Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in \cite{DM08}, is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.


Introduction
Kinetic models have been introduced in the last years for the mesoscopic description of collective behavior of agents/particles with applications in collective behavior of cell and animal populations, see [19,37,42] and the references therein for a general overview on this active field.These models usually include alignment, attraction and repulsion as basic bricks of interactions between individuals.We refer to [43,15,30,16,5,6,35,20,21] and the references therein for a derivation of the kinetic equation from the microscopic models.
In this work, we focus on the derivation of macroscopic equations for the collective motion of self-propelled particles with alignment and noise when a cruise speed for individuals is imposed asymptotically for large times as in [31,24,23,17,22].More precisely, in the presence of friction and self-propulsion and the absence of other interactions, individuals/particles accelerate or break to achieve a cruise speed exponentially fast in time.The alignment between particles is imposed via localized versions of the Cucker-Smale or Motsch-Tadmor reorientation procedure [26,36,34,18,19,41] leading to relaxation terms to the mean velocity modulated or not by the density of particles.By scaling the relaxation time towards the asymptotic cruise speed, or equivalently, penalizing the balance between friction and self-propulsion, this alignment interaction leads asymptotically to variations of the classical kinetic Vicsek-Fokker-Planck equation with velocities on the sphere, see [45,29,33,27,28,11,12].It was shown in [2] that particular versions of the localized kinetic Cucker-Smale model can lead to phase transitions driven by noise.Moreover, these phase transitions are numerically stable in this asymptotic limit converging towards the phase transitions of the limiting versions of the corresponding kinetic Vicsek-Fokker-Planck equation.
In this work, we choose a localized and normalized version of the Cucker-Smale model not showing phase transition.More precisely, let us denote by f = f (t, x, v) ≥ 0 the particle density in the phase space (x, v) ∈ R d × R d , with d ≥ 2. The standard self-propulsion/friction mechanism leading to the cruise speed of the particles in the absence of alignment is given by the term div v {f (α − β|v| 2 )v} with α, β > 0, and the relaxation toward the normalized mean velocity writes div v {f (v − Ω[f ])} cf.[29,45,25].Here, for any particle density f (x, v), the notation Ω[f ] stands for the orientation of the mean velocity Notice that we always have Let us remark that the standard localized Cucker-Smale model would lead to ρdiv v {f (v −u[f ])} while the localized Motsch-Tadmor model would lead to div v {f (v − u[f ])}.Our relaxation term towards the normalized local velocity Ω[f ] does not give rise to phase transition in the homogeneous setting on the limiting Vicsek-Fokker-Planck-type model on the sphere according to [27] and it produces a competition to the cruise speed term comprising a tendency towards unit speed.Including random Brownian fluctuations in the velocity variable leads to the kinetic Fokker-Planck type equation We include this equation in a more general family of equations written in a compact form as where for any density distribution f with V a general confining potential in the velocity variables and η > 0 (see Lemma 2.1 for more information on the type of potentials that we consider).In the particular example considered above we take We investigate the large time and space scale regimes of the kinetic tranport equation (1) with collision operator given by (2).Namely, we study the asymptotic behavior when ε → 0 of supplemented with the initial condition The rescaling taken in the kinetic transport equation ( 3) with confining potential V α,β can be seen as an intermediate scaling between the ones proposed in [12] and [1].The difference being that we have a relaxation towards the normalized mean velocity Ω[f ] rather than the mean velocity u[f ] as in [1,12].This difference is important since in the first case there is no phase transition in the homogeneous limiting setting on the sphere as we mentioned above, while in the second there is, see [27,2,12].In fact, in [12] the scaling corresponds to η = 1/ε in (3), that is the relaxation to the cruise speed is penalized with a term of the order of 1/ε 2 .Whereas in [1] the scaling correponds to η = ε, that is the cruise speed is not penalized at all.
The methodology followed in [12] lies within the context of measure solutions by introducing a projection operator onto the set of measures supported in the sphere whose radius is the critical speed r = α/β.These technicalities are needed because the zeroth order expansion of f ε lives on the sphere.This construction followed closely the average method in gyro-kinetic theory [8,9,10].
However, in our present case we will show in contrast to [12,1] that there are no phase transitions which is in accordance with the results obtained in [28] for the kinetic Vicsek-Fokker-Planck equation with analogous alignment operator on the sphere.A modified version of (1)-( 2) in which phase-transitions occur was studied in [2] whose analysis is postponed to a future work to focus here on the mathematical difficulties of the asymptotic analysis.Another difference in the present case is that the zeroth order expansion of f ε will be parameterized by Von Mises-Fisher distributions in the whole velocity space, that is f (t, x, v) = ρ(t, x)M Ω(t,x) (v), with ρ and Ω being, respectively, the density and the mean orientation of the particles.And where for any Ω ∈ S d−1 we define (see Sect. 2) and The main result of this paper is the asymptotic analysis of the singularly perturbed kinetic transport equation of Cucker-Smale type (3).The particle density ρ and the orientation Ω obey the hydrodynamic type equations given in the following result.
Theorem 1.1 Let f in ≥ 0 be a smooth initial particle density with nonvanishing orientation at any x ∈ R d .For any ε > 0 we consider the problem with initial condition At any (t, x) ∈ R + × R d the leading order term in the Hilbert expansion 4), where the concentration ρ and the orientation Ω satisfy with initial conditions The constants c 1 , c 2 are given by where e(c, r) = exp(rc/σ) exp(−(r 2 + 1)/(2σ) − V (r)/σ).
Our article is organized as follows.First, in Section 2 we state auxiliary results allowing us to discuss the kernel of the collision operator.Then in Section 3 we concentrate on the characterization of the collision invariants.We prove that the generalized collision invariants introduced in [29] coincide with the kernel of a suitable linearised collision operator.We explicitly describe the collision invariants in Section 4 and investigate their symmetries.Finally, the limit fluid model is determined in Section 5 and we analyse its main properties.

