LAGRANGIAN SOLUTIONS TO THE VLASOV-POISSON SYSTEM WITH L 1 DENSITY

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1 LAGRANGIAN SOLUTIONS TO THE VLASOV-POISSON SYSTEM WITH L 1 DENSITY ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA Abstract. The recently developed theory of Lagrangian flows for transport equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov-Poisson system with only integrable initial distribution function with finite energy. These solutions have a well-defined Lagrangian flow. An a priori estimate on the smallness of the superlevels of the flow in three dimensions is established in order to control the characteristics. 1. Introduction We consider the Cauchy problem for the classical Vlasov-Poisson system B t f ` v x f ` E v f 0, (1.1) fp0, x, vq f 0 px, vq, (1.) where fpt, x, vq ě 0 is the distribution function, t ě 0, x, v P R N, and Ept, xq x Upt, xq (1.3) is the force field. The potential U satisfies the Poisson equation x U ωpρpt, xq ρ b pxqq, (1.4) with ω `1 for the electrostatic (repulsive) case, ω 1 for the gravitational (attractive) case, and where the density ρ of particles is defined through ρpt, xq fpt, x, vqdv, (1.5) R N and ρ b ě 0, ρ b P L 1 pr N q is an autonomous background density. Since we are in the whole space, the relation (1.3) together with the Poisson equation (1.4) yield the equivalent relation Ept, xq ω x S N 1 x N pρpt, xq ρ bpxqq, (1.6) where the convolution is in the space variable. The Vlasov-Poisson system has been studied for long. Existence of local in time smooth solutions in dimension N 3 has been obtained in [15] after the results of [7]. Global smooth solutions have been proved to exist in [1] (and simultaneously in [17] with a different method), with improvements on the growth in time in [, 3]. These solutions need a sufficiently smooth Key words and phrases. Vlasov-Poisson system, Lagrangian flows, non BV vector fields, superlevels, weakly convergent initial data. 1

2 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA initial datum f 0. In particular, the following theorem is a classical result due to Pfaffelmoser [1]. Theorem 1.1. If N 3, let f 0 be a non-negative C 1 function of compact support defined on R 6. Then there are a non-negative f P C 1 pr 7 q and U P C pr 4 q, which tends to zero at infinity for every fixed t, satisfying equations (1.1)-(1.5) with ρ b 0. For each fixed t, the function fpt, x, vq has compact support. The solution is determined uniquely by the initial datum f 0. In a different spirit, global weak solutions were proved to exist in [6, 1, 14], with only f 0 P L 1 pr 6 q, f 0 log` f 0 P L 1, v f 0 P L 1, E 0 P L (and ρ b 0, ω `1). Related results with weak initial data have been obtained in [0, 16, 4]. In this paper we would like mainly to extend the existence result of [1] to initial data in L 1 with finite energy (in the repulsive case ω `1), avoiding the L log`l assumption. Our existence result is Theorem 8.4. It involves a well-defined flow. Even weaker solutions were considered in [5, 18, 19], where the distribution function is a measure. However, these solutions do not have well-defined characteristics. Our approach uses the theory of Lagrangian flows for transport equations with vector fields having weak regularity, developed in [13, 1, 4, 11, 5], and recently in [10,, 8]. It enables to consider force fields that are not in W 1,1 loc, nor in BV loc. In this context we prove stability results with strongly or weakly convergent initial distribution function. The flow is proved to converge strongly anyway. Our main results were announced in [9]. Related results can be found in [3].. Conservation of mass and energy We would like here to recall some basic identities related to the VP system. Integrating (1.1) with respect to v and noting that the last term is in v- divergence form we obtain the local conservation of mass B t ρpt, xq ` div x pjpt, xqq 0, (.1) where the current J is defined by Jpt, xq vfpt, x, vq dv. (.) R N Integrating again with respect to x, we obtain the global conservation of mass d d fpt, x, vqdxdv ρpt, xqdx 0. (.3) dt dt R N ˆR N R N Multiplying (1.1) by v, integrating in x and v, we get after integration by parts in v d v fpt, x, vqdxdv E vf dxdv 0. (.4) dt R N ˆR N R N ˆR N

3 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 3 Using (1.6) and (.1), one has Nÿ ˆ B ω x B t E ` Bx k S N 1 x N J k 0, (.5) k 1 or in other words B t E ω x p q 1 x div x J, (.6) which means that B t E is the the gradient component of ωj (Helmholtz projection). We deduce that E B t E dx ω R N E J dx. R N (.7) Using (.) in (.4), we obtain the conservation of energy» fi d dt R N ˆR N v fpt, x, vqdxdv ` ω Ept, xq ffi dxfl 0. (.8) R N The total conserved energy is the sum of the kinetic energy and of the potential energy multiplied by the factor ω 1. In particular, in the electrostatic case ω `1 we deduce from (.8) a uniform bound in time on both the kinetic and the potential energy, assuming that they are finite initially. In the gravitational case ω 1 it is not possible to exclude that the individual terms of the kinetic and potential energy become unbounded in finite time, while the sum remains constant. Indeed it is known that it does not happen in three dimensions as soon as f 0 is sufficiently integrable, but we cannot exclude this a priori for only L 1 solutions. Note that the assumption E 0 P L is satisfied in 3 dimensions as soon as ρ 0 ρ b P L 6{5. However, in one or two dimensions, for E 0 to be in L it is necessary that ş pρ 0 ρ b qdx 0, as is easily seen in Fourier variable. It is also necessary that ρ 0 ρ b has enough decay at infinity. Thus in one or two dimensions, in order to have finite energy, ρ b cannot be zero identically. 3. Regularity of the force field for L 1 densities 3.1. Singular integrals. Definition 3.1. A function K is a singular kernel of fundamental type in R N if the following properties hold: (1) K R N zt0u P C 1 pr N zt0uq. () There exists a constant C 0 ě 0 such that Kpxq ď C 0 x N, x P R N zt0u. (3.1) (3) There exists a constant C 1 ě 0 such that Kpxq ď C 1 x N`1, x P R N zt0u. (3.) (4) There exists a constant A ě 0 such that ˇ ˇ ď A, (3.3) R 1 ă x ăr Kpxqdx

