Cauchy theory for general Vicsek models in collective dynamics and mean-field limit approximations

In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity and nonlinear friction. Particularly, we include normalised kernels which display a singularity when the flux of particles vanishes. Thus, in terms of the Cauchy theory for the kinetic equation, we extend to more general interactions and complete the program initiated in Gamba et. al. (2016) (where the authors assume that the singularity does not take place) and in Figalli et. al. (2017) (where the authors prove that the singularity does not happen in the space homogeneous case). Moreover, we derive an explicit lower time of existence as well as a global existence criterion that is applicable, among other cases, to obtain a long time theory for non-renormalised kernels and for the original Vicsek problem without any a priori assumptions. On the second part of the paper, we also establish the mean-field limit in the large particle limit for an approximated (regularized) system that coincides with the original one whenever the flux does not vanish. Based on the results proved for the limit kinetic equation, we prove that for short times, the probability that the dynamics of this approximated particle system coincides with the original singular dynamics tends to one in the many particle limit.

1. Introduction 1.1. Motivation. The emergence of collective motions among a group of self-propelled agents is receiving a great deal of attention: flocks of birds [35], schools of fish [34], pedestrian dynamics [41], micro-swimmers [10]. Many models have been proposed to explain these collective behaviours. Among them, the Vicsek model [23,48] is one of the most studied ones. In this model all agents move at a constant speed while trying to adopt the same velocity as their neighbours, up to some noise.
In this introductory section we present the discrete dynamics for the Vicsek model as presented in [23] as well as its corresponding kinetic equation for the timeevolution of the distribution of the particles. Up to now, there were no complete rigorous results on the existence of solutions and the derivation of the kinetic equation (mean-field limit) for the Vicsek dynamics presented in [23]. The reason for this is the presence of a singularity in the dynamics that is reached when the local average velocity of the agents vanishes.
In contrast to [23], one can define a Vicsek model without a singularity [3,15,16,32], for different modelling choices. However, these different modelling choices for the Vicsek model have profound mathematical implications (like appearance of phase transitions). For this reason, in this article we investigate a general form of the Vicsek model that includes the forms presented in [23] (with a singularity), in [3,32] (without singularity) as well as further extensions of the model like the ones in [4,16,25,31]. We give conditions to have existence of solutions for the equations and investigate approximations to the mean-field limit for this general class of Vicsek models. All of these are detailed in the following sections.
1.2. Particular case: Vicsek model as in [23] and [3,32]. In what follows we assume that the agents move in a domain Ω with Ω being either the d-dimensional torus T d or the whole space R d for d 1. The orientations of the agents are given elements on the sphere S d−1 . Throughout the article, for a given vector ω ∈ R d we denote P ω ⊥ the orthogonal projection in R d onto (Rω) ⊥ . The time-continuous Vicsek model [23] considers N agents characterised by their positions X i,N (t) ∈ Ω, i = 1, . . . , N and orientations ω i,N (t) ∈ S d−1 over time t 0. The evolution of the system is given by the following Stratonovich stochastic differential equation, where c, ν, σ > 0 are positive constants: We explain next this system of equations. Eq. (1.1) describes the transport of the agents: agent i moves in the orientation ω i,N at speed c > 0. Eq. (1.2) gives the evolution of the orientations over time. It includes two competing forces. On one hand the first term in the form of a gradient represents organised motion: agents try to adopt the same orientation. The gradient ∇ ω is the gradient on the sphere and the term ω i represents the average orientation of the neighbours around agent i. Therefore, the first term in the right-hand-side of Eq. (1.2) is a gradient flow which relaxes the value of the orientation ω i,N towards the mean orientation of the neighbouring particles ω i . The constant ν > 0 gives the speed of this relaxation. We will comment later on the different choices to compute the average orientation ω i . These different choices give rise to the different models in [23] and [3,32].
On the other hand B i t , i = 1, . . . , N are N independent Brownian motions in R d and they introduce noise in the dynamics, driving particles away from organised motion. The constant σ > 0 gives the intensity of the noise. The symbol ′ • ′ is used to specify that Eq. (1.2) has to be understood in the Stratonovich sense. This ensures that ω i (t) ∈ S d−1 for all times where the solution is defined (this will be proven later). Now, formally at least, one can compute the time-evolution for the distribution of agents f = f (t, x, ω) in space x ∈ Ω and orientations ω ∈ S d−1 at time t 0 as the number of particles N → ∞ [23]. The dynamics for the distribution f is given by the following kinetic equation: where ∇ ω · denotes the divergence on the sphere S d−1 and where ω f (t, x) represents the average orientation of the particles around position x at time t. Notice that the projection term appears due to the fact that ∇ ω (ω · ω) = P ω ⊥ ω, for ω ∈ R d fixed.
One can show that the operator L on the right-hand-side of (1.3) can be rewritten as a non-linear Fokker-Planck operator: for any v ∈ S d−1 . The function M v is a probability density on the sphere named von-Mises distribution. Notice that its mean is in the orientation given by v. When σ → ∞ (large-noise limit) the von-Mises distribution converges to the uniform distribution on the sphere.
We comment now on the different choices to define the average orientation around an agent i, denoted by ω i in Eq. (1.2). The choice considered has profound implications in the dynamics of the particles and in the derivation of the kinetic equation (1.3) (mean-field limit) and the derivation of macroscopic equations (equations for the particle density and mean orientation). In the mathematics literature, two options were first considered given, in a compact way, by: The two options correspond to the cases α = 0 (as presented in [23]) or α = 1 (as presented in [3,32]). The kernel K 0 is an interaction kernel that represents the weights given to the neighbouring particles depending on their distance to particle i. A classical choice is to take K the indicator function of a ball of radius R > 0. The flux J N i is, indeed, an average of the orientations of the neighbouring particles. In the kinetic equation these choices define the operator ω f as: We compare next, the mathematical implications of these two choices: Case α = 1 (non-normalisation). If α = 1, the average orientation ω i = J N i / ∈ S d−1 is not a unit vector. This also holds in the kinetic equation (1.3) for ω f = J f / ∈ S d−1 in (1.7). However, the case α = 1 removes the singularity when |J f | = 0 or |J N i | = 0. In [32] the authors prove the well-posedness of the space-homogeneous kinetic equation in any Sobolev space and in [2] the authors prove the mean-field limit (here we will recover these results). As a counterpart, though, this choice for the average ω i makes the derivation of macroscopic equations for the density of the particles ρ = ρ(t, x) and the mean orientation ω = ω(t, x) of the agents more complex than in the case α = 0. Specifically, two equilibria exist for the mean particle orientation ω depending if J f = 0 or J f = 0 [15,16,32]. In loose terms, if J f = 0, then the von-Mises equilibria M J f is an equilibrium. The other equilibrium is given by the uniform distribution on the sphere, which corresponds to the case J f = 0 in M J f . In different spatial regions (depending on the particle density ρ) one of these two equilibria is stable. Consequently, this gives rise to a bifurcation or phase transition: in some spatial regions the mean orientation of the agents is ω = 0 (corresponding to disordered dynamics) and in other spatial regions the mean orientation is given by a unit vector |ω| = 1 (corresponding to ordered dynamics or flocking). The interested reader can find all the details in [4,15,16,32]. The presence of phase transitions enriches the dynamics, in the sense that allows for a wider variety of patterns to arise. Understanding phase transitions is also of great interest in the Physics community [48], who was the first one to introduce the Vicsek model. In particular, pattern formation has been studied through the simulation of particle simulations for the Vicsek model and its variations. Band formation arises under some parameter conditions [6,8,9]. Phase transitions could be key to explaining the emergence of this band formation.
