Lagrangian solutions to the Vlasov-Poisson system with a point charge

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [16], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [8] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.


Introduction and main results
We study the Cauchy problem associated with the Vlasov-Poisson system in the three dimensional space, where f : R + × R 3 × R 3 → R + stands for the non-negative density of particles in a plasma under the effect of a self-induced field E, while ρ : R + × R 3 → R + is the spatial density and γ ∈ {−1, 1} is a parameter which models the repulsive (γ = 1) or attractive (γ = −1) nature of the particles. We recall that the self-induced field E(t, x) is a conservative force. Therefore there exists a function U : R + × R 3 → R such that E(t, x) = ∇ x U (t, x), thus the Poisson equation −∆U = ρ is fulfilled. In other words, we can rewrite the system (1.1) as a Vlasov equation coupled with a Poisson equation, from which the name Vlasov-Poisson arises. From a physical viewpoint, the repulsive case represents the evolution of charged particles in presence of their self-consistent electric field and it is used in plasma physics or in semi-conductor devices. The attractive case describes the motion of galaxy clusters under the gravitational field with many applications in astrophysics. In this paper we focus on the repulsive case, by fixing γ = 1 in (1.1). In the last decades the Vlasov-Poisson system (1.1) has been largely investigated. Existence of classical solutions under regularity assumptions on the initial data goes back to Iordanski [21] in dimension one and to Okabe and Ukai [26] in dimension two. The three dimensional case has been addressed first by Bardos and Degond [6] for small initial data, and then extended to a more general class of initial plasma densities by Pfaffelmoser [28] and by Lions and Perthame [22]. Improvements in three dimensions have been obtained in [30,32,12,23,13]. Global existence of weak solutions has been studied by Arsenev [5] for bounded initial data with finite kinetic energy, while the global existence of renormalized solutions is due to Di Perna and Lions [17], assuming finite total energy and f 0 ∈ L log L(R 3 ×R 3 ). The latter assumption has been recently relaxed to f 0 ∈ L 1 (R 3 × R 3 ) in [3] and [7].
One might wonder what happens when f 0 / ∈ L 1 (R 3 × R 3 ). In this paper we shall address this question by assuming f 0 to be the sum of an integrable bounded plasma density and a Dirac mass. This is equivalent to studying the Cauchy problem associated with the following system: x−y |x−y| 3 ρ(t, y) dy , where the singular electric field F := F (t, x) is induced by a point charge located at a point ξ(t), whose evolution is given by the Newton equations: η(t) = E(t, ξ(t)) .
The model (1.2)-(1.3) has been recently introduced by Caprino and Marchioro in [10], where they have shown global existence and uniqueness of classical solutions in two dimensions. This result has been extended to the three dimensional case in [24] by Marchioro, Miot and Pulvirenti. Both [10] and [24] require that the initial plasma density does not overlap the point charge. This assumption has been relaxed in [16], where weak solutions of the system (1.2)-(1.3) have been obtained for initial data which may overlap the point charge, but do have to decay close to it. The price to pay is that the solution is no longer known to be unique and Lagrangian. In the following we will call Lagrangian solution a plasma density f and a trajectory (ξ, η) of the Dirac mass, both defined for t ∈ R + , such that f is transported by the Lagrangian flow (X, V ), solution to the ODE-system This is a finer physical structural information on the solution than the mere fact that f and (ξ, η) are weak solutions of (1.2)-(1.3).
In the framework of classical solutions, the Eulerian description and the Lagrangian evolution of particles given by the system of characteristics are completely equivalent. When dealing with weak or renormalized solutions, the correspondence between the Eulerian and Lagrangian formulations is non trivial and requires a careful analysis of the Lagrangian structure of transport equations with non-smooth vector fields. Indeed, without any regularity assumptions, it is not even clear whether the flow associated with the vector field generated by a weak solution exists.
In recent years the theory of transport and continuity equations with non-smooth vector fields has witnessed a massive amount of progress, also due to the large number of applications to nonlinear PDEs. In the seminal paper by DiPerna and Lions [17] the theory has been first developed in the context of Sobolev vector fields, with suitable bounds on space divergence and under suitable growth assumptions. This has been extended by Ambrosio [1] to the setting of vector field with bounded variation (BV ), roughly speaking allowing for discontinuities along codimension-one hypersurfaces. See also [4] for an up-to-date survey of this theory and its recent advances.
