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Journal of Hyperbolic Differential Equations Vol. 3, No. 1 (2006) 1–26 c World Scientific Publishing Company
COUPLING EULER AND VLASOV EQUATIONS IN THE CONTEXT OF SPRAYS: THE LOCAL-IN-TIME, CLASSICAL SOLUTIONS
´ CELINE BARANGER
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CEA/DIF, BP 12, 91680 Bruy` eres le Chˆ atel, France
[email protected]
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LAURENT DESVILLETTES
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Centre de Math´ ematiques et Leurs Applications CNRS UMR 8536, Ecole Normale Sup´ erieure de Cachan 61, Avenue du Pr´ esident Wilson, 94235 Cachan Cedex, France
[email protected]
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Received 12 Apr. 2005 Accepted 7 Oct. 2005
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Communicated by E. Tadmor
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Abstract. Sprays are complex flows made of liquid droplets surrounded by a gas. They can be modeled by the introducing a system coupling a kinetic equation (for the droplets) of Vlasov type and a (Euler-like) fluid equation for the gas. In this paper, we prove that, for the so-called thin sprays, this coupled model is well-posed, in the sense that existence and uniqueness of classical solutions holds for small time, provided the initial data are sufficiently smooth and their support have suitable properties. Keywords:
1. Introduction 27 29 31 33 35
In the framework of sprays (that is, gases in which droplets form a dispersed phase), couplings between an equation of fluid mechanics and a kinetic equation were introduced by Williams [18], cf. also [2]. In this modeling, the gas is described by macroscopic quantities depending on the time t and the position x: its density ρ(t, x) and its velocity u(t, x). The evolution of those quantities is ruled by a system of partial differential equations such as the Navier–Stokes or Euler (compressible or incompressible). We shall investigate here the case of the compressible Euler equation. In order to describe the dispersed phase (the droplets), we use their distribution function in the phase space (“pdf”): it is defined as f ≡ f (t, x, v) ≥ 0, density of 1
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droplets which at time t and point x have velocity v. This function is the solution of a kinetic equation. We concentrate here on the so-called “thin sprays” [13], in which the coupling between the gas and the droplets is made only through a drag term (whereas in so-called thick sprays, it is also made through the volume fraction). The system reads ∂t ρ + ∇x · (ρu) = 0, ∂t (ρu) + ∇x · (ρu ⊗ u + P (ρ)IdN ) = − mp F f dv, ∂t f + ∇x · (vf ) + ∇v · (F f ) = 0,
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13
(1.2) (1.3)
where mp is the mass of one single droplet (supposed to be a constant), mp F is the drag force, and P is the pressure. We consider here a gas which is isentropic, so that P depends on ρ only (and no equation of energy appears). We restrict ourselves to the case of perfect gases, that is P (ρ) = Aργ ,
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(1.1)
A > 0,
and γ > 1.
(1.4)
The drag force mp F is due to the resistance of the fluid to the motion of the droplets. It is possible to find in [13, 16] some analysis on the modeling of this term. One of the most standard formula is the following: 1 2 πr ρ(t, x)Cd |u(t, x) − v|(u(t, x) − v), 2 where r is the radius of the droplets (a constant in this work) and Cd is the drag coefficient. This coefficient is sometimes taken as 24 1 2/3 Cd = , 1 + Re Re 6 mp F =
15
17 19
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where Re = 2ρ|u−v|r is the Reynolds number and µ the dynamic viscosity of the µ fluid. This formula is used, for example, in [1]. We shall assume here (as in [3–6, 8]) 24 that Cd = Re , so that F =
Cµ (u(t, x) − v), r 2 ρl
where ρl is the (constant) density of the droplets (and where C is a generic constant). This assumption is reasonable as long as the Reynolds number is not too large. In many works (cf. [3–6, 8]), the viscosity is supposed to be constant, and the drag force becomes mp F = C(u(t, x) − v).
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(1.5)
It is, however, possible to consider also that the viscosity is proportional to the density of the fluid µ = ρν, with ν kinematic viscosity of the fluid (cf. [7]). This leads to a drag force mp F = Cρ(t, x)(u(t, x) − v).
(1.6)
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We shall make this assumption in this work. For the sake of simplicity, we shall also suppose that all the constants of the model are equal to 1 (taking other values does not lead to any difficulty). Let us comment a little bit on the modeling: depending on the physical context, the fluid equation can be the Euler or Navier–Stokes equation (with or without turbulent viscosity), compressible or incompressible. Note that though the drag force is proportional to the viscosity of the gas, the Euler equation (and not Navier– Stokes equation) is used in some realistic simulations (cf. [11]). Whenever the exchange of temperature is important in the study, one has to replace the isentropic Euler equation by the full Euler system (with 5 equations in dimension 3) and to add one extra variable (of temperature or internal energy) in the “pdf” of the droplets. When the volume occupied by the droplets is not neglegted in front of the volume occupied by the gas, one has to add a new unknown (α ≡ α(t, x)) representing the volumic fraction of gas: this leads to the theory of thick sprays (cf. [13]). One takes then into account various complex phenomena for the droplets (collisions, breakups, etc.). Here are the typical values of a computation on a thin-air mixture: the Reynolds 2 number relative to the gas is Regas = µρ LT ≈ 105 (L and T are the typical length and time scale). Therefore, no (molecular) diffusion is taken into account in the Euler ≈ 1. equation. The Reynolds number relative to the droplets is Re = 2rρl |u−v| µ The already existing mathematical studies on the fluid-kinetic coupling (in the context of sprays) concern models in which the fluid is described by its velocity u (but not its density ρ), and in which some diffusion is present. In [5], Domelevo and Roquejoffre show the existence and uniqueness of global smooth solutions of 1D Burger’s viscous equation when it is coupled with a kinetic equation. For the same system, but in the polydispersed case (that is, different radiuses of droplets are present), Domelevo proves the existence of solutions in [4]. Hamdache [8] shows the global existence and the large time behavior of the solution of a coupled system of Vlasov and Stokes equations (in all dimensions). Finally, in [7], Goudon studies the existence and uniqueness of smooth solutions to the coupling between the viscous Burger’s equation and a kinetic equation. In this work, we combine two ingredients in order to obtain the existence (and uniqueness) of solutions (locally in time) to our system (that is, (1.1)–(1.4) and (1.6)). On one hand, we use the classical theory of local (in time) solutions for symmetrisable hyperbolic systems of conservation laws (cf. [10, 17], for example), and on the other hand, the theory of characteristics for the control of H s norms of f and of its support (like in the works on the Vlasov–Poisson system, such as in [14]). It is easy to verify that our theorem also holds when (1.6) is replaced by (1.5), or when P is not a power function of ρ (but some well-behaved function). We think that the extension to systems in which an energy equation appears should also be not too difficult. Finally, polydispersion (when droplets have different radii) could certainly be taken into account.
