PARABOLIC LIMIT AND STABILITY OF THE VLASOV FOKKER PLANCK SYSTEM

Transcription

1 Mathematical Models and Methods in Applied Sciences Vol. 10, No. 7 ( c World Scientific Publishing Company PARABOLIC LIMIT AND STABILITY OF THE VLASOV FOKKER PLANCK SYSTEM F. POUPAUD Laboratoire J. A. Dieudonné, U.M.R C.N.R.S., Université Nice, Parc Valrose, F Nice Cedex 2, France J. SOLER Departamento de Matemática Applicada, Facultad de Ciencias, Universidad de Granada, Granada, Spain Communicated by P. Degond Received 9 November 1998 Revised 13 April 1999 In this paper the stability of the Vlasov Poisson Fokker Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analyzed. For the case in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, a parabolic limit equation is obtained for the mass density. Depending on the space dimension and on the hypothesis for the initial data, the convergence result in L 1 is weak and global in time or strong and local in time. 1. Introduction The aim of this paper is to study the limit behavior of the Vlasov Poisson Fokker Planck (VPFP system in terms of the parameter ε representing the scaled thermal mean free path and where we have assumed that the scaled thermal velocity is the inverse of the scaled thermal mean free path, see the Appendix for a discussion about the VPFP system and the physical constants involved. The VPFP ε system can be written in this context in terms of the scalar distribution of particles f ε (t, x, v 0, the mass density ρ ε (t, x and the potential Φ ε (t, x, with (t, x, v R +, N = 2 or 3, as follows: ε 2 f ε t + ε((v xf ε ( x Φ ε v f ε =L(f ε, (1.1 L(f ε def = v f ε +div v (vf ε =div v (e v2 /2 v ( e v 2 /2 f ε, (1.2 poupaud@unice.fr jsoler@ugr.es 1027

2 1028 F. Poupaud & J. Soler x Φ ε = θk N ρ ε, ρ ε = f ε dv, (1.3 f ε (0,x,v=f 0,ε (x, v, (1.4 where L defined as in (1.2 is the Fokker Planck operator and K N is the gradient of the fundamental solution of the Laplacian in dimension N. θ =+1and 1in the electrostatic and gravitational cases, respectively. The potential is given by θ 1 1 4π x ρ ε, N =3 Φ ε = θ 1 (1.5 2π log x ρ ε, N =2. We will assume that the sequence of initial data satisfies f 0,ε (1 + x + v 2 + ln f 0,ε d(x, v+ x Φ 0,ε 2 dx < C (1.6 for some constant C>0independent of ε. Depending on the dimension we will prove two kinds of results. In dimension N = 2 or 3 with the additional assumption that f 0,ε (x, ve (p 1 v 2 /2 d(x, v M for some p>n and some constant M independent of ε, we prove a strong convergence result locally in time. There exists T > 0 depending only on the constants M such that as ε 0, the concentration tends strongly to ρ in L q ((0,T ; L r ( and the solution f ε tends to (2π N/2 ρ(t, xe v 2 /2 in L q ((0,T ; L r ( for some q, r 1. Moreover, the limit concentration ρ solves the following drift diffusion equation ρ t div x( x ρ ρ x Φ = 0 (1.7 θ 1 1 ρ, N =3 4π x Φ= θ 1 (1.8 log x ρ, N =2, 2π ρ(0,x=ρ 0 (x, (1.9 ρ 0 being an accumulation point of ρ 0,ε = f 0,ε dv. In dimension N = 2 and for θ = 1, we can obtain a weak convergence result which is global in time. Thus, in this context we will prove that ρ ε ρ in C 0 ([0,T]; L 1 (R 2 weak T>0, where ρ solves (1.7 (1.9. The same result holds for θ = 1 under the additional hypothesis: lim sup x Φ ε (t, x 2 dx <. (1.10 ε 0, t R 2