In the context of the Vlasov-Poisson system with a Dirac mass considered in this paper ((1.2)-(1.3)) the system of characteristics is given by (1.4). The singular electric field F generated by the Dirac mass is not regular, and it does not even belong to any Sobolev space of order one or to the BV space. Therefore the theory of [17,1] cannot be directly applied to this case. However, a related theory of Lagrangian flows for non-smooth vector fields has been initiated in [15]. In a nutshell, the approach in [15] provides a suitable extension of Grönwall-like estimates to the context of Sobolev vector fields, by introducing a suitable functional measuring a logarithmic distance between Lagrangian flows. In addition, the theory in [15] has a quantitative character, providing explicit rates in the stability and compactness estimates, and it has been pushed even to situations out of the Sobolev or BV contexts of [17,1]. In particular, using more sophisticate harmonic analysis tools, the case when the derivative of the vector field is a singular integral of an L 1 function has been considered in [14]. This has been further developed in [8], allowing for singular integrals of a measure, under a suitable condition on splitting of the space in two groups of variables, modelled on the situation for the Vlasov-Poisson characteristics (1.4). This theory has been applied to the study of the Euler equation with L 1 vorticity [9] and of the Vlasov-Poisson equation with L 1 density [7]. The latter has also been studied in [3], using the theory of maximal Lagrangian flows developed in [2].
The purpose of this paper is to recover the relation between the Eulerian and the Lagrangian picture for solutions provided in [16] by exploiting the transport structure of the equation. In other words we aim to prove existence of Lagrangian solutions to the Vlasov-Poisson system (1.1) with γ = 1 and initial data f 0 + δ ξ 0 ⊗ δ η 0 , where f 0 satisfies the assumptions of [16].
Our main result is the following , such that the initial total charge and the total energy is finite. Assume that there exists m 0 > 6 such that for all m < m 0 the energy moments ) and ξ ∈ C 2 (R + ).
2. We observe that the hypothesis (1.5) is needed only to get a control on the electric field generated by the point charge (see Proposition 3.6). This means that the charge of the plasma has to be smaller than the charge associated with the Dirac mass. From the viewpoint of physics, this is a purely technical and too restrictive condition. In a forthcoming paper, we plan to remove this constraint.
3. When considering the Cauchy problem associated with (1.1) with γ = −1 (attractive case) and initial data f 0 + δ ξ 0 ⊗ δ η 0 , the whole strategy fails. This is due to a crucial change of sign in the total energy H and in H m . More precisely, the last two terms in (1.6) and the last term in (1.7), representing respectively the potential energy of the system and the potential energy per particle, come with a negative sign. This prevents to establish a control on the trajectory of the point charge as in Proposition 3.3 and to prove Proposition 3.7.
The simpler case of a system in which the particles in the plasma are interacting through a repulsive potential while the point charge generates an attractive force field has been treated in [11] in dimension two. Notice that, even in this case, the existence of solutions in three dimensions remains an interesting open problem.
4. Theorem 1.1 does not imply uniqueness of the Lagrangian solution. In analogy to [28], where uniqueness of compactly supported classical solutions of (1.1) has been proved, uniqueness of solutions to (1.2)-(1.3) which do not overlap with the point charge and have compact support in phase space has been established in [24]. In the context of weak solutions to (1.1), sufficient conditions for uniqueness have been proved in [22] and later extended to weak measure-valued solutions with bounded spatial density by Loeper [23]. Recently Miot [25] generalised the latter condition to a class of solutions whose L p norms of spatial density grow at most linearly w.r.t. p, then extended to spatial densities belonging to some Orlicz space in [20]. Unfortunately, it seems that none of these conditions apply to our setting and new ideas are needed.