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Note however that the well-posedness (even for small times) of systems for sprays (when no diffusion is present) is not obvious: (non diffusive) equations for diphasic flows are known to be linearly ill-posed (non hyperbolic) in certain regimes (cf. [9, 12]), and they have some similarity with equations for sprays. This similarity is most apparent for thick sprays: our method does not work in this case and the corresponding equations might be ill-posed. When some diffusion is present at the level of the fluid equations (that is, Euler is replaced by Navier–Stokes), the conjecture is that the corresponding system for thick sprays is well-posed (diphasic equations of Navier–Stokes type are known to be linearly well-posed (cf. [15])). It looks however quite difficult, even in this case, to prove rigorously the existence of local smooth solutions. Finally, we think that there is no hope to obtain (for general initial data) global smooth (H s ) solutions for the system we consider because the shocks created by the “Euler part” of the system have no reason to be smoothed by the “Vlasov part”, the coupling being made through source terms only. For having an idea on how the smoothness disappears at the level of the Euler equations, we refer, for example, to [10]. In all the sequel, we shall use the following notations (when h ≡ h(t, x), s ∈ N, T > 0, p ∈ [1, +∞] and α is a multi-index): |Dxα (h)|2 (t, x)dx, hs (t) = |α|≤s
RN
hs,T = max hs (t), 0≤t≤T
hLp,T = max h(t, ·)Lp . 0≤t≤T
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Sometimes, the same notations are used for f ≡ f (t, x, v) (with x replaced by (x, v)). In Sec. 2, we present our main result. Then, Sec. 3 is devoted to some (classical) preliminary results for the Euler and Vlasov equations taken separately. The rest of the paper presents the proof of our main theorem. In Sec. 4, we define an approximation scheme and show a priori estimates for its solution. The convergence of this scheme towards our equation is proven in Sec. 5. Finally, a few complementary results are presented in Sec. 6. 2. Main Theorem of Existence and Uniqueness Once all the constants have been eliminated from the equations, we end up with the following system (in dimension N ≥ 1): ∂t ρ + ∇x · (ρu) = 0, ∂t (ρu) + ∇x · (ρu ⊗ u + P (ρ)IdN ) =
RN
∂t f + ∇x · (vf ) + ∇v · (f (ρu − ρv)) = 0, γ
where P (ρ) = ρ , and γ > 1.
(2.1) f (ρv − ρu)dv,
(2.2) (2.3)
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We take as initial data: ∀x ∈ RN , ∀(x, v) ∈ R
N
×R , N
ρ(0, x) = ρ0 (x),
u(0, x) = u0 (x),
f (0, x, v) = f0 (x, v).
(2.4) (2.5)
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Finally, we define G = ]0, +∞[ × RN as the space in which (ρ, ρu) will take its values. We prove the following theorem.
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Theorem 2.1. We consider N ∈ N∗ , G = ]0, +∞[ × RN , s ∈ N such that s > N/2 + 1 and s ≥ N, and G1 , G2 open sets of G such that G1 ⊂ G2 , and such that G1 , G2 are compact. Let (ρ0 , ρ0 u0 ) : RN → G1 be functions satisfying ρ˜0 = ρ0 − 1 ∈ H s (RN ) and u0 ∈ H s (RN ). Let also f0 : RN × RN → R+ be a function of Cc1 (RN × RN ) ∩ H s (RN × RN ). Then, one can find T > 0 such that there exists a solution (ρ, ρu; f ) to system (2.1)–(2.5) belonging to C 1 ([0, T ] × RN , G2 ) × Cc1 ([0, T ] × RN × RN , R+ ). Moreover, ρ˜(= ρ − 1), u ∈ L∞ ([0, T ], H s (RN )) and f ∈ L∞ ([0, T ], H s (RN × RN )). Moreover, if (ρ1 , ρ1 u1 ; f1 ) and (ρ2 , ρ2 u2 ; f2 ) belong to C 1 ([0, T ] × RN , G2 ) × Cc1 ([0, T ] × RN × RN , R+ ), if they satisfy (2.1)–(2.5), and if they are such that ρ˜1 , ρ˜2 , u1 , u2 ∈ L∞ ([0, T ], H s (RN )), f1 , f2 ∈ L∞ ([0, T ], H s (RN × RN )), then ρ1 = ρ2 , u1 = u2 and f1 = f2 .
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Remark 2.2. This theorem shows the existence (and uniqueness) of solutions corresponding to a gas which is at rest at infinity (and of density 1), and which contains particles which are localized in a certain bounded domain. Proof of Theorem 2.1. In a first step, we shall restrict ourselves to initial data such that ρ˜0 and ρ0 u0 lie in Cc∞ (RN ), while f0 belongs to Cc∞ (RN × RN ). In Sec. 6.2, we shall explain how to regularize the initial data in order to obtain the result for all initial data described in Theorem 2.1. Sections 3–6 are devoted to the sequel of the proof of Theorem 2.1.
3. Preliminary Results 3.1. Symmetrisation
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We prove in this section the following proposition, which enables to obtain a symmetrized form for the Euler equation.
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Proposition 3.1. The system (2.1)–(2.2) can be written under the symmetrized form
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S(U )∂t U +
i
(SAi )(U )∂xi U = S(U )b(U, f ),
(3.1)
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where U =
“ ρ ” ρu ,
|ρu|2 (ρ) + P ρ2 S= ρu − ρ 3
(for i = 1, . . . , N ),
0 ρu1 ρui − ρ2 ρu2 ρui − ρ2 .. .
Ai = 2 P (ρ) − (ρui ) 2 ρ .. . ρuN ρui − ρ2
0 ρui ρ
(ρu) − ρ , IdN t
0
0
0
0
0
ρui ρ
0
0
0
..
0
0
0
0
0
0
0
0
.
0
1 ρu1 ρ ρu2 ρ .. . 2ρui ρ .. . ρuN ρ
0
0
0 0 0 0 .. 0
.
0 0 .. . , 0 .. . ρui ρ
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and
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Moreover, the symmetric definite positive matrix S(U ) is a smooth function of U satisfying
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0 f (ρv1 − ρu1 )dv b= f (ρv2 − ρu2 )dv . .. . f (ρvN − ρuN )dv
cIdN ≤ S(U ) ≤ c−1 IdN when U ∈ G1 (or G2 ), for some constant c > 0 (depending on G1 (or G2 )). Finally, all the matrices SAi (U ) are symmetric. Proof. The eigenvalues of S are 1 (ρu)2 +1 + λ1 = P (ρ) + 2 ρ2 1 (ρu)2 + 1 − λ2 = P (ρ) + 2 ρ2 λi = 1 for 3 ≤ i ≤ N + 1.
2 (ρu)2 (ρu)2 − 1 + 4 , P (ρ) + ρ2 ρ2 2 1 (ρu)2 (ρu)2 − 1 +4 2 , P (ρ) + 2 2 ρ ρ 1 2
(3.2)
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A simple computation shows that the matrices SAi are symmetric. It remains to 2 prove that S satisfies (3.2). This is due to the fact that λ1 ≤ P (ρ)+ (ρu) ρ2 +1, so that λ1 ≤ C. Moreover, λ1 λ2 = P (ρ), so λ2 ≥
P (ρ)
2
P (ρ)+ (ρu) +1 ρ2
and therefore λ2 ≥ C > 0.
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3.2. The transport equation
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Let ρ, u be any smooth functions. Then, the transport equations (2.3) and (2.5), which can be rewritten as ∂t f + v · ∇x f + (ρu − ρv) · ∇v f = N ρf,
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has a unique solution f (t, x, v) = f0 (X(0; x, v, t), V (0; x, v, t))e
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N ρ(X(τ ;x,v,t),τ )dτ
,
(3.3)
(3.4)
(3.5)
It is clear thanks to (3.3) that if f0 has a compact support, then f (t, ·, ·) will also have a compact support for all t. We denote sup (x,v)∈RN ×RN f (t,x,v)>0
|x|,
(3.6)
|v|.
