The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness

The Cucker-Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency e.g. to aggregate, flock or disperse. The paper examines the kinetic Cucker-Smale equation with a singular communication weight. Given a compactly supported measure as an initial datum we construct a global in time weak measure-valued solution in the space $C_{weak}(0,\infty;\mathcal{M})$. The solution is defined as a mean-field limit of the empirical distributions of particles, which dynamics is governed by the Cucker-Smale particle system. The studied communication weight is $\psi(s)=|s|^{-\alpha}$ with $\alpha \in (0,\frac 12)$. This range of singularity admits sticking of characteristics/trajectories. The second result concerns the weak--atomic uniqueness property stating that a weak solution initiated by a finite sum of atoms, i.e. Dirac deltas in the form $m_i \delta_{x_i} \otimes \delta_{v_i}$, preserves its atomic structure. Hence they coincide with unique solutions to the system of ODEs associated with the Cucker-Smale particle system.


Introduction
Flocking, swarming, aggregation -there is a multitude of actual real-life phenomena that from the mathematical point of view can be interpreted as one of these concepts. The mathematical description of collective dynamics of self-propelled agents with nonlocal interaction originates from one of the basic equations of the kinetic theory -Vlasov's equation from 1938. Recently it was noted that such models provide a way to describe a wide range of phenomena that involve interacting agents with a tendency to aggregate their certain qualities. This approach proved to be useful and the language of aggregation now appears not only in the models of groups of animals but also in the description of seemingly unrelated phenomena such as the emergence of common languages in primitive societies, distribution of goods or reaching a consensus among individuals [3,26,28,35]. The main class of kinetic equations associated with aggregation models reads as follows: where f = f (x, v, t) is usually interpreted as the density of those particles that at time t have position x and velocity v. The function k is the kernel of the potential governing the motion of particles. It is responsible for the non-local interaction between particles and depending on it the particles may exhibit various tendencies like to flock, aggregate or to disperse. The common properties of kernels k required in most models include Lipschitz continuity and boundedness. In such case the particle system associated with (1) is well posed, the characteristic method can be performed for (1) and one can pass from the particle system to the kinetic equation by the mean-field limit procedure. Our goal is to consider singular k that is neither Lipschitz continuous nor bounded and refine the mean-field limit to be applicable in such scenario. Singularity of the kernel admits possibility of sticking of characteristics, which causes a loss of backward uniqueness in time for (1). The present paper studies the case of the Cucker-Smale (CS) flocking model.
In [14] from 2007, Cucker and Smale introduced a model for the flocking of birds associated with the following system of ODEs: where N is the number of the particles while x i (t), v i (t) and m i denote the position and velocity of ith particle at time t and its mass, respectively. Function ψ : [0, ∞) → [0, ∞) usually referred to as the communication weight is nonnegative and nonincreasing and can be vaguely interpreted as the perception of particles. The communication weight plays the crucial role in our investigations. It characterizes mathematical difficulties and determines possible physical interpretations of the studied model.
As N → ∞ the particle system is replaced by the following Vlasov-type equation: which can be written as (1) with k(x, v) = vψ(|x|). As mentioned before we are considering (3) with a singular kernel ψ(s) = s −α for s > 0, ∞ for s = 0, α > 0. (4) Before we proceed with a more detailed statement of our goals let us briefly introduce the current state of the art for models of flocking and the motivations behind studying such models with singular kernels. The literature on aggregation models associated with Vlasov-type equations of the form (1) is rich thus we mention only a few examples of the most popular branches of the field. Here we find analysis of time asymptotics (see e.g. [21]) and pattern formation (see e.g. [20,34]) or analysis of the models with additional forces that simulate various natural factors (see e.g. [10,17] -deterministic forces or [13] -stochastic forces). The other variations of the model include forcing particles to avoid collisions (see e.g. [11]) or to aggregate under the leadership of certain individuals (see e.g. [12]). A well rounded analysis of a model that includes effects of attraction, repulsion and alignment is presented in [4]. The story of the CS model should probably begin with [36] by Vicsek et al., where a model of flocking with nonlocal interactions was introduced and it is widely recognized to be up to some degree an inspiration for [14]. Since 2007 the CS model with a regular communication weight of the form was extensively studied in the directions similar to those of more general aggregation models (i.e. collision avoiding, flocking under leadership, asymptotics and pattern formation as well as additional deterministic or stochastic forces -see [2,9,19,22,29,32]). Particularly interesting from our point of view is the case of passage from the particle system (2) to the kinetic equation (3), which in case of the regular communication weight was done for example in [23] or [24]. For a more general overview of the passage from microscopic to mesoscopic and macroscopic descriptions in aggregation models of the form (1) we refer to [5,15,16].
In the paper [23] from 2009 the authors considered the CS model with the singular weight (4) obtaining asymptotics for the particle system but even the basic question of existence of solutions remained open till later years. It turned out that system (2) possesses drastically different qualitative properties depending on whether α ∈ (0, 1) or α ∈ [1, ∞). More precisely in [1] the authors observed that for α ≥ 1 the trajectories of the particles exhibit a tendency to avoid collisions, which they used to prove conditional existence and uniqueness of smooth solutions to the particle system. On the other hand in [30] the author proved existence of so called piecewise weak solutions to the particle system with α ∈ (0, 1) and gave an example of solution that experienced not only collisions of the trajectories but also sticking (i.e. two different trajectories could start to coincide at some point). This dichotomy is an effect of integrability (or of the lack of thereof) of ψ in a neighborhood of 0. It is also the reason why the approach to the CS model should vary depending on α. One of the latest contributions to this topic is [6] where the authors showed local in-time well posedness for the kinetic equation (3) with a singular communication weight (4) and with an optional nonlinear dependence on the velocity in the definition of F( f ). They also presented a thorough analysis of the asymptotics for this model. The other more recent addition is [31], where the author proved existence and uniqueness of W 1,1 strong solutions to the particle system (2) with a singular weight (4) and α ∈ (0, 1 2 ).
Our present results are strongly dependent on regularity of solutions to the particle system, hence we assume, after [31], that throughout the paper α ∈ (0, 1 2 ). However, let us mention that these results (in particular Proposition 3.1) can be generalized to models of flocking with nonlinear dependence on the velocity in the alignment force term F( f ). Such models are considered for example in [6], where the force term takes the form In particular φ with large β reduces the impact of the singularity of ψ. This approach allows to push the singularity of the communication weight ψ up to 1. However, this interesting extension is outside of the scope of this paper.
As a final remark we suggest a possible application of this type of mathematical models. The phenomenon of sticking of trajectories gives a possibility of creation of Dirac measure solutions from regular distributions. Like in the case of [8] for the equations of attraction/repulsion. Here we think about formation of polymers from a solution of monomers. Still this qualitative nature of the CS model is an open question. However it would give a nice description of polymerization. Due to the possibility of sticking of trajectories the kinetic structure of the considered model is more complex than in [25] where the authors considered singular kinetic models that preserved L ∞ norm of the distribution of the particles.
In the model with regular weight its purpose is to suppress the distant interactions between particles. However from the modeling point of view it is often convenient to also amplify the local interactions, which was done for example in [27] by introducing a different nonsymmetric CS-type model known as the Mosch-Tadmor model. Singular communication weight in the CS model can also be viewed as a less effective yet easier to analyze way to emphasize the local interactions between particles.