Let us informally describe the main steps of our proof. We rely on the result in [24], which guarantees existence of a (unique) Lagrangian solution to the Cauchy problem for the Vlasov-Poisson system (1.2)-(1.3), provided that at initial time the plasma density has a positive distance from the Dirac mass and bounded support in the phase space. We therefore approximate the plasma density f 0 at initial time by a sequence f n 0 obtained by cutting off f 0 close to the Dirac mass in the space variable and out of a compact set in phase space. We use [24] to construct a Lagrangian flow (X n , V n ) and a trajectory for the Dirac mass (ξ n , η n ) corresponding to the initial data f n 0 and (ξ 0 , η 0 ). The assumptions of Theorem 1.1 together with the propagation of the moments H from [16] entail some additional integrability of the densities ρ n , which in turn implies uniform Hölder estimates on the electric fields E n . Moreover, assumption (1.5) allows to prove some uniform decay of the superlevels of the Lagrangian flows (X n , V n ), which combined with an extension of the Lagrangian theory developed in [8] gives compactness of the Lagrangian flows (X n , V n ). Finally, standard energy estimates guarantee the uniform continuity of the trajectories ξ n uniformly in n. All this enables us to pass to the limit in the Lagrangian formulation of the problem, eventually giving a Lagrangian solution corresponding to the initial plasma density f 0 .
One of the main technical difficulties of our analysis is the control on large velocities. In this work, this reflects in the necessity of some control on the superlevels of the Lagrangian flows. This was already an issue in [7] and here the situation is made even more complicated by the presence of the singular field generated by the point charge. We tackle this problem by weighting superlevels with the measure given by the initial distribution of charges f 0 (x, v) dx dv (see Lemma 4.1). In this way the control on the superlevels can be proven exploiting virial type estimates on the time integral of the electric field generated by the diffuse charge and evaluated in the point charge (see Proposition 3.6). This carries the physical meaning that it is only relevant to control the flow starting from points in the support of the initial density of charge.
In connection to the theory of [8], this weighted estimates manifests in the presence of the density h = f 0 in the functional (2.12) measuring the compactness of the flows. Moreover, in contrast to [7], which was based on the isotropic analysis of [14], here we strongly rely on the anisotropic theory of [8] in which some components of the gradient of the velocity field are allowed to be singular integrals of measures, accounting for the presence of the point charge.
The plan of the paper is the following: in Section 2 we present and prove the key theorem on Lagrangian flows; in Section 3 we recall some useful properties related to solutions of the Vlasov-Poisson system; in Section 4 we give the proof of Theorem 1.1, which follows from compactness arguments by using the results established in Section 2 and 3.

Lagrangian flows
Consider a smooth solution u to a transport equation in with initial data Z(t, t, z) = z. Thus the solution can be expressed as u(t, z) = u 0 (Z(0, t, z)).
For simplicity from now on we will consider the initial time t in (2.1) fixed and denote the flow Z(s, t, z) by Z(s, z).
In this paper we deal with flows of non-smooth vector fields. In order to extend the usual notion of characteristics to our case, we extend the definition of regular Lagrangian flows in a renormalized sense by introducing a reference measure with bounded density. This turns out to be convenient in the estimates involving the superlevels of the flow (see Lemma 4.1). is a µ-regular Lagrangian flow in the renormalized sense starting at time t relative to b if we have the following: (1) The equation (3) There exists a L ≥ 0, called compressibility constant, such that, for every s ∈ [t, T ], We have denoted with L 0 loc the space of measurable functions endowed with the local convergence in measure, by log log L loc the space of measurable functions u such that log(1 + log(1 + |u| 2 )) is locally integrable, and by B the space of bounded functions. When the reference measure µ is not explicitly specified, the spaces under consideration are endowed with the Lebesgue measure.
Remark 1. Our definition of µ-regular Lagrangian flow slightly differs from the one in [8].
On the one hand we change the reference measure from the Lebesgue measure to µ. On the other hand we consider a different class of β's, which grow slower at infinity. (2.4)

Setting and result of [8]
We summarize here the regularity setting and the stability estimate of [8]. We say that a vector field b satisfies (R1) if b can be decomposed as . Notice that this hypothesis leads to an estimate for the decay of the superlevels of a regular Lagrangian flow. In fact Lemma 3.2 of [8] tells us that, if b satisfies (R1) and Z is a regular Lagrangian flow associated with b starting at time t, with compressibility constant L, then L d (B r \ G λ ) ≤ g(r, λ) for any r, λ > 0, where g depends only on L, b 1 L 1 ((0,T );L 1 (R d )) and b 2 L 1 ((0,T );L ∞ (R d )) and satisfies g(r, λ) ↓ 0 for r fixed and λ ↑ ∞.