(3.7)
and VM (t) =
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0
dX (t; x, v, s) = V (t; x, v, s), dt X(s; x, v, s) = x, dV (t; x, v, s) = (ρu)(t, X(t; x, v, s)) − ρ(t, X(t; x, v, s))V (t; x, v, s), dt V (s; x, v, s) = v.
XM (t) = 17
Rt
where the characteristic curves X(t; x, v, s), V (t; x, v, s) are defined by
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f (0, x, v) = f0 (x, v),
sup (x,v)∈RN ×RN f (t,x,v)>0
In other words, Supp f (t, ·, ·) ⊂ B(0, XM (t)) × B(0, VM (t)). 4. The Construction Scheme We recall that we denote U = t (ρ, ρu). According to Sec. 3.1, the system (2.1)–(2.3) can be written as S(U )∂t U +
N
(SAi )(U )∂xi U = S(U )b(U, f ),
(4.1)
i=1
∂t f + ∇x · (vf ) + ∇v · (f (ρu − ρv)) = 0, 21
where S satisfies (3.2).
(4.2)
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We also recall that up to Sec. 6.2, we suppose that the initial data are such that
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U0 − U0 ∈ Cc∞ (RN ),
f0 ∈ Cc∞ (RN × RN ),
(4.3)
where U0 = 10 . In this section, we write an inductive approximation of the system. Then, we show a priori estimates on a time interval [0, T∗ ], where T∗ > 0 is the same for all the steps of the approximation. We note that since s > N/2 + 1 > N/2, H s (RN ) is embedded in L∞ (RN ) (the inclusion being continuous). Using the fact that U0 takes its values in G1 (such that G1 ⊂ G2 ), we can find R ≡ R(G1 , G2 , s, U0 ) > 0 (defined once and for all) such that for any function U , if U − U0 s ≤ R, then U takes its values in G2 . Finally, Supp f0 ⊂ B(0, XM (0)) × B(0, VM (0)), with the notations (3.6) and (3.7). We define by induction the quantities θk > 0 and (U k , f k ) in this way: • θ0 = +∞, and for t ∈ [0, θ0 [, (U 0 (t), f 0 (t)) = (U0 , f0 ). • We now suppose that θk > 0 is defined, together with the smooth functions U k , f k on the time interval [0, θk [. We suppose moreover that ∀t ∈ [0, θk [, x ∈ RN , one has U k (t, x) ∈ G2 . Then, we define (U k+1 , f k+1 ) on [0, θk [ as the unique smooth solution of the linear system S(U k )∂t U k+1 +
N
(SAi )(U k )∂xi U k+1 = S(U k )b(U k , f k ),
(4.4)
i=1
U k+1 (x, 0) = U0 (x), ∂t f
k+1
+ ∇x · (vf
k+1
) + ∇v · (f
k+1
(ρ u − ρ v)) = 0, k k
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k+1
k
(0, x, v) = f0 (x, v).
(4.5) (4.6) (4.7)
Finally, we introduce θk+1 > 0 as the supremum of the times θ < θk satisfying U k+1 (t, x) ∈ G2 for all t ∈ [0, θ[, x ∈ RN . Note that since the system (4.4) is linear, symmetric and has smooth coefficients (on [0, θk [), it admits a unique smooth solution U k+1 (on [0, θk [). Moreover f k+1 is the unique smooth solution of a linear Vlasov equation with smooth coefficients (it can be explicitly computed by the method of characteristics as in Sec. 3.2). Finally, θk+1 > 0 since U k+1 is smooth and U0 ∈ G1 . All of this ensures that the induction is well-defined. Then, one can observe that the support (in the x and v variables) of U k − U0 and f k are compact (and depend on k and t ∈ [0, θk [). This property is true for k k = 0 (thanks to (4.3)) and it can then be proven by induction. We denote by XM k k and VM the quantities XM and VM related to f (defined by (3.6) and (3.7)). We finally define Tk as the supremum of the times T ∈ [0, θk [ such that U k − U0 s,T ≤ R,
(4.8)
f s,T ≤ 2f0 s , k
f k
L∞ ,T
≤ 2f0
L∞
(4.9) ,
(4.10)
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k XM (t) ≤ 2XM (0), k (t) VM
(4.11)
≤ 2VM (0),
(4.12)
and such that Tk+1 ≤ Tk . It is clear (by induction) that for all k ∈ N, Tk > 0. We now prove the decisive a priori estimate, namely the existence of T∗ > 0 such that ∀k ∈ N, Tk ≥ T∗ . Proposition 4.1. We consider initial data such that (4.3) holds and define the sequences θk , U k , f k by (4.4)–(4.7), and Tk by (4.8)–(4.12). Then one can find T∗ > 0 which depend only upon G1 , G2 , s, U0 and f0 such that ∀k ∈ N, Tk ≥ T∗ . Proof. We recall that (on the time interval [0, Tk [) f k and U k − U0 are smooth and have a compact support, so that we can manipulate them (and in particular perform integrations by parts) without taking care of their behaviour at infinity. In the sequel, we use C for any constant (C(f0 ) for any constant depending only on f0 , etc.). Though R depends only on G1 , G2 , s and U0 , we keep its dependence in the various constants, for the sake of readability of the proof. The proof of Proposition 4.1 is divided in three steps. Step 1. For all k ≥ 0, ∂t U k+1 s−1,Tk+1 ≤ C(s, G2 , R, U0 , f0 ).
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(4.13)
Proof of Step 1.1. Using Eq. (4.4), we get for t ∈ [0, Tk+1 [, ∂t U k+1 s−1 (t) ≤ Ai (U k )∂xi U k+1 s−1,Tk+1 + b(U k , f k )s−1,Tk+1 ,
(4.14)
i
19
where b(U k , f k )s−1,Tk+1
0 = Dα (ρk f k vdv) − Dα (ρk uk f dv) |α|≤s−1 0,T
.
k+1
21
We use the following (classical) result (cf. [10]): If h, g ∈ H s (RN ) ∩ L∞ (RN ) and |α| ≤ s, then Dα (hg)L2 ≤ C(s)(hL∞ gs + gL∞ hs ).
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We get (for |α| ≤ s)
(4.15)
k + f k vdv D (b(U , f ))0 ≤ C(s) ρk L∞ vdv f α
k
k
s
k k k k + ρ u L∞ f dv + f dv s
L∞
ρk s
L∞
ρ u s . k k
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For t ∈ [0, Tk [, we can bound by R the H s norm of U k − U0 , so k + f k vdv f Dα (b(U k , f k ))0 ≤ C(s, R) vdv ∞ L s k k + . f dv + f dv L∞
s
For t ∈ [0, Tk [, we can also bound the support of f , and its H s and L∞ norms: k f k dv ∞ ∞ ≤ f L 1Supp f k dv ∞ k
L
f k vdv
L
k N ≤ 2N f k L∞ (VM ) ≤ C(f0 ),
L∞
f k vdv
s
k N +1 ≤ 2N f k L∞ (VM ) ≤ C(f0 ), 2 α k = vDx (f )dv dx
(4.16) (4.17)
|α|≤s
k N/2+1 ) f k s ≤ C(f0 ), ≤ C(VM f k dv ≤ (V k )N/2 f k s ≤ C(f0 ). M s
1
3
5
Finally, we obtain b(U k , f k )s,Tk ≤ C(s, R, f0 ),
(4.18)
b(U k , f k )s−1,Tk+1 ≤ C(s, R, f0 ).