Case α = 0 (normalisation). The first mathematical works on the Vicsek model correspond to this case [23]. The choice α = 0 comes with the difficulty of dealing with singularities when the flux J N i = 0 (another example of collective dynamics with singularities, different from the Vicsek model can be found in [45] for the Kuramoto model). However, assuming that the flux J f in the kinetic equation does not vanish along the dynamics, then the macroscopic equations can be obtained more easily than in the case α = 1 because there is a unique equilibria: the von-Mises distribution in (1.5). Then the macroscopic equations for the particle density ρ = ρ(t, x) and mean orientation ω = ω(t, x) ∈ S d−1 correspond to the Self-Organised Hydrodynamics (SOH) given in [23]. This was the first formal derivation of the SOH dynamics. See also [38,50] as well as [37] for later rigorous results. In this scenario a rigorous mean-field limit to derive the kinetic equation was missing as well as a Cauchy theory for the particle dynamics and the kinetic equation. Here we will not prove the mean-field limit starting from a particle system of the form (1.1) -(1.2), but from a modified system that we term 'approximated particle system'. This system does not have a singularity when the flux vanishes and therefore, it can be thought of as a regularization of the original particle dynamics. Proving the meanfield limit for particle systems with non-regular coefficients or with singularities is in general a difficult problem and, to the authors knowledge, only very few results exist and focus on specific problems (see e.g. [36]). Here, we will prove that with a probability which tends to one in the many-particle limit, for short times, solutions of the approximated particle system are also solutions of the Vicsek particle system. We investigate these questions in the present article: Theorem 1.1 (Cauchy theory for general Vicsek models with normalisation and mean-field from approximated particle dynamics). Suppose that α = 0 (normalised case). Suppose that the kernel K is Lipschitz, bounded and that f (0, x, ω) satisfies assumptions (i) and (ii) in Theorem 2.4. Then, there exists a unique local-in-time solution to the kinetic equation (1.7). Moreover, the kinetic equation equation (1.7) can be obtained as the mean-field limit of an 'approximated' particle dynamics whose solutions are, with a probability which tends to one in the many-particle limit, also solutions to the particle Vicsek dynamics (1.1)-(1.2).
The precise definition of 'solution', of the approximated particle system and what we mean with 'a probability which tends to one in the many-particle limit' will be explained in Th. 2.4 and in Sec. 2.3, respectively. As mentioned before, the difficulty in proving the previous theorem is to show that there exists at least some time interval t ∈ [0, T ] such that J f = 0 and that for N large enough the probability that |J N i | > 0 goes to 1 as N → ∞ for all i = 1, . . . , N and t ∈ [0, T ]. To prove a lower bound in the flux J f = 0 is, therefore, key. Indeed, in [33] the authors manage to prove well-posedness for the kinetic equation (1.3) assuming precisely that a priori all solutions satisfy |J f |(t, x, ω) a > 0 for all times, positions and orientations. In [30] the authors prove well-posedness in the space homogeneous setting as long as initially |J f 0 | > 0 (plus some other assumptions). This has been followed by further extensions in [39]. The space-homogeneous case corresponds to spatially local kernels, i.e., K(x − y) = δ 0 (x − y) (delta distribution).
To conclude this part, the choice on how to define the average orientation ω i has profound mathematical and modelling implications (like dealing with singularities or with phase transitions). For this reason, in this article we will consider a general class of Vicsek models to encompass different modelling choices -existing or yet to be developed.

1.3.
General forms of the Vicsek model considered. As we saw in the previous section, modelling choices in the Vicsek model may produce important differences in the mathematical properties of the equations. For this reason, in this article we consider a wide class of Vicsek models based not only on the choice of the averaged orientation (1.2) but also on the particular shape of the friction ν, the viscosity σ and the interaction kernel K. Specifically, we will allow ν and σ to be functions of f (the solution to the kinetic equation) at the kinetic level and functions of the empirical distribution (see Eq. (2.16)) at the particle level. The interaction kernel K can take the general form K = K(t, x, x * , ω, ω * ). The precise assumptions on ν, σ and K can be found in Section 2.1.
As a consequence, the results presented here apply to a wide breath of Vicsektypes models, like the ones in [15,16,25,31]. In [15,16] the authors consider functions ν = ν(|J f |), σ = σ(|J f |) that depend on the flux of the particles; in [31] the author considers a kernel K that is not isotropic; in [25] the authors consider a Vicsek-type model for alignment of discs. Beyond this, the approach presented here can be adapted to investigate other models like the ones in [4,12,13,18,17,19,20,21,22,24,42]. In [13] the authors couple the Vicsek model with a Kuramoto model. In [42] the author considers a Vicsek model with two species having different velocities. The models in [4,18,19,20,21] describe collective motion based on nematic alignment (where particles align by adopting the same direction of motion and not necessarily the same orientation). BGK versions of Vicsek-type models are considered in [12,17]. In [22] the authors couple the Vicsek model with Stokes equations to model micro-swimmers. Finally, in [24] a model for the persistent turning walker with curvature 'alignment' is presented.
