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Mean-field limits for some Riesz interaction gradient flows

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open.


Introduction
We consider the energy of a system of N particles in the Euclidean space R d (d ≥ 1) interacting via (repulsive) Riesz pairwise interactions: where the interaction kernel is given by with c d,s > 0 some normalization constants. We note that the Coulomb case corresponds to the choice s = d − 2, d ≥ 2. Particle systems with more general Riesz interactions as considered here are extensively motivated in the physics literature (cf. for instance [2,20]), as well as in the context of approximation theory with the study of Fekete points (cf. [14] and the references therein). Recently, a detailed description of such systems beyond the mean-field limit in the static case was obtained in [21], and also in [17] for non-zero temperature. In the present contribution, we are rather interested in the dynamics of such systems, and more precisely in a rigorous justification of the mean-field limit of their gradient flow evolution as the number N of particles tends to infinity, which has indeed remained an open problem whenever s ≥ d − 2, s > 0, d ≥ 2.
We thus consider the trajectories x t i,N driven by the corresponding flow, i.e. the solutions to the following system of ODEs: where (x • i,N ) N i=1 is a sequence of N distinct initial positions. Since energy can only decrease in time and since the interaction is repulsive, particles cannot collide, and moreover it is easily seen that a particle cannot escape to infinity in finite time; from these observations and from the Picard-Lindelöf theorem, we may conclude that the trajectories x t i,N are smooth and well-defined on the whole of R + := [0, ∞). As the number of particles gets large, we would naturally like to pass to a continuum description of the system, in terms of the particle density distribution. For that purpose, we define the empirical measure associated with the point-vortex dynamics: 2) and the question is then to understand the limit of µ t N as N ↑ ∞. More precisely, assuming convergence at initial time formal computations lead us to expect under fairly general assumptions µ t N * − ⇀ µ t for all t ≥ 0, where µ t is a solution to the following nonlocal nonlinear PDE on R + × R d , sometimes called the fractional porous medium equation: This equation in the weak sense just means the following: µ ∈ L 1 loc (R + ; L 1 (R d )), g s * µ ∈ L 1 loc (R + ; W 1,1 loc (R d )), µ∇g s * µ ∈ L 1 loc (R + ; L 1 (R d )), and, for all φ ∈ C ∞ (R + ; C ∞ c (R d )) such that φ(t, ·) = 0 for all t > 0 large enough, As far as existence issues as well as basic properties of the solutions of (1.3) are concerned, we refer to [9,8] for d − 2 < s < d, s ≥ 0, to [19,12,1,29] for s = d − 2, d ≥ 2, and to [10] for 0 ≤ s < d − 2. See also Proposition 2.3 below.
In the case of logarithmic interactions s = 0, this expected mean-field limit result was essentially first proven (in arbitrary dimension) by Schochet [27] based on his simplification [26] of the proof of Delort's theorem [11] on existence of weak solutions to the 2D Euler equation with initial nonnegative vorticity in H −1 . Schochet's original paper [27] was actually only concerned with the mean-field limit for a particle approximation of the 2D Euler equation, but the same argument directly applies to the present setting. However, due to a possible lack of uniqueness of L 1 weak solutions to equation (1.3), Schochet [27] could only prove that the empirical measure µ t N converges up to a subsequence to some solution of (1.3). The key idea, which only holds for logarithmic interactions, consists in exploiting some logarithmic gain of integrability to find uniform bounds on the number of close particles, which allows to directly pass to the limit in the equation and conclude by a compactness argument.
In the case 0 ≤ s < d − 2, d ≥ 3, the complete mean-field limit result (not restricted to a subsequence) was proven more recently by Hauray [15] (see also [10]), but his method, based on a control of the infinite Wasserstein distance, cannot be adapted at all to higher powers s ≥ d − 2. In the 1D case, Berman and Önnheim [3] obtained a similar result for the whole range 0 ≤ s < 1, in the framework of Wasserstein gradient flows, but their method cannot be extended to higher dimensions.
