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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-19 Wien, Austria Convergence of the Vlasov{Poisson System to the Incompressible Euler Equations Yann Brenier Vienna, Preprint ESI 678 (1999) March 1, 1999 Supported by Federal Ministry of Science and Transport, Austria Available via

2 CONVERGENCE OF THE VLASOV-POISSON SYSTEM TO THE INCOMPRESSIBLE EULER EQUATIONS Yann Brenier Resume On etudie la convergence du systeme de Vlasov-Poisson vers les equations d'euler des uides incompressibles dans deux regimes asymptotiques : la limite quasi-neutre et la limite gyrocinetique. Abstract The convergence of the Vlasov-Poisson system to the incompressible Euler equations is investigated in two asymptotic regimes: the quasi-neutral limit and the gyrokinetic limit. A paraitre dans Comm. PDEs Institut Universitaire de France, et Laboratoire d'analyse numerique, Universite Paris 6, France, brenier@ann.jussieu.fr 1

3 We consider the displacement of an electronic cloud generated by the local dierence of charge with a uniform neutralizing background of nonmoving ions. The equations are given by the Vlasov-Poisson system, with a coupling constant = ( 2 )2 where is the (constant) oscillation period of the electrons. In the so-called quasi-neutral regime, namely as!, the current is expected to converge to a solution of the incompressible Euler equations, at least in the case of a vanishing initial temperature. This result is proved by adapting an argument used by P.-L. Lions [Li] to prove the convergence of the Leray solutions of the 3d Navier-Stokes equation to the so-called dissipative solutions of the Euler equations. For this purpose, the total energy of the system is modulated by a test-function. An alternative proof is given, based on the concept of measure-valued (mv) solutions introduced by DiPerna and Majda [DM] and already used by Brenier and Grenier [BG], [Gr2] for the asymptotic analysis of the Vlasov-Poisson system in the quasi-neutral regime. Through this analysis, a link is established between Lions' dissipative solutions and Diperna-Majda's mv solutions of the Euler equations. A second interesting asymptotic regime, still leading to the Euler equations, known as the gyrokinetic limit of the Vlasov-Poisson system, is obtained when the electrons are forced by a strong constant external magnetic eld and has been investigated by Grenier [Gr3], Golse and Saint-Raymond [GSR]. As for the quasi-neutral limit, we justify the gyrokinetic limit by using the concepts of dissipative solutions and modulated total energy. 1 Formal analysis 1.1 The Vlasov-Poisson system After suitable normalizations, the Vlasov-Poisson system reads (see [BR] for example) t f :r x f? r x :r f = ; (1) IR d f(d) = 1? (2) where (x; ) 2 IR 2d is the position/velocity variable, with d = 1; 2 or 3, f(t; x; ) the electronic density, (t; x) 2 IR the electric potential and > the coupling constant between the Vlasov equation (1) and the Poisson equation (2). To complete this system, initial conditions f(; x; ) = f (x; ) (3) 2

4 and d periodicity in x are prescribed. Up to a change of sign, we call charge and current the two rst moments (t; x) = f(t; x; d); J(t; x) = f(d): (4) Electrons are called cold electrons when the temperature, proportional to j? J j2 f(t; x; d); (5) vanishes. The conservation of total energy reads 1 2 jj2 f(t; dx; d) 2 jr(t; x)j2 dx (6) (where integrals in x are performed on the unit cube [; 1] d ), and the conservation laws for charge and current are t f(d) r x : f(d) = (7) (or, equivalently because of t r x : f(d) r x : f(d) t ); (8) = r : (r r)? 2 r(jrj2 ): f(d) r (9) By computing the divergence of the last equations and using the Poisson equation,?(@ tt 1)? r 2 x : f(d) = (1) =?r 2 : (r r) 2 (jrj2 ) is obtained for the electric potential. The mathematical analysis of the Vlasov-Poisson system is now well known, in particular after the recent contributions of Batt and Rein [BR], Lions and Perthame [LP], Pfaelmoser [Pf], etc... Global existence and uniqueness of smooth solutions have been proved for smooth initial data f (x; ), suciently decaying at innity in. Then, all the formal computations we have performed are fully justied. 3