(R2) We want to consider a vector field b(t, z) such that its regularity changes with respect to different directions of the variable z ∈ R d , that is we consider R d = R n 1 × R n 2 and z = (z 1 , z 2 ) with z 1 ∈ R n 1 and z 2 ∈ R n 2 . We denote with D 1 the derivative with respect to z 1 and D 2 the derivative with respect to z 2 . Accordingly we denote b = (b 1 , b 2 )(s, z) ∈ R n 1 ×R n 2 and Z = (Z 1 , Z 2 )(s, z) ∈ R n 1 × R n 2 . Therefore we assume that the elements of the matrix Db, denoted as (Db) i j , are in the form where -S i jk are singular integral operators associated with singular kernels of fundamental type in R n 1 (see [31]), We have denoted by We recall the main theorem from [8].
Theorem 2.3. Let b andb be two vector fields satisfying assumption (R1), where b satisfies also (R2), (R3). Fix t ∈ [0, T ] and let Z andZ be regular Lagrangian flows starting at time t associated with b andb respectively, with compressibility constants L andL. Then the following holds. For every γ, r, η > 0 there exist λ, C γ,r,η > 0 such that The constants λ and C γ,r,η also depend on: • The equi-integrability in L 1 ((0, T ); L 1 (R n 1 )) of all the m i jk which belong to this set, as well as the norm in L 1 ((0, T ); M(R n 1 )) of the remaining m i jk (where these functions are associated with b as in (R2)), • The norms of the singular integrals operators S i jk , as well as the norms of γ i jk in L ∞ ((0, T ); L q (R n 2 )) (associated with b as in (R2)), • The compressibility constants L andL.

Flow estimate in the new setting
We are going now to state a variant of this theorem, where (R1) and (R2) are replaced by (R1a) and (R2a) below. The dimension d will be here equal to 2N , instead of n 1 + n 2 , and the variable z will be in the form We consider the following assumptions, that are adapted to our setting of the Vlasov-Poisson system with a point charge: (R1a) For all µ-regular Lagrangian flow Z : [t, T ] × R 2N → R 2N relative to b starting at time t with compression constant L, and for all r, λ > 0, where G λ denotes the sublevel of the flow Z defined in (2.2).
and where b 2 is such that for every j = 1, . . . , N , where S jk are singular integrals of fundamental type on R N and m jk ∈ L 1 ((0, T ); M(R N )).
Proof. The proof follows the same line as in Theorem 2.3 (see [8]), with some modifications due to the different hypotheses. Given δ 1 , δ 2 > 0, let A be the constant 2N × 2N matrix We consider the following functional depending on the two parameters δ 1 and δ 2 , with δ 1 ≤ δ 2 : In order to improve the readability of the following estimates, we will use the notation " " to denote an estimate up to a constant only depending on absolute constants and on the bounds assumed in Theorem 2.4, and the notation " λ " to mean that the constant could also depend on the truncation parameter for the superlevels of the flow λ. The norm of the measure m however will be written explicitly.
Step 1: Differentiating Φ δ 1 ,δ 2 . Differentiating with respect to time and taking out of the integral the L ∞ norm of h, we get Then we set Z(s, x, v) = Z andZ(s, x, v) =Z and we estimate After a change of variable along the flowZ in the first integral, and noting that δ 1 ≤ δ 2 , we further obtain Step 2: Splitting the quotient. Using the special form of b from (R2a) and the action of the matrix A −1 , we have Step 3: Definition of the function U. Using assumption (R2a), we can now use the estimate of [14] on the difference quotient of b 2 , where U for fixed s is given by with M j a certain smooth maximal operator on R N x .