Very recently, in the context of the 2D Gross-Pitaevskii and parabolic Ginzburg-Landau equations, Serfaty [28] proposed a new way of proving such mean-field limits 1 , based on a Gronwall argument for the so-called modulated energy, which is some adapted measure of the distance to the (postulated) limit. This idea of proof originates in the relative entropy method first introduced by Yau [31] for hydrodynamic limits (see e.g. [23] and the references therein for later developments), the modulated Hamiltonian method used by Grenier [13] for boundary layer problems, and the modulated energy method designed by Brenier [6] for the quasi-neutral limit of the Vlasov-Poisson system. The advantage of this method is to be completely global, bypassing the need for a precise understanding of the microscopic dynamics. It relies on the regularity of the solution to the limiting equation, and exploits its stability properties. As will be seen, however, we are able to apply this method in the present context only in dimensions 1 and 2 and for s not too large. More precisely, we treat in 1D the whole range 0 ≤ s < 1 as in [3], while in 2D we treat but the case 0 ≤ s < 1, which is new and in particular completes Schochet's partial result [27] in the logarithmic case. Our main result, for which we need an additional regularity assumption on the limiting equation (cf. Remark 1.2(a) below), is as follows: Theorem 1.1. Let d = 1 or 2, and 0 ≤ s < 1. Let µ • be a probability measure, and in the case s = 0 also assumé for some T > 0 and some σ > 2 − d + s. In the case s = 0, d = 1, also assume ∇µ ∈ L ∞ ([0, T ]; L p (R d )) for some p < ∞. Let µ • N * − ⇀ µ • as above, assume the convergence of the initial energy , which remains a compactly supported probability measure for all times (see Proposition 2.3 below). As far as the additional regularity assumption is concerned, as explained in Section 2.3, it has been proven to hold with T = ∞ in the case 0 ≤ s ≤ d − 2, d ≥ 2, and at least up to some time T > 0 in the case d − 2 < s ≤ d − 1, s ≥ 0, for sufficiently smooth initial data µ • , but all other cases remain unsolved, and the additional regularity assumption in Theorem 1.1 above is then crucially needed.
In the 2D Coulomb case s = 0, d = 2, as this regularity problem is solved (cf. [19,Theorem 1]), the conclusion of Theorem 1.1 above holds in that case with T = ∞ (that is, with L ∞ ([0, T ]; ·) replaced by L ∞ loc (R + ; ·)), whenever we have µ • ∈ P(R 2 ) ∩ C σ (R 2 ) for some σ > 0, and´R 2 log(2 + |x|)|µ • (x)|dx < ∞. This completes Schochet's partial result [27]. As the regularity problem is further solved in short time in Sobolev spaces in the case d − 2 < s ≤ d − 1, s ≥ 0 (cf. [30]), the conclusion of Theorem 1.1 above holds for some T > 0 (depending on initial data) in the case 0 < s ≤ 1, d = 2, whenever µ • ∈ P(R 2 ) ∩ H σ (R 2 ) for some σ > 2, and also in the 1D logarithmic case s = 0, d = 1, whenever µ • ∈ P(R) ∩ H σ (R) for some σ > 3/2, and A closer look at the proof actually shows the following quantitative statement, where the distance between µ t N and µ t is measured in terms of the modulated energy: for all t ∈ [0, T ], we have for some C t > 0 x =y (c) The well-preparedness assumption (1.4) for the initial positions (x • i,N ) N i=1 , N ≥ 0, is statistically relevant, in the sense that it is automatically satisfied almost surely if, for each N , the positions (x • i,N ) N i=1 are chosen to be independent R d -valued random variables with law µ • . This easily follows from the strong law of large numbers, together with the bound (for s > 0) (d) We may also add to the energy a potential V , thus turning H N intoH N : then all the arguments may be directly adapted, as long as the corresponding limit equation admits a regular enough solution.
(e) In dimension d = 2, we could also consider a mix between the gradient flow (1.1) and its conservative counterpart, that is, replacing (1.1) by the following system of ODEs, for i = 1, . . . , N , where we have also added a potential V as in item (d) above. If α > 0, then all the arguments may again be directly adapted, as long as the corresponding limit equation admits a regular enough solution. (Note that the same proof can a priori not work for the choice α = 0, since in Step 2 of the proof of Proposition 2.8 below some term cannot be estimated directly and needs instead to be absorbed using the negative diffusion term, which would however vanish in the case α = 0.) Notation. Denote by B(x, r) the ball of radius r centered at x in R d , and set B r := B(0, r). We also use the notation a ∧ b = min{a, b} and a ∨ b = max{a, b} for all a, b ∈ R. The space of probability measures on R d is denoted by P(R d ), and, for all σ > 0, C σ (R d ) stands as usual for the Hölder space C ⌊σ⌋,σ−⌊σ⌋ (R d ), while C σ c (R d ) denotes the subspace of compactly supported functions. In the sequel, C denotes any positive constant only depending on d and s. We denote by C t any positive constant only depending on d, s and on time t, such that C t ≤ C T for all t ∈ [0, T ] and all T > 0. We also denote x y and x t y for x ≤ Cy and x ≤ C t y, respectively, and we use the notation x ≃ y if both x y and y x hold. Finally, we denote by o a (1) a quantity that goes to 0 when the parameter a goes to its limit, uniformly with respect to other parameters, and we write o 2 Proof of Theorem 1.1

Strategy of the proof
Translating the idea of [28] in the present setting (see also [6]), the clue of the proof of Theorem 1.1 comes from the following stability estimate, which we first present for simplicity in the Coulomb case.