5 1.2 The quasi-neutral regime The asymptotic analysis! is dicult and only partial results have been obtained, in particular by Grenier in [Gr1], [Gr2], [Gr3] (see also [Br]). The oscillatory behaviour of the linear part of equation (1) is one of the main diculties. Let us start by a purely formal analysis of the limit!. The Poisson equation (2) becomes f(d) = 1 (11) and we get from equations (8), t r x : f(d) r x : f(d) = (12) f(d) r = : (13) For the potential, we nd? = r 2 x : f(d): (14) For perfecty cold electrons, the probablity measure (in ) f(t; x; ) is a delta function, which exactly means f(t; x; ) = (? J(t; x)); (15) since J is the current and the charge is identically equal to 1. In this particular case, we obtain r:j = t J r : (J J) r = ; (17) which is nothing but the classical Euler equations for an incompressible uid (with velocity J and pressure ), for which we refer to [AK], [Ch], [Li], [MP]... The case of cold electrons is precisely the one for which we get a rigorous asymptotic result in the present paper. 4

6 2 The convergence result Theorem 2.1 Let T > and J (x) be a given divergence-free, d periodic in x, square integrable vector eld. Assume the initial data f (x; ) to be smooth, d periodic in x, nicely decaying as! 1, with total mass 1. In addition, we assume f(x; )d = 1 o( 1=2 );! ; (18) in the strong sense of the space H?1 (IR d = d ) and j? v (x)j 2 f (x; )dxd! jj? v (x)j 2 dx; (19) for all square integrable, divergence-free, d periodic, vector eld v. Then, up to the extraction of a sequence n!, the divergence-free component of the current J converges in C ([; T ]; D (IR d = d )) to a dissipative solution J 2 C ([; T ]; L 2 (IR d = d )? w) of the Euler equations, in the sense of Lions [Li], with initial condition J. In particular, if J is smooth and d = 2 (or d = 3 and T small), the entire family (without extraction of any subsequence) converges to the unique smooth solution of the Euler equations with J as initial condition. Remark 1 Following Lions [Li], we say that J is a dissipative solution with initial condition J if, for all smooth vector divergence-free vector elds v on [; T ] IR d = d, almost every t 2 [; T ], jj(t; x)? v(t; x)j 2 dx 2 t exp( t s 2jjd(v())jjd)( jj (x)? v(; x)j 2 dx exp( t 2jjd(v())jj)d (2) A(v)(s; x):(v? J)(s; x))dsdx; where d(v) is the symmetric part of Dv = ((Dv) ij ) = (@ j v i ) d ij (v) = 1 2 (@ x i v xj v i ); (21) 5

7 jjd(v(t))jj is the supremum in x of the spectral radius of d(v)(t; x), and A(v) is the acceleration operator A(v) t v (v:r)v: (22) Notice a slight change of denition with respect to [Li], since here we use the spectral radius of the entire matrix d(v), not only its negative part. Remark 2 The quasi-neutrality assumption (18) exactly means, because of (2), jr (; x)j 2 dx! : (23) Assumption (19) means that the electrons are cold and the initial current converges to J. Indeed, we have (take v = J and v = ) j? J (x)j 2 f (x; )dxd! ; (24) 3 Proofs jj 2 f (x; )dxd! jj (x)j 2 dx (25) The proof is a simple adaptation of the way that Lions follows in [Li] to show the convergence of Leray solutions of the Navier-Stokes equations to the so-called dissipative solutions of the Euler equations. To do that, the total energy of the system is modulated by a test-function. 3.1 Control of the modulated total energy Let us compute the time derivative of the total energy of the Vlasov-Poisson system, modulated by a test function (t; x)! v(t; x); d periodic, divergencefree in x, 1 Hv(t) = 2 j? v(t; x)j2 f (t; x; )dxd 2 jr (t; x)j 2 dx: (26) Let us temporarily drop the index. Because of the total energy conservation, we have, for the charge and the current J, d dt H v(t) = d dt 1 2 jv(t; x)j2 (t; x)dx? t (J(t; x):v(t; x))dx: (27)