3 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1029 Let us remark that the system of equations (1.7, (1.8 may blow up in finite time in the case θ = 1. Therefore (1.10 cannot be satisfied in general. This assumption prevents, in some sense, gravitational collapse. The study of the stability of solutions of the VPFP system with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, and in particular the case analyzed in this paper in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, is interesting from different point of views. Besides its intrinsic mathematical interest and its applications in numerical simulations, the asymptotic limits studied here allow one to establish some links between kinetic models and macroscopic (or fluid models. This kind of results is also related to diffusion approximation techniques which have been widely used in various contexts: transport equation of neutronics, 2 radiative transfer, 3 and semiconductors physics. 10,12 The techniques used in this paper are mainly based on the control of the kinetic and potential energies, the entropy of the system and also of some moments associated with the density. This implies, via the Dunford Pettis theorem, the weak L 1 ( compactness of the sequence {ρ ε } ε>0 for every time t 0. In the 2-D case this allows, due to the antisymmetry property of the Poisson kernel, to pass to the limit in the nonlinear term and, hence, in the continuity equation for the density. This cancellation property for the singularity in the Poisson kernel was similarly observed in the study of the existence of solutions for the 2-D Euler equations in fluid mechanics, see Ref. 14. For θ = ±1 andn= 2 or 3, we also obtain strong L 1 ( compactness for {f ε } ε>0 and {ρ ε } ε>0, but only locally in time, by means of a weighted norm for the particle distribution involving the associated Maxwellian. From this estimate we get a bound for ρ ε, independent of ε, insomel p (, with p>nand, as a consequence, a uniform bound in space for the force field x Φ ε and for t Φ ε. Let us summarize the literature concerning the existence results for the VPFP problem. Classical solutions have been studied by H. D. Victory and B. P. O Dwyer 18 who proved the existence of a locally in time smooth solution to the problem (1.1 (1.4. G. Rein and J. Weckler 13 gave sufficient conditions to prove the global existence of classical solutions in the three-dimensional case. In the more general setting of weak solutions, we can mention the works by H. D. Victory 17 and J. A. Carrillo and J. Soler 7 with initial data in L p spaces. F. Bouchut studied in Refs. 4 and 5 the regularity of the weak solutions of this system. He proved the existence and uniqueness of smooth solutions defined globally in time and for a large class of initial data. A. Majda and Y. Zheng 19 andg.majda,a.majdaandy. Zheng 11 obtained the existence of global measure solutions in the 1-D case by using the relationship with the two-dimensional Euler equation with vortex sheet initial data and constructed some relevant explicit solutions, which show the phenomena of singularity formation in finite time. Recently, J. A. Carrillo and J. Soler in Refs. 8 allow for measures in Morrey spaces as initial data and prove the existence of a

4 1030 F. Poupaud & J. Soler locally in time weak solution. Finally, J. A. Carrillo and J. Soler introduced in Ref. 9 the concept of functional solution and proved the global existence of a functional solution when the initial data are only Radon measures with bounded variation. The rest of the paper is organized as follows: In Sec. 2 we obtain the a priori estimates on the system which imply the weak L 1 compactness for the sequences {f ε } ε>0 and {ρ ε } ε>0.insec.3westudythestrongl q compactness for the above sequences and obtain uniform bounds for x Φ ε which are local in time. Section 4 is devoted to show how the weak L 1 compactness of {ρ ε } ε>0 is enough to pass to the limit for N = 2. In Sec. 5 we obtain the parabolic limit equation. Finally, in Sec. 6 we motivate the problem under consideration through the analysis of the VPFP and its physical constants: The scaled thermal velocity and the scaled thermal mean free path. 2. A P riori Estimates We start by defining the concept of weak solution to the problem (1.1 (1.4. Let Q T =[0,T.Givenf 0 L 1 ( we will say that the pair (Φ ε,f ε is aweaksolutiontothevpfp ε problem if (a f ε L 1 (, Φ ε is given by (1.5, (b f ε x Φ ε L 1 loc (R2N, (c for any Ψ C0 (Q T, we have ( f ε ε 2 Ψ Q T t + ε((v xψ ( x Φ ε v Ψ (v v Ψ + v Ψ d(t, x, v = f 0,ε (x, vψ(0,x,vd(x, v. (2.11 In this paper we will assume that the initial condition is smooth enough in order to have existence and uniqueness of a smooth solution to the VPFP ε problem for any fixed ε, see Refs. 4 and 5. The properties of solutions that we will study in this section can rigorously be obtained from (2.11 by combining the formal arguments to be exposed here with the choice of an appropriate sequence of test functions in (2.11 for every studied property. Since a similar rigorous approach that the one given in Refs. 1 and 6 can be easily adapted for the properties studied in this section, we omit the sometimes tedious and standard regularization procedure in order to give the main ideas. We refer to Refs. 1 and 6 to complete the proofs. The first result gives us the mass conservation property as well as an equation for the kinetic energy, the potential energy and the entropy of the system, i.e. for the free energy functional. Lemma 2.1. Let f ε be a smooth solution of the VPFP ε system. Then, we have (a The total mass of the system, i.e. the L 1 ( norm of f ε, is preserved.