Proof. Proposition 2.3(ii) below yields ∇(h 1 − h 2 ) ∈ L ∞ (R + ; L 2 (R d )). Combining this with the additional boundedness assumptions, all integration by parts arguments in the sequel may be justified. Using the equations for µ t 1 and µ t 2 , the time derivative of the left-hand side of (2.1) can be computed as follows: 2) The first term in the right-hand side is nonpositive, so it suffices to estimate the second one. Using the relations −∆h t i = µ t i , i = 1, 2 (which hold with a unit factor for the suitable choice of the normalization constant c d,d−2 > 0), may be rewritten à la Delort using the stress-energy tensor: where the divergence of a 2-tensor here denotes the vector whose coordinates are the divergences of the corresponding columns of the tensor. Combining this with an integration by parts, we find The inequality 2|ab| ≤ a 2 + b 2 and an integration by parts then yield so that the result (2.1) follows from (2.2) and a Gronwall argument.
We are also interested here in the non-Coulomb case d − 2 < s < d, s ≥ 0, and hence, just as in [21], we need to use the extension method of Caffarelli and Silvestre [7] (cf. Section 2.2 below) in order to find a similar Delort-type formula as in (2.3) of the proof above, and then repeat the same integration by parts argument, thus circumventing the fact that the Riesz kernel is not the convolution kernel of a local operator. This allows to prove the same estimate as above in all cases 0 ∨ (d − 2) ≤ s < d with g d−2 replaced by g s (cf. Lemma 2.2 below).
This stability estimate gives us a control of the H −1 -distance (or the H −(d−s)/2 -distance, for general 0 ≤ s < d) between µ t 1 and µ t 2 in terms of the initial distance, up to a factor that only depends on the regularity of µ t 2 in the form of ∇ 2 h t 2 L ∞ . We would then like to replace µ t 2 by the smooth solution µ t and to replace µ t 1 by µ t N . However, the corresponding distance would then be infinite because of the presence of Dirac masses, and moreover µ t N does not exactly satisfy the limiting equation (1.3). The idea of the proof of Theorem 1.1 consists in finding a suitable way of adapting the proof above to that setting.
First, the natural way of giving a meaning to this divergent distance between µ t N and µ t simply consists in excluding the diagonal terms, thus considering the (renormalized) modulated energy The goal is then to compute the time-derivative ∂ t E N (t), and trying to adapt the proof of the stability estimate above to bound it by CE N (t) for some constant C > 0, up to a vanishing additive error. However, at the end of the proof above, the use of the inequality 2|ab| ≤ a 2 + b 2 is clearly not compatible with the removal of the diagonal terms. To solve this main issue, the crucial idea is due to Serfaty [28]: regularizing the Dirac masses at a (small) scale η so that the diagonal terms become well-defined and diverge only as η ↓ 0, we need to try to construct around the particle locations small balls that contain most of the divergent η-approximate energy, so that excluding diagonal terms essentially amounts to restricting the η-approximate integrals to outside these small balls. Using the same approximation argument as in [28] to be allowed to restrict all integrals to outside these balls, the end of the proof above is then easily adapted, using the inequality 2|ab| ≤ a 2 + b 2 only on the restricted domain. In this way, for any 0 under some mesoscopic regularity assumption on the distribution of the particles in time (cf. Proposition 2.8 below). Finally, in the case s < 1 (hence our limitation to that regime), these conditions can be directly checked using a modification of the ball construction introduced by [24,16] for the analysis of the Ginzburg-Landau vortices (cf. Section 2.7 below).

Extension representation for fractional Laplacian
We recall here the extension representation for the fractional Laplacian by Caffarelli and Silvestre [7] (we follow notation of [21, Section 1.2]). Let 0 ∨ (d − 2) < s < d be fixed. For a finite Borel measure µ on R d , the associated Riesz potential (−∆) −(d−s)/2 µ can be written (up to a constant) as (2.5) We denote coordinates in R d × R by (x, ξ), and we denote by µδ R d ×{0} the Borel measure on R d × R defined as follows: Extending The normalization constant c d,s is chosen exactly to satisfy this property with a unit factor (for an explicit value, see Step 1 of the proof of Lemma 2.11 below).