8 Elementary calculations lead to d dt H v(t) =? d(v)(t; x) : (? v(t; x)) (? v(t; x))f(t; x; )dxd (28) d(v)(t; x) : r(t; x) r(t; x)dx A(v)(t; x):((t; x)v(t; x)? J(t; x))dx where d(v) is the symmetrized gradient of v dened by (21) and A(v) is the acceleration operator (22). Thus, we get, after rising index, d dt H v(t) 2jjd(v(t))jjH v(t) A(v)( v? J )dx; (29) where Hv is dened by (26) and jjd(v(t))jj is the supremum in x of the spectral radius of d(v)(t; x). We deduce, after integrating (29) in t, t exp( t s H v(t) H v() exp( 2jjd(v())jjd)( t 2jjd(v())jj)d (3) A(v)(s; x):( v? J )(s; x))dsdx: In particular, in the case v =, we recover the total energy bound 1 H(t) = 2 jj2 f (t; x; )dxd 2 jr (t; x)j 2 dx H(): (31) Remark Here we use the spectral radius of the entire matrix d(v) and not only its negative part (as in Lions' denition for dissipative solutions of the Euler equations). Indeed, in the right-hand side of (28), the rst and the second terms involve d(v) with opposite signs! 3.2 A priori bounds The assumptions on the initial conditions and equations (18), (7), imply that f (t; x; )dxd = (x)dx = 1; (32) 7

9 jj 2 f(x; )dxd jr (; x)j 2 dx! jj (x)j 2 dx: (33) >From (31), we deduce that jj 2 f (t; x; )dxd jr (t; x)j 2 dx C: (34) Thus J is bounded in L 1 ([; T ]; L 1 (IR d = d )) since ( jj (t; x)jdx) 2 jj 2 f (t; x; )ddx f (t; x; )ddx C: Up to the extraction of a sequence ( n ), we can assume that J has a vague limit J, in the sens of (Radon) measures on [; T ] IR d = d. Similarly, from (32), (7) and (34), we get that (t; x) converges to 1 in C ([; T ]; D (IR d = d )) and therefore in the vague sense of measures. Let us now consider the convex functional of (Radon) measures K(; m) = sup b? 1 2 jb(t; x)j2 (dtdx) b(t; x):m(dtdx); where b spans the space of all continuous functions from [; T ] IR d = d to IR d and, m respectively denote nonnegative and vector-valued measures on [; T ] IR d = d. When (t; x) = 1 (the Lebesgue measure), we simply obtain 2K(; m) = jm(t; x)j 2 dtdx; if m is a square integrable function and 1 otherwise. Functional K is lsc with respect to the vague convergence of measures. Since, for each nonnegative function z 2 C ([; T ]), jj 2K(z ; zj (t; x)j ) = 2 (t; x) z(t)dtdx we deduce that jj 2 f (t; x; )z(t)dtdxd C 2K(z; zj) C T z(t)dt; T z(t)dt; which exactly means that J belongs to L 1 ([; T ]; L 2 (IR d = d )). >From (8), we get that J is divergence-free in x and, from (9), t J is bounded in 8

10 L 1 ([; T ]; D (IR d = d )), since J is divergence-free (which allows us to ignore r in (9), although this term could be of size O(?1=2 )). It follows that the vague limit J(t; x) of J (t; x) is a divergence-free vector eld belonging to C ([; T ]; L 2 (IR d = d )? w). For the same reasons, the divergence-free (or solenoidal) part of J converges toward J, not only in the vague sense of measures, but also in C ([; T ]; D (IR d = d )). 3.3 Convergence We can rewrite (29) in weak form? Hv (t)z (t)dt? z()hv () A(v)( v? J )(t; x)z(t)dtdx; 2jjd(v(t))jjHv (t)z(t)dt (35) for all test function z in D ([; T [), where Hv (t) is dened by (26). Let us introduce jj h (t; x)? v(t; x) (t; x)j 2 v(t) = 2 dx (36) (t; x) = sup b [? 1 2 jb(x)j2 (t; x) b(x):(j? v )(t; x)]dx; where b spans the space of all continuous functions from IR d = d to IR d, which is, for each xed t, a convex fuction of J (t; :) and (t; :). (It is a just a modulated version of functional K, with a test function v.) By Cauchy-Schwarz inequality, we have 1 h v(t) 2 j? v(t; x)j2 f (t; x; )dxd Hv(t): The a priori bound previously obtained show that, for xed v, Hv (t) and h v(t) are bounded functions in L 1 ([; T ]) and, up to the extraction of a sequence ( n ), respectively converge, in the weak-* sense, to some limits H v (t) and h v (t), with H v h v. Since! 1 and J! J in the vague sense of measures, by convexity of the functional dened by (36), we get jj(t; x)? v(t; x)j 2 dx 2h v (t): (37) The assumptions on the initial conditions mean 2Hv () = j? v(; x)j 2 f (x; )dxd jr (; x)j 2 dx! 2H ;v (38) 9