4 4 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA for every 0 ă R 1 ă R ă 8. Theorem 3. (Calderón-Zygmund). A singular kernel of fundamental type K has an extension as a distribution on R N (still denoted by K), unique up to a constant times Dirac delta at the origin, such that ˆK P L 8 pr N q. Define Su K u, for u P L pr N q, (3.4) in the sense of multiplication in the Fourier variable. Then we have the estimates for 1 ă p ă 8 Su L p pr N q ď C N,p pc 0 ` C 1 ` ˆK L 8q u L p pr N q, u P L p X L pr N q. (3.5) If K is a singular kernel of fundamental type, we call the associated operator S a singular integral operator on R N. We define then the Fréchet space RpR N q X mpn, 1ăpă8 W m,p pr N q and its dual R 1 pr N q Ă S 1 pr N q, where S 1 pr N q is the space of tempered distributions on R N. Since all singular integral operators are bounded on RpR N q, by duality we can define the operator S also R 1 pr N q Ñ R 1 pr N q. In particular it enables to define Su for u P L 1 pr N q or for u a measure. The result Su is in R 1 pr N q Ă S 1 pr N q. 3.. The split vector field. Let ρpt, xq P L 8 pp0, T q; L 1 pr N qq. We denote by bpt, x, vq pb 1, b qpt, x, vq pv, Ept, xqq v, ω x p q 1 x pρpt, xq ρ b pxqq (3.6) the associated vector field on p0, T q ˆ R N ˆ R N. Then the Vlasov equation can be written in the form of the transport equation B t f ` b x,v f 0. In the following subsections we establish bounds on the vector field b Local integrability. For L 1 densities, we have the weak estimates from the Hardy-Littlewood-Sobolev inequality: ˇ ˇˇˇˇˇ p q 1 pρpt, xq ρ b pxqqˇˇˇˇˇˇm N 1 N pr N q ď 1 ˇˇˇ x 1 N S N 1 ρpt, xq ρ b pxq ˇ ˇˇ (3.7) N ˇM N 1 pr N q ď c N ρpt, xq ρ b pxq L 1 pr N q, where u M p pr N q sup γą0 γ L N ptx P R N s.t. upxq ą γuq 1{p. It follows that E L 8 pp0,t q;m N 1 N ď c N ρ ρ b L pr N qq 8 pp0,t q;l 1 pr N qq, (3.8) and using the inclusion M N N 1 pr N q Ă L p loc prn q for 1 ď p ă N N 1 we conclude that b P L 8 pp0, T q; L p loc prn x ˆ R N v qq for any 1 ď p ă N N 1, since v P L p loc prn x ˆ R N v q for any p.

5 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY Spatial regularity. Since b 1 v is smooth, the only non-trivial gradient is the one of b E, indeed the differential matrix of the vector field is given by ˆ ˆDx b Db 1 D v b 1 0 Id. (3.9) D x b D v b D x E 0 We have by (3.6) pd x Eq ij B xj E i ωb x i x j pp x q 1 pρ ρ b qq for 1 ď i, j ď N. (3.10) It is well-known that the operator Bx i x j p q 1 x is a singular integral operator. Its kernel is K ij pxq 1 ˆ B xi S N 1 x N, (3.11) it is given outside of the origin by ˆ 1 K ij pxq S N 1 N x ix j δij x N` x N B xj, for x P R N zt0u. (3.1) The kernel satisfies the conditions of Subsection 3.1, and ˆK ij pξq ξ i ξ j { ξ. Thus (each component of) D x E is a singular integral of an L 8 pp0, T q; L 1 pr N qq function Time regularity. According to (.6), B t E is a singular integral of the current J defined by (.). Using the bounds available for solutions with finite mass and energy fpt, q L 1 pr N x ˆR N v q, v fpt, x, vqdxdv ď C, (3.13) and since v ď 1 ` v, we get that J P L 8 pp0, T q; L 1 pr N x qq. Hence B t E is a singular integral of an L 8 pp0, T q; L 1 pr N x qq function. In particular, B t E P L 8 pp0, T q; S 1 pr N qq. (3.14) 4. Lagrangian flows Suppose that f is a smooth solution to (1.1)-(1.5) with fp0, x, vq f 0. Then f is constant along the characteristics pxps, t, x, vq, V ps, t, x, vqq, which solve the system of equations $ dx & ps, t, x, vq V ps, t, x, vq, ds (4.1) % dv ps, t, x, vq Eps, Xps, t, x, vq, V ps, t, x, vqq, ds with initial data Xpt, t, x, vq x and V pt, t, x, vq v. Thus the solution can be expressed as fpt, x, vq f 0 pxp0, t, x, vq, V p0, t, x, vqq. In order to extend this notion of characteristics to non-smooth solutions, we define regular Lagrangian flows, which are defined in an almost everywhere sense. Definition 4.1. Let b P L 1 loc pr0, T s ˆ RN ; R N q, and t P r0, T q. A map Z : rt, T s ˆ R N Ñ R N is a regular Lagrangian flow starting at time t for the vector field b if