1.4. Aims of the paper. In this paper, for a general form of the Vicsek model, we aim at: (i) establishing the well-posedness, at least locally in time, for the kinetic model (Theorem 2.4); existence of solutions will be proven in Lebesgue spaces; (ii) rigorously proving that the kinetic equation can be derived as the mean-field limit of some 'approximated' agent-based dynamics in the limit N → +∞ (Theorems 2.10, 2.11, 2.12). Also show that with a probability which tends to one in the many-particle limit realizations of the 'approximated' particle system are solutions of the original Vicsek system (Theorem 2.14); (iii) deriving a criterion that characterises the global-in-time well-posedness of these systems (points (a) and (b) in Theorem 2.4) and apply it to several interesting cases (Section 3.3.1). Along the way we shall also open the discussion to long-time behaviour and construct a free energy but we shall not investigate further (Section 3.3.2). We aim at giving constructive proofs rather than weak-compactness arguments and at working in Lebesgue spaces rather than more regular Sobolev spaces. Our strategy is thus to prove a fixed point contraction theory for the kinetic equation (2.5) and, to prove the mean-field limit, using a coupling approach popularised by Sznitmann [3,5,7,47].
1.5. Structure of the paper. The paper is structured as follows. Section 2 describes our main results, first in the kinetic framework in Section 2.2 and second for the particle dynamics and its mean-field limit in Section 2.3. The proofs for the results in Section 2.2 are given in Section 3 and the proofs of the results in Section 2.3 are given in Section 4. In the Appendix A the reader can find known results on stochastic differential equations and mean-field limits that are used in Section 4 and that have been added here for the sake of completeness. The results in Appendix A are mainly from [5].

2.
Main results and strategy 2.1. Functional framework and notations. First let us give some notations for functional spaces. We shall work on Lebesgue spaces indexed by the variable into consideration: for any p in [1, +∞) we denote and for a time-dependent function for any T > 0 Finally we denote the several variable Lebesgue spaces for any q and r in [1, +∞) with direct modifications for p, q or r being +∞. Note that when two indexes are the same we shall use the short-hand notation L p x L p ω = L p x,ω .
For the mean-field limit results, we will consider the space H = (Ω × R d × P 2 (Ω × R d )), where P 2 (Ω × R d ) is the space of probability measures in Ω × R d with finite second-order moment, i.e., µ ∈ P 2 (Ω × R d ) if it fulfils For µ, µ ′ ∈ P 2 (Ω × R d ), the 2-Wasserstein distance is given by The distance W 2 induces the topology of weak convergence of measures and the convergence of all the moments of order up to 2 [5] (see also [49, p. 83]). The space H is a metric space with the distance d H given by The hypotheses we shall make on the nonlinearities are the following.
(H1) The interaction kernel K is Lipschitz, regular and bounded in all the variables. More precisely, it is assumed that K(t, x, x * , ω, ω * ) ∈ W 1,∞ t,x * W 2,∞ ω * L ∞ x,ω , and in the case Ω = R d we further assume that K ∈ L ∞ t,x,ω L 2 x * ,ω * ; (H2) The viscosity satisfies that σ : L 2 [0,T ],x,ω −→ L ∞ [0,T ],x,ω is bounded from below and Lipschitz uniformly for any T > 0 in the sense there exists σ 0 , σ ∞ and σ lip such that for any T > 0: (H3) The friction is local in time and Lipschitz that is (H4) For the mean-field limit results we will assume further that σ, ν and ∇ ω σ are Lipschitz and bounded in W 2 .
• Note that (H2) and (H3) are satisfied when σ and ν do not depend on f , which is the case in most models so far. Ultimately, one would want σ(f ) to be local in time and Lipschitz, like ν(f ). We think that this could be achieved with standard parabolic regularity methods for more regular initial data and σ(f ). The main issue for closing a fixed point argument is the speed of convergence to 0 of the gain of regularity of the solutions f that we did not manage to quantify, see Proposition 3.3 and Remark 3.4.
• In applications, typically the functions ν = ν(t, x, ω, f ) and σ = σ(t, x, ω, f ) are of the form for some function ν, and analogously for σ. For example, in [23] the authors consider And in [16] the authors consider ν and σ of the form with J f given by (2.4).

2.2.
Kinetic point of view: strategy and results. We will consider the following kinetic equation for the distribution function f = f (t, x, ω) for (t, x, ω) ∈ R + ×Ω×R d (with Ω either the d-dimensional torus T d or the full space R d ): where ∇ ω , ∇ ω · are the gradient and divergence operators on the sphere; c > 0 is a positive constant and ν, σ, Ψ are given functions. There exists an interaction kernel K : From the flux J we define the functional Ψ as where |·| stands for the norm in R d . Note that when α = 0 we talk about a normalised operator since |Ψ[f ]| = 1 whereas for α = 1 we have a complete kernel operator. The orthogonal kernel non-linearity (2.5) we tackle is more general than the gradient-type interaction (2.9). Furthermore, it is also more physically relevant when one wants to derive the Kolmogorov-Vicsek type of kinetic equation from particle systems behaviour [2,3]. The main issue for the kinetic part is the possible degeneracy of |Ψ[f ]| α as well as the nonlinearity of the dissipativity and viscosity that does not necessarily compensate the aformentionned degeneracy, unlike existing works in the literature. The key part of our strategy is to provide first a quantification of the possible time of degeneracy of the nonlinearity combined to a study of the dependencies over σ and ν at a linear level. Remark 2.3. In the literature, typically, the operator K is of the form K(x, x * , ω, ω * ) = K(x − x * , ω * ). In this case it holds that Therefore, in this case the Kolmogorov-Vicsek equation (2.5) can be rewritten as a gradient-type interaction There exists a time T max > 0, independent of p, and a unique weak solution f in , preserves the mass M 0 and one of the following holds Remark 2.5. Of important note are the following consequences.
• Our Cauchy theory does not stand on any a priori assumption of the solutions and provide an explicit lower bound for T max (see where K ∞ is given by (3.17) and M 0 is as given in the theorem above.
In the sequel we will define c * = c * (T ) as • The theorem above includes all the previous results made in L 2 or L ∞ , both in the non-homogeneous and the homogeneous case (it suffices to consider The global existence criterion offers direct global existence for non-normalized interactions α = 0 but it also gives global existence for the original Vicsek equation with spatially-homogeneous kernel K(t, x, x * , ω, ω * ) = ω * and fully homogeneous viscosity and negative friction (i.e. only f and time-dependent). Section 3.3.1 describes several general cases where global existence happens in the problematic and purely normalised case α = 0.