11 where we set H ;v = 1 2 jj (x)? v(; x)j 2 dx: (39) Then, we can pass to the limit in (35) to get? H v (t)z (t)dt? z()h ;v By integrating in t, we get Thus t t exp( exp( t s t s A(v)(v? J)(t; x)z(t)dtdx: H v (t) H ;v exp( 2jjd(v())jjd)( h v (t) H ;v exp( 2jjd(v())jjd)( t 2jjd(v(t))jjH v (t)z(t)dt (4) 2jjd(v())jj)d (41) A(v)(s; x):(v? J)(s; x))dsdx: t 2jjd(v())jj)d (42) A(v)(s; x):(v? J)(s; x))dsdx and, therefore, (2) holds true, which concludes the proof. 4 An alternative proof Let us sketch an alternative proof, which can be seen as a natural extension of the analysis made in [BG] (stationary case) and [Gr2] (general case) to study the defect measures of the Vlasov-Poisson system in the quasi-neutral regime. After adaptating the proof (which requires an a priori L 1 bound for f, which is not acceptable in the framework of the present paper), we can show 1) the existence of f(t; x; ), a nonnegative measure f in (x; ) 2 IR d = d IR d, measurable in t, as the vague limit of f, with enough tightness in to allow the zero and rst order moments in (namely the charge and the current) to pass to the limit; 2) the existence of K (t; x; ) and E (t; x; ), two defect measures in (x; ) 2 IR d = d S d?1, measurable in t, that correspond respectively to the defect of kinetic and potential energies; 3) the existence of two defect electric elds E (t; x) and E? (t; x) 2 L 1 ([; T ]; L 2 (IR d = d )), taking into account the temporal oscillations of the electric eld generated by (1); 4) the convergence of the solenoidal part 1

12 R of J toward J = f(d) in C ([; T ]; D (IR d = d )). This is enough to enforce 1) the conservation in time of the total energy with defects 2H(t) = jj 2 f(t; dx; d) (je (t; x)j 2 je? (t; x)j 2 )dx; ( K E )(t; dx; d) (43) R 2) the following properties for the current J(t; x) = f(t; x; d) : r:j = ; (44) where Q t J r : Q = ; (45) f(d) ( K? E )(d) (46)?E E? E? E? : (Note the change of sign between K E and K? E when we switch from the energy conservation to the current conservation.) >From these relations, we deduce that the weak-* L 1 limit of the modulated total energy Hv(t) is given by 2H v (t) = Thus, we directly get d dt H v(t) =? j? v(t; x)j 2 f(t; dx; d)? (je (t; x)j 2 je? (t; x)j 2 )dx: ( K E )(t; dx; d) (47) d(v)(t; x) : (? v(t; x)) (? v(t; x))f(t; dx; d) (48) d(v)(t; x) : ( K? E )(t; dx; d) d(v)(t; x) : (E E E? E? )(t; x)dx 2jjd(v(t))jjH v (t) and we conclude as in the rst proof. A(v)(t; x):(v(t; x)? J(t; x))dx A(v)(t; x):(v(t; x)? J(t; x))dx; (49) 11

13 5 Comparison of dissipative and mv solutions to the Euler equations Our analysis makes a link between Lions' concept of dissipative solutions [Li] and Diperna-Majda's concept of measure-valued solutions (\mv solutions) [DM], both introduced to describe the vanishing viscosity limit of the Navier- Stokes equations R [Li]. If we get back to [DM], we obtain, as before, two limits f, J = f(d), and a kinetic defect measure K (the only relevent defect measure when approaching the Euler equations from the Navier-Stokes side and not from the Vlasov-Poisson side). We get for J (44) and (45) with, Q = f(d) K (d): (5) In addition, the total kinetic energy, including def ects, namely : = jj 2 f(t; dx; d) K (t; dx; d) (51) is decaying in time. Thus, after the same kind of manipulations we already used, we see that J is a dissipative solution of the Euler equations. Thus, the mv solutions are not as dierent from the dissipative solutions as they look. Anyway, the concept of dissipative solutions clarify the relationship between mv solutions and classical solutions, which was not discussed in [DM]. 6 The gyrokinetic limit There is a second asymptotic regime of the Vlasov-Poisson system leading to the Euler equations, the so-called gyrokinetic limit. We consider, as in [Gr3] (see also the included references) or in [GSR] (with a dierent scaling), the eect of a large external magnetic eld. If this magnetic eld is parallel to the third coordinate x 3, we get the following two-dimensional (in both x and ) Vlasov-Poisson t f :r x f 1 (?r? ):r f = ; (52) = 1? ; (t; x)dx = 1; (53) where x 2 IR 2 = 2, 2 IR 2, and? = (? 2 ; 1 ) is the additional term due to the external magnetic eld. We assume the total mass of to be equal to one at time to enforce global neutrality. The total energy is still conserved and, here, dened by 12