5 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1031 (b The following equation is verified by (Φ ε,f ε : ( ( d v 2 dt R 2 +lnf ε f ε d(x, v+ θ x Φ ε 2 dx 2 2N = h ε 2 d(x, v, (2.12 where h ε def = ε 1 ( v f ε +2 v fε. Proof. The mass conservation f ε (t,, L 1 ( = f 0,ε L 1 ( (2.13 follows formally by integrating Eq. (1.1 in v, which gives the continuity equation for the mass density t ρ ε + 1 ε div xj ε =0, (2.14 in the sense of distributions, where j ε is the current density defined by j ε = vf ε dv and then integrating in x. On the other hand, the equation for the balance of energy can be deduced multiplying (1.1 by v 2, integrating the result with respect to x and v and then using the divergence theorem. Thus we obtain ε 2 d v 2 dt 2 f ε d(x, v+ε (v x Φ ε f ε d(x, v = N f ε d(x, v v 2 f ε d(x, v. (2.15 The second term on the left-hand side of (2.15 can be written as follows: ( (v x Φ ε f ε d(x, v = Φ ε div x vf ε dv dx. Then, taking into account the continuity Eq. (2.14 for ρ ε we find (v x Φ ε f ε d(x, v =ε Φ ε t ( f ε dv dx. Now, thanks to (1.3 we obtain (v x Φ ε f ε d(x, v =ε θ d x Φ ε 2 dx. R 2dt 2N

6 1032 F. Poupaud & J. Soler Using this equality, (2.15 becomes ε 2 1 ( d v 2 f ε d(x, v+θ x Φ ε 2 dx 2 dt = N f ε d(x, v v 2 f ε d(x, v. (2.16 Finally, the balance of entropy identity is formally obtained by multiplying (1.1 by ln f ε, integrating in x and v and using again the divergence theorem which yields ε 2 d f ε ln f ε d(x, v dt = v f ε 2 1 d(x, v+n f ε d(x, v. R f 2N ε (2.17 Adding (2.16 and (2.17 we have ( ε 2 d dt = ( v 2 2 +lnf ε f ε d(x, v+ θ x Φ ε 2 dx 2 (2Nf ε v 2 f ε v f ε 2 1fε d(x, v. (2.18 Then, since Nf ε d(x, v = v v f ε d(x, v, we can write the second member of (2.18 as 1 vf ε + v f ε 2 d(x, v. R f 2N ε Now use the identity ( f ε 1 v f ε =2 v fε we find the announced result (2.12. Previous lemma allows one to deduce, due to the negativity of the right-hand side of (2.12, that the free energy functional ( E ε (t def v 2 = R 2 +lnf ε f ε d(x, v+ θ x Φ ε 2 dx 2 2N is bounded from above. However, to conclude that the kinetic energy, the potential energy and the entropy of the VPFP ε system are uniformly bounded with respect to t [0,T], T > 0, we must prove that the above functional is also bounded from below. For that we need the following lemma: Lemma 2.2. We have d x x f ε d(x, v = dt x ε 1 j ε (t, x dx, (2.19 where the current density j ε (t, x = vf ε (t, x, v dv can be estimated by ε 1 j ε (t, L 1 ( 1 h ε 2 (t, x, v dx dv R 2 f 0,ε L 1 (. (2.20 2N

7 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1033 Proof. We first multiply (1.1 by x and integrate in x RN v to obtain (2.19. The current density can be written in terms of h ε as follows: j ε = ε h ε fε dv, (2.21 which implies (2.20 using the mass conservation. We are now ready to obtain our uniform estimates. Lemma 2.3. Assume θ =1and that the initial distribution of particles verifies that the sequences (1 + v 2 + x +lnf 0,ε f 0,ε L 1 ( and Φ 0,ε L 2 ( are bounded. Let us also assume for the case θ = 1 the hypothesis (1.10. Let T be an arbitrary positive constant. (a The following quantities are bounded for any t [0,T], with bounds which are independent of ε and t, f ε ln f ε d(x, v, ( x + v 2 f ε d(x, v, Φ ε (t, x 2 dx. (b The function h ε is uniformly bounded (with respect to ε in L 2 ([0,T] and ε 1 j ε in L 2 ((0,T; L 1 (. Proof. From Lemma 2.1 we deduce that E ε (t+ h ε 2 (s, x, v ds dx dv E ε (0. (2.22 (0,T Define ln f ε def = max{ ln f ε, 0} and ln + f ε def = max{ln f ε, 0}, which gives ln f ε =ln + f ε ln f ε. (2.23 To obtain a bound from below for E ε (t, we split the domain into two parts {f ε > e x /2 e v 2 /4 } and {f ε e x /2 e v 2 /4 }. In the first part we have: ( x f ε ln f ε 2 + v 2 f ε 4 and in the second part f ε ln f ε C f ε Ce x /4 e v 2 /8, where C =max( y ln (y is a universal constant. Therefore we can obtain E ε (t+ x f ε (t, x, v dx dv ( x 2 + v 2 4 +ln+ f ε (t, x, v f ε (t, x, v dx dv K,