Main goal -the CS model with a singular communication weight
We aim at solving the issue of well posedness for (3) with the singular weight (4) and initial data from the class of Radon measures. The goal is twofold: -Prove existence and analyze continuous dependence on the initial data. The existence is obtained by approximating measure solutions to (3) by solutions to particle system (2) using the mean-field limit, similarly to [23]. The key obstacle is a lack of sufficient information about the continuous dependence for solutions to particle system (2), thus we are not allowed to apply the standard approach. To our best knowledge the most that can be assumed is solvability of (2) in the W 1,1 -class that has been proved in [31]. Therefore by results from [31] we restrict our considerations to α ∈ (0, 1 2 ) and modify the mean-field limit procedure to that regularity. -Prove the weak-atomic uniqueness property to system (3). It means that any weak solution is unique and corresponds to a solution to the particle system (2) provided it initiates from a finite sum of Dirac's deltas m i δ x i (t) ⊗ δ v i (t) . Thus any atomic solution is preserved by kinetic equation (3), and since it is generated by particle system (2), it is unique. This result is a step in the direction of stability of solutions to (3). We elaborate further on the difficulties in obtaining stability in Remark 3.2.
The paper is organized as follows. In section 2 we provide the preliminary definitions and tools required throughout the paper, in particular we introduce the weak formulation for (3). In section 3 we state the main result along with the overview of the proof. In section 4 we present the proof of the existence part, while in section 5 we establish the weak-atomic uniqueness of the solutions. The paper is closed with Appendix A where one can find more technical/tedious elements of proofs and a simple proof of uniqueness to particle system (2).

Preliminaries and notation
In this section we present the basic toolset and definitions of considered problems. Let Ω ⊂ R d be an arbitrary domain with d ∈ N. By W k,p (Ω) we denote the Sobolev's space of the functions with up to kth weak derivative belonging to space L p (Ω), while by C(Ω) and C 1 (Ω) we denote the space of continuous and continuously differentiable functions, respectively. Hereinafter, Ra). Throughout the paper letter C denotes a generic positive constant that may change in the same inequality and usually depends on other constants that are of less importance from the point of view of estimates.