6 6 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA (1) for a.e. z P R N the map s ÞÑ Zps, zq is an absolutely continuous integral solution of 9 βpsq bps, βpsqq for s P rt, T s with βptq z. () There exists a constant L independent of s such that L N pzps, q 1 paqq ď LL N paq for every Borel set A Ă R N. The constant L is called the compressibility constant of Z. Definition 4.. Define the sublevel of the flow as the set G λ tz P R N : Zps, zq ď λ for almost all s P rt, T su. (4.) 5. Stability estimate for Lagrangian flows We summarize the main result from [10] in the following regularity setting of the vector field in arbitrary dimension. We say that a vector field b satisfies (R1) if b can be decomposed as bpt, zq 1 ` z b 1 pt, zq ` b pt, zq (5.1) where b 1 P L 1 pp0, T q; L 1 pr N qq, b P L 1 pp0, T q; L 8 pr N qq. We assume also that b satisfies (R): for every j 1,..., N, mÿ B zj b S jk g jk (5.) k 1 where S jk are singular integrals of fundamental type on R N and g jk P L 1 pp0, T q; L 1 pr N qq. Moreover, we assume condition (R3), that is b P L p loc pr0, T s ˆ RN q, for some p ą 1. (5.3) We recall the following stability theorem from [10], where we denote by B r the ball with center 0 and radius r in R N. Theorem 5.1. Let b, b be two vector fields satisfying (R1), b satisfying also (R), (R3). Fix t P r0, T q and let Z and Z be regular Lagrangian flows starting at time t associated to b and b respectively, with compression constants L and L. Then the following holds. For every γ ą 0, r ą 0 and η ą 0 there exist λ ą 0 and C γ,r,η ą 0 such that L N `B r X t Zps, q Zps, q ą γu ď C γ,r,η b b L 1 pp0,t qˆb λ q ` η for all s P rt, T s. The constants λ and C γ,r,η also depend on The equi-integrability in L 1 pp0, T q; L 1 pr N qq of g jk coming from (R), The norms of the singular integral operators S jk from (R), The norm b L p pp0,t qˆb λ q corresponding to (R3), The L 1 pl 1 q and L 1 pl 8 q norms of the decompositions of b and b in (R1), The compression constants L and L.

7 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 7 We would like now to state a variant of this theorem, where (R1) and (R) are replaced by (R1a) and (Ra). We consider the following weakened assumption (R1a): for all regular Lagrangian flow Z : rt, T s ˆ R N Ñ R N relative to b starting at time t with compression constant L, and for all r, λ ą 0, L N pb r zg λ q ď gpr, λq, with gpr, λq Ñ 0 as λ Ñ 8 at fixed r, (5.4) where G λ denotes the sublevel of the flow Z, defined in (4.). We next consider a splitting of the variables, R N z R N x ˆ R N v, and we denote Zpt, x, vq px, V qpt, x, vq. Noticing the special form (3.6), we assume the condition that b satisfies (Ra): bpt, x, vq pb 1, b qpt, x, vq pb 1 pvq, b pt, xqq, (5.5) with b 1 P LippR N v q, (5.6) and where b involves singular kernels only in the first set of variables x, that is for every j 1,..., N, mÿ B xj b S jk g jk, (5.7) k 1 where S jk are singular integrals of fundamental type on R N and g jk P L 1 pp0, T q; L 1 pr N qq. Theorem 5.. Let b, b be two vector fields satisfying (R1a), b satisfying also (Ra), (R3). Fix t P r0, T q and let Z and Z be regular Lagrangian flows starting at time t associated to b and b respectively, with compression constants L and L, and sublevels G λ and Ḡλ. Then the following holds. For every γ ą 0, r ą 0 and η ą 0, there exist λ ą 0 and C γ,r,η ą 0 such that L N `B r X t Zps, q Zps, q ą γu ď C γ,r,η b b L 1 pp0,t qˆb λ q ` η for all s P rt, T s. The constants λ and C γ,r,η also depend on The equi-integrability in L 1 pp0, T q; L 1 pr N qq of g jk coming from (Ra), The norms of the singular integral operators S jk from (Ra), The Lipschitz constant of b 1 from (Ra), The norm b L p pp0,t qˆb λ q corresponding to (R3), The rate of decay of L N pb r zg λ q and L N pb r zḡλq from (R1a), The compression constants L and L. Proof. We summarize the modifications of the proof of Theorem 5.1 from [10]. The main added difficulty is the singular integral operators on R N x instead of R N. More general situations with singular integrals of measures instead of L 1 functions are considered in [8]. Let z px, vq P R N ˆ R N. We estimate the quantity ˆ Φ δ psq log 1 ` Zps, zq szps, zq dz. (5.8) δ B rxg λ XG s λ