Step 4: We can estimate the L p (Ω) norm of Ψ by considering the first element of the minimum and changing variables along the flows: (2.14) Considering now the second element of the minimum and eq.n (2.13), we can also bound the M 1 (Ω) pseudo-norm of Ψ (where M 1 is the Lorentz space): From Theorem 2.10 in [8], we know Step 5: Interpolation. We have now the ingredients to apply the Interpolation Lemma 2.2 in [14], which allows to bound the norm in L 1 (Ω) of Ψ using Ψ L p (Ω) and |||Ψ||| M 1 (Ω) as follows: .

Uniqueness, stability and compactness
In this subsection we use the result obtained in Theorem 2.4 to show uniqueness, stability, and compactness of the regular Lagrangian flow.  • The measure of the superlevels associated with Z n in hypothesis (R1a) is bounded by some functions g n (r, λ) which go to zero uniformly in n as λ → ∞ at fixed r, • The sequence {L n } is equi-bounded.
Then the sequence {Z n } converges to Z locally in measure with respect to µ in R 2N , uniformly in s and t.
Proof. We setb = b n andZ = Z n in Theorem 2.4, then there exist two positive constants λ and C γ,r,η , which are independent of n, such that for all s ∈ [0, T ] it holds In particular, for any r, γ > 0 and any η > 0, we can choosen large enough so that µ(B r ∩ {|Z(s, ·) − Z n (s, ·)| > γ}) ≤ 2η for all n ≥n and s ∈ [t, T ], which is the thesis. • The measure of the superlevels associated with Z n in hypothesis (R1a) is bounded by some functions g n (r, λ) which go to zero uniformly in n as λ → ∞ at fixed r, • For any compact subset K of R 2N , is equi-bounded in n and s, t, • For some p > 1 the norms b n L p ((0,T )×Br ) are equi-bounded for any fixed r > 0, • The norms of the singular integral operators associated with the vector fields b n (as well as their number m) are equi-bounded, • The norms of m n jk in L 1 ((0, T ); M(R N )) are equi-bounded in n. Then as n → ∞ the sequence {Z n } converges to some Z locally in measure with respect to µ, uniformly with respect to s and t, and Z is a regular Lagrangian flow starting at time t associated with b.
Proof. We apply Theorem 2.4 with b = b n andb = b m . Observe that the compressibility constants L andL of the same theorem are equal to 1. Indeed b andb are divergence free as they both satisfy assumption (R2a). Hence we have for any r, γ > 0 µ(B r ∩ {|Z n (s, ·) − Z m (s, ·)| > γ}) → 0 as m, n → ∞, uniformly in s, t.
Thus it follows that Z n converges to some Z ∈ C([t, T ]; L 0 loc (R 2N , dµ)) locally in measure with respect to µ, uniformly in s, t. The uniformity in n and s, t of the bound (2.23) implies Z ∈ B([t, T ]; log log L loc (R 2N , dµ)). We notice that conditions (2) and (3) in Definition 2.1 are satisfied, since thanks to (R2a) the vector fields b n are divergence free. We are left with the proof of condition (1). Observe that a β ∈ C 1 (R 2N ) can be approximated by a sequence of β ǫ ∈ C 1 c (R 2N ), therefore it suffices to show condition (1) for this latter class of functions. To this end we want to perform the limit in n of equation (2.2) written for Z n and b n . From the convergence in measure of Z n to Z and the fact that Z n and Z lie in a fixed ball B r (the support of β ǫ ) it follows the convergence in distributional sense of β ǫ (Z n ) to β ǫ (Z) and of β ′ ǫ (Z n ) to β ′ ǫ (Z). While using the uniform bound of b n L p ((0,T )×Br ) and Lusin's Theorem, we get convergence in L 1 loc of b n (Z n ) to b(Z). Thus we have convergence in the sense of distribution to equation (2.2).
The above compactness statement does not directly translate into an existence result for Lagrangian flows, since in general it is not trivial to find a sequence b n approximating b as in the hypotheses of Corollary 2.7. This is due to the fact that the function g(r, λ) in Lemma 4.1 does not depend only on bounds on the vector field, but also on bounds on the density of charge. We are able to do this in the specific case of the flow associated with the Vlasov-Poisson equation (solution to (1.4)) and therefore we postpone this to Section 4.

Useful estimates
In this Section we recall some well known a priori estimates on physical quantities related to the Vlasov-Poisson equation and we adapt them to the context of the system (1.2)-(1.3).
where C is a constant depending only on s.