(4.19)
and (in particular)
Then,
Ai (U k )∂xi U k+1 s−1,Tk+1 =
Dα (Ai (U k )∂xi U k+1 )0,Tk+1
|α|≤s−1
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depends only on derivatives of U of order ≤ s. We recall that U k+1 − U0 s,Tk+1 ≤ R, and we use the following (classical) result (cf. [10]): if u → g(u) is a smooth function on G2 and if x → u(x) is a continuous function with values in G2 such that u ∈ L∞ (RN ) ∩ H s (RN ), then for |α| ≤ s (and s ≥ 1), k+1
Dα g(u)L2 ≤ C(s) sup
sup |Dβ g(u)| us−1 L∞ us .
(4.20)
u∈G2 |β|≤s−1
We obtain (for |α| ≤ s−1) thanks to the Sobolev embedding H s−1 (RN ) ⊂ L∞ (RN ), Dα (Ai (U k )∂xi U k+1 )0,Tk+1 ≤ Dα ((Ai (U k ) − Ai (U0 ))∂xi U k+1 )0,Tk+1 + Ai (U0 )Dα (∂xi U k+1 )0,Tk+1 ≤ C(s)(Ai (U k ) − Ai (U0 )s−1,Tk+1 ∂xi U k+1 L∞ ,Tk+1 + Ai (U0 )L∞ ,Tk+1 ∂xi U k+1 s−1,Tk+1 ) ≤ C(s, G2 , R, U0 ). Putting together (4.14), (4.19) and (4.21), we get (4.13).
(4.21)
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Then, we turn to the Step 2. For all k ≥ 0, T ∈ [0, inf(1, Tk+1 )[, sup U k+1 − U0 s (t) ≤ T c(G2 )−1 C(s, R, G2 , U0 , f0 ).
3
5
(4.22)
t∈[0,T ]
Proof of Step 2. We begin with an abstract (classical) lemma which enables to obtain energy estimates for solutions of symmetrized hyperbolic systems. Lemma 4.2. Let S ≡ S(t, x), Ai ≡ Ai (t, x) be smooth matrices (on [0, T ]) such that S and SAi are symmetric. We suppose moreover that c IdN ≤ S(t, x) ≤ c−1 IdN for some c > 0. Then all (smooth and compactly supported) vectors W ≡ W (t, x) and F ≡ F (t, x) satisfying the system S∂t W + SAi ∂xi W = F, (4.23) i
W (x, 0) = W0 (x),
(4.24)
can be estimated in the following way: for all t ∈ [0, T ], t 1 −1 W 0 (t) ≤ c ∂xi (SAi ) W 0 (τ )dτ W0 0 + ∂t S + ∞ 2 0 i L ,T t
+ 0
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9
F 0 (τ )dτ .
(4.25)
Proof of Lemma 4.2. By multiplying (4.23) by t W and by integrating over x ∈ RN , we obtain 1 1 t t ∂t W SW dx = W ∂t S + ∂xi (SAi ) W dx + t W F dx. (4.26) 2 2 i Then, (4.25) is obtained by differentiating t W SW dx and by using the estimate t W SW ≥ ct W W . We now turn back to the proof of Step 2. We study W k+1 = U k+1 − U0 . Then, is solution of W S(U k )∂t W k+1 + (SAi )(U k )∂xi W k+1 = S(U k )b(U k , f k ) + H k , k+1
i
W k+1 (x, 0) = 0,
11 13
15
with H k = − i (SAi )(U k )∂xi U0 . We recall that W k+1 is smooth (C ∞ ) and has a compact support in [0, Tk+1 ] × RN . We look for an estimate on the norm H s of W k+1 . We denote Wα = Dα W k+1 (for |α| ≤ s). The function Wα satisfies S(U k )∂t Wα + (SAi )(U k )∂xi Wα = S(U k )Dα (S −1 (U k )H k + b(U k , f k )) + Fα , i k
with Fα = S(U )
i (Ai (U
k
)∂xi Wα − Dα (Ai (U k )∂xi W k+1 )).
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We use formula (4.26) in order to obtain 1 1 t t k k k ∂t Wα S(U )Wα dx = Wα ∂t S(U ) + ∂xi (SAi )(U ) Wα dx 2 2 i t Wα S(U k )Dα (S −1 (U k )H k + b(U k , f k ))dx + t + Wα Fα dx. Up to time Tk , U k takes its values in G2 , on which S and SAi are smooth (more precisely, one can bound the derivatives (of any order) of those matrices by a constant depending on G2 only). We also recall that thanks to the Sobolev inequalities, H s−1 (RN ) ⊂ L∞ (RN ). We estimate ∂t U k L∞ ,Tk ≤ C(s)∂t U k s−1,Tk ≤ C(s, G2 , R, U0 , f0 ) from (4.13), ∂xi U k L∞ ,Tk ≤ C(s)∂xi U k s−1,Tk ≤ C(s)(U k − U0 s,Tk + U0 − U0 s,Tk ) ≤ C(s, R, U0 ), 1
so that (since Tk+1 ≤ Tk ) ∂t S(U k ) + ∂xi (SAi )(U k )L∞ ,Tk+1 ≤ C(s, G2 , R, U0 , f0 ).
(4.27)
i
3 5
We now use the following (classical) inequality (cf. [10]): if h ∈ H s (RN ), ∇h ∈ L∞ (RN ), g ∈ H s−1 (RN ) ∩ L∞ (RN ) and |α| ≤ s, Dα (hg) − hDα (g)0 ≤ C(s)(∇hL∞ gs−1 + gL∞ hs ).
(4.28)
Then, for |α| ≤ s, Ai (U k )∂xi Wα − Dα (Ai (U k )∂xi W k+1 ) = (Ai (U k ) − Ai (U0 ))Dα (∂xi W k+1 ) − Dα ((Ai (U k ) − Ai (U0 ))∂xi W k+1 ), and (according to (4.28)) Fα 0,Tk+1 ≤ S(U k )L∞ ,Tk+1 C(s)
(D(Ai (U k ) − Ai (U0 ))L∞ ,Tk+1
i
× ∂xi W k+1 s−1,Tk+1 + ∂xi W k+1 L∞ ,Tk+1 Ai (U k ) − Ai (U0 )s,Tk+1 ) ≤ C(s, G2 , R, U0 ).
(4.29)
According to (4.15), we get (for |α| ≤ s) S(U k)Dα (S −1 (U k)H k)0,Tk+1 ≤ S(U k)L∞ ,Tk ≤ C(s, G2 )
Dα (Ai (U k)∂xi U0 )0,Tk
i
(Ai (U k) − Ai (U0 )L∞ ,Tk ∂xi U0 s,Tk
i
+ ∂xi U0 L∞ ,Tk Ai (U k ) − Ai (U0 )s,Tk + C∂xi U0 s,Tk ) ≤ C(s, G2 , R, U0 ).
(4.30)
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13
Finally, thanks to (4.18), (and for |α| ≤ s) S(U k )Dα (b(U k , f k ))0,Tk+1 ≤ C(G2 )C(s, R, f0 ).