Bounded-Lipschitz distance
The standard tool used in the studies of Vlasov-type models are Wasserstein metrics. They provide convenient topologies on the space of Radon measures and are often used in the research on CS model. The general definition of Wasserstein distances is relatively complex, however Wasserstein-1 (or Kantorovich-Rubinstein) distance is, in the sense explained in [33] p. 26, equivalent to the easily defined bounded-Lipschitz distance. Due to such convenient representation we use only the bounded-Lipschitz distance but it is worth to note that in similar cases (e.g. [6]) other Wasserstein metrics can be applied.
where the supremum is taken over all bounded and Lipschitz continuous functions g, such that g ∞ ≤ 1 and Lip(g) ≤ 1.
In the above definition g ∞ and Lip(g) represent the L ∞ norm and Lipschitz constant of g.
We also need to distinct between spaces of measures with different topologies i.e. we denote M = M(Ω) = (M, T V) as the space of finite Radon measures defined on Ω with total variation topology and we denote (M, d) as the space of finite Radon measures defined on Ω with bounded-Lipschitz distance topology. The importance of the space (M, d) comes from the prime difference between the bounded-Lipschitz distance and the total variation. Namely, for x 1 x 2 , In our considerations a crucial role is played by There is a convenient relation between the weak * topology in (C b (Ω)) * and the topology generated by the bounded-Lipschitz distance on M that we present below.
where |µ| is the total variation measure of µ. Proof. The proof of Proposition 2.1 is a modification of the proof of Theorem 2.7 from [18].
Remark 2.1. Since in our considerations Ω is compact, our situation is more straightforward than in a general case (e.g. in [18]). In such case the so called weak convergence of measures (otherwise known as the narrow convergence) is equivalent to the weak * convergence in (C b (Ω)) * .
In our considerations we deal only with nonnegative measures, which equipped with the bounded-Lipschitz distance are a complete metric space.
The following corollary is the very reason for which the bounded-Lipschitz distance is applied. It serves us as a topology with pointwise sequential compactness for measure-valued functions. What we mean is that if  Proof. Since the supports of µ n are uniformly bounded then there exist compact set K ⊂⊂ Ω ⊂⊂ R d such that n supp µ n ⊂ K.
Thus we may treat {µ n } n∈N as a sequence of measures defined on a compact Ω. Then (M, T V) is isomorphic to (C b (Ω)) * , which is a separable normed vector space then by Banach-Alaoglu theorem the set {µ n : n = 1, 2, ...} is sequentially weakly * compact in (C b (Ω)) * . Therefore there exists a measure µ ∈ (C b (Ω)) * such that up to a subsequence µ n converges to µ weakly * in (C b (Ω)) * . Proposition 2.1 implies that the weak * (C b (Ω)) * convergence is equivalent to the convergence in d(·, ·) if only {µ n } n∈N is tight (which is true since µ n vanish on Ω \ K). Thus µ n converges to µ in d(·, ·). Finally since by Proposition 2.2 (M + , d) is a complete space we conclude that actually µ ∈ (M + , d) and the proof is finished.
Lastly we present a useful lemma related to the bounded-Lipschitz distance.
Lemma 2.1. Let d(·, ·) be the bounded-Lipschitz distance. Then for any µ, ν ∈ M and any bounded and Lipschitz continuous function g, we have Proof. The proof of this lemma belongs to the standard theory and can be found, for example, in [23].

Measure-valued solutions to the kinetic equation
We introduce the following weak formulation for (3): We say that f is a weak solution to (3) with the initial data 3. The following identity holds: and Lipschitz continuous and φ has a compact support in t ;

For each pair of concentric balls B((x
then there exists T * ∈ [0, T ], such that There is a natural question of the correspondence between solutions to (3) in the sense of Definition 2.3 and solutions to (2). The answer to this question is to some merit positive, which we explain below. Let (2). For this system let (x, v) be a sufficiently smooth 2 solution. Then the function is a solution of (3) in the sense of Definition 2.3 with the initial data f 0 . Indeed, if we plug f defined in (10) into (6), by a simple use of a chain rule, we obtain The converse assertion that a solution to (3) in the sense of Definition 2.3 corresponds to a solution of (2) is also true provided that the initial data are of the form (9). However, the proof is much more involved and it is in fact the second part of the paper.

Definition 2.4. We say that f is an atomic solution if it is of form (10).
Definition 2.5. In case of solutions of particle system (2) we say that ith and jth particles collide at the time t if and only if x i (t) = x j (t) and we say that they stick together at the time t if and Remark 2.3. Throughout the paper whenever we consider a solution of system (2) we assume that the number of particles is constant in time. However if in our proofs (particularly in Section 5) any two particles stick together, then we tend to treat them as a single particle with a bigger mass, thus reducing the total number of particles in the system. This is justified by the fact that according to [31] solutions to (2) with α ∈ (0, 1 2 ) are unique (see Theorem 3.2).
Remark 2.4. Point 5 of Definition 2.3 requires some explanation. Its purpose is to establish a local control over the propagation of the support of f . Basically if we can divide the support of f 0 into two parts of distance R − r, then in some small time interval [0, T * ] the distances between those parts is no lesser than R−r 4 .
Remark 2.5. In Section 5 we frequently test our weak solution by various test functions that at the first glance may seem not admissible. In particular we test with functions with derivatives in x and v not necessarily Lipschitz continuous. This is however correct since by a simple density argument we may test (6) by C 1 functions. Moreover we are allowed to test (6) by functions that are not compactly supported in time. In such case we get a version of (6) with both endpoints of the time interval, The justification of the above equation is standard and can be found in the proof of Proposition 3.1,(v) in Appendix A.