14 1 2 jj2 f (t; dx; d) 1 2 jr (t; x)j 2 dx (54) (notice that the magnetic eld is not involved). In addition, we t r:j = ; t J r x : f (d) (56) = 1 (? r? J ): By combining (56) and (55), we also t (?? r:j )? r:( r ) (57) =? r:(r x : f (d)): Formally, as goes to zero, we expect for the limits, J and, the self-consistent system t r:j = ; (58)?r? J = ; = 1? ; (59) which is nothing but the Euler equations written in the so-called vorticity formulation, with? 1 standing for the vorticity and for the streamfunction. The limit! has been successfully investigated in [Gr3] for monokinetic data and small time, as well as in [GSR] for a dierent scaling and global weak solutions of the Euler equation in Delort's sense (see [De]). We can perform the same kind of analysis as for the quasi-neutral limit, and show : Theorem 6.1 Let T > and J (x) =?? r be a given divergence-free, 2 periodic in x, square integrable vector eld. Assume the initial data f (x; ) to be smooth, 2 periodic in x, nicely decaying as! 1, with total mass 1. In addition, we assume jj 2 f (x; )ddx! ; (6) jr (; :)?? J (x)j 2 dx! : (61) 13

15 Then, up to the extraction of a sequence n!,?? r converges in C ([; T ]; L 2 (IR 2 = 2 )? w) to a dissipative solution J of the Euler equations with initial condition J. In particular, if J is smooth, the entire family converges to the unique smooth solution of the Euler equations with J as initial condition. To prove this result, we use the same technique as for the quasi-neutral limit by introducing a modulated total energy, dened in the following way. Given a smooth divergence-free vector eld v(t; x) =?? r (t; x), we set 1 Hv(t) = 2 j? v(t; x)j2 f (t; x; )dxd 2 jr(? )(t; x)j 2 dx: (62) A straightforward but lengthy calculation (using (56) in a crucial way, see the details in the appendix), leads to d dt H v(t) =? d(v)(t; x) : (? v(t; x)) (? v(t; x))f (t; x; )dxd (63) d(v)(t; x) : r(? )(t; x) r(? )(t; x)dx A(v)(t; x):( (t; x)v(t; x)? J (t; x))dx A(v)(t; x):(v(t; x)? r (t; x))dx where d(v), A(v) are still dened by (21), (22). We also get the following bounds : r is bounded in and 1=2 J are bounded in L 1 ([; T ]; L 2 (IR 2 = 2 )); L 1 ([; T ]; L 1 (IR 2 = 2 )) (because of the conservation of charge and energy). Next, r is bounded in L 1 ([; T ]; D (IR 2 = 2 )): Indeed, for all smooth vector eld g(x), because of (2), g(x): (t; x)r (t; x)dx = g:r 14

16 ( 1 2 jr j 2 r:g (r :r)g:r )dx Cjjgjj C 1 (IR 2 = 2 : ) Then, because of (57),?? r:j is compact in C ([; T ]; D (IR 2 = 2 )): Since? r:j = ( 1=2 ) in L 1 ([; T ]; D (IR 2 = 2 )), we deduce that, and therefore r, are also compact in C ([; T ]; D (IR 2 = 2 )). Thus, we conclude that, up to the extraction of a sequence n!, H v and r converge to some limits H v and r, respectively in L 1 ([; T ]) weak-* and C ([; T ]; L 2 (IR 2 = 2 )? w): Then, we can pass to the limit in (62) and (63) to get jr(? )(t; x)j 2 dx H v (t); (64)? H v (t)z (t)dt? z()h ;v 2jjd(v(t))jjH v (t)z(t)dt (65) A(v):(v? r)(t; x)z(t)dtdx; for all smooth nonnegative z(t) compactly supported in t < T, where H ;v = jr(? (; :))(x)j 2 dx (66) is, by assumption, the limit of H v(). By integrating in t, we get t exp( t s t H v (t) H ;v exp( 2jjd(v())jj)d (67) 2jjd(v())jjd)( A(v)(s; x):(v? r)(s; x))dsdx; and, nally, by using (64), we conclude that?? r is a dissipative solution of the Euler equations with initial condition J =?? r, which concludes the proof. 15