8 1034 F. Poupaud & J. Soler where K = C e x /2 e v 2 /4 dx dv. On the other hand, using Lemma 2.2 we get x f ε (t, x, v dx dv x f 0,ε (x, v dx dv + 1 h ε 2 (s, x, v ds dx dv + t 2 (0,t R 2 f 0,ε L 1 (. 2N Now from the last two estimates and from (2.22 we get ( x R 2 + v 2 4 +ln+ f ε (t, x, v f ε (t, x, v dx dv 2N + 1 ( h ε 2 (s, x, v ds dx dv K + t L 2 (0,t R 2 + x f 0,ε + E ε (0 2N 1 ( which leads to the desired results. The estimate for ε 1 j ε is obtained from (2.21 by the Cauchy Schwartz inequality. Remark 2.1. Let us note that in our context (assuming that x 2 f 0,ε L 1 ( we cannot assure that the inertial momentum x 2 f ε d(x, v remains bounded for t>0. The equation for the inertial momentum reads ε 2 d ( x 2 f ε d(x, v =2ε x vf ε dv dx (2.24 dt and it seems difficult to obtain a uniform bound with respect to ε. We have the following compactness result: Lemma 2.4. Under the same assumption as in Lemma 2.3 the sequence ρ ε (t, x = fε (t, x, v dv lies in a compact set of C 0 ([0,T]; L 1 ( weak for any time T>0. Proof. From the estimates provided by Lemma 2.3 we classically deduce that for any time t 0 f ε (t,, andρ ε (t, belong to some weakly compact set of L 1 ( and L 1 ( respectively. The continuity equation t ρ ε + 1 ε div xj ε =0, implies that t ρ ε is a bounded sequence of L 2 (0,T;W 1,1 (. Therefore for any function ϕ C 0 (RN, the integral ρ ε (t, x ϕ(x dx belongs to a compact set of C 0 ([0,T]. Let g L (, then we can approximate g uniformly on B R \ Θ by smooth functions (thanks to Egoroff theorem, where B R is the ball of radius R with R arbitrarily large and where Θ is a set of measure arbitrarily small. Therefore using that ρ ε (t, belongs to a weakly compact set of L 1 (, the integrals

9 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1035 ρ ε (t, x g(x dx can be uniformly (with respect to ε and t [0,T] approximated by integrals of the form ρ ε (t, x ϕ(x dx. It leads to the desired result. The above compactness property in L 1 ( is not enough to pass to the limit on the nonlinear term of Eq. (1.1. Note that we cannot use L p, p>1, a priori estimates because they all depend on ε, see also Refs. 7 and 17. In the next two sections we will give some results that provide some extra compactness properties which will depend on the dimension N and on the hypothesis on the initial data. 3. Strong Convergence in a Bounded Time Interval In this section we will obtain bounds for ρ ε in some L p, p>n, independent of ε which implies that x Φ ε is a bounded sequence in L ((0,T. These estimates give us strong convergence locally in time. With this in mind, we define the norm ( 1/p def f p = f p e (p 1 v 2 /2 d(x, v. (3.25 Lemma 3.1. Assume that e (1 1/p v 2 /2 f 0,ε is a bounded sequence in L p (, with p>n.then, there exists a finite T > 0 and a constant C>0independent of ε such that (a The distribution of particles verifies (b The following estimate holds for the density. (c The potential verifies f ε (t, p C, t [0,T ]. (3.26 ρ ε (t, L p ( C, t [0,T ] (3.27 x Φ ε (t, L ( <C, t [0,T ]. (3.28 Proof. Let H be a convex regular function to be precised. If we multiply the righthand side of (1.1 by H (e v 2 /2 f ε, we have L(f ε H (e v 2 /2 f ε dv = e v 2 /2 H (e v 2 /2 f ε e v 2 /2 v f ε 2 dv. We define q H (f ε,f ε as follows: q H (f ε,f ε := 2 /2 H ( ( e v 2 /2 v f ε e v 2 /2 2 f ε dv. (3.29 e v