8 8 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA Differentiating with respect to time, we get Φ 1 δ psq ď bps, Zps, zqq sbps, Zps, s zqq δ ` Zps, zq szps, dz zq B rxg λ X s G λ L ď s δ bps, q sbps, q L 1 pb λ q " bps, Zps, zqq ` bps, Zps, s zqq ` min, δ B rxg λ XG s λ b 1 ps, Zps, zqq b 1 ps, s Zps, zqq Zps, zq szps, zq Using the special form of b from (Ra), we obtain ` b ps, Zps, zqq b ps, Zps, s * zqq Zps, zq szps, dz. zq Φ 1 δ psq ď L s δ bps, q sbps, q L 1 pb λ q " bps, Zps, zqq ` bps, Zps, s zqq ` min, δ B rxg λ XG s λ b 1 pv ps, zqq b 1 p s V ps, zqq V ps, zq sv ps, zq ` b ps, Xps, zqq b ps, Xps, s * zqq Xps, zq sxps, dz zq L ď s δ bps, q sbps, q L 1 pb λ q ` Lippb 1 ql N pb r q " bps, Zps, zqq ` bps, Zps, s zqq ` min, δ B rxg λ XG s λ b ps, Xps, zqq b ps, Xps, s * zqq Xps, zq sxps, dz. zq (5.9) Using assumption (Ra), we can use the estimate of [10] on the difference quotients of b, b ps, Xps, zqq b ps, s Xps, zqq Xps, zq sxps, zq ď U ps, Xps, zqq ` U ps, s Xps, zqq, (5.10) where U ps,.q P M 1 pr N q for fixed s is indeed given by Nÿ mÿ U ps, q M j ps jk g jk ps,.qq, (5.11) j 1 k 1 with M j a smooth maximal operator on R N. Next, we can define the function Z ps, x, vq U ps, xq1 px,vqp Ď Bλ, (5.1) and we notice that since the above integrals are over G λ X s G λ, we can replace the right-hand side of (5.10) by Z ps, Zps, zqq`z ps, s Zps, zqq. Then, for given ε ą 0, we can decompose g jk g 1 jk `g jk, with }g1 jk } L 1 pp0,t qˆr N q ď ε and }g jk } L pp0,t qˆr N q ď C ε, g jk having support in a set A ε of finite measure. This gives rise to two type of terms Z 1 and Z. Since all the Z terms have

9 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 9 compact support in v, this allows to perform the same estimates as in [10]. We finally get L N pb r X t Zpτ, q szpτ, q ą γuq 1 ď logp1 ` γ{δq τ t Φ 1 δ psq ds ` L N pb r zg λ q ` L N pb r z s G λ q. (5.13) By choosing λ large first, then ε small, and finally δ small, we conclude the proof as in [10]. 6. Control of superlevels In order to apply Theorem 5., we need to satisfy (R1a). Therefore, we seek an upper bound on the size of B r zg λ The case of low space dimension. We first recall the following lemma from [10]. Lemma 6.1. Let b : p0, T q ˆ R N Ñ R N be a vector field satisfying (R1). Then b satisfies (R1a), where the function gpr, λq depends only on L, b 1 L 1 pp0,t q;l 1 pr N qq, b L 1 pp0,t q;l 8 pr N qq. This lemma allows us to control the superlevels of b in 1 or dimensions. Proposition 6.. Let b be the vector field in (3.6), with E P L 8 pp0, T q; L pr N qq. For N or N 1, b satisfies (R1), hence also (R1a). Proof. It is clear that v 1 ` x ` v P L8 t pl 8 x,vq, (6.1) and Ept, xq Ept, xq 1 ` x ` v 1 ` x ` v 1 Ept, xq v ď Ept,xq ` 1 ` x ` v 1 v ą Ept,xq Ẽ1 ` Ẽ. (6.) Clearly Ẽ P L 8 t pl 8 x,vq, and if N, Ẽ 1 P L 8 pp0, T q; L 1 x,vq since Ept, xq 1 ` x ` v 1 v ď Ept,xq dxdv R ˆR ˆ 1 (6.3) ď Ept, xq v dv dx π Ept, xq dx. R v ď Ept,xq R In the case N 1, we have directly that Ept, xq{p1 ` v q P L 8 pp0, T q; L x,vq. 6.. The case of three space dimensions. The condition (R1) being not satisfied in 3 dimensions, we need an estimate on Z in order to control the superlevels. For getting this we integrate in space a function growing slower at infinity than logp1 ` Z q (this corresponding to the case (R1)). Proposition 6.3. Let b be as in (3.6) with N 3, E P L 8 pp0, T q; L pr N qq, satisfying (.6) with J P L 8 pp0, T q; L 1 pr N qq. Furthermore, assume that ω `1, ρ ě 0, and ρ b P L 1 X L p pr 3 q for some p ą 3{. Then (R1a)