Main result
The main result of the paper is the following. The part of the proof concerning the issue of existence follows from analysis of approximation by atomic solutions originating from sums of Dirac's deltas, which correspond in the sense of Remark 2.2 to solutions of (2). The main idea behind this approach is twofold. First, we have better regularity of solutions of (2) for α < 1 2 . It was connected to results from [31], where we proved that for 0 < α < 1 2 , system (2) admits a unique W 1,1 ([0, T ]) solution (x, v), which by Remark 2.2 corresponds to a solution of (3) in the sense of Definition 2.3. Since in fact α ∈ (0, α 0 ) for some α 0 < 1 2 , we can in fact prove that (x, v) is bounded in W 1,p ([0, T ]) for some p > 1. Such bound provides equicontinuity of sequences of solutions of (2), which allows to extract a convergent subsequence. The second element of the proof is to change the way of looking at the alignment force term The uniqueness part of Theorem 3.1 is explained and proved in section 5. The main idea is to analyze possible support of measure-valued solutions for initial atomic configurations. It is related to point 5 of Definition 6 of weak solutions to (3). We emphasize that we derive the propagation of the support described in point 5 from the behavior of the approximate solutions. The question whether these properties are exhibited by a wider class of possible weak solutions is out of the scope of this paper. Such generalization would require an essentially different approach to construction of solutions.
Remark 3.1. Throughout the remaining part of the paper we assume without a loss of generality that the total mass of f 0 equals to 1.
Let us give an overview of the proof of existence. Suppose that f 0 is a given, compactly supported measure belonging to M and assume without a loss of generality that where B(R 0 ) is a ball centered at 0 with radius R 0 . For such f 0 we take f 0,ǫ ∈ M of the form which corresponds to the initial data (x 0,ǫ , v 0,ǫ ) to a particle system (2). Moreover we assume that and that the support of f 0,ǫ is contained in B(2R 0 ). The existence of such approximation is standard (we refer for example to the beginning of section 6.1 in [23] for the details). Now suppose that (x n ǫ , v n ǫ ) is a solution to (2) with the communication weight subjected to the initial data (x 0,ǫ , v 0,ǫ ), which by Remark 2.2 means that is a solution of (3) with the initial data f 0,ǫ . Our goal is to converge with ǫ to 0 and with n to ∞ to obtain a solution f of equation (3) subjected to the initial data f 0 .
The proof of existence can be summarized in the following steps: Step 1. Given T > 0, for each ǫ and n, we prove existence of a solution f n ǫ corresponding to the initial data f 0,ǫ and satisfying various regularity properties.
Step 2. We take a sequence f n = f n ǫ for ǫ = 1 n . Due to the conservation of mass and the regularity proved in step 1 we extract a subsequence f n k converging in Step 3. We converge with each term in the weak formulation for f n k to the respective term in the weak formulation for f . This can be easily done for each term except the alignment force term i.e. the term Step 4. Analysis of the alignment force term is not straightforward. Formally we multiply an L p function by a measure, thus the result is not well defined at the very first sight. We replace (17) with an n k -independently regular modification of the form The error between the alignment force term and it's substitute will be controlled in terms of m and uniformly with respect to n k .
Step 5. For such subsequence we converge with the substitute alignment force term to Step 6. We are then left with converging with the substitute alignment force term to the original alignment force term i.e. with m → ∞. We show that F( f ) is a measure with respect to the measure d f = f dxdv.
Step 7. We finish the proof by making sure that each and every point of Definition 2.3 is satisfied by our candidate for the solution.
Remark 3.2. Before we proceed further, let us compare our strategy to the mean-field limit used for the CS model with regular weight like in [23]. For Lipschitz continuous ψ, through standard argumentation, one can easily show well-possedness for the particle system (2). This includes Lipschitz continuous dependence of solutions with respect to the perturbations of the initial data. Such stability can in turn be translated into Lipschitz continuous dependence of measure-valued solutions with respect to perturbations of initial measures. Thus inequality can be obtained, where f 1 and f 2 are two solutions of the type (10) with initial data f 0,1 and f 0,2 of the form (9). Then stability ensures possibility to extract a convergent subsequence as presented after Remark 13. If one aims to apply such reasoning to the case with singular weight it turns out that the constant C(T ) from (18) depends on the Lipschitz constant of ψ on [δ, ∞), where δ is the lower bound of the distances between particles. However even for the initial data, the distance between particles δ converges to 0 as ǫ → 0. Thus the Lipschitz constant of ψ on [δ, ∞) converges to infinity and so does C(T ). There are ways to overcome this difficulty at least locally in time like in [6], but for a global in time result stability of the type (18) does not seem to work since particles may eventually collide causing a blowup of C(T ). For the discussion on the possibility of collisions between particles in the singular case of α ∈ (0, 1) we refer to [30], while in case of α ≥ 1 we refer to [7].
Let us state some various properties of the approximative solutions f n ǫ . It is in fact the first step of the proof (as presented above) but since it is self-contained and quite lengthy we will present it in a form of separate proposition the proof of which can be found in Appendix A.
Proposition 3.1. Given T > 0. Let f 0,ǫ be of the form (9). Then for each n = 1, 2, ..., there exists a unique solution f n ǫ to kinetic equation (3) that corresponds 3 to a smooth and classical solution (x n , v n ) of particle system (2). Moreover there exist n and ǫ independent constants M > 0 and p > 1, such that the following conditions are satisfied: (i) For all t ∈ [0, T ] and all n and ǫ the total mass of f n ǫ i.e. the value R 2d f n ǫ dxdv is equal to 1.
(ii) The support of f n ǫ is contained in a ball B(R), where R := 2R 0 (T + 1).
(v) For each Lipschitz continuous and bounded g : Remark 3.3. Point (iii) of Proposition 3.1 implies in particular that the sequence (x n ǫ , v n ǫ ) is uniformly bounded in W 1,p ([0, T ]). We mention this to keep the continuity with the idea of the proof presented at the beginning of this section.

