1) is hyperbolic. Regarding the bubbling regime system (4.9), we see that this system consists in a drift-diffusion equation for the particle density ρcoupled with a system on (n, j =nu) which is clearly hyperbolic. Alternatively, one can write an equivalent system to (4.9) by the standard change of variable r=ρeΦobtaining:            ∂tr+ divx(ru) = r(u·∇xΦ) + eΦdivx¡e−Φ∇xr¢, ∂tn+ divx(nu) = 0, ∂t(nu)+Divx(nu ⊗u)+∇x(χp(n)+re−Φ)+(sign(α)n+re−Φ)∇xΦ=0, for which the first order term gives a hyperbolic system. FLUID-PARTICLE INTERACTION MODELS 25 5.2. Nonlinear Stability. Both limit systems: the flowing and the bubbling regime systems keep the remarkable property of the original Vlasov- Euler system. Both systems have a natural entropy which implies the nonlinear stability of the corresponding steady states. Following the ideas of Section 3, it is easy to guess the following free energy functional: (5.2) Ffr(ρ, n, u) = ZΩh(n+β−2ρ)|u|2 2+χΠ(n)+ρln ρ+(αβ2n+ρ)Φidx, for the flowing regime system (4.6). It can be checked easily that: (5.3) d dtFfr(ρ(t), n(t), u(t)) = 0 since we have the following identities: •Entropy of the particles d dt ZΩ ρln ρdx=ZΩ u·∇xρdx. •Entropy of the fluid d dt ZΩ Π(n) dx=ZΩ u·∇xp(n) dx. •Potential energy of the particles and fluid d dt ZΩ (αβ2n+ρ)Φ dx=ZΩ (αβ2n+ρ)u·∇xΦ dx. •Kinetic energy of the particles and fluid d dt ZΩ (n+β−2ρ)|u|2 2dx=−ZΩ u·∇x(χp(n) + ρ) dx −ZΩ (αβ2n+ρ)u·∇xΦ dx. Therefore, analogously to Section 3, one considers the free energy functional Ffr(ρ, n, u) that has a unique minimizer in L1 +(Ω)×L1 +(Ω)×L2(Ω,R3) with total particles and fluid masses equal to MPand MFrespectively. This minimizer by Theorem 1 is given by (ρS, nS, uS= 0) where ρSis given by (5.4) ρS(x) = ZR3 fSdξ=MP e−Φ(x) ZΩ e−Φ(x)dx =Zρ MPe−Φ(x) and nSgiven by (3.6) where η=β. 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Carrillo ICREA and Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain. E-mail:[email protected] Thierry Goudon Labo. Paul Painlev´e, UMR 8524, C.N.R.S.–Universit´e Sciences et Technologies de Lille, Cit´e Scientifique, F-59655 Villeneuve d’Ascq cedex, France. E-mail:[email protected]