10 1036 F. Poupaud & J. Soler Proceeding in the same way with the other terms in Eq. (1.1 we first find for the nonlinear one the following estimate: ( x Φ ε v f ε H (e v 2 /2 f ε dv = x Φ ε f ε H x Φ ε 2 /2 H ( e v 2 /2 f ε e v (e v 2 /2 f ε v ( e v 2 /2 f ε dv e v 2 /2 f ε v ( e v 2 /2 f ε dv ( 1/2 x Φ ε fεh (e 2 v 2 /2 f ε e dv v 2 /2 q H (f ε,f ε 1/2. (3.30 The last estimate has been obtained by using the Cauchy Schwartz inequality with the measure e v 2 /2 H (e v 2 /2 dv. Similarly for the other terms we have f ε R t H ( e v 2 /2 f ε dv = H (e v 2 /2 f ε e v 2 /2 dv (3.31 t N and (v x f ε H (e v 2 /2 f ε d(x, v = (v x H (e v 2 /2 f ε e v 2 /2 d(x, v =0. (3.32 Therefore, combining (3.29 (3.32 we have that a solution of (1.1 satisfies d H(e v 2 /2 f ε e v 2 /2 d(x, v+ 1 dt R ε 2 q H (f ε,f ε dx 2N 1 ( ε xφ ε L ( fε 2 H ( 1/2 e v 2 /2 f ε e dv v 2 /2 q H (f ε,f ε 1/2 dx 1 2ε 2 q H (f ε,f ε dx + 1 ( 2 xφ ε 2 L ( fε 2 H f ε e v 2 /2 e v 2 /2 d(x, v. Then, we obtain d H(f ε e v 2 /2 e v 2 /2 d(x, v+ 1 dt R 2ε 2 q H (f ε,f ε dx 2N 1 ( 2 xφ ε (t, 2 2 ( L ( f ε e v 2 /2 H f ε e v 2 /2 e v 2 /2 d(x, v. (3.33

11 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1037 We choose, for 1 <p<,h(τ=τ p. Then, using the norm (3.26, (3.33 becomes d dt f ε(t, p 1 4 p(p 1 xφ ε (t, 2 L ( f ε(t, p, (3.34 wherewehaveusedthepositivityofq H (f ε,f ε. On the other hand, using Hölder s inequality it is straightforward to find that ρ ε (t, Lp ( (2π N/(2p f ε (t, p, where p is the conjugate exponent of p. As a consequence of the Hardy Littlewood Sobolev theorem (see Ref. 16, for p>n, we find x Φ ε (t, L ( c(p( ρ ε (t, L1 ( + ρ ε (t, Lp (. Therefore, combining the above estimates in (3.34 we have d dt f ε(t, p c(p( f 0,ε L1 ( + f ε (t, p 2 f ε (t, p. (3.35 From (3.35 we deduce the announced result (a. These bounds ensure that ρ ε is uniformly bounded in L (0,T ;L 1 ( L p (, with p>n. As a consequence, we have (3.28. Let us now give some consequences of Lemma 3.1 which will be useful in Sec. 5 to deduce the limit equation and that improve the estimates in Lemma 2.2. Lemma 3.2. Under the hypothesis of Lemma 3.1, we have (a Thesequenceoffunctions e v 2 /4 v (f ε e v 2 /2 = O(ε in L 2 ((0,T. (b The current density j ε is of order ε in L 2 ((0,T. Proof. We first remark that for any 1 q p we have (fe v2 /2 q fe v2 /2 + (fe v2 /2 p which yields q 1 + p = L 1 ( + p. Therefore it follows from the mass conservation and from Lemma 3.1 that f ε (t,, 2 is uniformly bounded with respect to ε and t [0,T ]. Then Eq. (3.33 with H(τ =τ 2 implies that q H (f ε,f ε is of order ε 2 in L 1 ((0,T. It leads to Lemma 3.2(a. Hence, we have that ( j ε = f ε e v 2 /2 e v 2 /2 dv is O(ε inl 2 ((0,T. v We now remark that f ε e v 2 /4 is bounded in L 2 ((0,T andthat ε t f ε + v x f ε =div v g ε,