Proposition 3.2 (Mass and energy conservation). Let
be respectively the total mass and the total energy associated with the system (1. Proof. It follows from direct inspection by performing the time derivative of M (t) and H(t).
As a consequence of Proposition 3.2, we observe that if the energy H(t) is assumed to be initially finite, then it is bounded for all times. This ensures in particular that the velocity of the Dirac mass located at ξ(t) is finite. Proof. We observe that H(t) is a sum of positive terms. Notice that here we are heavily using the electrostatic nature of the particles in the plasma. In the gravitational case, the total energy has a nonpositive term. By Proposition 3.
Then there exists a constant C > 0, which only depends on m, such that . (3.6) Proof. By definition of ρ we have . (3.7) Fix R > 0 and split the integral in the v variable into two pieces: By optimising in R in the last line of the above inequality, we get . (3.8) We plug (3.8) in (3.7) and we obtain (3.9) (1.2). Assume the total energy to be initially finite, then ρ(t, ·) ∈ L 1 ∩ L 5/3 (R 3 ) and E(t, ·) ∈ L q (R 3 ), for any Proof. The bound ρ(t, ·) ∈ L 5/3 (R 3 ) follows by Proposition 3.4 for m = 2. The estimate on the electric field is a consequence of Proposition 3.1 for s = 1 and s = 5 3 .
The following two propositions regard specifically the case in which we deal with a Dirac mass and their proof relies on the condition that the total charge M (0) has to be strictly less than one. This is the only reason why we need to assume (1.5) in Theorem 1.1. (3.10) Proof. For s ∈ [0, T ], consider (X(s, x, v), V (s, x, v)) solution to the characteristic system (1.4) with initial data (x, v). We now use the shorter notation (X(s), V (s)) and compute Then we obtain (3.11) By integrating the above expression w.r.t. time and the measure f 0 (x, v) dx dv, we get (3.12) The first term in the r.h.s. of (3.12) can be bounded as follows where we used Hölder inequality and the conservation of mass and energy in the latter estimate. The second term in (3.12) is bounded by means of Hölder inequality and Proposition 3.5: (3.14) We use (3.13) and (3.14) in the r.h.s. of (3.12) and we obtain that concludes the proof since M (0) < 1.
(ii) There exists m 0 > 6 such that for all m < m 0 Then there exists a global weak solution (f, ξ) to the system (1. Moreover, for all t ∈ R + and for all m < min(m 0 , 7), where C and c only depend on the initial data. Let f 0 and (ξ 0 , η 0 ) be the initial data of system (1.2), satisfying the hypotheses of Theorem 1.1. We consider the approximating initial densities given by Thanks to [24], this choice ensures existence and uniqueness of f n and (ξ n , η n ), solutions to the Vlasov-Poisson system (1.2)-(1.3). Moreover f n is a Lagrangian solution, i.e.

(4.4)
From now on the abstract measure µ of Section 2 will be set as µ = f 0 L 2N , where f 0 is the initial density of our problem. In order to apply Corollary 2.7, we need then the approximating vector fields b n (s, x, v) = (v, E n (s, x) + F n (s, x)) to satisfy hypotheses (R1a), (R2a), and (R3) "uniformly" in n (with equi-bounds on the quantities involved) and the bound (2.23). Furthermore we set the dimension N equal to 3.
Proof of (R1a) + equibound: control of superlevels In [8] a control on the superlevels was obtained using hypothesis (R1) which provided an upper bound on the integral of log(1 + |Z|). Without assumption (R1), we need estimates on |V | 2 in order to control the superlevels. This requires integrating a function which grows slower than log(1 + |V |) at infinity. Furthermore, differently from [7], we will bound the superlevels of Z with respect to the measure µ = f 0 L 6 . For the sake of clarity we will use the notation f 0 (B) to indicate the measure µ of a set B ⊆ R 6 . The result is the following lemma, whose proof is postponed to Subsection 4.2.