Remark 3.4.
It is worthwhile to note that since by (iii) from Proposition 3.1 the derivative of velocityv is uniformly integrable, then Moreover the function ω is independent of i and n.  The proof of the above theorem can be found in [31]. Moreover its existence part follows also directly from Theorem 3.1 and Remark 2.2. Furthermore our argumentation from Section 5 can be used to simplify the uniqueness part of the proof of Theorem 3.2. We present the simplified proof in Appendix A for the sake of completeness.

Proof of Theorem 3.1 (existence)
In this section we follow the steps presented in the previous section and prove the existence part of Theorem 3.1.
Step 1. From the very beginning we fix T > 0. Proposition 3.1 and Remark 2.2 ensure the existence of f n ǫ with properties (i)-(v) from Proposition 3.1. We solve particle system (2) with initial data (14) in the time interval [0, T ] under assumption that the communication weight is in form (15). By Proposition 3.1 we are ensured that for some p > 1 Step 2. We take ǫ = 1 n and denote f n := f n 1 n . Since f n is of the form (16) it is clear that For each n the function f n may be treated as a mapping from [0, T ] into the metric space (M + , d).
For the purpose of showing that f n has a convergent subsequence we use Arzela-Ascoli theorem. We make sure f n is a bounded and equicontinuous sequence of functions with a relatively compact pointwise sequences f n (t). Uniform boundedness of f n is implied by the conservation of mass, while relative compactness of f n (t) follows from the uniform boundedness of f n (t) in TV topology and Corollary 2.1. Finally in order to prove equicontinuity of f n we take arbitrary s, t ∈ [0, T ] and arbitrary Lipschitz continuous, bounded function g with Lip(g) ≤ 1 and g ∞ ≤ 1 and use estimation (v) from Proposition 3.1 to write Point (v) of Proposition 3.1 states that functions t → d dt R 2d g f n (t)dxdv are uniformly bounded in L p ([0, T ]) for some p > 1, which in particular means that they are uniformly integrable. On the other hand it implies that the function ω is a good modulus of uniform continuity for the left-hand side of (19). Now since this estimation does not depend on the choice of g (only on the choice of Lip(g)), it is also valid for the supremum over all g, which implies that d( f n (s), f n (t)) ≤ ω(|s − t|).
The above inequality proves that the sequence of functions t → f n (t) is equicontinuous as a mapping from [0, T ] to (M, d) (recall the bounded-Lipschitz distance defined in (2.1)). Thus the sequence f n satisfies the assumptions of Arzela-Ascoli theorem. Therefore there exists f ∈ L ∞ (0, T ; M + ), such that up to a subsequence d( f n , f ) ∞ → 0.

By (ii) from Proposition 3.1 it implies that the support of f is included in B(R).
Step 3. After a brief look at the weak formulation for f n i.e. (6), we understand that since f n → f in L ∞ (0, T ; (M + , d)), then in particular for φ ∈ G, we have f 0 φ(·, ·, 0)dxdv and the only problem is with the second term on the left-hand side of (6) i.e. the alignment force term Step 4. To deal with the problem of convergence with the alignment force term we replace it in the following manner However, as mentioned at the beginning of Section 3, instead of looking at (20) as an integral of a product of F n ( f n ) with f n , we are going to see it as an integral of with respect to the measure g n dµ n dt and a similar identity holds for Moreover we have Therefore for Furthermore, integrating with respect to dµ n reveals that which by Proposition 3.1,(iv) implies that the sequence L n is uniformly bounded in L p ([0, T ]) for some p > 1 and thus -it is uniformly integrable which further implies that To estimate II we introduce the set B t (m, n) of those pairs (i, j) such that |x n i (t) − x n j (t)| ≤ m − 1 α . Then by Hölder's inequality with exponent q = 1 θ , for some arbitrarily small θ > 0, we have By Proposition 3.1, (iv) the first multiplicand on the right-hand side of (26) is uniformly bounded, which implies that Estimations (23) and (27) imply that Step 5. Our next goal is to ensure that the convergence holds for each m and each φ ∈ G. Let us fix φ ∈ G and m = 1, 2, ... . For g m defined in (21), we have Furthermore, again by Fubini's theorem and since for each x, v the function (y, w) → g m (x, y, v, w) is Lipschitz continuous and bounded with Lip(g m ) + g m ∞ ≤ C 1 for some ) then by Lemma 2.1 we have Similarly also II → 0 with n → ∞. This concludes the proof of convergence (28).
Step 6. At this point after converging with n to infinity we are left with the weak formulation for f that reads as follows: for all m = 1, 2, ... and all φ ∈ G with J(m) → 0 as m → ∞.
Therefore it suffices to show that By Fubini's theorem for (31) provided that the integral on the right-hand side of (31) is well defined. Therefore to show (30) it suffices to prove that g m → g in L 1 with respect to measure dµ. To prove this we first show that g m → g a.e. with respect to the measure dµ. Clearly the convergence holds on and it suffices to show that the set A c = {(x, v, y, w, t) : x = y, v w} is of measure dµ zero. We have ψ m ≡ m on A c and thus