10 10 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA holds, where the function g depends only on L, T, E L 8 t pl x q, J L 8 t pl 1 x q, }p q 1 ρ b } L 8, and one has gpr, λq Ñ 0 as λ Ñ 8 at fixed r. Proof. Step 1.) Let Z : rt, T s ˆ R 3 ˆ R 3 Ñ R 3 ˆ R 3 be a regular Lagrangian flow relative to b starting at time t, with compression constant L and sublevel G λ. Denoting Z px, V q, we have the ODEs # Xps, 9 x, vq V ps, x, vq, (6.4) 9V ps, x, vq Eps, Xps, x, vqq. Recalling that E x U, one has V ps, x, vq B s V ps, x, vq B s V ps, x, vq Eps, Xps, x, vqq B s Xps, x, vq B s rups, Xps, x, vqqs ` B t Ups, Xps, x, vqq. (6.5) This computation is indeed related to the form of the Hamiltonian for (1.1), H v { ` Upt, xq. We are going to bound the superlevels of V ps, x, vq. We claim that B r sup sprt,t s 1 ` log 1 ` V ps, x, vq { α dxdv ď A, (6.6) where 0 ă α ă 1{3, and for some constant A depending on L, T, r, α, and on the norms E L 8 t pl xq, J L 8 t pl 1 xq, }p q 1 ρ b } L 8. Assume for the moment that this holds. From the lower bound B r sup sprt,t s 1 ` log 1 ` V ps, x, vq { α dxdv ě L 6 pb r z r G λ qp1 ` logp1 ` λ {qq α, (6.7) with G r λ the sublevel of V, we get that L 6 pb r zg r A λ q ď p1 ` logp1 ` λ {qq α. (6.8) Next, we remark that by the first equation in (6.4), whenever px, vq P G r λ one has Xps, x, vq ď x ` s t λ, and Zps, x, vq ď x ` p1 ` T qλ. Thus for λ ą r, one has B r zg λ Ă B r zg r pλ rq{p1`t q, which enables to conclude the proposition (for λ ď r we can just bound L 6 pb r zg λ q by L 6 pb r q). Step.) By Step 1, it is enough to prove that we have a decomposition ˆ 1 ` log ˆ1 ` V ps,, q α ď f 1`f P L 1 pr 3 xˆr 3 vq`l 8 pr 3 xˆr 3 vq, (6.9) for px, vq P B r, where f 1, f are independent of s P rt, T s. Let Then βpyq p1 ` log p1 ` yqq α, for y ě 0. (6.10) β 1 αp1 ` logp1 ` yqqα 1 pyq, 1 ` y 0 ă β p1 ` logp1 ` yqqα 1 pyq ď p1 ` yq. (6.11)

11 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 11 Using (6.5), we compute j ˆ V ps, x, vq B s β ˆ V ps, x, vq B s rups, Xps, x, vqqs ` B t Ups, Xps, x, vqq β 1 j ˆ V ps, x, vq B s Ups, Xps, x, vqqβ 1 ˆ V ps, x, vq ` Ups, Xps, x, vqqβ V ps, x, vq Eps, Xps, x, vqq ˆ V ps, x, vq ` B t Ups, Xps, x, vqqβ 1. Thus, integrating between t and s, ˆ α V ps, x, vq 1 ` log ˆ1 ` αups, Xps, x, vqq 1 ` 1 ` log 1 ` V ps,x,vq 1 α V ps,x,vq (6.1) ˆ α αupt, xq ` 1 α ` 1 ` log ˆ1 ` v 1 ` v 1 ` log 1 ` v s ˆ V pτ, x, vq ` "Upτ, Xpτ, x, vqqv pτ, x, vq Epτ, Xpτ, x, vqqβ t `B t Upτ, Xpτ, x, vqqβ 1 ˆ V pτ, x, vq * dτ. (6.13) Step 3.) Since Ept, q P L pr 3 q, we have by the Sobolev embedding that Upt, q P L 6 pr 3 q. Thus clearly Upt, xq P L 6 pr 3 1 ` v x ˆ R 3 vq Ă L 1 pr 3 x ˆ R 3 vq ` L 8 pr 3 x ˆ R 3 vq. (6.14) Next, since ω `1 and ρ ě 0, one has U U ρ U ρb, with U ρ ě 0. Thus U ě }U ρb } L 8. Thus the first three terms in the expansion (6.13) are upper bounded in L 1 pr 3 x ˆR 3 vq `L 8 pr 3 x ˆR 3 vq. It remains to estimate the integral. We can bound it by Φ 1 ` Φ, with T Φ 1 : ˇ ˆ V pτ, x, vq ˇUpτ, Xpτ, x, vqqv pτ, x, vq Epτ, Xpτ, x, vqqβ t ˇˇˇˇ dτ, (6.15) T Φ : ˇ ˆ V pτ, x, vq ˇBtUpτ, Xpτ, x, vqqβ 1 ˇˇˇˇ dτ. (6.16) t Note that Φ 1, Φ are independent of s. We estimate Φ 1 in L 3{ pr 3 x ˆ R 3 vq Ă L 1 pr 3 x ˆ R 3 vq ` L 8 pr 3 x ˆ R 3 vq. Passing the L 3{ norm under the integral and