Preliminaries
Plugging into (3) the Hilbert expansion we obtain at the leading order whereas to the next order we get where dQ f denotes the first variation of Q with respect to f .The constraint (7) leads immediately to the equilibrium Indeed, by using the identity we can recast the operator Q as We denote by S d−1 the set of unit vectors in R d .For any Ω ∈ S d−1 , we consider the weighted spaces and The nonlinear operator Q should be understood in the distributional sense, and is defined for any particle density f = f (v) in the domain We introduce the usual scalar products and we denote by | • | M Ω , • M Ω the associated norms.We make the following hypotheses on the potential V .We assume that for any Ω ∈ S d−1 we have Clearly (11) holds true for the potentials V α,β .Notice that in that case 1 ∈ L 2 M Ω and |1| M Ω = 1 for any Ω ∈ S d−1 .Moreover, we need a Poincaré inequality, that is, for any Ω ∈ S d−1 there is A sufficient condition for (12) to hold comes from the well-known equivalence between the Fokker-Planck and Schrödinger operators (see for instance [7]).Namely, for any Ω ∈ S d−1 we have The operator Using classical results for Schrödinger operators (see for instance Theorem XIII.67 in [44]), we have a spectral decomposition of the operator H Ω under suitable confining assumptions.
Let us note that the spectral gap of the Schrödinger operator H Ω is the Poincaré constant in the Poincaré inequality (12).Notice also that the hypotheses in Lemma 2.1 are satisfied by the potentials V α,β , and therefore (12) holds true in that case.It is easily seen that the set of equilibrium distributions of Q is parametrized by d parameters as stated in the following result.
and therefore there is We are done if we prove that Ω[f ] = Ω.If Ω = 0, it is easily seen that where we have used the radial symmetry of V , O ξ ξ = −ξ and O ξ Ω = Ω.We deduce that Obviously, we have for any