be the µ-regular Lagrangian flow relative to b starting at time t, with sublevel G λ . Assume M (0) < 1. Then, for all r, λ > 0, we have , H(0), and g(r, λ) ↓ 0 for r fixed and λ ↑ ∞. Notice that this lemma holds also for the regularized problem (system (1.2)-(1.3) with initial density f n 0 ). Therefore we have, for all r, λ > 0, where g n converges to zero for r fixed and λ ↑ ∞. Moreover, this convergence is uniform in n. Indeed the proof of Lemma 4.1 entails the functions g n to be increasing with respect to the norms of E n , U n , F n , f n , and with respect to H n (0). These quantities are in turn all bounded by the same quantities without the index n. Therefore, due to the choice of the initial densities of the regularized problem, we have where g n (r, λ) depends on the norms of E, U , F , f and on H(0), and tends to zero as λ → ∞ uniformly in n. Moreover the last two terms tend to zero as n → ∞ by Lebesgue's Dominate Convergence Theorem. Hence we have, for any fixed ǫ, r > 0, that there exist λ > 0 and N ∈ N such that f 0 (B r \G n λ ) ≤ ǫ (4.7) for each n ≥ N .
Proof of (R2a): spatial regularity x), we observe that the Lipschitz constants of b n 1 and b 1 are trivially equi-bounded. We are left to show that the derivatives of b n 2 and b 2 are singular integrals of fundamental type on R 3 of finite measures, and that the norms of the kernels associated with the singular integral operators and those of the measures in L 1 ((0, T ); M(R 3 )) are equi-bounded. We compute, outside of the origin, Therefore ∂ x j (b 2 ) i is a singular integral of the finite measure ρ + δ ξ(t) , with kernel The kernel satisfies conditions of Def.2.13 in [14], therefore it is a singular kernel of fundamental type. Similarly we have ∂ x j (b n 2 ) i = K ij (·) * (ρ(t, ·) + δ ξn(t) ), hence also ∂ x j (b n 2 ) i are singular integrals of finite measures, with equi-bounded kernels and equi-bounds on the measures' norms.

Proof of (R3)
We shall prove now that the L p -norms of b and b n in (0, T ) × B r are equi-bounded, for some p > 1 and for any fixed r > 0. Through an easy computation we notice that the M 3/2 -pseudo-norms of F and F n are equi-bounded and uniform in t: Similarly we have that the L 1 -norms of F and F n are equi-bounded in (0, T ) × B r for any r > 0: sup Furthermore Propositions 3.1 tells us that E and E n belong to L ∞ ((0, T ); M 3/2 (R 3 )), with the respective pseudo-norms which are equi-bounded in n. Therefore the second component of the vector fields b and b n (i.e. E + F , E n + F n ) are equi-bounded in the space Since v ∈ L p loc ((0, T ) × R 3 ) for any p, we conclude that b, b n belong to L p loc ((0, T ) × R 3 ) for any 1 ≤ p < 3 2 , with uniform bound on the norms.
Proof of the equi-boundedness of (2.23) We observe that |Z n | ≤ |X n | + |V n | ≤ |x| + (1 + T )|V n | . Thus it suffices to prove the equi-boundedness of (4.16) for the regularised flow V n . This is a byproduct of the proof of Lemma 4.1, where we show that the constant A depends on quantities which are uniformly bounded in n.

Conclusion of the proof of Theorem 1.1: existence of Lagrangian solutions to the Vlasov-Poisson system
Let f 0 be as in Theorem 1.1. In order to prove existence of a Lagrangian solution to system (1.2)-(1.3), we use a compactness argument. For each n, we consider the initial datum f n 0 defined in (4.1), which converges to f 0 . The result in [24] ensures existence and uniqueness of the classical Lagrangian solution f n , (ξ n , η n ) to the Vlasov-Poisson system with point charge where (ξ n (t), η n (t)) evolves according to   ξ n (t) = η n (t) , η n (t) = E n (t, ξ n (t)) , (ξ n (0), η n (0)) = (ξ 0 , η 0 ) .

(4.10)
Therefore, there exists a unique flow Z n = (X n , V n ) : . From Subsection 4.1, there exists Z such that Z n → Z in measure, with respect to µ = f 0 L 6 . Therefore we define a density f which is the push forward of the initial data f 0 through the limiting flow Z, i.e.