(4.31)
We now use estimate (4.21) for t ∈ [0, Tk+1 [. We get t 1 −1 Wα 0 (0) + ∂t S + ∂xi (SAi ) Wα 0 (τ )dτ Wα 0 (t) ≤ c(G2 ) ∞ 2 0 i L ,Tk+1 t (Fα 0 + S(U k )Dα (S −1 (U k )H k )0 + 0 + S(U k )Dα (b(U k , f k ))0 )dτ . By using (4.27)–(4.31), we obtain t −1 Wα 0 (0) + C(s, G2 , R, U0 , f0 ) Wα 0 (τ )dτ Wα 0 (t) ≤ c(G2 ) + 0
0
t
(C(s, G2 , R, U0 ) + C(s, G2 , R, U0 ) + C(G2 )C(s, R, f0 ))dτ .
Summing for all |α| ≤ s these estimates, we end up (for t ∈ [0, Tk+1 [) with k+1 −1 W s (t) ≤ c(G2 ) W k+1 s (0) + C(s, G2 , R, U0 , f0 ) t × W k+1 s (τ )dτ + c(G2 )−1 tC(s, R, G2 , U0 , f0 ). 0
Thanks to Gronwall’s lemma, for all t ∈ [0, T ] with T ≤ Tk+1 , W k+1 s (t) ≤ c(G2 )−1 (W k+1 s (0) + T C(s, R, G2 , U0 , f0 )) × ec(G2 ) 3
−1
C(s,R,f0 ,U0 ,G2 )T
.
When T ≤ 1 and since W k+1 (0) = 0, we end up with (4.22). We now turn to the Step 3. The following inequalities hold for all T ∈ [0, inf(1, Tk+1 )[, f k+1 L∞ ,T ≤ f0 L∞ eC(G2 )T , √ f k+1 s,T ≤ 2eC(G2 )T f0 s + T C(s, R, f0 , G2 , U0 ), sup t∈[0,T ]
k+1 VM (t)
≤ VM (0)e
C(G2 )T
+ C(G2 )T,
k+1 (t) ≤ XM (0) + C(G2 , f0 )T. sup XM
t∈[0,T ]
(4.32) (4.33) (4.34) (4.35)
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Proof Step 3. Using notations similar to those of Sec. 3.2, we define the characteristics dX k+1 (t; x, v, s) = V k+1 (t; x, v, s), dt X k+1 (s; x, v, s) = x, dV k+1 (t; x, v, s) = (ρk uk )(X k+1 (t; x, v, s), t) dt − ρk (X k+1 (t; x, v, s), t)V k+1 (t; x, v, s),
(4.36)
V k+1 (s; x, v, s) = v. 1
(4.37)
Estimate (4.32) is a direct consequence of the formula: f k+1 (t, x, v) = f0 (X k+1 (0; x, v, t), V k+1 (0; x, v, t))e
Rt 0
N ρk (X k+1 (τ ;x,v,t),τ )dτ
.
Then, writing in an implicit way (4.36)–(4.37), we obtain t k+1 (t; x, v, s) = x + V k+1 (τ ; x, v, s)dτ, X V
k+1
(t; x, v, s) = e
s R − st ρk (X k+1 (τ ;x,v,s),τ )dτ
t
+
e−
Rt τ
ρ(X
k+1
v
(˜ τ ;x,v,s),˜ τ )d˜ τ
(ρk uk )(X k+1 (τ ; x, v, s), τ )dτ
s
so that V k+1 (0; x, v, t) = ve− +
R0 t
0
ρk (X k+1 (τ ;x,v,t),τ )dτ
e−
R0 τ
ρk (X k+1 (˜ τ ;x,v,t),˜ τ )d˜ τ
(ρk uk )(X k+1 (τ ; x, v, t), τ )dτ.
t
3
Then, |v| ≤ |V k+1 (0; x, v, t)|e
Rt 0
ρk L∞ (τ )dτ
t
+ 0
ρk uk L∞ (τ )e
Rt τ
ρk L∞ (˜ τ )d˜ τ
dτ.
(4.38)
Since k+1 VM (t) =
sup f k+1 (t,x,v)>0
=
|v| sup
f0k+1 (X k+1 (0;x,v,t),V k (0;x,v,t))>0
≤
5
we get (for t ∈ [0, T [, T ∈ [0, Tk+1 [) k+1 VM (t) ≤ VM (0)e
7
sup |X k+1 (0;x,v,t)|≤XM (0) |V k+1 (0;x,v,t)|≤VM (0)
Rt 0
ρk L∞ (τ )dτ
+ 0
t
|v|,
ρk uk L∞ (τ )e
Then, we obtain (4.34) by noticing that T ≤ 1.
|v|
Rt τ
ρk L∞ (˜ τ )d˜ τ
dτ.
(4.39)
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k+1 k We proceed similarly for XM . Using the formula for VM , we get
Rτ
t
k
k+1
(0; x, v, t) + ve− t ρ (X (s;x,v,t),s)ds dτ 0 t τ Rτ k+1 e− τ˜ ρ(X (s;x,v,t),s)ds (ρk uk )(X k+1 (˜ τ ; x, v, t), τ˜)d˜ τ dτ, +
x=X
k+1
0
t
so that |x| ≤ |X
k+1
(0; x, v, t)| + |V
k+1
(0; x, v, t)|
t R τ
e
0
ρk L∞ (s)ds
dτ
0
t R Rt k t k ρk uk L∞ (τ )e τ ρ L∞ (s)ds dτ e τ ρ L∞ (s)ds dτ 0 0 τ R t τ k ρk uk L∞ (τ ) e τ˜ ρ L∞ (s)ds d˜ τ dτ, +
t
+
0
0
and (for t ∈ [0, T [, T ∈ [0, Tk+1 [)
k+1 (t) XM
0
1 3 5 7
t R τ
k
≤ XM (0) + VM (0) e 0 ρ L∞ (s)ds dτ 0 t R t Rt k t k k k ρ L∞ (s)ds τ ρ u L∞ (τ )e dτ e τ ρ L∞ (s)ds dτ + 0 0 τ R t τ k ρk uk L∞ (τ ) e τ˜ ρ L∞ (s)ds d˜ τ dτ. (4.40) + 0
Using the fact that T ≤ 1, we get (4.35). It remains to prove estimate (4.33) on the H s norm of f k+1 . For this, we take α derivatives with respect to x and β derivatives with respect to v (with |α|+|β| ≤ s) of f k+1 . As a result, we get a coupled system of Vlasov equations whose characteristic fields are the same as those of the equation satisfied by f k+1 , and whose righthand side contains derivatives of order ≤ |α| + |β| of f k+1 , and derivatives of order ≤ |α| of U k . The equation for a derivative of arbitrary order of f k+1 writes: Dxα Dvβ f k+1 (t, x, v) = Dxα Dvβ f0 (X k (0; x, v, t), V k (0; x, v, t)) t R Rt t k k e τ Cα,β ρ (X(˜τ ;x,v,t),˜τ )d˜τ × e 0 Cα,β ρ (X(τ ;x,v,t),τ )dτ + 0
× Bα,β (X (τ ; x, v, t), V (τ ; x, v, t))dτ, k
9
k
where Cα,β ∈ R (in fact it is always some positive multiple of N ) and where Bα,β is a linear combination of the Dxα Dvβ f k+1 , with |α |+ |β | ≤ |α|+ |β|, whose coefficients are themselves linear combinations of v and Dxγ U k , with |γ| ≤ |α|.