12 1 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA changing pxpτ, x, vq, V pτ, x, vqq to px, vq, this gives (up to a factor L) 3{ Upτ, xqv Epτ, xq 1 α dxdv R 3 R ˇ 1 ` v 3 1 ` log 1 ` v ˇ ď Upτ, xqepτ, xq 3{ 3{4 dv dx 9{4 R 3 R 3 1 ` v 1 ` log 1 ` v ď c Upτ, q 3{ Epτ, L 6 pr 3 q 3{ q L pr 3 q. 3p1 αq{ Thus Φ 1 P L 3{ pr 3 x ˆ R 3 vq. Step 4.) For Φ, we notice that E satisfies (.6) and E x U, thus B t U ωp x q 1 div x J. (6.17) Since Jpτ, q P L 1 pr 3 xq, we deduce by the Hardy Littlewood Sobolev inequality that B t Upτ, q M 3{ pr 3 q ď c Jpτ, q L 1 pr 3 q. (6.18) Therefore, we estimate B t Upτ, Xpτ, x, vqq V pτ,x,vq ˇˇˇ 1 ` 1 ` log 1 ` ď L {3 B t Upτ, xq ˇˇˇ 1 ` v 1 ` log ď L {3 B t Upτ, xq ˇˇˇ 1 ` v 1 ` log ď L {3 B t Upτ, q M 3{ pr 3 x q V pτ,x,vq 1 ` v 1 ` v R 3 1 ` v 1 α 1 α 1 α ˇ ˇˇ ˇ ˇˇ ˇ ˇˇ ˇM 3{ pr 3 xˆr 3 vq ˇM 3{ pr 3 xˆr3 v q ˇL3{ pr 3 v;m 3{ pr 3 xqq dv 3{ 1 ` log 1 ` v 3p1 αq{ ď c Jpτ, q L 1 pr 3 q, (6.19) where the last integral is convergent since 3p1 αq{ ą 1. From the inclusion M 3{ pr 3 x ˆ R 3 vq Ă L 1 pr 3 x ˆ R 3 vq ` L 8 pr 3 x ˆ R 3 vq, and integrating (6.16) over B r, we get (6.9) as desired. 7. Renormalized solutions and Lagrangian solutions We recall the different notions of weak solutions for the Vlasov-Poisson system. We shall always assume that 1 ď N ď 3, and we consider an initial datum f 0 P L 1 pr N x ˆR N v q, f 0 ě 0. We introduce first renormalized solutions, following [1, 14]. {3

13 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 13 Definition 7.1. We say that f P L 8 pp0, T q; L 1 pr N x ˆ R N v qq, f ě 0, is a solution to the Vlasov equation (1.1) in the renormalized sense if for all test functions β P C 1 pr0, 8qq with β bounded, we have that B t βpfq ` v x βpfq ` div v Ept, xqβpfq 0, (7.1) in D 1 pp0, T q ˆ R N x ˆ R N v q. We next introduce the notion of Lagrangian solutions. Definition 7.. Let be given a vector field bpt, x, vq pv, Ept, xqq as in (3.6) for some ρ P L 8 pp0, T q; L 1 pr N qq, ρ ě 0, and ρ b P L 1 pr N q. We assume that E P L 8 pp0, T q; L pr N qq, and that (.6) holds with J P L 8 pp0, T q; L 1 pr N qq. We assume furthermore that either N 1 or, or N 3 and ω `1, ρ b P L p pr 3 q for some p ą 3{. We consider regular Lagrangian flows Z as in Definition 4.1, except that now s P r0, T s instead of s P rt, T s (forwardbackward flow), and with compression constant L independent of t P r0, T s. Then according to Subsections 3.3, 3.4, Proposition 6., Proposition 6.3, the vector field b satisfies assumptions (R1a), (Ra), (R3). Therefore, Theorem 5. yields the uniqueness of the forward-backward regular Lagrangian flow Z px, V q. The whole theory of [10] then applies indeed, with very little modifications in the proofs. In particular there is existence and uniqueness of the forward-backward regular Lagrangian flow, with compression constant 1, and stability. We can thus define in accordance with [10] a Lagrangian solution f to the Vlasov equation (1.1) by fpt, x, vq f 0 Xps 0, t, x, vq, V ps 0, t, x, vq, for all t P r0, T s, (7.) for arbitrary f 0 P L 1 pr N x ˆR N v q. It verifies in particular f P Cpr0, T s; L 1 pr N x ˆ R N v qq, and it is indeed also a renormalized solution. Definition 7.3. We define a Lagrangian solution to the Vlasov-Poisson system as a couple pf, Eq such that (1) f P Cpr0, T s; L 1 pr N x ˆR N v qq, f ě 0, v f P L 8 pp0, T q; L 1 pr N x ˆR N v qq, () Ept, xq is given by the convolution (1.6) with ρpt, xq ş fpt, x, vqdv, ρ b P L 1 pr N q, ρ b ě 0 (and if N 3, ω `1, ρ b P L p pr 3 q for some p ą 3{), (3) E P L 8 pp0, T q; L pr N x qq, (4) The relation (.6) holds with Jpt, xq ş vfpt, x, vqdv, (5) f is a Lagrangian solution to the Vlasov equation, in the sense of (7.). 8. Existence of Lagrangian solutions 8.1. Compactness. In this subsection we prove two compactness results, Theorems 8. and 8.3, for families of Lagrangian solutions to the Vlasov- Poisson system, with strongly or weakly convergent initial data. Lemma 8.1. Let gpxq Then for any 1 ă p ă x x N for x P R N, and denote by τ h gpxq gpx ` hq. N N 1, }τ h gpxq gpxq} L p pr N q ď c h α, (8.1)