12 1038 F. Poupaud & J. Soler where g ε = x Φ ε f ε + 1 ε e v 2 /2 v (f ε e v 2 /2 belongs to a bounded set of L 2 ((0,T. Using as in Ref. 10 a mean compactness argument we obtain that ρ ε belongs to a bounded set of L 2 ((0,T ; H α ( for some α>0. Then the continuity equation gives that t ρ ε belongs to a bounded set of L 2 ((0,T ; H 1 (. It follows that ρ ε belongs to a compact set of L 2 loc ([0,T ]. In view of (2.19 we also have x ρ ε bounded in L ((0,T ; L 1 ( so we have for instance that ρ ε belongs toacompactsetofl q ((0,T ; L 1 ( for any 1 q<. Since the sequence ρ ε is bounded in L ((0,T ; L p ( we finally obtain by interpolation that ρ ε belongs toacompactsetofl q ((0,T ; L r ( for any 1 q< and any 1 r<p.to conclude this part we summarized the compactness results in Lemma 3.3. Under the hypothesis of Lemma 3.1, and up to subsequences we have (a The concentration strongly converges ρ ε ρinl q ((0,T ; L r ( for any 1 q< and any 1 r<p. (b The nonlinear term pass to the limit ρ ε x Φ ε ρ x Φ weakly in L q ((0,T ; L r ( for any 1 q< and any 1 r<p, where x Φ= θk N ρ. (c The distribution function tends strongly to a Maxwellian f ε (2π N/2 ρ(t, xe v 2 /2 in L q ((0,T ; L r (, with 2 q<, 2 r<p. It remains to prove the convergence of f ε. We first need the following: Lemma 3.4. There is a positive constant K>0such that for any ϕ L 1 ( which satisfies ψ(v = v (ϕe v 2 /2 e v 2 /4 [L 2 ( ] N we have ϕ L1 ( ϕ(v dv + K ψ [L 2 ( ] N. We use the above estimate for ϕ ε = f ε (2π N/2 ρ ε e v 2 /2. Using Lemma 3.2 we get that ϕ ε is an O(ε inl 2 ((0,T x ;L 1 ( v. Since ϕ ε is bounded in L ((0,T ; L p (, it also vanishes in L r ((0,T ; L q ( with 2 r< and 2 q<p. This concludes the proof of Lemma 3.3. Proof of Lemma 3.4. By a standard density argument we can reduce the proof to the case where ϕ C0 (. By using Taylor formula with integral remainder, we have ϕ(ve v 2 /2 = ϕ(v 0 e v0 2 / ψ(v 0 + σ(v v 0 (v v 0 e v0+σ(v v0 2 /4 dσ.

13 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1039 Therefore multiplying by e ( v 2 + v 0 2 /2 and integrating with respect to v 0 we get (2π N/2 ϕ(v = ϕ(v 0 dv 0 e v 2 /2 1 + ψ(v 0 + σ(v v 0 (v v 0 e v0+σ(v v0 2 /4 ( v 2 + v 0 2 /2 dσdv 0. 0 Now, integrating with respect to v we have ϕ(v dv ϕ(v 0 dv 0 +(2π N/2 1 ψ(v 0 + σ(v v 0 v v 0 e v0+σ(v v0 2 /4 ( v 2 + v 0 2 /2 dσdv 0 dv. 0 By convexity we have v 0 + σ(v v 0 2 (1 σ v σ v 2 v v 2,thuswe obtain ϕ(v dv ϕ(v 0 dv 0 1 +(2π N/2 ψ(v 0 + σ(v v 0 v v 0 e ( v 2 + v 0 2 /4 dσdv 0 dv. 0 (3.36 We use the change of variables u = v v 0,w=v 0 +σu=σv+(1 σ v 0.We have dudw = dv 0 dv and taking into account u 2 + w 2 3( v v 2 wehave e ( v 2 + v 0 2 /4 e ( u 2 + w 2 /12. It yields to the estimate 1 ψ(v 0 + σ(v v 0 v v 0 e ( v 2 + v 0 2 /4 dσdv 0 dv 0 ψ(w u e ( w 2 + u 2 /12 dudw. The last estimate together with (3.36 leads to the result. Remark 3.1. The above result as the spaces of convergence for f ε arenotatall optimal. However it suffices for our purpose. 4. Global in Time Weak Convergence for N=2 From Lemma 2.4 we have that up to a subsequence ρ ε ρ in C 0 ([0,T]; L 1 (R 2 weak. (4.37 Then the L 2 estimate provided by Lemma 2.3 gives x Φ ε x Φ= θk 2 ρin L ((0,T; L 2 ( weak. (4.38