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We write down explicitly the term B3,0 , which contains the terms of highest order (of derivatives) of U k (among those for which |α| + |β| ≤ 3): B3,0 = 3∇x ρk ∇xx f k+1 − 3(∇x (ρk uk ) − v∇x ρk )∇xxv f k+1 + 3∇xxρk ∇x f k+1 − 3(∇xx (ρk uk ) − v∇xx ρk )∇xv f k+1 + ∇xxx (ρk )f k+1 − (∇xxx (ρk uk ) − v∇xxx ρk )∇v f k+1 .
(4.41)
We look for the L2 norm of Dxα Dvβ f k+1 . We have (for T ∈ ]0, Tk+1 [ and t ∈ [0, T [)
(Dxα Dvβ f k+1 (t, x, v))2 dxdv ≤ 2 (Dxα Dvβ f0 (X k (0; x, v, t), V k (0; x, v, t)))2 Rt
k
k
× e2 0 Cα,β ρ (X (τ ;x,v,t),τ )dτ dxdv t R t k k e τ Cα,β ρ (X (˜τ ;x,v,t),˜τ )d˜τ +2 0
2 × Bα,β (X (τ ; x, v, t), V (τ ; x, v, t))dτ dxdv. k
1
k
(4.42)
In the first integral of (4.42), we use theR change of variables (x, v) → t k k (X k (0; x, v, t), V k (0; x, v, t)), whose Jacobian is e 0 N ρ (X (τ ;x,v,t),τ )dτ . We obtain Rt k 2(Dxα Dvβ f0 (X k (0; x, v, t), V k (0; x, v, t)))2 e2 0 Cα,β ρ (X(τ ;x,v,t),τ )dτ dxdv Rt k ≤ 2(Dxα Dvβ f0 (X, V ))2 e 0 Cα,β ρ (X(τ ;x,v,t),τ )dτ dXdV ≤ 2etC(G2 ) Dxα Dvβ f0 20 .
(4.43)
We now estimate the second integral. Using Cauchy–Schwarz’ inequality on the square of the time integral:
t R t
eτ
2
Cα,β ρk (X(˜ τ ;x,v,t),˜ τ )d˜ τ
0
≤ 2 × ≤ 2te
0 tC(G2 )
k
Bα,β (X (τ ; x, v, t), V (τ ; x, v, t))dτ t R t k e τ Cα,β ρ (X(˜τ ;x,v,t),˜τ )d˜τ dτ
0 t R t
e
2 k
τ
Cα,β ρk (X(˜ τ ;x,v,t),˜ τ )d˜ τ
t e
Rt τ
2 Bα,β (X k (τ ; x, v, t), V k (τ ; x, v, t))dτ
Cα,β ρk (X(˜ τ ;x,v,t),˜ τ )d˜ τ
0
2 × Bα,β (X k (τ ; x, v, t), V k (τ ; x, v, t))dxdvdτ.
dxdv
dxdv
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Then we use the change of variables (x, v) → (X k (τ ; x, v, t), V k (τ ; x, v, t)), whose Rt k N Jacobian is e τ ρ (X(˜τ ;x,v,t),˜τ )dτ . We obtain in this way 2 t R t k C ρ (X(˜ τ ;x,v,t),˜ τ )d˜ τ k k α,β eτ Bα,β (X (τ ; x, v, t), V (τ ; x, v, t))dτ dxdv 2 0
≤ 2tetC(G2 )
t
2 (X, V )dXdV dτ. Bα,β
0
(4.44)
We now show that the L∞ norms of ∇x f k+1 and ∇v f k+1 are bounded. We notice that ∂t (∇v f k+1 ) + v · ∇x (∇v f k+1 ) + (ρk uk − ρk v) · ∇v (∇v f k+1 ) = 2N ρk (∇v f k+1 ) − ∇x f k+1 , so that ∇v f k+1 (t, x, v) = e
Rt
−
0
2N ρk (X k (τ ;x,v,t),τ )dτ
t R t
e
τ
∇v f0 (X k (0; x, v, t), V k (0; x, v, t))
2N ρk (X k (s;x,v,t),s)ds
0
× ∇x f k+1 (X k (τ ; x, v, t), V k (τ ; x, v, t), τ )dτ 1
and therefore (for t ∈ [0, T [) ∇v f k+1 L∞ (t) ≤ ∇v f0 L∞ eC(G2 )t + eC(G2 )t
0
t
∇x f k+1 L∞ (τ )dτ.
With the same kind of arguments (and noticing that U k has its derivatives in x of first order in L∞ since it is bounded in H s ), we get (for t ∈ [0, T [) ∇x f k+1 ∞ (t) ≤ ∇x f0 ∞ eC(G2 )t + C(s, R)f0 ∞ teC(G2 )t t + C(f0 , s, R)eC(G2 )t ∇v f k+1 ∞ (τ )dτ. 0
3
Thanks to Gronwall’s lemma, we obtain (for t ∈ [0, T [, T ≤ 1): ∇x f k+1 L∞ (t) + ∇v f k+1 L∞ (t) ≤ C(s, R, f0 , G2 ).
5
7 9
We now prove that for τ ∈ [0, T [: 2 (X, V )dXdV ≤ C(s, R, f0 , G2 , U0 ). Bα,β
(4.45)
(4.46)
We write down in a detailed way the proof in the case N = 3, s = 3 (the most important for applications). A summary of the proof in the general case is presented in Sec. 6.3. In this case (N = 3, s = 3), the most representative and complex term (for |α| + |β| ≤ 3) is B3,0 . The other terms Bα,β can be treated analogously. Since 3 > 32 + 1, H 3 (R3 ) and H 2 (R3 ) are embedded in L∞ (R3 ). Moreover, H 1 (R3 ) is
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embedded in L6 (R3 ) (the inclusions being continuous). We recall that on [0, Tk+1 [, U k − U0 3 ≤ R. Therefore, U k L∞ ,Tk+1 ≤ C(R, U0 ), ∇x U k L∞ ,Tk+1 ≤ C(R, U0 ), ∇xx U k L6 ,Tk+1 ≤ C(R, U0 ). 1
In the same way, since H 1 (R6 ) is embedded in L3 (R6 ), and f k+1 satisfies (4.9) and (4.10), we have:
3
∇xv f k+1 L3 ,Tk+1 ≤ C(f0 ).
5
k+1 Moreover, thanks to (4.12), f k+1 has a compact support in v, given by VM . We now examine each term appearing in (4.41), for some t ∈ [0, T [ (t is not explicitly written down in the sequel). The terms containing derivatives of first order of U k are the simplest:
3∇x ρk ∇xx f k+1 − 3(∇x (ρk uk ) − v∇x ρk )∇xxv f k+1 0 k+1 ≤ ∇x U k L∞ (∇xx f k+1 0 + (1 + VM )∇xxv f k+1 0 )
≤ C(R, U0 , f0 ). Then, we treat the terms containing derivatives of second order of U k : 1/2 ∇xx U k ∇xv f k+1 0 = (∇xx U k )2 (∇xv f k+1 )2 dv dx ≤
1/6 3/2 1/3 k+1 2 (∇xx U ) dx ) dv dx (∇xv f k 6
≤ ∇xx U L6 k
(∇xv f
×
1/2 1Suppv f k+1 dv
k+1 3
) dv
1/3 dx
k+1 1/6 ≤ C(R, U0 )C(f0 )(VM )
≤ C(R, U0 , f0 ), 7
(4.47)
and similarly v∇xx U k ∇xv f k+1 0 ,
∇xx U k ∇x f k+1 0 ≤ C(R, U0 , f0 ).