Besides the issue of well-posedness, it is of great interest to understand the large time behaviour of the solutions. We recall that Section 3.3.1 proves global existence for different types of interactions and it also exhibits a free energy which decreases along the flow. One cannot expect a general theory since it heavily depends on the shape of the kernel K but one can still wonder if there are equilibria and if they are attractive. For this purpose, kinetic equations with gradient non-linearities (2.9) are often used because one can extract an explicit free energy functional decreasing along time. Hence leading to the existence of equilibria and a hope for an asymptotic study of the solutions. This has been tackled in [32,14,16] where the authors exhibited a decreasing energy functional when σ and ν are solely functions of |J f |. Associated to the latter is an energy dissipation allowing to dig out explicitely the equilibria in the spatially homogeneous setting. It appears that a free energy is also underlying in the general equation: (2.5) enjoys a free energy functional that decreases along the flow with an explicit energy dissipation. Namely, dxdω.
which will be proven, in Section 3.3, to satisfy along the flow Remark 2.6. Let us emphasize that considering the spatially homogeneous case K = ω * , (2.13) and (2.14) are the ones obtained in [14] with |J f | 2 instead of |J f | (in case of constant viscosity and friction we also recover the original Doi-Onsager free energy [43,26]). For general gradient kernel interactions ψ[f ] (2.9) that are symmetric in ω and ω * the free energy (2.13) becomes for ν and σ being solely functions of ψ[f ], f L 2 x,ω (which equals |J f | 2 ). Hence offering a new view on the gradient structure where the natural dependencies are, in fact, One can immediately see that the energy dissipation vanishes on (2.15) and when it vanishes so does the right-hand side of the kinetic equation ( x,ω) and we recover generalised von Mises equilibria of the previous [32, 14,16]. Hence, as mentionned earlier, a gradient structure pops out naturally but does not solely concern P ω ⊥ [Ψ[f ]] as first considered in (2.9). It however is not the purpose of this article and we did not investigate further.

2.3.
Microscopic point of view and mean-field limit for the Vicsek kinetic equation: strategy and results. We will prove that the kinetic equation can be obtained as the mean-field limit of some 'approximated' (regularized) particle system. When there is normalization (α = 0), we will show that realizations of the approximated dynamics are also solutions of the Vicsek particle system with a probability which tends to one in the many-particle limit (Th. 2.14). Note, however, that for the case α = 0, we will not prove the mean-field limit for the kinetic equation starting from particle systems of the form (1.1)-(1.2), nor its well-posedness. We consider a system of N particles given by their positions Notice that we shall consider that the equations are written for v ∈ R d rather than in ω ∈ S d−1 , but we shall later prove that the velocities are restricted to the sphere with radius c, thus recovering the expected spherical dynamics. We recall the empirical distribution of the process ( As mentioned before, in this article we will prove results on two types of particle systems. The first one we will call 'general particle Vicsek' and the second one 'approximated particle dynamics'. We define the general particle Vicsek as the particle system given by the following Stratonovich stochastic differential equation: where Id is the identity matrix; ((B i t ) t ) i=1,...,N are independent Brownian motions in R d . The symbol ′ • ′ denotes that the stochastic differential equation (2.17) is in Stratonovich convention. The precise way in which the function Ψ is extended to V t ∈ R d is explained in Sec. 4.2, see system (4.5).
The 'approximated particle system' is given by similar equations where the difference is that the functional Ψ is replaced by a functional τ ε 0 that will be made precise later: for some ε 0 > 0 and so that it is well defined in H (i.e., without singularities). In particular, in the system above we will have that Because τ ε 0 does not present any singularities when |J| = 0, the approximated particle system (2.18) can be thought of a regularization of the Vicsek particle system (2.17). Associated to the approximated particle system we define the approximated kinetic equation given by Observe that when τ ε 0 is substituted by Ψ in the approximated kinetic equation (2.20), we obtain the Vicsek kinetic equation (2.5).
First, we will show the well-posedness for the approximated particle system (Th. 2.10) and that in the mean-field limit gives the Vicsek kinetic equation (for short times) (Th. 2.11 and Cor. 2.12).
Remark 2.8. Notice that the term (2.18b) in system (2.17) appears so that we obtain the kinetic equation (2.5). This is just a technicality: system (2.17) in Itô's convention corresponds to: See also Appendix A.2 for more details. Without the extra term (2.18b) we would obtain a kinetic equation where the operator in ω is of the form (see Eq.
But with the extra term the operator ω in the kinetic equation is of the form which is the operator that we are dealing with in (2.5). Notice that if we wanted to obtain just a factor ∆ ω (σ(f )f ) then the constant in front of the extra term (2.18b) should be −1/2 rather than 1/2. Notice also that if σ = σ(t, x, f ) does not depend on ω, then the extra term (2.18b) does not appear. This is the case, for example, [16]. The extra term (2.18b) has the effect of relaxing V i,N towards the value taken by ∇ v σ. It can be rewritten equivalently as a gradient where the gradient is on the sphere.
As announced before, the dynamics described by our agent-based model indeed force the velocities to have unit norm, as shown by the next lemma.
is a solution to the approximated particle dynamics (2.18) or the Vicsek particle dynamics (2.17).
for all times where the solution is defined.
Proof of Lemma 2.9. It is a direct check that using Stratonovich chain rule (see, for example, [29, p. 122]) and the fact that The potential degeneracy of Ψ breaks the Lipschitz regularity of the interactions and thus prevents standard agent-based well-posedness results to apply. There exist results for non-Lipschitz interactions [2] but their singularities differ from the one at stake in the present article.
In the spirit of [3], we thus shall prove well-posedness of the system hand-in-hand with the mean-field limit with the aid of an auxiliary process which mixes microscopic dynamics with the mesoscopic distribution function f t . The main difference with respect to [3] is that we need to deal the fact that the noise coefficient σ is not constant. Let us explicit the auxiliary process for the approximated particle system (X where the initial data and the Brownian processes (B i t ) t 0 are the same ones as in (2.18). Notice that all the processes (X i,N t , V i,N t ) are independent by construction and they all have the same law f t .
We will prove the following two results for the approximated particle system (2.18) and the approximated kinetic equation (2.20). Along with this well-posedness result, we rigorously show its mean-field limit towards the approximated kinetic equation (2.20).