14 1040 F. Poupaud & J. Soler Let us remark that a priori the product x Φ ρ does not even define a distribution. However, the above convergences are enough to pass to the limit in the nonlinear term x Φ ε ρ ε = ( 1 2π x x 2 ρ ε ρ ε, thanks to a cancellation property. Indeed in a weak sense we have to study T x Φ ε (t, xρ ε (t, xϕ(t, x dt dx 0 R 2 = 1 T x y 2π x y 2 ρ ε(t, yρ ε (t, xϕ(t, x dt d(y, x, 0 R 4 where ϕ C0 ([0,T R2. Note that due to the antisymmetry of the kernel K 2 we can write T x Φ ε (t, xρ ε (t, xϕ(t, x dt dx 0 R 2 = 1 2π T 0 R 4 (x y(ϕ(t, x ϕ(t, y x y 2 ρ ε (t, yρ ε (t, x dt d(y, x. Since ϕ is regular, in particular Lipschitz, the function (x y(ϕ(t, x ϕ(t, y x y 2 is bounded. Then the above expression for the nonlinear term allows one to pass to the limit with the only property of the weak L 1 (R 2 convergence of the sequence {ρ ε } ε>0,fort [0,T], T>0. Indeed, thanks to Lemma 2.4, we have ρ ε (t, x ρ ε (t, y ρ(t, x ρ(t, y in C 0 ([0,T]; L 1 (R 2 R 2 weak. Then, we have x Φ ε ρ ε = ( 1 2π ( x 1 x 2 ρ ε ρ ε 2π x x 2 ρ ρ = x Φρ in D, (4.39 where the product is defined as a distribution by x Φρϕdx def = 1 (x y(ϕ(x ϕ(y R 2π 2 R x y 2 ρ(t, yρ(t, x d(y, x ( for any test function ϕ. This cancellation property of the 2-D Poisson kernel was used previously in the framework for the study of weak solutions for the Euler equations, see S. Schochet 14 and references therein.

15 Parabolic Limit and Stability of the Vlasov Fokker Planck System The Parabolic Limit Equation From the results in Sec. 2 we have that and f ε (1 + x + v 2 + ln f ε is bounded in L (0,T;L 1 (, (5.41 x Φ ε is bounded in L (0,T;L 2 ( (5.42 h ε = 1 ε (2 v fε + v f ε is bounded in L 2 ((0,T, (5.43 for all T>0. We also recall that the current density is O(ε inl 1 ((0,T, which is given in Lemma 2.2. We will try to obtain the convergence properties of j ε /ε, asε 0, to obtain the parabolic limit from the continuity equation for ρ ε. Then, multiplying Eq. (1.1 by v/ε, we find that ε t j ε +div x v vf ε dv + x Φ ε ρ ε = 1 ε j ε (5.44 is satisfied in the sense of distributions. Taking into account that v v f ε dv = ρ ε I N, where I N is the identity matrix of,wehave v vf ε dv ρ ε I N = ε h ε v f ε dv = O(ε inl 2 ((0,T; L 1 (. (5.45 Then we obtain div x v vf ε dv x ρ in D ((0,T. In view of the results of the preceding parts (cf. Lemma 3.3, (4.39 and (4.40 we can pass to the limit in (5.44 by obtaining lim ε 0 ε 1 j ε = x ρ x Φρ in D ((0,T. Taking into account this relation in the continuity equation of ρ ε,weobtain that ρ verifies in the sense of distributions. t ρ div x ( x ρ + x Φρ = 0 (5.46

16 1042 F. Poupaud & J. Soler Since ρ ε lies in a compact set of C 0 (0,T;L 1 ( weak (cf. Lemma 2.4, we also have that ρ 0,ε (x = f 0,ε (x, v dv ρ(0,x (5.47 which gives the Cauchy data. We also get: x φ ε K N ρ. (5.48 Then, we have proved the following result Theorem 5.1. Assume that f 0,ε (1 + x + v 2 + ln f 0,ε d(x, v+ x Φ 0,ε 2 dx is a bounded sequence. Then, we have (a For θ =1and N =2,up to a subsequence, {ρ ε } ε>0 converges in C 0 ([0, ; L 1 (R 2 weak uniformly on bounded intervals towards a solution ρ of (5.46, (5.48 with initial data given by (5.47. In this case the product x Φ ρ is defined by (4.40. The same result is still valid for θ = 1 assuming the following hypothesis: lim sup ε 0,t x Φ ε (t, x 2 dx <. R 2 (5.49 (b If the initial data also satisfies e (1 1/p v 2/2 f 0,ε is a bounded sequence of L p (, with N = 2 or 3 and p > N, then there exists a finite T > 0 such that up to a subsequence the concentration converges strongly ρ ε ρinl q ((0,T ; L r ( for any 1 q< and any 1 r<p. The distribution {f ε } ε>0 converges in L q ((0,T ; L r ( towards (2π N/2 ρ(t, xe v 2 /2 for 2 q< and 2 r<p,and the concentration ρ solves (5.46, (5.48 with initial data given by (5.47. Appendix. The VPFP System and the Physical Constants The idea of this section is to write the VPFP system in terms of the physical constants: The scaled thermal velocity and the scaled thermal mean free path. This will allow one to study the behavior of solutions with respect to these constants, see Ref. 15. Consider the VPFP system in the case of charged particles interacting through electrostatic forces. f t +(v xf+ q m ( xφ v f =L(f, (A.1 ε 0 x Φ= θρ, ρ(t, x = f(t, x, v dv (A.2 f(0,x,v=f 0 (x, v, (A.3