(4.48)
The last term contains derivatives of third order of U k . We use here the L∞ norm of f k+1 or ∇v f k+1 as obtained in (4.45), for example: ∇xxx U k ∇v f k+1 0 ≤ ∇v f k+1 L∞ ∇xxx U k 0
(4.49)
≤ C(s, R, f0 , G2 )C(R, U0 ).
(4.50)
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Finally, we use estimates (4.47)–(4.49) in order to obtain B3,0 0,T ≤ C(s, R, f0 , G2 , U0 ).
3
(4.51)
Using (4.42)–(4.44) and (4.51), we end up with the estimate (for t ∈ [0, T [): f k+1 2s,t ≤ 2eC(G2)t f0 2s + 2t2 eC(G2 )t C(s, R, f0 , G2 , U0 ).
5 7 9
19
(4.52)
Remembering that T ≤ 1, we get (4.33). We now conclude the proof of Proposition 4.1. We see that if T∗ ∈ ]0, 1] satisfies T∗ c(G2 )−1 C(s, R, G2 , U0 , f0 ) ≤ R (in (4.22)), eC(G2 )T∗ ≤ 2 (in (4.32)), √ C(G )T 2 ∗ 2e ≤ 32 (in (4.33)), T∗ C(s, R, f0 , G2 , U0 ) ≤ 12 f0 s (in (4.33)), eC(G2 )T∗ ≤ 32 (in (4.34)), C(G2 )T∗ ≤ 12 VM (0) (in (4.34)), C(G2 , f0 )T∗ ≤ XM (0) (in (4.35)), then Tk+1 ≥ T∗ .
11
5. Passing to the Limit
13
We now pass to the limit when k → ∞ in (4.4) and (4.6). As suggested in [10], we study U k+1 − U k 0,T∗∗ for some T∗∗ ∈ ]0, T∗ [. We show the
15
Proposition 5.1. We consider initial data such that (4.3) holds and define the sequence θk , U k , f k by (4.4)–(4.7), and T∗ thanks to Proposition 4.1. Then one can find T∗∗ ∈ ]0, T∗ [, such that (for k ≥ 2) 1 k 1 U − U k−1 0,T∗∗ + U k−1 − U k−2 0,T∗∗ , 4 4 ≤ C(G2 , s, R, f0 )U k−1 − U k−2 0,T ∗∗ .
U k+1 − U k 0,T∗∗ ≤ f k − f k−1 0,T∗∗
(5.1) (5.2)
Proof. Note first that (for k ≥ 2), the function U k+1 −U k is solution of the system: S(U k )∂t (U k+1 − U k ) + (SAi )(U k )∂xi (U k+1 − U k ) i
= b(U k , f k ) − b(U k−1 , f k−1 ) + Fk , where 17
Fk = (S(U k−1 ) − S(U k ))∂t U k +
((SAi )(U k−1 ) − (SAi )(U k ))∂xi U k .
i
Moreover, U (0, x) − U (0, x) = 0. Thanks to Lemma 4.2 (formula (4.25)), we can write (when t ∈ [0, T∗ [) k+1
U k+1 − U k 0 (t) ≤ c(G2 )−1 + 0
t
k
1 ∂t S(U k ) + ∂xi (SAi )(U k )L∞ ,T∗ 2 i
(Fk 0 (τ ) + b(U , f ) − b(U k
k
k−1
,f
k−1
0
t
U k+1 − U k 0 (τ )dτ
)0 (τ ))dτ .
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We use then Gronwall’s lemma. For all t ∈ [0, T∗ [, according to (4.17), c(G2 )−1
k
P
k
U k+1 − U k 0 (t) ≤ c(G2 )−1 e 2 ∂t S(U )+ i ∂xi (SAi )(U )L∞ ,T∗ T∗ t × (Fk 0 (τ ) + b(U k , f k ) − b(U k−1 , f k−1 )0 (τ ))dτ 0
≤ c(G2 )−1 ec(G2 )
−1
C(s,G2 ,R,U0 ,f0 )T∗
× (Fk 0,T∗ + b(U , f ) − b(U k
1
k
T∗ k−1
, f k−1 )0,T∗ ).
(5.3)
Using estimates (4.8), (4.13) and the fact that S and SAi are smooth on G2 (more precisely, their derivatives are bounded by a constant C(G2 )), we get Fk 0,T∗ ≤ C(s, G2 , R, U0 )U k − U k−1 0,T∗ .
3
(5.4)
Moreover, for t ∈ [0, T∗ [ (and without writing t explicitly) b(U k , f k ) − b(U k−1 , f k−1 )0 k k k−1 k−1 k k k−1 k ≤ (ρ u − ρ u ) f dv − (ρ − ρ ) f vdv 0 k−1 k−1 k k−1 k−1 k k−1 + (ρ u ) (f − f )dv − ρ )vdv (f − f 0 k k k k−1 k−1 k k k−1 ≤ u 0 + 0 f dv ∞ ρ u − ρ f vdv ∞ ρ − ρ L L k k−1 + ρk−1 L∞ (f k − f k−1 )vdv . (f + ρk−1 uk−1 L∞ − f )dv 0
Then,
0
(f k − f k−1 )dv ≤ (4VM (0))N/2 f k − f k−1 0 0
≤ C(f0 )f k − f k−1 0 , (f k − f k−1 )vdv ≤ C(f0 )f k − f k−1 0 . 0
Finally, b(U k , f k ) − b(U k−1 , f k−1 )0 ≤ C(f0 )U k − U k−1 0 + C(G2 , f0 )f k − f k−1 0 .
(5.5)
Then, we note that ∂t (f k − f k−1 ) + v · ∇x (f k − f k−1 ) + (ρk−1 uk−1 − ρk−1 v) · ∇v (f k − f k−1 ) = N ρk−1 (f k − f k−1 ) + ∇v · (f k−1 ((ρk−2 uk−2 − ρk−2 v) − (ρk−1 uk−1 − ρk−1 v))). Moreover, at t = 0, f k (x, v, 0) = f k−1 (x, v, 0) = f0 (x, v). So t R t k−1 k−1 k k−1 )(t, x, v) = e τ N ρ (X (s;x,v,t),s)ds (f − f 0
× B(X k−1 (τ ; x, v, t), V k−1 (τ ; x, v, t), τ )dτ,
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with B = ∇v · (f k−1 ((ρk−2 uk−2 − ρk−2 v) − (ρk−1 uk−1 − ρk−1 v))) = (ρk−2 uk−2 − ρk−1 uk−1 ) · ∇v f k−1 − (ρk−2 − ρk−1 )(N f k−1 + v · ∇v f k−1 ). →
Using Cauchy–Schwarz’ inequality and the change of variables (x, v) (X k−1 , V k−1 ), we get for all t ∈ [0, T∗ [, t R t k−1 k−1 e τ N ρ (X (s;x,v,t),s)ds ((f k − f k−1 )(t, x, v))2 dxdv = 0
× B(X ≤ ×
2
k−1
(τ ; x, v, t), τ )dτ t R t k−1 k−1 e τ N ρ (X (s;x,v,t),s)ds dτ
0 t R t
e
τ
(τ ; x, v, t), V
k−1
dxdv
N ρk−1 (X k−1 (s;x,v,t),s)ds
0
× B 2 (X k−1 (τ ; x, v, t), V k−1 (τ ; x, v, t), τ )dτ dxdv t ≤ teC(G2 )t B 2 (x, v, τ )dxdvdτ. 0
1
2
∞
In order to bound B in L , we use the L bound on f k−1 and its derivative with respect to v obtained in (4.45). Then, k−1 (t))N (∇v f k−1 L∞ B 2 (x, v, τ )dxdv ≤ (2VM k−1 + f k−1 L∞ + VM (τ )∇v f k−1 L∞ )2 U k−1 − U k−2 20 (τ )
≤ C(s, R, f0 , G2 )U k−1 − U k−2 20 (τ ).