Characterization of the collision invariants
In [29], the following notion of generalized collision invariant (GCI) has been introduced.
In order to obtain the hydrodynamic limit of (3), for any fixed (t, x) ∈ R + × R d , we multiply (8) by a function v → ψ t,x (v) and integrate with respect to v yielding The above computation leads naturally to the following extension of the notion of collision invariant, see also [12].
We are looking for a good characterization of the linearized collision operator L f and its adjoint with respect to the leading order particle density f .Motivated by (7), we need to determine the structure of the equilibria of Q which are given by Lemma 2.2.By Lemma 2.2, we know that for any (t, x) ∈ R + ×R d , there are (ρ(t, x), The evolution of the macroscopic quantities ρ and Ω follows from ( 8) and (13), by appealing to the moment method [3,4,13,14,38,39].Next, we explicitly determine the linearization of the collision operator Q around its equilibrium distributions.For any orientation Ω ∈ S d−1 ∪ {0} we introduce the pressure tensor and the quantity We will check later, see Lemma 3.2, that the pressure tensor M Ω is symmetric. where 2. The formal adjoint of L f is given by 3. We have the identity Note that div v refers to the divergence operator acting on matrices defined as applying the divergence operator over rows. Proof.
1.By standard computations we have and Therefore we obtain 2. We have 3. For any i ∈ {1, ..., d} we have , and therefore, since Notice that at any (t, we deduce, thanks to (13), a balance for the macroscopic quantities ρ given by the relationship When taking as collision invariant the function h(t, x, v) = 1, we obtain the local mass conservation equation As usual, we are looking also for the conservation of the total momentum, however, the nonlinear operator Q does not preserve momentum.In other words, v is not a collision invariant.Indeed, if f = ρM Ω is an equilibrium with nonvanishing orientation, we have , and therefore v is not a collision invariant.
We concentrate next on the resolution of ( 14).We will use the notation ∂ v ξ = ∂ξ i ∂v j for the Jacobian matrix of a vector field ξ and div v for the divergence operator in v of both vectors and matrices with the convention of taking the divergence over the rows of the matrix.With this convention, we have for all smooth functions g, vector fields ξ, η, and matrices A. We now focus in finding a parameterization of the kernel of the operator L ⋆ f .Lemma 3.1 Let f = ρM Ω be an equilibrium of Q with nonvanishing orientation.The following two statements are equivalent: for some vector W ∈ ker(M Ω − σc 1 I d ).
Moreover, the linear map 1 induces an isomorphism between the vector spaces ker(L ⋆ f )/ ker W and ker(M Ω − σc 1 I d ), where ker W is the set of the constant functions.
Notice that thanks to (19) the coefficient c 1 does not depend upon Ω ∈ S d−1 In order to determine all the collision invariants, we focus on the spectral decomposition of the pressure tensor M Ω for any Ω ∈ S d−1 .In particular, the next lemma will imply the symmetry of the pressure tensor.Proof.Let us consider {E 1 , . . ., E d−1 } an orthonormal basis of (RΩ) ⊥ .By using the decomposition We claim that the following equalities hold true Formula ( 21) is obtained by using the change of variable and therefore we have which implies (21).For the formulae (22) with i = j, we appeal to the orthogonal transformation After this change of variable we deduce that and also

Thanks to the equality
Coming back to (20) we obtain We are done if we prove that Notice that, using ( 10): and therefore By Lemmas 3.1 and 3.2 the computation of the collision invariants is reduced to the resolution of (18) for any W ∈ (RΩ) ⊥ .Hence, if we denote by E 1 , E 2 , . . ., E d−1 any orthonormal basis of (RΩ) ⊥ , we obtain a set of d − 1 collision invariants This set of collision invariants forms a basis for the ker(L ⋆ f ).In the next section we will characterize this set of collision invariants and provide and easy manner to compute them (see Lemma 4.1).
We conclude this section by showing that in our case the set of all GCIs of the operator Q coincide with the kernel of the operator L ⋆ f .
Theorem 3.1 Let M Ω be an equilibrium of Q with nonvanishing orientation Ω ∈ S d−1 .The set of collision invariants of Q associated to M Ω coincides with the set of the generalized collision invariants of Q associated to Ω.

Proof.
Let ψ = ψ(v) be a generalized collision invariant of Q associated to Ω.We denote by {e 1 , . . ., e d } the canonical basis of R d .For any We deduce that there is a vector W = ( W1 , . . ., Wd ) ∈ R d such that for any f and thus implying that ψ satisfies (18) with the vector W = ( Conversely, let ψ = ψ(v) be a collision invariant of Q associated to M Ω .By Lemmas 3.1, 3.2 we know that there is W ∈ (RΩ) ⊥ such that Multiplying by any function f such that (