From this we deduce a mean-field limit result for the Vicsek kinetic equation: Proof. Take T 0 ∈ [0, T ) and define Cor. 2.12 implies the mean-field limit of the approximated dynamics (2.18) towards the Vicsek kinetic equation (2.5) for short times (in the case α = 0, then T = +∞ and the mean-field limit holds for all times).  1 in [2] for more details. In particular, in [2] it is shown the following upper bound in 2-Wasserstein distance where ε(N) → 0 as N → ∞ (by Th. 2.11). The function f (1) denotes the first marginal of the N particle system. They also show that for any Lipschitz map ϕ Under more regularity assumptions, one can obtain explicit estimates on ε(N). See Th. 10 in Ref. [5] (Sec. 1.3.4).
2.3.1. The Vicsek particle system when α = 0. All the previous results for the meanfield limit correspond only to the approximated system (2.18), which is not singular when α = 0. When the norm of the flux for all t ∈ [0, T ] and all i = 1, . . . , N, the approximated particle dynamics (2.18) coincides with the Vicsek particle dynamics (2.17) for t ∈ [0, T ]. When α = 0 (normalized case), every realization of the 'approximated' particle system (2.18) such that |J(t, X i,N t , V i,N t , µ N t )| > ε 0 for all t ∈ [0, T ] and all i = 1, . . . N is also a solution to the Vicsek particle system. The next result shows that for short times (2.25) happens with a probability which tends to one in the many-particle limit: Theorem 2.14 (Lower bound on the flux for the normalized case). Suppose that we are under the assumptions of Th. 2.4 and assumption (H4). Consider a time T < T 1 (where T 1 is defined in (3.20)) and c * = c * (T ) > 0 given by (2.11). Then, for all t ∈ [0, T ] and all ε 0 ∈ (0, c * ) it holds that T,ε 0 as the event such that: T,ε 0 = {for all t ∈ [0, T ] and all i = 1, . . . , N, J (t, X i,N t , V i,N t , µ N t ) > ε 0 }. Then, as a consequence of Th. 2.14, it holds that for all ε 0 ∈ (0, c * ) and so with a probability which tends to one in the many-particle limit, in the sense of (2.27), realizations of the approximated particle dynamics (2.18) will have nonzero flux for short times and they will also be solutions to the general Vicsek particle system (2.17).
Remark 2.15. Proving the well-posedness and the rigorous mean-field limit for the Vicsek particle system (2.17) in the normalized case (α = 0) seems to at least require (using the current strategy) to show |J(t, x, v, µ N t )| > 0 a.s., for all N large enough, for all times in some interval and all values of (x, v). It is not clear if this is even true. This would require some kind of generalization of a strong law of large numbers. Such strong law of large numbers exists for infinite exchangeable sequences as a result of de Finetti's theorem (see [40]). However, to the best of our knowledge, there is not an analogous result for a hierarchy of finite exchangeable sequences.
3. The mesoscopic framework: kinetic differential equation 3.1. Linear equation: dependence on coefficients and averaging positivity. The differential operator ∇ ω is the tangential gradient -also called Günter derivatives -on the sphere S d−1 . It can be related to the standard gradient on R d by where ∇ v is the euclidean gradient for functions from R d to R d . First of all let us emphasize that the tangential gradient ∇ ω displays very different behaviour than the usual gradient. We give here three formulas that we shall use throughout the proofs. We refer the interested reader to [27,1] for an introduction on tangential -Günterderivatives and to [44,Appendix II] for the specific calculus of the following formulas (proven in dimension 3 but immediately extendable in dimension d). Integration by parts is allowed but generate an additional term in the direction of the orientation ω: It also holds the following explicit tangential derivatives for each coordinate ω i of ω = (ω 1 , . . . , ω d ) on the sphere As mentioned in the introduction, (2.5) can be viewed as a nonlinear Fokker-Planck equation on the torus with local and nonlinear viscosity σ and drift ν. The purpose of this section is to study the associated linear equation when the nonlinearities are considered as given parameters. In other words we consider here the following differential problem where σ and Ψ are given functions. The issue of existence and uniqueness for (3.3) has been solved for instance in [11, Appendix A] for velocities in R or R 2 for constant σ with Ψ and ∇ ω · Ψ being in L ∞ x,ω . However their methods are directly applicable for velocities in S d−1 as shown in [33, Lemma 4.1] for constant σ. The case of non-constant σ is a straightforward adaptation of their proofs if σ belongs to L ∞ t,x,ω and is uniformly bounded from below by a constant σ 0 > 0 (note that one could also adapt to tangential derivatives standard Galerkin type methods for linear parabolic equations [28,Chapter 6], as done in [32] in the spatially homogeneous case). We thus omit the proof and state: x,ω uniformly bounded from below: σ(t, x, ω) σ 0 > 0 and let Ψ and ∇ ω · Ψ be in L ∞ t,x,ω with the orthogonal property P ω ⊥ (Ψ) = Ψ. Then for any f 0 in L 2 x,ω there exists a unique f in the space Moreover, if f 0 is non-negative then f is non-negative.
The next Section 3.1.1 is dedicated to L p bounds and gain of regularity for solutions to (3.3) whilst Section 3.1.2 studies the dependences on the viscosity and friction. To conclude, Section 3.1.3 tackles the issue of an explicit lower bound on the vanishing time for velocity averaging densities.
3.1.1. L p bounds and gain of regularity. The issue of L p x,ω estimates and gain of regularity had already been investigated in previous cited references for particular σ and Ψ and we provide here an adapted version that fits our general setting. e Cp(σ,Ψ)t f 0 L p x,ω , and it also gains regularity in ω when p < +∞ in the following sense: Proof of Proposition 3.2.
Step 1: L p bounds and gain of regularity. Note that since ∇ ω and Ψ are orthogonal to ω, we can perform on (3.3) integration by parts on S d−1 as if we were working with classical derivatives. Indeed, the orthogonality to ω exactly cancels the additional term in the tangential derivatives integration by parts formula (3.1). It yields for any p 2, We use that the first integrand on the right-hand side can be written as a divergence in x : ∇ x · (cω |f | p ) whereas the second one can be written as (p − 1)σ(t, x, ω) |f | p−2 (∇ ω f ) 2 . This leads to the following upper bound x,ω . Note that we used Cauchy-Schwarz and Young inequality for any η > 0. Choosing From which we deduce thanks to Grönwall lemma that for all t 0 (3.5) x,ω ds e Cpt f 0 L p x,ω . The latter is exactly the standard bounds in Proposition 3.2 for p in [2, +∞), obtained as an a priori estimate. The belonging to L p for f follows standard methods (also used in [33]) where our kinetic equation is approximated by a linear iterative scheme for which each solution f n is in L p and (3.5) holds at each step. Then the uniqueness of the kinetic solution implies that f n → f and taking the limit in (3.5) shows that f belongs to L p and satisfies (3.5).