In the case for the suitable choice of the normalization constant c 1,0 , so the above again holds with Using this extension representation, we may now directly adapt the stability estimate of Lemma 2.1 to the non-Coulomb case: Proof. By Lemma 2.1, we only need to consider the case d − 2 < s < d, Combining this with the boundedness assumptions, all integration by parts arguments in the sequel may be justified. Just as in (2.2), the time derivative of the left-hand side of (2.6) is The first term in the right-hand side is nonpositive, so it suffices to estimate the second one. Using the relations − div(|ξ| γ ∇h t i ) = µ t i δ R d ×{0} , for i = 1, 2, we find the following proxy for the Delort-type formula (2.3): for all Combining this with an integration by parts, we obtain Hence, arguing as in Lemma 2.1, an integration by parts yields

Properties of the fractional porous medium equation
Let us first state for weak solutions to (1.3) an existence result as well as some basic properties. We refer to [9,8] for d − 2 < s < d, s ≥ 0, and to [19,1,29,4] [10,Section 4], but there existence is proven only for short times, while in the present repulsive context it can easily be extended to all times, using that no blow-up can occur in finite time. 2 Proposition 2.3 (Existence for the fractional porous medium equation). Let 0 ≤ s < d. ( where the left-hand side remains finite. Moreover, for all t ≥ 0, where both sides remain finite. Also, if µ • is compactly supported, then µ t remains compactly supported for all t ≥ 0. In the case d−2 < s < d, s ≥ 0, uniqueness remains an open problem: it has been obtained in dimension 1 by [5] (integrating the equation with respect to x and then considering viscosity solutions), but in higher dimensions no result is known (cf. [9,8]). Nevertheless, as a consequence of the stability result of Lemma 2.2, we easily find that uniqueness of bounded weak solutions always follows from the existence of a smoother solution, so the problem is somehow reduced to a regularity question: Corollary 2.4 (Weak-strong uniqueness for the fractional porous medium equation).
Proof. Let µ t be a weak solution to (1.3) as in the statement, and let ν t denote another weak solution to (1.3) in This proves ∇g s * µ t = ∇g s * ν t , and hence, applying the operator − div(|ξ| γ ·) to both sides, µ t = ν t for all t ∈ [0, T ]. We may argue similarly in the Coulomb case s = d − 2, d ≥ 2.
As the following lemma shows, the required boundedness of ∇ 2 h t is implied by a sufficient amount of Hölder regularity for µ t .
Proof. Without loss of generality we may assume µ ∈ C ∞ c (R d ), as the claimed result then follows by an obvious approximation argument. We first prove that for any µ ∈ C ∞ c (R d ) the Riesz potential h µ satisfies (2.10). We only argue for the second gradient ∇ 2 h µ L ∞ , the other part being similar and easier. Let we find by multiple integrations by parts and hence, for any 2 − d + s < σ < 1, the result (2.10) again follows from a direct computation. As far as the additional L p -boundedness is concerned, it is a direct consequence of the L p -boundedness of Riesz transforms for 1 < p < ∞, simply noting that we have Motivated by these considerations, we would like to prove at least that the regularity of the initial condition is conserved along the flow, so that the boundedness of ∇ 2 h t would be ensured by the above lemma for sufficiently smooth initial data. In the Coulomb case s = d − 2, d ≥ 2, any weak solution µ t to (1.3) can be shown to belong [19,Theorem 1], which is easily rewritten in any dimension). By a similar but easier argument, the corresponding result can also be proven to hold in the case 0 ≤ s < d − 2, d ≥ 3. In the case d − 2 < s ≤ d − 1, s ≥ 0, a recent result [30] shows that there exists a time T > 0 (depending on initial data) such that any weak solution µ t to (1.3) belongs to In the case d − 1 < s < d, however, even such a short-time regularity result is unknown. This is why we needed to add some regularity assumption in the statement of Theorem 1.1.

Modulated energy and elementary properties
Let 0 ≤ s < d, and let µ • , µ • N , µ t , µ t N be as in the statement of Theorem 1.1, for some T ∈ (0, ∞). Let N ≥ 1. Since µ • N is assumed to have bounded energy, and since the energy is decreasing along the flow, we find If 0 < s < d, since g s is nonnegative, this proves If s = 0, g 0 changes sign and some more work is then needed: noting that by symmetry As g 0 (u) + c −1 d,0 u 2 ≥ 0 for all u, this proves that (2.11) also holds in the case s = 0. Next, we recall the truncation procedure introduced in [21], which serves to make energies finite without removing the diagonal. For fixed N ≥ 1, let η > 0 be small enough such that 2η < 1 ∧ η N . Then define where δ (η) z denotes the uniform unit Dirac mass on the sphere ∂B(z, η). Denote for simplicity h t N,η := g s * µ t N,η , and use the same notation for their extensions to R d × R as in Section 2.2. Also define g s,η := g s (η) ∧ g s . Noting that by symmetry − div(|ξ| γ ∇g s,η ) = δ Let us now introduce our notation for the small balls around the particle locations, which we will be crucially using in the proof: for all R > 0, let B t N (R) denote a union of disjoint balls with total radius R = m r t m,N and such that x t i,N ∈ B t N (R) for all 1 ≤ i ≤ N . These balls will be carefully chosen in Section 2.7 below.