Identification of the collision invariants
In this section we investigate the structure of the collision invariants of Q associated to an equilibrium distribution f = ρM Ω .By Lemmas 3.1, 3.2, we need to solve the elliptic problem for any W ∈ (RΩ) ⊥ .We appeal to a variational formulation by considering the continuous bilinear symmetric form a Ω : and the linear form l : The above hypothesis is obviously satisfied by the potentials V α,β .We say that ψ ∈ H 1 M Ω is a variational solution of (23) if and only if Proposition 4.1 A necessary and sufficient condition for the existence and uniqueness of variational solution to (23) is Proof.The necessary condition for the solvability of ( 23) is obtained by taking θ = 1 (which belongs to H 1 M Ω thanks to (11)) in (24) leading to (25).This condition is satisfied for any W ∈ (RΩ) ⊥ since we have The condition (25) also guarantees the solvability of (23).Indeed, under the hypotheses ( 11), (12), the bilinear form a Ω is coercive on the Hilbert space H1 M Ω := {χ ∈ H 1 M Ω : ((χ, 1)) M Ω = 0}, i.e. we have: Applying the Lax-Milgram lemma to (24) with ψ, θ ∈ H1 M Ω yields a unique function ψ ∈ H1 Actually, the compatibility condition l(1) = 0 allows us to extend (26) to H 1 M Ω .This follows by applying (26) with θ = θ −((θ, 1)) M Ω , for θ ∈ H 1 M Ω .Moreover, the uniqueness of the solution for the problem on H1 M Ω implies the uniqueness, up to a constant, of the solution for the problem on H 1 M Ω .
As observed in (13), the fluid equations for ρ and Ω will follow by appealing to the collision invariants associated to the orientation Ω ∈ S d−1 , for any W ∈ (RΩ) ⊥ .When W = 0, the solutions of ( 23) are all the constants, and we obtain the particle number balance (16).Consider now W ∈ (RΩ) ⊥ \{0} and ψ W a solution of (23).Obviously we have where ψW is the unique solution of ( 23) in H1 M Ω .It is easily seen, thanks to (16) and the linearity of (15), that the balances corresponding to ψ W and ψW are equivalent.Therefore for any W ∈ (RΩ) ⊥ it is enough to consider only the solution of ( 23) in H1 M Ω .From now on, for any W ∈ (RΩ) ⊥ , we denote by ψ W the unique variational solution of (23) verifying The structure of the solutions ψ W , W ∈ (RΩ) ⊥ \ {0} comes by the symmetry of the equilibrium M Ω .Analyzing the rotations leaving invariant the vector Ω, we prove as in [12] the following result.
where t O denotes the transpose of the matrix O.
Proof.First of all notice that t OW ∈ (RΩ) ⊥ \ {0}.We know that ψ W is the minimum point of the functional It is easily seen that, for any orthogonal transformation O of R d leaving the orientation Ω invariant, and any function z ∈ H1 M Ω , we have, by defining Moreover, we obtain with the change of variables v ′ = Ov and using that M Ω (v) = M Ω (v ′ ): Finally, we deduce that for any z ∈ H1 M Ω , implying that ψ W • O = ψt OW .The computation of the collision invariants {ψ W : W ∈ (RΩ) ⊥ \ {0}} can be reduced to the computation of one scalar function.For any orthonormal basis {E 1 , . . ., E d−1 } of (RΩ) ⊥ we define the vector field F = d−1 i=1 ψ E i E i .This vector field does not depend upon the basis {E 1 , . . ., E d−1 } and has the following properties, see [12].Lemma 4.1 The vector field F does not depend on the orthonormal basis {E 1 , . . ., E d−1 } of (RΩ) ⊥ and for any orthogonal transformation O of R d , preserving Ω, we have F • O = OF .There is a function χ such that and thus, for any i ∈ {1, ..., d − 1}, we have Proof.Let {F 1 , . . ., F d−1 } be another orthonormal basis of (RΩ) ⊥ .The following identities hold and therefore For any orthogonal transformation of R d such that OΩ = Ω we obtain thanks to Proposition 4.2 where, the last equality holds true since and consider Notice that E • Ω = 0, |E| = 1.When d = 2, since the vector F (v) is orthogonal to Ω, there exists a function Λ = Λ(v) such that If d ≥ 3, let us denote by ⊥ E, any unitary vector orthogonal to E and Ω. Introducing the orthogonal matrix and thus from which it follows that ⊥ E • F (v) = 0, for any vector ⊥ E orthogonal to E and Ω.Hence, there exists a function Λ(v) such that We will show that Λ(v) depends only on v • Ω/|v| and |v|.Indeed, for any d ≥ 2, and any orthogonal transformation O such that OΩ = Ω we have implying that Λ(Ov) = Λ(v), for any v ∈ R d \ (RΩ).We are done if we prove that Λ The equality OE = E ′ implies that Ov = v ′ and therefore Λ(v ′ ) = Λ(Ov) = Λ(v), showing that there exists a function χ such that Λ In the last part of this section we concentrate on the elliptic problem satisfied by the function (c, r) → χ(c, r) introduced in Lemma 4.1.Even if ψ E i are eventually singular on RΩ, it will be no difficulty to define a Hilbert space on which solving for the profile χ.We again proceed using the minimization of quadratic functionals.
Proof.For any i ∈ {1, . . ., d − 1}, let us consider Observe that if h = χ, then ψ E i ,h coincides with ψ E i .Note that generally ψ E i ,h are not collision invariants, but perturbations of them, corresponding to profiles h.In this way, the minimization problem (29) will lead to a minimization problem on h, whose solution will be χ.Notice that once that M Ω .We know that ψ E i is the minimum point of J E i on H1 M Ω and therefore A straightforward computation shows that 2  , and which is equivalent, thanks to the Poincaré inequality (12) to Based on formula (19), we have 2 dc dr} and therefore we consider the Hilbert space 2 dc dr for g and h in H d and the norm given by The expression J E i (ψ E i ,h ) writes as functional of h Coming back to (29) and using (27), we deduce that χ ∈ H d and J(χ) ≤ J(h) for any h ∈ H d .
Therefore, by the Lax-Milgram lemma, we deduce that χ solves the problem (28).