Finally, for t ∈ [0, T∗ [ (and remembering that T∗ ≤ 1), t k k−1 2 C(G2 )t f − f 0 (t) ≤ te C(s, R, f0 , G2 ) U k−1 − U k−2 20 (τ )dτ 0
≤ t2 C(s, R, f0 , G2 )U k−1 − U k−2 20,t .
(5.6)
Then, thanks to (5.5) and (5.6), b(U k , f k ) − b(U k−1 , f k−1 )0,T∗ ≤ C(s, R, f0 , G2 )(U k − U k−1 0,T∗ + U k−1 − U k−2 0,T∗ ). Using now (5.3), (5.4) and (5.7), we end up (for t ∈ [0, T∗ ], and T∗ ≤ 1) with U k+1 − U k 0 (t) ≤ C(s, R, f0 , U0 , G2 )T∗ × (U k − U k−1 0,T∗ + U k−1 − U k−2 0,T∗ ).
(5.7)
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1
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We choose T∗∗ ∈ ]0, T∗ [ in such a way that C(s, R, f0 , U0 , G2 )T∗∗
3 We explain here briefly how to prove estimate (4.33) in the case when N = 3 (or s = 3). The only estimate that we did not prove already in this general setting is (4.46) (with |α| + |β| ≤ s). In order to do so, we notice that all terms appearing in Bα,β (still with |α| + |β| ≤ s) are of the form v p Dm U k Dl f k+1 , with D denoting any derivative, p ∈ {0, 1}, and m, l ∈ N, m + l ≤ s + 1, m ≤ s. First, we consider the case l ∈ {0, 1}. Then, using (4.10) and (4.45), we have Dl f k+1 L∞ ,Tk+1 ≤ C(s, R, f0 , G2 ). So, v p Dm U k Dl f k+1 0 ≤ Dl f k+1 L∞ ,Tk+1 1/2 × (Dm U k )2 (v p )2 1Suppv f k+1 dv dx ≤ C(s, R, f0 , G2 )C(f0 )Dm U k 0 ,
15 17
and it is bounded since U k − U0 s ≤ R (on [0, Tk+1 ]) and m ≤ s. Secondly, we consider the case l ≥ 2, so that m ≤ s − 1. But U k − U0 s ≤ R, so thanks to Sobolev inequalities, Dm U k
2N
L N −2(s−m) ,Tk+1
≤ C(R, U0 ).
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Then, using (4.9) and (4.10), and still thanks to Sobolev inequalities, Dl f k+1
3
2N
L N −s+l ,Tk+1
≤ C(f0 ).
(6.2)
In the previous equation, one has to replace the exponent of the Lebesgue space by ∞ if it is nonpositive. Therefore, using the compact support (in v) of f k+1 given by (4.12), 1/2 p m k l k+1 m k 2 p l k+1 2 v D U D f 0 = (D U ) ) dv dx (v D f ≤
m
k
(D U ) ×
2N N −2(s−m)
N −2(s−m) 2N dx
(v p Dl f k+1 )2 dv
s−m N
N 2(s−m)
dx
≤ D U m
k
×
l k+1
(D f
2N
L N −2(s−m)
(|v |1Suppv f k+1 ) p
2N N −2(s−m)
≤ C(R, U0 )C(f0 )Dl f k+1 5 7 9
25
N
L s−m
)
N s−m
dv
N 2(s−m) −1
dv
s−m N dx
.
Then, according to estimate (6.2), the norm in this last equation is bounded as soon N 2N as s−m ≤ N −s+l , or, equivalently, N + l + 2m ≤ 3s. Remembering that this has to hold for m, l ∈ N such that m + l ≤ s + 1, m ≤ s − 1, we see that this is true as soon as s ≥ N . In this way, we can prove that all the terms appearing in Bα,β are bounded in L2 , so that (4.46) holds. References
11 13 15 17 19 21 23 25
[1] A. A. Amsden, P. J. O’Rourke and T. D. Butler, Kiva-II, a computer program for chemical reactive flows with sprays, Tech. report Los Alamos National Laboratory (1989). [2] R. Caflisch and G. Papanicolaou, Dynamic theory of suspensions with brownian effects, SIAM J. Appl. Math. 43 (1983). [3] J. F. Clouet and K. Domelevo, Solutions of a kinetic stochastic equation modeling a spray in a turbulent gas flow, Math. Models Methods Appl. Sci. 7 (1997) 239–263. [4] K. Domelevo, Long time behaviour for a kinetic modelling of two-phase flows with thin sprays and point particles, Preprint de TMR-project “Asymptotics Methods in Kinetic Theory” (2001). [5] K. Domelevo and J.-M. Roquejoffre, Existence and stability of travelling waves solutions in a kinetic model of two phase flows, Commun. Partial Differential Equations 24(1–2) (1999) 61–108. [6] K. Domelevo and M.-H. Vignal, Limites visqueuses pour des syst`emes de type FokkerPlanck-Burgers unidimensionnels, C. R. Acad. Sci. Paris S´ er. I Math. 332(9) (2001) 863–868.
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[7] T. Goudon, Asymptotic problems for a kinetic model of two-phase flow, Proc. Roy. Soc. Edinburgh Sect. A 131(6) (2001) 1371–1384. [8] K. Hamdache, Global existence and large time behaviour of solutions for the VlasovStokes equations, Japan J. Indust. Appl. Math. 15(1) (1998) 51–74. [9] B. L. Keyfitz, R. Sanders and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dynam. Syst. B3 (2003) 541–563. [10] A. Majda, Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables (Springer-Verlag, 1984). [11] R. Motte, A numerical method for solving particle-fluid equations, in Proc. Conf. Trends in Numerical and Physical Modeling for Industrial Multiphase Flows (Carg`ese, France, 2000). [12] M. Ndjinga, A. Kumbaro, F. De Vuyst and P. Laurent-Gencoux, Influence of the interfacial forces on the hyperbolicity of the two-fluid model, in 5th Int. Symp. Multiphase Flow, Heat Mass Transfer and Energy Conversion (Xi’an China, July 3–6 2005). [13] P. J. O’Rourke, Collective drop effects on vaporizing liquid sprays. PhD thesis, Los Alamos National Laboratory (1981). [14] K. PfaffelMoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95(2) (1992) 281–303. [15] D. Ramos, Quelques r´esultats math´ematiques et simulations num ´eriques d’´ecoulements r´egis par des mod`eles bifluides, PhD thesis, Ecole Normale Sup´erieure de Cachan (2000). [16] W. E. Ranz and W. R. Marshall, Evaporization from drops, part I-II, Chem. Eng. Prog. 48(3) (1952) 141–180. [17] D. Serre, Syst`emes de lois de Conservation. I (Fondations. Diderot Editeur, Paris, 1996). [18] F. A. Williams Combustion Theory (Benjamin Cummings, 1985).