Step 2: Mass conservation and L ∞ bounds. Now let us suppose that f 0 is non-negative, which implies that f is non-negative at all time by Theorem 3.1.
because, again, the integration by parts (3.1) does not generate additional terms in the direction of ω. This concludes the preservation of the L 1 x,ω -norm.
The L ∞ x,ω estimates follows straight from the limit p goes to +∞, since f (t) belongs to L 1 x,ω .
3.1.2. Dependence on the coefficients. Our strategy to tackle the non-linear equation is via a fixed point argument. We thus need to understand how solutions to (3.3) differ from one another when they are associated to different coefficients σ and Ψ.
The main issue in establishing estimate on f 1 − f 2 L p x,ω of two different solutions relies on the fact that the gain of regularity proved in Proposition 3.2 is higly nonlinear as soon as p > 2. Moreover, we did not manage to quantify the way the gain of regularity vanishes at initial time so we cannot close a direct L ∞ t fixed point method. We shall thus only study the dependence on coefficient in L 2 t,x .
3. Let f 0 be a non-negative function in L 1 x,ω ∩ L 2 x,ω . Suppose that σ 1 , σ 2 and Ψ 1 , Ψ 2 satisfy the assumptions of Theorem 3.1. Let f 1 and f 2 be the solutions of (3.3) associated respectively to the coefficient (σ 1 , Ψ 1 ) and (σ 2 , Ψ 2 ) with initial datum f 0 . Then the following holds for any t 0, Remark 3.4. The main difficulty with dealing with local-in-time σ(f ) appears in the second term on the right-hand side above. Indeed, from Proposition 3.2 we see x,ω vanishes when t goes to 0, thus almost allowing to close a contraction fixed point argument. As we shall see later on, we however need an explicit independence over σ (other than inf σ and sup σ) to avoid any nonlinear breakdown of contraction theorem in short times. As a result a L 2 ([0, T ], L 2 x,ω ) setting is required to get an extra integration in time. This explains hypothesis (H2), but one could also ask for more regularity for σ and f 0 to explicitly estimate the convergence to 0 of x,ω with parabolic regularity. Proof of Proposition 3.3. To shorten notations we will use Using the algebraic identity ab − cd We start by bounding the terms in f 1 − f 2 as in Proposition 3.2 and we obtain where σ + 0 = inf{σ 1 + σ 2 }. We bound the two terms inside the integral on the right-hand side. First thanks to an integration by parts (3.1).

Cauchy-Schwarz with Young's inequalities yields
x,ω . (3.7) The second integrand in (3.6) is dealt with in the same way and we get x,ω . (3.8) We then plug (3.7) and (3.8) into (3.6) and get 1 2 x,ω on which we apply Grönwall lemma to obtain x,ω ds (3.9) To conclude we apply Proposition 3.2 to bound the second term on the right-hand side, which directly gives the expected result.
Then the following holds Proof of Proposition 3.5. Since f 0 is non-negative and belongs to L 1 x,ω ∩ L p x,ω we deduce from Theorem 3.1 and Proposition 3.2 that f (t) is non-negative and belongs to L 1 x,ω ∩ L p x,ω for all time. Hence, we can multiply (3.3) by K(x, x * , ω, ω * ) and integrate by parts (3.1). Here again note that there are no additional terms along ω * since Ψ and ∇ ω * are both orthogonal to ω * . This yields where we used the conservation of mass along time. Summing over i we obtain The latter implies that which concludes the proof.
3.2. The local-in-time nonlinear Cauchy theory. The present section is devoted to the proof of Theorem 2.4 thanks to the linear study we presented in Sec.
3.1. We shall prove existence, uniqueness and global existence criterion to (3.12) Proof of Theorem 2.4. We fix p in [2, +∞] and f 0 0 in L 1 x,ω ∩ L p x,ω . We recall some assumptions of Theorem 2.4: The strategy is to apply a contraction fix point argument so we start by defining a complete metric space on which we shall work. For any M M 0 and T > 0 we define Note that since J[f ] is an integral over (x * , ω * ) against K and because K is L 2 x * ,ω * (by (H1)), a Cauchy sequence for the L 2 -norm in Γ M T indeed converges in Γ M T . For a function g in Γ M T we have that g belongs to L 2 [0,T ],x,ω so by Fubini's theorem g belongs to L 2 ([0, T ], L 2 x,ω ) and for almost every t in [0, T ] the function g(t, ·, ·) belongs to L 2 x,ω . Therefore σ(g(t)) and ν(g(t)) are defined almost everywhere in [0, T ] and with hypotheses (H2) and (H3) one has Defining and where C α > 0 only depends on α and equals 1 when α = 1. Therefore thanks to Theorem 3.1 we are able to define associated to the initial datum f 0 .
We now show that for a specific T , Φ is in fact a contraction from Γ M T to Γ M T . Due to Proposition 3.2 and since f 0 belongs to L 2 x,ω we see that Φ(g) belongs to Moreover, Proposition 3.5 gives that almost everywhere We defined K ∞ in (3.10) and we recall it Therefore if we choose and we thus proved that for any 0 < T < T 0 , Φ maps Γ M T to itself.
It remains to prove that Φ is a contraction on Γ M T for T sufficiently small. By hypothesis (H2) we have Also, thanks to the Lipschitz property (H3) of the friction ν and the kernel form of x,x * ,ω,ω * , M, α, Ψ 0 and ν lip . This inequality comes from the algebraic identity ab − cd = (a−c)(b+d) 2 We apply Proposition 3.3 to see that for any g 1 and g 2 in Γ M Note that we used Proposition 3.2 to obtain explicit constants. To conclude it suffices to choose because then for any g 1 and g 2 in Γ M T we have proved which implies that Φ is a contraction on Γ M T .
We thus proved the local existence and uniqueness of a solution f to the nonlinear kinetic equation (2.5) x H 1 ω ) due to Proposition 3.2 since σ(f ) and ν(f ) are well-defined and satisfy the required assumptions, as we saw above. At last, thanks to the global in time Cauchy theory for the linear equation given by Theorem 3.1 we see that if lim < +∞, then we can apply our fixed point argument starting at T max and therefore T max must be +∞. Which concludes the proof of Theorem 2.4.