As already announced, for all N ≥ 1, we will consider the following modulated energy where D := {(x, x) : x ∈ R d } denotes the diagonal. This quantity can be thought of as a natural renormalization of the H −(d−s)/2 -distance in the presence of Dirac masses. Its main property is as follows: Lemma 2.6 (Modulated energy). For all t ≥ 0, if the sequence (µ t N ) N is tight, then the following two conditions are equivalent: Proof. Property (ii) clearly implies (i) (and even E N (t) → 0), so it suffices to check the converse. Assume that lim sup N E N (t) ≤ 0. By tightness, up to extraction of a subsequence, the Prokhorov theorem gives µ t N * − ⇀ ν t for some ν t ∈ P(R d ). For any K > 0, we may writë and hence, successively passing to the limits N ↑ ∞ and K ↑ ∞, we find (2.14) Combining this with convergence µ t N * − ⇀ ν t and with assumption lim sup N E N (t) ≤ 0, we obtain The result then follows, noting that µ t has bounded energy by Proposition 2.3, that ν t has bounded energy by (2.14), and noting that for any two Radon measures µ, ν with finite energy we havê In the case of bounded weak solutions µ t 1 , µ t 2 to (1.3) as given by Proposition 2.3, the following identity follows from an integration by parts and was crucially used in the proofs of Lemmas 2.1 and 2.2 (cf. (2.4) and (2.8)): Now we would need a corresponding identity in the context of the modulated energy E N (t). Since ∇h t N does not belong to L 2 (R d ) or L 2 (R d × R, |ξ| γ dxdξ), a regularization is then needed. Besides the modulated energy E N , we thus define the following η-approximation, based on the truncation introduced above: An integration by parts then yields the following proxy for identity (2.15), showing that the difference between the modulated energy E N (t) and its approximation E N,η (t) just comes from the diagonal terms (which are indeed excluded in E N (t) but not in E N,η (t)). We refer to [21, Section 2.1] for a detailed proof.

Gronwall argument on the modulated energy
By Lemma 2.6, in order to prove convergence µ t N * − ⇀ µ t as well as convergence of energies, up to tightness issues, it suffices to check that lim sup N E N (t) ≤ 0. This is achieved by a Gronwall argument. From now on we focus on the Riesz case d − 2 < s < d, s ≥ 0. The Coulomb case s = d − 2, d ≥ 2 can be treated in exactly the same way, but is actually easier since it does not require to use the extension representation of Section 2.2. Proposition 2.8. Let d − 2 < s < d, s ≥ 0. Let µ • be a probability measure such that equation (1.3) admits a solution µ t that belongs to L ∞ ([0, T ]; P(R d ) ∩ C σ (R d )) for some T > 0 and some σ > 2 − d + s. In the logarithmic case s = 0, d = 1, also assume that´R log(2 + |x|)|µ • (x)|dx < ∞, and ∇µ ∈ L ∞ ([0, T ]; L p (R)) for some p < ∞.

16)
and let µ t N be defined by (1.1)-(1.2). Assume that, for all t ∈ [0, T ], the collection B t N (R t N ) can be chosen with R t N → 0 in such a way that

17)
and, denoting g + s (t) := c −1 d,s t −s for s > 0 and g + 0 (t) : Remark 2.9. In the ideal case when all particles remain well-separated, that is with a minimal distance η N of order N −1/d , then, taking B t N (R t N ) to be the union of balls of radius R t N /N centered at the points On the other hand, neglecting interactions between particles, hence focusing on the (divergent) self-interactions, we formally find so that condition (2.17) would amount to requiring g s (R t N /N )/N ≪ 1, which is thus just the same as condition (2.18). In other words, for s > 0, both conditions would then take the form R t N ≫ N −(1−s)/s , which is compatible with R t N → 0 only if s < 1. In Section 2.7, we prove that a consistent choice of the small balls B t N (R t N ) is indeed possible whenever 0 ≤ s < 1. To go beyond the restriction s < 1 via this approach, we would need to modify Proposition 2.8, in particular by refining the (blind) approximation argument used in Step 2 of the proof below, in order to relax the smallness condition for the total radius R t N → 0. To do that, precise microscopic information on the particle dynamics would become needed. Getting a handle on such information seems however to be a difficult task and is not pursued here.