Thus either
The proofs of Step 4 and Step 5 remain true if we substitute g m and g n with |g m | and |g n | respectively 5 . Therefore also the respective convergences hold for |g m | and |g n |, yielding Moreover for each m and n, we have Now, (34) implies that for each m we may choose n big enough, so that Furthermore, by (33) for such n we have |g n |dµ n dt ≤ |J(m)| and finally by estimation (iii) from Proposition 3.1 and thus for some positive constant C 2 . Therefore (32) and (35) imply that A c |w − v||∇ v φ|dµ = 0 and since the function |w − v| is positive on A c , then by a standard density argument A c is of measure µ zero and we have proved that

Moreover by Fatou's lemma
Therefore the function (x, y, v, w, t) → ψ(|x − y|)|w − v||∇ v φ| belongs to L 1 (dµ). This function is a proper dominating function for ψ m (|x − y|)(w − v)∇ v φ and by the dominated convergence theorem we have (30) and the proof of step 6 is finished.
Step 7. Let us now wrap up the proof and compare Definition 2.3 with what we were able to prove about f . We took an arbitrary initial data f 0 ∈ M + and proved existence of f ∈ L ∞ (0, T ; M + ). Moreover in step 2 using estimates (ii) and (v) from Proposition 3.1 we proved that actually supp f ⊂ B(R) and (point 1 of Definition 2.3). Point 2 of Definition 2.3 is an immediate consequence of (ii) from Proposition 3.1, while point 3 was the main focus of all the steps of the proof and it was finally proved in step 6. Point 4 of Definition 2.3 follows from (36) and Fubini's theorem. As a consequence of the weak formulation for f we conclude that also ∂ t f ∈ L p (0, T ; (C 1 (B(R))) * ). We are left with point 5 of Definition 2.3. Suppose that B(R) and B(r) are two concentric balls, such that (7) is satisfied. Then the construction of f 0,n ensures that n and for sufficiently large n we have r + 1 n < r + R−r 8 < R − R−r 8 . Translating it according to (16) we write that in the set I of those i that ( . By (ii) and (iii) from Proposition 3.1 (and in particular by Remark 3.4), for each i ∈ I and for each sufficiently big n, we have the n independent bounds: The above bounds, for sufficiently small t imply that (x n i (t), v n i (t)) ∈ B(r + R−r 6 ) as long as i ∈ I. Similarly for i I in a sufficiently small neighborhood of t = 0, we have ( . Therefore for sufficiently large n and sufficiently small t. Thus we may pass to the limit with n → ∞ to obtain (8). This finishes the proof of the existence part of Theorem 3.1.