Hydrodynamic equations
After identifying the collision invariants, we determine the fluid equations satisfied by the macroscopic quantities entering the dominant particle density f (t, x, v) = ρ(t, x)M Ω(t,x) (v).As seen before the balance for the particle density follows thanks to the collision invariant ψ = 1.The other balances follow by appealing to the vector field F (cf. Lemma 4.1) and the details are given in Sect.5.1.

Proof of Theorem 1.1
Applying ( 13) with ψ = 1 leads to (16).For any (t, x) ∈ R + × R d we consider the vector field By the definition of F (t, x, •), we know that It remains to compute and Ω(t, x).By a direct computation we obtain It is an easy exercise to show that the integral |P f v| dv vanishes and that the following relationship holds Therefore, by taking into account that ∂ t Ω • Ω = 0, the equality (31) becomes Similarly we write (for any smooth vector field ξ(x), the notation ∂ x ξ stands for the Jacobian matrix of ξ, i.e. (∂ x ξ) i,j = ∂ Notice that in the above computations we have used ( t ∂ x Ω)Ω = 0 and for any i, j, k ∈ {1, . . ., d − 1}.Combining ( 30), ( 32) and ( 33 where

Properties of the hydrodynamic equations
Let us start by noticing that the system (5)-( 6) is hyperbolic as a consequence of Theorem 4.1 in [32].On the other hand, the orientation balance equation (34) propagates the constraint |Ω| = 1.Indeed, let Ω = Ω(t, x) be a smooth solution of (34), satisfying |Ω(0, •)| = 1.Multiplying by Ω(t, x) we obtain (Ω(t, x) • ∇ x )|Ω| 2 = 0 , implying that |Ω| is constant along the characteristics of the vector field c 2 Ω(t, x) • ∇ x and thus |Ω(t, •)| = |Ω(0, •)| = 1, for all t ≥ 0. The rescaled equation (3) can be considered as an intermediate model between the equations introduced in [1] and [12] when there are no phase transitions.In [12], the authors considered a strong relaxation towards the 'terminal speed' (or cruise speed).Whereas in [1] the authors do not impose a penalization on the self-propelled/friction term.Our result could be applied to obtain the results in [12] without resorting to measures supported on the sphere by doing a double passage to the limit.First, passing to the limit ε → 0 in (3), taking V = V α,β , we obtain ( 5)- (6).Afterwards, we rescale V to λ V in the system ( 5)-( 6) and study the limit when λ → ∞.This amounts to study the behavior of the coefficients The proof of this result is a direct application of the Laplace method, see for instance [40].As an immediate consequence of Lemma 5. , where r 0 is the minimum of the potential function V α,β (r).Let us note that the asymptotic study of the coefficient c 1,λ when λ → ∞ can also be performed using Lemma 5.1 for more general potentials than V α,β (r).In particular, we could also consider smooth potentials V (|v|) having a unique global minimum r 0 such that V ′ (r) < 0, for 0 < r < r 0 , and V ′ (r) > 0, for r > r 0 .On the other hand, the asymptotic study of c 2,λ could be addressed following similar techniques as in [32], however, we do not dwell upon this matter here and leave it for a future work.