Proof. By the regularity assumption for µ t , Lemma 2.5 ensures that we have (∇ x h t , ∇ 2 x h t ) L ∞ t 1, and also, in the case s = 0, d = 1, ∇ 2 x h t L p t 1 for some p < ∞. We split the proof into four steps.
Step 1: Time-derivative of E N (t) and modulated stress-energy tensor. In this step, we prove equality and where the modulated stress-energy tensor T t N = (T t;kl N ) d+1 k,l=1 is defined as follows: for all 1 ≤ k, l ≤ d, Moreover, as checked at the end of this step, the integrals in (2.19) are summable: more precisely, we prove that Although the second term in the right-hand side of (2.19) is nonpositive, we do not bound it by 0 yet, contrarily to what is done in the proof of Lemmas 2.1 and 2.2, since it will be useful in Step 2 below to absorb some error terms.
Using the equations satisfied by µ t and by the trajectories x t i,N , and noting that the gradient ∇h t is given by where the principal value may only be omitted for s < d−1, we find the following expression for the time-derivative of the modulated energy E N (t) defined in (2.13): Let us rearrange the terms as follows: and note that the last four terms in the right-hand side may be combined to yield the following simpler expression: .
In the distributional sense on R d , using canonical regularizations, we may alternatively write where S t N = (S t;k N ) d k=1 and, for all 1 ≤ k ≤ d,

Now note the following algebraic identity in the distributional sense on
where we have set This proves the (Delort-type) identity for all 1 ≤ k ≤ d, and the conclusion (2.19) then follows from (2.21), (2.22) and an integration by parts. We now turn to the claimed integrability of the modulated stress-energy tensor T t N . We first consider the case d − 2 < s < d, s > 0. For that purpose, we begin with the bound Let us compute the integral over R d × R. Denoting for simplicity c yz := (y + z)/2 and q := s + 1, we decompose, for all y = z,ˆR where I 1 yz :=ˆ| x−y|≤ 1 2 |y−z| (|x − y| 2 + 1) −q/2 (|x − z| 2 + 1) −q/2 dx, Using that |x − y| ≤ 1 2 |y − z| implies |x − z| ≥ 1 2 |y − z|, we may estimate and similarly for I 2 yz . Moreover, and also, since d − 2q < 0 follows from the choice s > d − 2, s ≥ 0, This proves, for all y = z, and hence by scalingˆR so that we obtain, as by definition γ = q − d ∈ (−1, 1), Splitting the integrals over ξ into the part where |ξ| ≤ |y − z| and that where |ξ| > |y − z|, and noting that q > 1 follows from s > 0, we find Combining this with (2.25) finally yieldŝ and hence, by assumption (2.16), since both the particle and the mean-field energies are decreasing along the flow (see Proposition 2.3 for the mean-field energy), We now briefly consider the case s = 0, d = 1 (hence γ = 0, q = 1). Let 1 < p < ∞ be such that ∇ 2 h t L p t 1. Arguing as above, we obtain and similarly, by the Hölder inequality, for 1/p + 1/p ′ = 1, p ′ > 1, Splitting the integral over ξ into the part where |ξ| ≤ |y − z| ∨ 1 and that where |ξ| > |y − z| ∨ 1, we may then estimateˆR so that the conclusion now easily follows just as in the case s > 0.
Step 2: Approximation argument. For all t ≥ 0 and all R ∈ (0, 1), applying [25, Proposition 9.6], there exists a smooth approximation v t of the function ∇h t ∈ C 0,1 (R d ; R d ) such that v t is constant on each ball of the collection and also satisfies ∇v t L p t 1 for some p < ∞ in the case s = 0, d = 1. In this step, we prove the following estimate: where o R (1) denotes a quantity that goes to 0 as R ↓ 0.
Using relation (2.24) as well as the integrability properties of T t N , we may decompose the first term in the right-hand side of (2.19) as follows: It remains to estimate the last term in the right-hand side of (2.28). Denoting for simplicity w t := ∇h t − v t , we may decompose For the first term in the right-hand side of (2.29), we simply have by (2.26) As far as the second term is concerned, choosing σ > s + 1 − d, 0 ≤ σ < 1, and recalling that µ t remains bounded by assumption, we find by (2.26) Combining these two bounds with (2.29), using (2.26) once again, we obtain, for R ↓ 0, and thus, noting thatˆR we find Hence, for all ǫ ∈ (0, 1), using inequality R|a| ≤ ǫa 2 + (4ǫ) −1 R 2 , we obtain and the result (2.27) then follows from (2.28) and (2.19), choosing ǫ > 0 small enough (depending on t).