17 Parabolic Limit and Stability of the Vlasov Fokker Planck System 1043 where L(f = µ τ v ( e v 2 /2µ v ( e v 2 /(2µ f, θ =1,mis the particle mass, ε 0 the permitivity of vacuum, τ the relaxation time and where µ is the thermal velocity. There is a microscopic variation of v which is µ and a macroscopic mean velocity associated with the distribution of particles f given by vf dv u 0 = RN. fdv Hence, we choose a scaling such that v µv, x Rx, t Tt, with R/T = u 0. To adimensionalize the Poisson equation, we introduce a characteristic value of concentration M and a characteristic variation of the potential Φ 0 over a typical length R. We perform the change of unknowns choosing Φ 0 = 1 ε 0 MR 2,toobtain f M µ 3 f, Φ Φ 0 Φ, x Φ=θρ. We remark that we control only two constants (the rest are physical constants M, which depends on the size of the initial data and R (or u 0. We are now ready to adimensionalize the Fokker Planck equation by using the rescaling µ α = for the scaled thermal velocity, and β = τ µ R for the scaled thermal mean free path. Then, our system reads f t + α(v xf+ 1 β ( xφ v f = α β L(f, u 0 x Φ=θρ, (A.4 (A.5 f(0,x,v=f 0 (x, v, (A.6

18 1044 F. Poupaud & J. Soler where L(f = v (e v 2 /2 ( v e v 2 /2 f. The same result holds (with different physical constants for massive particles interacting through gravitational forces. In this case θ = 1. Now, the idea is to study the stability of solutions to the VPFP system with respect to α and β. As soon as we choose α =1/β, Eq. (A.4 becomes Eq. (1.1, which has been studied in this paper. The analysis of solutions to the system (1.1 (1.5 as β goes to zero leads to the parabolic limit of the VPFP system. The hyperbolic limit consists of assuming that α = 1 and β 0. We conjecture that in this case we will find the following limit behavior f (2π N/2 ρ(t, xe v xφ 2 /2, where ρ(t, x satisfies the following continuity equation with non-bounded energy: ρ t + x(ρ x Φ = 0, which will be studied in a forthcoming publication. Acknowledgments This work was partially supported by DGES-MEC (Spain, Project PB , by TMR-European Union, contract ERB FMBX-CT , and by the Erwin Schrödinger Institute (Wien, Austria. The second author would like to express his gratitude to José L.López and José A. Carrillo for some useful discussions. References 1. L. L. Bonilla, J. A. Carrillo and J. Soler, Asymptotic behaviour of the initial boundary value problem for the three dimensional Vlasov Poisson Fokker Planck system, SIAM J. Appl. Math. 57 ( C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size of a transport operator, Trans. Amer. Math. Soc. 284 ( C. Bardos, F. Golse, B. Perthame and R. Sentis, The non-accretive radiative transfert equation, existence of solutions and Rosseland approximation, J. Funct. Anal. 88 ( F. Bouchut, Existence and uniqueness of a global smooth solution for the VPFP system in three dimensions, J. Funct. Anal. 111 ( F. Bouchut Smoothing effect for the nonlinear VPFP system, J. Differential Equations 122 ( F. Bouchut and J. Dolbeaut, On long asymptotics of the Vlasov Fokker Planck equation and of the Vlasov Poisson Fokker Planck system with Coulombic and Newtonian potentials, Differential Integ. Equations 8 ( J. A. Carrillo and J. Soler, On the initial value problem for the VPFP system with initial data in L p spaces, Math. Methods Appl. Sci. 18 (

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