The aim of this subsection is to verify that the above defined f is indeed a solution to (1.2)-(1.3). In other words, we want to perform the limit n → ∞ in (4.9)-(4.10) and get (1.2)-(1.3). This will conclude the proof of Theorem 1.1. To this end we observe that, up to subsequences: • f n ⇀ f weakly in L 1 x,v and weakly * in L ∞ x,v , uniformly in t. Indeed, f n 0 → f 0 in L 1 x,v and Z n → Z in measure µ. Since the latter limit is uniform in s and t, we define the inverse of the flow Z −1 n (t, s, x, v) := Z n (s, t, x, v) and observe that Z −1 n → Z −1 in measure and therefore µ-a.e., uniformly in t. Given ϕ ∈ C c (R 3 × R 3 ), we can estimate The first term in the r.h.s. converges to zero, since Z n → Z µ-a.e. The second term also converges to zero because ϕ is bounded and f n 0 → f 0 in L 1 x,v . Moreover, since f n is equi-bounded in L 1 x,v ∩ L ∞ x,v , uniformly in t, we obtain weak convergence in L 1 x,v and weak * convergence in L ∞ x,v of f n to f , uniformly in t.
• ρ n ⇀ ρ weakly in L 1 x . It follows from the weak L 1 x,v convergence of f n to f . Moreover, thanks to Remark 2, ρ n ⇀ ρ weakly in L s x , for some s > 3.
• ∂ t f n converges to ∂ t f in D ′ and v · ∇ x f n converges to v · ∇ x f in D ′ .
• E n → E uniformly. This is a consequence of Proposition 3.7. Indeed, the r.h.s. of equation (3.15) is uniformly bounded in n. Therefore, by Proposition 3.4, ρ n L m+3 3 is uniformly bounded and Proposition 3.1 yields {E n } n equi-Hölder. Ascoli-Arzelà Theorem guarantees the existence of a uniformly convergent subsequence. The limit couple (E, ρ) satisfies E(t, x) = x−y |x−y| 3 ρ(t, y) dy, since E ∈ M 3/2 and decays at infinity, while ρ ∈ L s , for some s > 3.
, and by the facts that E n → E uniformly and f n ⇀ f weakly in L 1 x,v . We are left with the part of the system (4.9)-(4.10) which involves the point charge. In particular, we define γ n (t) = (ξ n (t), η n (t)) (4.11) and set (ξ(t), η(t)) := lim n→∞ γ n (t) . (4.12) Observe that the limit in (4.12) exists. Indeed, γ n (t) is equi-Lipschitz because of the following estimate: where Lip(γ n ) is the Lipschitz constant of γ n . Proposition 3.3 yields a uniform bound on the first term in the r.h.s. of (4.13), that combined with the uniform bounds on E n proved in this subsection, implies γ n equi-Lipschitz. By Ascoli-Arzelà Theorem, there exists a subsequence {(ξ n k (t), η n k (t))} k which converges uniformly to (ξ(t), η(t)). To perform the limit in (4.9)-(4.10), we observe that • (ξ n (t),η n (t)) → (ξ(t),η(t)). Indeed, (ξ n (t), η n (t)) converges to (ξ(t), η(t)) uniformly and 14) The first term in the r.h.s. of (4.14) converges to zero uniformly. As for the second term, we use that Combining the facts that E n → E, ξ n → ξ and E is uniformly continuous, the last line in (4.15) vanishes as n → ∞.
• F n → F in L 1 x, loc . Indeed, F n → F pointwise, by the uniform convergence of ξ n (t) to ξ(t) up to subsequences, and F n , F ∈ L 1 loc (R 3 ). Therefore, we conclude by Dominated Convergence's Theorem.
• F n · ∇ v f n → F · ∇ v f in D ′ . This follows by rewriting F n · ∇ v f n = div v (F n f n ) and F · ∇ v f = div v (F f ), and by the facts that F n → F in L 1 loc (R 3 ) and f n *
For Φ 4 we compute  Since the denominator of the integrand is bounded, we can estimate the above quantity as follows: where in the last inequality we used Proposition 3.6. Thus, condition (4.16) is satisfied and the proof is completed thanks to (4.17).