Proof of Theorem 3.1 (weak-atomic uniqueness)
In what follows we aim at proving that if initial configuration f 0 is an atomic measure, i.e. it satisfies (9), then solution f in the sense of Definition 2.3 is of the form (10), and it is unique. We will base the proof on a very careful analysis of the local propagation of the support of f that comes from point 5 of Definition 2.3. What, we basically need, is that any amount of the mass f that is separated from the rest of the mass remains separated at least for some time. It is required to refine this property by adding a control over the shape in which the support in the x and v coordinates propagates. The difficulty comes from the fact that in the case of the particle system the position x i of ith particle changes with its own unique velocity v i . However in the case of the kinetic equation characteristics are not well defined.
Step 1. By point 1 in Definition 2.3 it is sufficient to prove the proposition only in an arbitrarily small neighborhood of t = 0. Let f 0 be of the form (9). Our first task is to restrict f 0 to small balls with one particle (say ith particle) in R 2d . Then we will use the local propagation of the support to prove that the mass that initially formed the ith particle remains atomic in some right-sided neighborhood of t = 0. Since for number of atoms N, we have a finite number of initial positions and velocities of the particles (x 0,i , v 0,i ) for i = 1, ..., N, which implies that there exists R 1 > 0 such that for all r 0 < R 1 , we have At this point let us concentrate on one atom, we fix i. We aim at showing that there exists T * such that in [0, T * ] for some R d valued functions x i and v i . We emphasize that r 0 and T * (r 0 ) can be chosen to be arbitrarily small. Identity (38) implies that for any 0 < r < r 0 , we have We have then f D η = f D . All these properties allow to state the following equation satisfied by f D on [0, T * ]: This equation is satisfied in the same sense that (6) from Definition 2.3. To prove that f D is indeed of form (39) we introduce with the initial data (x a (0), v a (0)) = (x 0,i , v 0,i ). Condition (40) ensures that the right-hand side of (42) 2 is smooth and thus (42) has exactly one smooth solution in [0, T * ]. Our goal is to show that f D is supported on the curve (x a (t), v a (t)) and that in fact (39) holds with (x i (t), v i (t)) ≡ (x a , v a ).
Since this feature will hold for all atoms, the whole f will then be atomic.
Step 2. In the next step we characterize possible evolution of the support of the weak solution to (41).
Lemma 5.1. Let f be a weak solution to (3) in the sense of Definition 2. 3. Assume further that f has the structure of f = f D + f C and fulfills the weak formulation of (41), and for some given (x 0 , v 0 ). Then for any R > 0 there exists T * , such that for all t ∈ [0, T * ], with ǫ := √ 2R(R + |v 0 |), which can be arbitrarily small depending on smallness of R.
To prove Lemma 5.1 it is required to show the following result.
for some given (x 0 , v 0 ) and R > 0 and all t ∈ [0, T * ]. Then It means that the support in the x-coordinates propagates in a cone defined by the ball B Proof of Lemma 5.2. Without a loss of generality we assume that (x 0 , v 0 ) = (0, 0). The boundedness of the support in the v-coordinates is trivial and thus we focus on the support in the x-coordinates. Suppose that x 1 ∈ R d and ρ > 0 are such that We test (41) by φ 2 and integrate over the time interval [0, T * ], obtaining Since the first term on the left-hand side of the above equality is nonnegative, we have But for the interior of the support of φ, we have ρ − Rt > |x − x 1 | and by (43) R > |v|. It implies that This way we proved that in the complement of the support in x of f (t) lay all the balls centered outside of supp f 0 and with a radius equals to ρ − Rt, which implies (44).
Proof of Lemma 5.1. We base the proof on Lemma 5.2. First we establish proper R and T * . Since f D 0 is concentrated in one point (x 0 , v 0 ) then for arbitrarily small ρ Now, Definition 2.3 point 5 ensures that there exist R(ρ) and T * (ρ) such that in [0, T * ] and R can be chosen arbitrarily small (then also T * is small but still positive). We fix such R and T * and note that we may apply Lemma 5.2 on [0, T * ]. Without a loss of generality we assume that x 0 = 0 and test (41) with the function φ 2 , where By (11), we have On the support of f D , we have |v − v 0 | ≤ R and by Lemma 5.2 it holds Hence, in view of definition of ǫ, we conclude Therefore the integrand on the right-hand side of (47) is nonnegative, which means that it has to be equal to 0, which further implies that By the definition of φ it follows that f D (t) vanishes outside of the cone balls tB x (v 0 , ǫ) × R d . The lemma is proved.
Step 3. In this part we show that f initiated by a state of (37) stays indeed atomic for all time.
Proposition 5.1. Let f be a solution to (41) in the sense of Definition 2.3. Then if f 0 is of form (9) then f is an atomic solution (of form (10)) and it is unique.
Proof. We show separately for each of atoms that each initial particle generates a mono-atomic solution (at least locally in time). Finiteness of number of atoms allows to conclude that the whole solutions is atomic. Hence, we study (41) with a mono-atomic initial data located in (x 0 , v 0 ).
We test (41) by (v − v a (t)) 2 getting First we deal with III. By symmetry of f D ⊗ f D with respect to (x, v) and (y, w), we have Next let us take a closer look at II. By the definition of F( f C ) Now we compare II 2 with I: The main problem with estimating the right-hand side of the above inequality lays in the estimation of This is the place where the separation of supports explained by Lemma 5.1 comes into play.
Both (x a (t), v a (t)) and (x, v) are in the support of f D , while (y, w) is in the support of f C . Thus (40) implies that either We handle the above two cases separately.
In case (50) it is clear that for some constant L = L(r 0 ) > 0, since ψ is smooth outside of any neighborhood of 0.
In case of (51) we are actually in a situation when at t = 0 multiple particles are situated in the same spot with different velocities i.e. f C is divided into two parts f C 1 and f C 2 . The first part submits to the same bounds as (50) while for the second, f C 2 , we have Thus, initially f C 2 is concentrated in the same position as f D but with different velocities. In this case we apply Lemma 5.1 multiple times (once for f D and multiple times for each f C 2 j ). Even though Lemma 5.1 is written for solutions of (6) we may still apply it for f D and each of f C 2 j , since the proof does not involve directly the dependence on v. Therefore, by Lemma 5.1, we have . At this point we fix R > 0 and T * from Lemma 5.1, so that ǫ is small enough that Again, we used that the number of all atoms is finite. If so, then also |x − y| > tC(R) and |x a (t) − y| > tC(R) for x ∈ supp f D and y ∈ supp f C 2 . Therefore in such case (ψ(|s|) = s −α and ψ ′ (|s|) ∼ s −1−α ) which by Gronwall's lemma and the fact that A(t) ∼ t −1/2−α is integrable in a neighborhood of t = 0 (restriction α ∈ (0, 1 2 ) is used here again) implies Thus on [0, T * ] we have x ≡ x a and v ≡ v a on the support of f D , which is exactly equivalent to (39).
We have proved f D is mono-atomic. Then repeating the procedure for all atoms (the number is finite) we conclude that f is atomic on a time interval [0, T * ] with possibly smaller, but positive T * > 0. This procedure works till the first moment of sticking of an ensemble of particles.
As a final remark we explain the case of the sticking some particles in a finite time, say T 1 . The above considerations prove uniqueness and atomic structure of the solutions for the time interval (0, T 1 ) without sticking of particles, and we want to reach T 1 . The regularity of the weak solution guarantees that ∂ t f ∈ L p (0, T ; (C 1 (B(R))) * ) with