3.3.
Global existence and decay of the free energy. The present section focuses on some important cases where one is ensured to have globally defined solution to the fully non-linear kinetic equation (2.5). Then we get a look into the free energy and its energy dissipation.  [14,16] where ν(f ) = ν(|J f |) with ν(y) y bounded and is generalised here for instance to any ν(f ) = ν(|J [f ]|) with ν(y) y bounded near zero.
The normalised case of kernel with one coordinate with a strict sign. In this paragraph we assume that the kernel K(t, x, x * , ω, ω * ) is such that one of its coordinate K i has a strict sign in the following sense: there exists a positive constant It is a direct verification since, thanks to the positivity of f , either The Vicsek kernel operator. We study the special case when K(t, x, x * , ω, ω * ) = ω * and normalised α = 0, as in [14,16,33,30] for the spatially homogeneous equation. Note that here we also obtain the global in time for the non-spatially homogeneous setting. We also assume that ν and σ do not depend on x nor ω and that the friction ν is negative (standard in all previous studies).
Contrary to the setting above, the present kernel can degenerate in the sense that one non-zero coordinate of K can possibly vanish later on but giving rise to another non-zero coordinate. As a consequence, the full |K| will not vanish in finite time. First we explicitly write As before let us consider the computation below for 0 t < T max and recall (??) which in the present setting yields Thanks to the properties of the tangential derivatives we can compute directly with (3.2) Note that we use the fact that the friction is negative. Using Grönwall inequality we conclude that |J (t)| 2 |J (0)| 2 e −2(d−1)σ 0 t . These computations were already done in an homogeneous regularised setting in [30] and our local theory allows us to carry them out directly. Such a lower bound thus imply the non-vanishing of |J (t)| in finite time and thus leading again to global in time solution in this setting.

3.3.2.
Decay of the free energy. We recall our definitions for the free energy and the energy dissipation (2.13) − (2.14) dxdω.
Let f 0 , σ, ν and K be as in Theorem 2.4. We now establish the decay of F [f ](t): It comes from direct computations. Indeed since ∇ x/ω [1 + lnf ] = ∇ x/ω f f it follows by the integrating by parts formula (3.1), Hence, as claimed above.

Mean-field limit: proofs of the results in Section 2.3
Let us first describe the strategy we are about to implement. At the core, we will use the coupling approach popularised by Sznitmann [47] (which differs from classical BBGKY approaches [36]). We will prove theorems 2.10 and 2.11. We will follow the same steps in [3] and the results in [5].
4.1. Preliminaries: functional framework and notations. In this section we will work in the space H defined in Section 2.1 and with the 2-Wasserstein distance also defined there. We will need some results on Wasserstein distances in the sequel, that we summarise next (these results and proofs can be found in [49]). In our setting, the Wasserstein distance of order 1 is given by |z − u| π(dz, du) ; π ∈ P(Ω × R d ) 2 with marginals m and p .
This distance can be expressed in duality form using the Kantorovich-Rubinstein distance We will use the function γ as a substitute of the variable V (t) in some parts of the equations. We do this to be able to apply known results of existence an uniqueness of solutions for stochastic differential equations that require the coefficients to be Lipschitz and bounded (see Th. A.2). We will prove in a second stage that along the dynamics it holds that |V (t)| = 1, therefore γ(V (t)) = V (t) and we recover the terms in the original equation. Secondly, we define a functional τ 0 as We will show that τ 0 is Lipschitz and bounded in the whole H: Proof of Lemma 4.1. We notice, first, that since γ is bounded, then K(x, γ(v), m) is bounded in all the variables and, therefore, r f (z) is bounded in H. Now, let f, g ∈ P 2 (Ω × R d ) and fix z ∈ Ω × R d , we show next that r f := r(z, f ) is Lipschitz in f . Consider z, z ′ ∈ Ω × R d and for z = (x, v), denote z = (x, γ(v)). Now, we consider (the integrals are in Ω × R d ) where in the second line we used the reverse triangle inequality; in the third line we have used that R 2d f = 1 and that K is Lipschitz; in the fourth line we used that γ is Lipschitz and choose an arbitrary z 0 ∈ R 2d ; the fifth line is given by (4.1); and the last inequality follows from (4.2). Therefore, we conclude that r = r f (z) is Lipschitz in H.
Next, we also show that τ 0 is Lipschitz and bounded in H. First, whenever r ε 0 we have that (α + (1 − α)r) −1 is also Lipschitz and bounded for all α ∈ [0, 1], therefore, τ 0 is product of Lipschitz and bounded functions, hence it is Lipschitz and bounded.
With the function τ 0 we define the non-singular particle dynamics as: (this is exactly System (2.18) substituting τ 0 by the function τ ε 0 ) and the nonsingular auxiliary process is given by: (again this is exactly System (2.22) after substituting τ 0 by τ ε 0 ). Analogously, we also consider the non-singular kinetic equation in Ω × S d−1 given by Note that under the assumptions of Theorem 2.4, (4.7) has global existence and uniqueness of solutions in the spaces stated in Theorem 2.4.

Remark 4.2.
Notice that we are assuming that ν, σ and ∇ v σ are bounded and Lipschitz for all v ∈ R d rather than v ∈ S d−1 (see hypothesis (H4)). However, we just need that ν, σ and ∇ v σ to be Lipschitz and bounded in a neighbourhood of |v| = 1, as we can use regularising arguments as the one done to define the functional τ 0 . 14. Let f t be solution to the Vicsek kinetic equation (2.5). By the proof of Theorem 2.4 we know that for T < T 1 (with T 1 given in Eq. (3.20)) it holds that where ε(N) → 0 as N → ∞. If (4.9) holds, then we have that where we used the following inequality: which follows from the triangular inequality (||a| − |b|| |a − b| for all a, b ∈ R).
The following bound holds by (4.8). Using this inequality in combination with (4.10) we deduce that To bound this last quantity we will consider the path solutions to the auxiliary particle system (2.22), i.e, the path solutions (X ∈ P(C).
Since the space C with the norm w ∞ := sup t∈[0,T ] |w t | is a Banach space, one can define the Wasserstein distance on P(C): 1/2 m ∈ P 2 (C × C) with marginals m 1 and m 2 .