Step 3: Modification with η-approximations. In the definition (2.20) of T t N , the diagonal terms were excluded. In order to apply inequality 2|ab| ≤ a 2 + b 2 to T t N as in the proof of Lemmas 2.1 and 2.2, we would need to add these diagonal terms explicitly. Moreover, η-approximations then become needed in order to avoid the divergence of the corresponding diagonal terms that will appear after application of the above-mentioned inequality. More precisely, we prove in this step By the choice of v t to be constant on each ball of the collection B t N (R) and the bound on ∇v t , (2.27) becomes Denote for simplicity and define, for all 1 ≤ k ≤ d + 1 and all 1 ≤ l ≤ d, T t;k,d+1 N,η = 0 and Also noting that definition (2.23) may be rewritten as Combining this with (2.32) yields, for all From (2.31), we then deduce, for all η > 0 small enough such that The result (2.30) then follows, using inequality 2|ab| ≤ a 2 + b 2 in the form of Step 4: Conclusion. In this step, we show that The statement of Proposition 2.8 immediately follows from this inequality, with the suitable choice of R = R t N , together with a simple Gronwall argument.
By Lemma 2.7, inequality (2.30) may be rewritten as follows: or equivalently, expanding the square, The last term in the first line of (2.34) is easily estimated as follows, using the notation (2.12) for the union B t N (R) of small balls, while the term in the second line of (2.34) is, in the case s > 0, In the case s = 0, d = 1 (so γ = 0), denoting ρ t i,N := d(x t i,N , ∂B t N (R)), and applying the Hölder inequality with 1/p + 1/p ′ = 1, where p < ∞ is chosen in such a way that ∇v t L p t 1, the term in the second line of (2.34) is and hence, by the choice p ′ > 1, The result (2.33) then follows from inequality (2.34) together with (2.35) and with (2.36) or (2.37).

Bypass of tightness issues
If we know that the sequence (µ t N ) N is tight, Lemma 2.6 then allows us to conclude with the desired convergence µ t N * − ⇀ µ t , while tightness can easily be checked under the additional assumption that the initial measures µ • N 's are well localized in the sense that lim sup N´| x| 2 dµ • N < ∞. However, in the spirit of [28, Section 4.3.5], the following refinement of Lemma 2.6 shows that much more information may be directly extracted from the fact that E N (t) ≤ C t o N (1), so that in particular tightness is obtained a posteriori without any additional assumption. − ⇀ µ t . In particular, (µ t N ) N is tight and Lemma 2.6 then implies the convergence of the energy. Proof. By assumption, Proposition 2.8 yields E N (t) t o N (1). We split the proof into three steps.
Step 1: Strong convergence outside small balls. In this step, we provë and hence, for any 1 ≤ p ≤ 2, the Hölder inequality implies for all R > 0 Applying Lemma 2.7 and expanding the square, we may decompose as follows the L 2 (R d × R, |ξ| γ dxdξ)-norm of ∇(h t N,η − h t ) outside the small balls B t N : Applying Proposition 2.8 in the form of E N (t) t o N (1), and using assumption (2.17), this turns intö Now arguing just as in (2.35), we find and hence¨( Passing to the limit η ↓ 0 in this inequality, and noting that ∇h t N,η → ∇h t N in the distributional sense, the result (2.38) follows.
Step 2: Neglecting the contribution inside small balls. The contribution inside the small balls B t N is of course infinite, since ∇h t N does not belong to L 2 (R d ×R, |ξ| γ dxdξ). However, we show that it is small in L p loc (R d ; L 2 (R, |ξ| γ dξ)) for p small enough. More precisely, for any 1 ≤ p < 2d/(s + d), we show that we have for all R > 0 A direct computation of the integral over ξ yields As for each i the integral over x ∈ B R is clearly bounded above by the same integral over x ∈ B R (x t i,N ), we obtain , and hence, for any 1 ≤ p < 2d/(s + d), Now, for any 1 ≤ p < 2d/(s + d), choosing any p < q < 2d/(s + d), the Hölder inequality yieldŝ The result (2.40) follows from this and from the Hölder inequality in the form Step 3: Conclusion. Combining (2.39) and (2.40), for any 1 ≤ p < 2d/(s + d), we may conclude, for all R > 0, This proves ∇h t N → ∇h t in L p loc (R d ; L 2 (R, |ξ| γ dξ)) for any 1 ≤ p < 2d/(s + d). Applying the operator − div(|ξ| γ ·) to both sides, we deduce µ t N → µ t in the distributional sense on R d × R, so the result is proven.