A Appendix
Proof of Proposition 3.1. The existence and uniqueness part as well as points (i) and (ii) are no different than in the case of regular weight and we will not prove them here. Their proofs can be found in the literature (see for instance [23] or [30]). Thus it remains to prove (iii)-(v).
(iii) − (v) First, assuming for notational simplicity that (x n , v n , N n , m n i ) = (x, v, N, m i ) let us prove a particularly useful estimate. Let 1 < p < q be given numbers satisfying additional conditions that will be specified later. For each n = 1, 2, ..., velocity v n (denoted by v) is absolutely continuous on [0, T ] and thus by (2) 2 , we have Inequality (57) is obtained by Young's inequality with exponent q p while (58) follows by Young's inequality with exponent 2 p . In both of the above inequalities we also use the assumption that N i=1 m i = 1.
Furthermore recalling that ψ 2q p n (s) ≤ ψ 2q p (s) = |s| −λ , where λ := 2qα p , integral A can be estimated as follows: However, the above estimation is valid only if λ < 1, which means that q p · 2α < 1 and such condition can be easily satisfied if α < 1 2 and 1 < p < q are small enough. By point (ii) we have |v| ≤ R and |x| ≤ R. This leads to the concluding estimation of A, which reads: Now we will apply the above calculation (particularly estimations (58) and (59)) in the effort to prove (iii) and (iv). For (iii) let us assume that p = q = 1 8 . We sum (58) over i = 1, ..., N to get  ) and (iii) is proved for some sufficiently small p > 1. In order to prove (iv) we take 1 = p < q in (58), which leads us to a very similar result to (61) and to the end of the proof of (iv).
Let us prove (v). Fix n = 1, 2, ... and a bounded, Lipschitz continuous function g = g(x, v). Then according to Definition 2.3, for t ∈ [0, T ), ǫ > 0 and χ ǫ,t (s) :=          1 for 0 ≤ s ≤ t − ǫ − 1 2ǫ (s − t − ǫ) for t − ǫ < s ≤ t + ǫ 0 for t ǫ < s the function φ(s, x, v) := χ ǫ,t (s)g(x, v) ∈ G is a good test function in the weak formulation for each f n . Thus we plug φ into (6) Since t → R 2d f n gdxdv, t → R 2d f n χ ǫ,t v∇gdxdv and t → R 2d F n ( f n ) f n χ ǫ,t ∇ v gdxdv are integrable functions (for fixed n and g), then converging with ǫ → 0 leads to the following equation holding for a.a t ∈ [0, T ): f 0 gdxdv, 8 Note that (57) remains true also for p = q = 1. where , v n i (t)) + There are two options: x 0,i x 0, j , then the difference is bounded from below by a constant (on a short time interval), or x 0,i = x 0, j and v 0,i v 0, j . Then particles travel in different cones and by Lemmas (5.1) and (5.2) we find that the difference is bounded by t. Hence at least for a short time ψ(|x i (t) −x j (t)|) ≤ C + Ct −α Next we examine ψ(|x i −x j |) − ψ(|x i − x j |). Here we have again two cases. The first one holds as x 0,i x 0, j . Then for short time interval In the second case x 0,i = x 0, j and v 0,i v 0, j . Then by Lemma 5.1 supp x i (t),x i (t) ∈ x 0,i + tB(v 0,i , ǫ) and supp x j (t),x j (t) ∈ x 0,i + tB(v 0, j , ǫ), Next, trivially from (64) we find sup τ≤t |δx i (τ)| ≤ t sup τ≤t |δv(τ)|.
The solutions are unique. The above procedure works till the time of the first sticking of a group of particles, say at time T 1 . Then the regularity in time (v ∈ W 1,p (0, T )) implies that all solutions are uniquely extended till time T 1 . And then we can restart our procedure with initial configuration x i (T 1 ), v i (T 1 ), taking into account that sticking particles create a new one with an appropriate mass.