17 7 Appendix In this appendix, we prove the crucial identity (63). Because of the conservation of energy, we get from denition (62) : d dt H v = I 1 I 2 I 3 I 4 I 5 I 6 I 7 ; where index has been dropped and We have Thus 2I 5 = 2I 1 = I 3 =? jvj t t jvj 2 ; I 6 =? I 3 = =? =? I 7 = = v:(r : d(v) : d(v) : 2I 2 = v:@ t J; I 4 t jvj 2 ; J:@ t v; r:@ t r ; I 7 t =? r:j =? f) f? f t v:r? v:r r :@ t r: v:? J J:r d(v) : r r? I 7 : where Q 1 =? I 3 I 7 = Q 1 Q 2 Q 3 Q 4 Q 5 Q 6 Q 7 d(v) : Q 2 =? (? v) (? v)f Dv : J v; Q 3 =? d(v) : r(? ) r(? ); Dv : v J 16

18 Q 4 = Q 6 = Then, = 1 2 Next, we observe that FROM VLASOV-POISSON TO EULER Dv : r r; Q 5 = Dv : v v; Q 7 =? Dv : r r Dv : r r : Q 3 j v i J j v i r:jjvj 2 =? 1 t jvj 2 =?I 1 : I 2 I 4 Q 2 Q 6 = and I 5 I 6 Q 4 Q 5 Q 7 = A(v):(v? J) A(v):(v? r) R; where R 1 = R 3 = R = R 1 R 2 R 3 R 4 ; Dv : (?v v) =? R 2 = v i v j v i = ; Dv : r(? ) r ; Dv : (r r?? r v): Since v =?? r and (? ) 2 =?1,, we get = R 3 = (r? v) : (v r? r v) = = Similarly, after setting =?, = (r v) : (?v? r r? v) (r r ) : (v r? r v) (r r) : (v r? r v) = : R 2 = (?r? r ) : r r = (r v) : r r (r r ) : r? r 17

19 = FROM VLASOV-POISSON TO EULER (r r) : r? r = Thus, R = and we nally get d dt H v = Q 1 which is the desired result. r( 1 2 jr j2 ):? r = : A(v):[v? r (v? J)]; Acknowledgments completed. The author would also like to thank Francois Murat for his The author thanks Francois Golse and Emmanuel Grenier for many stimulating discussions concerning the gyrokinetic limit. This work has been supported by the program on Charged Particle Kinetics at the Erwin Schroedinger Institute. References [AK] V.I.Arnold, B.Khesin, Topological methods in hydrodynamics, Springer, New York, [BR] J.Batt, G.Rein, Global classical solutions of the periodic Vlasov- Poisson system, C.R. Acad. Sci. Paris 313 (1991) [Br] Y.Brenier, Quelques lois of conservations issues de modeles cinetiques, Journees Equations aux Derivees Partielles (Saint-Jeande-Monts, 1995), Exp. No. I, Ecole Polytech., Palaiseau, [BG] Y.Brenier, E.Grenier, Limite singuliere du systeme de Vlasov- Poisson, C.R. Acad. Sci. Paris, 318 (1994) [Ch] J.-Y.Chemin, Fluides parfaits incompressibles, Asterisque 23 (1995). [De] [DM] J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer.Math. Soc. 4 (1991) DiPerna, R., and Majda, A., Oscillations and concentrations in weak solutions of the incompressible uid equations, Comm. Math. Phys. 18 (1987) [GSR] F.Golse, L.Saint-Raymond, L'approximation centre-guide pour l'equation de Vlasov-Poisson 2D, C.R. Acad. Sci. Paris, 327 (1998)

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