One can check that for m, m ′ ∈ P(C) it holds that where the limit is consequence of Lem. A.4 (notice that following the proof in [5] this lemma can be applied to random variables in C). As a consequence we have that E sup From these estimates, we have that since ε(N) → 0 as N → ∞ by Th. 2.11. Finally, combining (4.11), (4.12) and (4.13), we conclude (4.9).

4.4.
Proof of Proposition 4.3. The proof of Proposition 4.3 follows closely the methodology in [3] and [5] which are based on Sznitman approach [47]. The main difference is that the interaction rate ν and the noise coefficient σ are considered to be constant in [3]. In particular, this gives rise to an extra term in the equations (coming from (2.18b)).
Step 1. Regularised version of the non-singular dynamics The non-singular particle dynamics (4.5) written in Itô's convention corresponds to (see Section A.2 for more details): And the Itô formulation for the non-singular auxiliary process (4.6) is given by Remark 4.4. Notice that the solutions of (4.14) and (4.15) fulfil |V t | 2 = |V 0 | 2 in the velocities for all times where the solution is defined (this is shown as in Lemma 2.9).
Step 2. Existence and uniqueness for the regularised particle system -Proof of part (i) of Theorem 2.10 for (4.5).
We consider now a regularised version of Systems (4.14) and (4.15) using two functions τ 1 and τ 2 both Lipschitz and bounded and satisfying: if |v| 1/2.
With these functions we defined the regularised particle dynamics as Remark 4.5. Notice that the functions τ 0 , τ 1 and τ 2 are introduced to regularise the original system, in the sense that we obtain a new system where all the coefficients are Lipschitz and bounded in H. This regularity allows us to apply classical results of existence of solutions and convergence, as we will see next. At the same time, when |V | = 1 and |J| ε 0 we recover the approximated particle equations.
Firstly, we have by Lemma 4.1 that τ 0 is a Lipschitz bounded function and, moreover, all the coefficients are also Lipschitz bounded (using that the product of Lipschitz bounded functions is Lipschitz bounded, see also Remark 4.2). So we have existence and uniqueness of pathwise solutions for (4.16) (see [5,Theorem 1.2], which, for completeness we have added in a simplified form in Theorem A.2 in the Appendix.).
Secondly, one can check as in Lemma 2.9 that |V i,N t | = |V i,N 0 | = 1. Therefore, the solution to the regularised system (4.16) is also a solution to the non-singular system (4.5).
Step 3. Existence and uniqueness for an auxiliary regularised process of (2.22) -Proof of part (ii) of Theorem 2.10 for (4.6). Similarly as before, we consider a regularised version of the non-singular auxiliary process (4.6) given by: Remark 4.6. The control in the particle position is immediate from the one in the velocities: Appendix A. Some known results on SDE For the sake of completeness, we introduce in this section known results for stochastic differential equations that we used in the main part of the document.
A.1. Results on existence of solutions and large-particle limits. The results and proofs from this section can be found in a more general form in [5]. The original approach to proving the large-particle limit is due to Sznitmann and can be found in [47].
General stochastic differential equations. Consider a filtered probability space (Ω, F , (F t ) t 0 , P) and an (F t ) t 0 -Brownian motion B taking values in R m . Consider the stochastic differential equation giving the evolution for Z t ∈ R p , for t 0: with initial data Z(t = 0) = Z 0 ∈ R p , where the coefficients a : R p → R p×m and b : R p → R p are Lipschitz and bounded. (This is the form taken by the SDE (4.16) with p = 2d and Z t = (X t , V t ).) Definition A.1 (See Definition 1.1 in [5]). An (F t ) t 0 -adapted continuous process (Z t ) 0 t T is a solution to (A.1) if Theorem A.2 (Adapted from Theorem 1.2 [5] ). Let us assume that Z 0 ∈ L 2 is independent of B and that the coefficients b and a are Lipschitz and bounded. Then, there exists a unique solution of (A.1). Moreover, Z t ∈ L 2 for all t < T , with T finite.
Results for non-linear SDE.
We consider a nonlinear SDE of the form where L(Z) denotes the distribution or law of the random element Z. Here we also assume that b : R p × P 2 (R p ) → R p and a : R p × P 2 (R p ) → R p×m are bounded and Lipschitz. Specifically, in this case to be Lipschitz means that for all z, z ′ ∈ R d and all µ, µ ′ ∈ P(R d ) it holds that |(b, a)(z, µ) − (b, a)(z ′ , µ ′ )| c(|z − z ′ | + W (2) (µ, µ ′ )), for some constant c (see Section 4.1 for more details). System (A.2) is the form taken by the system auxiliary system (4.17).
Theorem A.3 (From Theorem 1.7 in [5]). Let us assume that Z 0 ∈ L 2 is independent of B, and that the coefficients b and a are bounded and Lipschitz. Then, there exists a unique solution to (A.2).
For ϕ ∈ C 2 b (R p ), Itô's formula applied to the process Z t gives (see Remark 1.8 in [5]): ϕ(Z t ) = ϕ(Z 0 ) + where the exponent T denotes the transpose and D 2 denotes the Hessian matrix.
Lemma A.4 (Law of large numbers: Lemma 1.9 in [5]). Let µ ∈ P 2 (R p ), and let (Z i ) i∈N be a sequence of independent identically distributed random variables with common law µ. For each N 1 denote by µ N the empirical distribution associated to the first N elements of the sequence, i.e.
Then, it holds that Let z i ∈ R p and z ′ i ∈ R p for i = 1, . . . , N, it holds ([5, (1.24)]) Particle approximations (from [5, Section 1.3.4]). where µ N s is the empirical distribution of the N particles. Assume that a, b are bounded and Lipschitz in R p × P 2 (R p ). Then, it holds that A.2. Stratonovich to Ito's convention. We use the results presented in [29] as well as [46,V.30,Theorem (30.14)]. Particularly, if Z t is solution to the Stratonovich ∂a ik ∂z j (z, t)a jk (z, t).
In our case a : R 2d → R 2d×d with where 0 d×d is a d × d zero-matrix and With this we have that c : R 2d → R 2d with c i = 0 for i = 1, . . . , d since γ jk = 0 for all j = 1, . . . , d and any k = 1, . . . , d. Now, for i = d + 1, . . . , 2d we have that Using that η jk (x, v) = α(x, v) δ j=k − v j v k |v| 2 , we compute the previous expression and obtain that