Ball construction
In this section, we make the heuristics of Remark 2.9 rigorous, showing that for 0 ≤ s < 1 the collection B t N (R t N ) can indeed be chosen with R t N → 0 in such a way that both conditions (2.17) and (2.18) are satisfied. Let us first describe the construction that we will use for the collection B t N (R), R > 0. This is precisely the same construction as the one used e.g. in [25,Chapter 4], which was introduced by [24,16] for the analysis of the Ginzburg-Landau vortices. We first consider N disjoint small balls centered at the points x t i,N 's with equal radii (smaller than η N /2), and we grow their radii by the same multiplicative factor. At some point during this growth process, two (or more) balls may become tangent to one another. We then merge them into a bigger ball: if tangent balls are of the form B(a i , r i ), we merge them into B( i a i r i / i r i , i r i ). If the resulting ball intersects other balls, we proceed to another merging, and so on, until all the balls are again disjoint. Then again we grow all the resulting radii by a multiplicative factor, etc., and we stop when the total radius R is the one desired.
As we will see, condition (2.18) is easily checked directly from the construction above, so we may focus on the validity of condition (2.17). Hence we need to study integrals of the form´B t N (R)×R |ξ| γ |∇h t N,η | 2 for R > 0. For that purpose, the basic tool is then the following crucial lower bound, which is a refinement of [21, Lemma 2.2]. In the sequel, for x ∈ R d and t > 0, we denote by B ′ (x, t) the ball of radius t centered at (x, 0) in R d × R, and we set B ′ t := B ′ (0, t). Lemma 2.11 (Embryo of a lower bound). Let R > r, let (z i ) k i=1 be a collection of points inside the ball B r , and let (z k+i ) l i=1 be a collection of points outside the ball B R . Then dxdξ ≥ k 2 (g s (r) − g s (R)). (2.41) The same remains true if point charges are smeared out on small spheres around them, i.e. if g s is replaced by g s,η with η < d({z i } k+l i=1 , B R \ B r ). In particular, for any z 1 , any R > η > 0, and any collection (z i ) 1+l i=2 of points outside the ball B(z 1 , R + η), ∇g s,η (x − z i , ξ) 2 dxdξ ≥ g s (η) − g s (R). (2.42) With this result at hand, as in [25,Chapter 4], we may now deduce the following lower bound estimate for the energy on the balls of the collection B t N (R). For logarithmic interactions (thus in particular for the Ginzburg-Landau vortices, as treated in [25,Chapter 4]), a particularly simple additive structure shows up, simplifying computations a lot; here we show that the same result still holds for all s ≤ 1.
Proposition 2.12 (Lower bound). Let R > 0. If s ≤ 1, then, for all 0 < η < η N ∧ (R/N ), B t N (R)×R |ξ| γ |∇h t N,η | 2 ≥ 1 N (g s (η) − g s (R/N )). (2.44) Proof. We prove that, for all R > 0, if B(y, r) is a ball belonging to the collection B t N (R) and contains n of the particles x t i,N 's, thenˆB ′ (y,r) |ξ| γ |∇h t N,η | 2 ≥ n N 2 (g s (η) − g s (R/N )). (2.45) The desired result (2.44) indeed follows from summing the corresponding inequalities (2.45) associated with each ball B(y, r) of the collection B t N (R), noting that B ′ (y, r) ⊂ B(y, r) × R. We prove (2.45) by induction: we first show that it holds when B(y, r) contains only one particle x t i,N , and then that it is preserved through the growth process.
First, suppose that B(y, r) is a ball of B t N (R) and contains only one particle x t i,N . By definition we must have B(y, r) = B(x t i,N , r) and x t j,N / ∈ B(y, r + η) for all j = i. Lemma 2.11 in the form of (2.42) then yieldŝ B ′ (y,r) |ξ| γ |∇h t N,η | 2 ≥ 1 N 2 (g s (η) − g s (r)).
On the other hand, suppose that B(y i , r i ), i = 1, . . . , k, are k disjoint balls of B t N (R − ) for some R > 0, suppose that each of them satisfies (2.45), and suppose that these balls are merged by the growth process into a larger ball B(y, r), which is then disjoint of all other balls of the collection B t N (R). Denoting by n i the number of points in B(y i , r i ), we then find B ′ (y,r) n i (g s (η) − g s (R/N )), so that B(y, r) also satisfies (2.45). This completes the proof.
We are now in position to prove that both conditions (2.17) and (2.18) may be satisfied whenever s < 1, thus finishing the proof of Theorem 1.1. On the other hand, since by definition N i=1 B(x t i,N , R/N ) ⊂ B t N (R), we may estimate Therefore, both conditions (2.17) and (2.18) are satisfied if we choose R t N such that 1 N g + s (R t N /N ) ≪ 1, and the result follows.