14 14 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA with α 1 N ` N{p ą 0, and where c depends on N, p. Proof. Fix h P R N, h 0. For x ą h, we have for all 0 ď θ ď 1, x ` θh ě x θ h ą x {, thus we have Then we estimate 8 h p x Np dx c N h p x ą h τ h gpxq gpxq ď h sup gpx ` θhq 0ďθď1 h ď h sup 0ďθď1 c N x ` θh N ď c N h x N. (8.) r N 1 Np dr c N h p p h qn Np Np N c N,p h p Np`N. Next, for x ď h, we write 1 τ h gpxq gpxq ď x ` h N 1 ` 1, (8.3) x N 1 and clearly x ď h ď ˆ 1 x ` h pn 1qp ` 1 x pn 1qp dx y ď3 h dy y pn 1qp c N 3 h since the last integral is convergent for p ă 0 r N Np`p 1 dr c N,p h N Np`p, (8.4) N N 1. Theorem 8.. Let pf n, E n q be a sequence of Lagrangian solutions to the Vlasov-Poisson system satisfying and v fnpt, x, vqdxdv ` f 0 n Ñ f 0 in L 1 pr N x ˆ R N v q, (8.5) E n pt, xq dx ď C, for all t P r0, T s. (8.6) Then, up to a subsequence f n converges strongly in Cpr0, T s; L 1 pr N x ˆR N v qq to f, E n converges in Cpr0, T s; L 1 loc prn qq to E, and pf, Eq is a Lagrangian solution to the Vlasov-Poisson system with initial datum f 0. Moreover, the regular forward-backward Lagrangian flow Z n ps, t, x, vq converges to Zps, t, x, vq locally in measure in R N ˆ R N, uniformly in s, t P r0, T s. Proof. Step 1.) (Equi-integrability) Because of (7.) and (8.5) we have }f n pt,, q} L 1 pr N ˆR N q }f 0 n} L 1 pr N ˆR N q ď M. (8.7) Then because of the bounds (8.7), (8.6), and applying Propositions 6., 6.3, one has for any r ą 0 L N tpx, vq P B r : sup 0ďsďT Z n ps, t, x, vq ą γu Ñ 0, as γ Ñ 8, uniformly in t, n. (8.8)

15 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 15 Since the sequence fn 0 is uniformly equi-integrable, and since Z n is measurepreserving, we have by (7.) and (8.8) that f n pt, q is equi-integrable, uniformly in t, n. Consequently, ρ n pt, q is also equi-integrable, uniformly in t, n. Using the bound (8.6), vf n pt, q is also equi-integrable, and therefore J n pt, q is also equi-integrable, uniformly in t, n. Step.) (Compactness of the field) In order to prove that E n pt, xq Ñ Ept, xq in L 1 loc pp0, T q ˆ RN x q, we first look at the compactness in x. Denote by τ h E n pt, xq E n pt, x ` hq. Then using (1.6), τ h E n pt, q E n pt, q L p pr N q ď c τ x h x N x (8.9) x N ρ n pt, q ρ b p q L 1 pr N q. L p pr N q Thus according to Lemma 8.1, we get for any 1 ă p ă N N 1 that E n pt, x ` hq E n pt, xq L p pr N q Ñ 0, as h Ñ 0, uniformly in t, n. (8.10) Then, because of (.6), we have that B t E n is bounded in L 8 pp0, T q; S 1 pr N qq. Applying Aubin s lemma, we conclude that E n is compact in L 1 loc pr0, T s ˆ R N x q. Thus after extraction of a subsequence, E n pt, xq Ñ Ept, xq strongly in L 1 pp0, T q; L 1 loc prn x qq. Step 3.) (Convergence of the flow) Because of the bound (8.6), one has E P L 8 pp0, T q, L pr N qq. Also, using the uniform bounds on ρ n, J n in L 8 pp0, T q; L 1 pr N qq and the uniform equiintegrability obtained in Step 1, one has up to a subsequence ρ n Ñ ρ, J n Ñ J in the sense of distributions, with ρ, J P L 8 pp0, T q; L 1 pr N qq. We can pass to the limit in (1.6) and (.6). Therefore, b pv, Eq satisfies the assumptions (R1a), (Ra), (R3) and Definition 7. applies. According to [10, Lemma 6.3], since (8.8) holds, and ρ n are equi-integrable, we deduce the convergence of Z n to Z locally in measure in R N x ˆ R N v, uniformly with respect to s, t P r0, T s, where Z is the regular forward-backward Lagrangian flow associated to b. Step 4.) (Convergence of f) Using the convergence (8.5), we can apply [10, Proposition 7.3], and we conclude that f n Ñ f in Cpr0, T s; L 1 pr N x ˆ R N v qq, where f is the Lagrangian solution to the Vlasov equation with coefficient b and initial datum f 0. It follows that ρ n Ñ ρ ş fdv in Cpr0, T s; L 1 pr N x qq. By lower semi-continuity, we get from (8.6) that v fpt, x, vqdxdv ` Ept, xq dx ď C, for all t P r0, T s. (8.11) The bound (8.6) gives also that J n Ñ J ş vfdv in Cpr0, T s; L 1 pr N x qq. Therefore, pf, Eq is a Lagrangian solution to the Vlasov-Poisson system. Using (1.6), we get that E n Ñ E in Cpr0, T s; L 1 loc prn qq, which concludes the proof. Theorem 8.3. Let pf n, E n q be a sequence of Lagrangian solutions to the Vlasov-Poisson system satisfying f 0 n á f 0 weakly in L 1 pr N x ˆ R N v q, (8.1)

16 16 ANNA BOHUN, FRANÇOIS BOUCHUT, AND GIANLUCA CRIPPA and the bound (8.6). Then, up to a subsequence f n converges in Cpr0, T s; weak L 1 pr N x ˆ R N v qq to f, E n converges in Cpr0, T s; L 1 loc prn qq to E, and pf, Eq is a Lagrangian solution to the Vlasov-Poisson system with initial datum f 0. Moreover, the regular forward-backward Lagrangian flow Z n ps, t, x, vq converges to Zps, t, x, vq locally in measure in R N ˆR N, uniformly in s, t P r0, T s. Proof. It is the same as that of Theorem 8., except the last step 4. Instead we apply [10, Proposition 7.7] and conclude that f n á f in Cpr0, T s; weak L 1 pr N x ˆ R N v qq, where f is the Lagrangian solution to the Vlasov equation with coefficient b and initial datum f 0. It follows that ρ n á ρ ş fdv in Cpr0, T s; weak L 1 pr N x qq. By lower semi-continuity, we get again from (8.6) the energy bound (8.11). The bound (8.6) also enables to conclude that J n á J ş vfdv in Cpr0, T s; weak L 1 pr N x qq. Therefore, pf, Eq is a Lagrangian solution to the Vlasov-Poisson system. Using (1.6) and the compactness estimate (8.10), we get that E n Ñ E in Cpr0, T s; L 1 loc prn qq, which concludes the proof. 8.. Existence. We conclude this section by the existence of Lagrangian solutions to the Vlasov-Poisson system for initial datum in L 1 with finite energy, in the repulsive case. Theorem 8.4. Let N 1, or 3, and let f 0 P L 1 pr N x ˆ R N v q, f 0 ě 0. Define ρ 0 and E 0 by ρ 0 pxq f 0 px, vqdv, E 0 pxq ω x S N 1 x N pρ0 pxq ρ b pxqq, (8.13) with ω `1 (repulsive case), ρ b P L 1 pr N q, ρ b ě 0, and in the case N 3 ρ b P L p pr 3 q for some p ą 3{. Assume that the initial energy is finite, v f 0 px, vqdxdv ` E 0 pxq dx ă 8. (8.14) Then there exists a Lagrangian solution pf, Eq to the Vlasov-Poisson system defined for all time, having f 0 as initial datum, and satisfying for all t ě 0 v fpt, x, vqdxdv ` Ept, xq dx ď v f 0 px, vqdxdv ` E 0 pxq dx. (8.15) Proof. We use the classical way of getting global weak solutions to the Vlasov-Poisson system, i.e. we approximate the initial datum f 0 by a sequence of smooth data fn 0 ě 0 with compact support. We approximate also ρ b by smooth ρ n b ě 0 with compact support (with ş pρ 0 n ρ n b qdx 0 if N 1, ). It is possible to do that with the upper bounds limsup v fnpx, 0 vqdxdv ď v f 0 px, vqdxdv, nñ8 (8.16) limsup Enpxq 0 dx ď E 0 pxq dx. nñ8 Then, for each n, there exists a smooth classical solution pf n, E n q with initial datum fn, 0 to the Vlasov-Poisson system, defined for all time t ě 0. Note that

17 LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L 1 DENSITY 17 we can alternatively consider a regularized Vlasov-Poisson system. Since ω `1, the conservation of energy (.8) gives for all t ě 0, v f n pt, x, vqdxdv` E n pt, xq dx v fnpx, 0 vqdxdv` Enpxq 0 dx. (8.17) The couple pf n, E n q is in particular a Lagrangian solution to the Vlasov- Poisson system, for all intervals r0, T s. We can therefore apply Theorem 8.. Extracting a diagonal subsequence, we get the convergence of pf n, E n q to pf, Eq as stated in Theorem 8., where pf, Eq is a Lagrangian solution to the Vlasov-Poisson system defined for all time, with f 0 as initial datum. The bound (8.11), together with (8.16), gives (8.15). Let us end with a remark on measure densities. When considering a sequence of solutions to the Vlasov-Poisson system, the vector fields b n pv, E n q have a gradient in px, vq of the form ˆD1 b Db n 1 n D b 1 ˆ n 0 Id D 1 b n D b, (8.18) n Spρ n ρ b q 0 where the index 1 stands for x, for v, and where S is a singular integral operator. If we require only that D 1 b n converges in the sense of distributions to D 1 b Spρ ρ b q, for some measure ρ P M pr N q, then we are in the setting of [8]. If ρ n is uniformly bounded in L 1 pp0, T q; M pr N qq, and b n Ñ b strongly in L 1 pp0, T q; L 1 loc prn x ˆ R N v qq with b satisfying (8.18), we conclude that Z n Ñ Z strongly, where Z is the regular Lagrangian flow associated to b. However, we are not able to define the push forward (7.) of a measure f 0. This prevents from applying the context of [8] to the Vlasov-Poisson system with measure data. References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158, 7-60, 004. [] L. Ambrosio, M. Colombo, A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields, preprint 014. [3] L. Ambrosio, M. Colombo, A. Figalli, On the Lagrangian structure of transport equations: the Vlasov-Poisson system, preprint 014. [4] L. Ambrosio, G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Transport equations and multi-d hyperbolic conservation laws, 3-57, Lect. Notes Unione Mat. Ital., 5, Springer, Berlin, 008. [5] L. Ambrosio, G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144, , 014. [6] A.A. Arsen ev, Existence in the large of a weak solution of Vlasov s system of equations (Russian), Z. Vycisl. Mat. i Mat. Fiz. 15, , 76, [7] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations 5, , [8] A. Bohun, F. Bouchut, G. Crippa, Lagrangian flows for vector fields with anisotropic regularity, preprint 014. [9] F. Bouchut, G. Crippa, Equations de transport à coefficient dont le gradient est donné par une intégrale singulière, In séminaire EDP, Ecole Polytechnique, , exposé no 1.

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