Fluid models with phase transition for kinetic equations in swarming

We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.


Introduction
This paper concerns the derivation of fluid models for populations of self-propelled individuals, with alignment and noise [18,23,24] starting from their kinetic description. The alignment between particles is imposed by relaxing the individuals velocities towards the mean velocity [19,20,25,31,33,37]. We refer to [38,16,29,17,6,7,32,21,22] and the references therein for a derivation of kinetic equations for collective behavior from microscopic models.
We concentrate on models with phase transition [2,3,36,26,27,30,41]. We 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 self-propulsion and friction mechanism writes div v {f ∇ v V (| · |)}, where v → V (|v|) is a confining potential. When considering V α,β (|v|) = β |v| 4 4 − α |v| 2 2 , with α, β > 0, we obtain the term div v {f (β|v| 2 − α)v} see [12,13] and also [9,10,11] for results based on averaging methods in magnetic confinement. The relaxation towards the mean velocity is given by div v {f (v − u[f ])} cf. [28], where for any particle density the notation u[f ] stands for the mean velocity Including noise with respect to the velocity variable, we obtain the Fokker-Planck type equation When considering large time and space scales in (1), we are led to the kinetic equation We investigate the asymptotic behavior of the family (f ε ) ε>0 , when ε becomes small. We expect that the limit density f (t, x, ·) = lim εց0 f ε (t, x, ·) is an equilibrium for the interaction mechanism Q(f (t, x, ·)) = 0, (t, x) ∈ R + × R d .
For any u ∈ R d we introduce the notations Actually the function Z depends only on σ and |u|, see Proposition 2.1, and thus we will write Z = Z(σ, l = |u|). Notice that for any smooth particle density f and any u ∈ R d we have leading to the following representation formula Multiplying by f /M u[f ] and integrating by parts with respect to the velocity imply that any equilibrium satisfies Recall that u[f ] is the mean velocity, and therefore we impose Notice that Φ u is left invariant by any orthogonal transformation preserving u. Consequently, we deduce (see Proposition 2.1) that R d f (v) v dv is parallel to u, and therefore the constraint (3) fix only the modulus of the mean velocity, and not its orientation (which remains a free parameter). Our first important observation gives a characterization to find the bifurcation diagram of stationary solutions of Q(f ) = 0. We prove that M u is an equilibrium if and only if l = |u| is a critical point of Z(σ, ·), cf. Proposition 2.1. Moreover, several values for |u|, or only one are admissible, depending on the diffusion coefficient σ. In that case we will say that a phase transition occurs. Notice that in this work we do not distinguish between phase transitions and bifurcation points. For any particle density f = f (v), the notation Ω[f ] stands for the orientation of the mean velocity u[f ], if u[f ] = 0 Finally, for any (t, x) ∈ R + × R d , the limit particle density is a von Mises-Fisher distribution f (t, x, v) = ρ(t, x)M |u|Ω(t,x) (v) parametrized by the concentration ρ(t, x) = ρ[f (t, x, ·)] and the orientation Ω(t, x) = Ω[f (t, x, ·)]. We identify a class of potentials v → V (|v|) such that a phase transition occurs and we derive the fluid equations satisfied by the macroscopic quantities ρ, Ω. More exactly we assume that the potential v → V (|v|) satisfies lim |v|→+∞ |v| 2 2 + V (|v|) |v| = +∞ (4) (such that Z is well defined) and belongs to the family V defined by: there exists σ 0 > 0 verifying 1. For any 0 < σ < σ 0 there is l(σ) > 0 such that Z(σ, l) is stricly increasing on [0, l(σ)] and strictly decreasing on [l(σ), +∞[; 2. For any σ ≥ σ 0 , Z(σ, l) is strictly decreasing on [0, +∞[.
The first important result in this work shows that potentials in V have a phase transition at σ = σ 0 as shown in Section 2.
Remark 1.1 The potential V (|v|) = β |v| 4 4 − α |v| 2 2 belongs to the family V as shown in [40,3,36] in any dimension. Theorem 1.1 Assume that the potential v → V (|v|) satisfies (4), belongs to the family V defined above and that 0 < σ < σ 0 . Let us consider (f ε ) ε>0 satisfying Therefore, at any (t, x) ∈ R + ×R d the dominant term in the Hilbert expansion The constant c ⊥ is given by c ⊥ = R + r d+1 π 0 cos θ χ(cos θ, r) e(cos θ, r, l(σ)) sin d−1 θ dθdr l(σ) R + r d π 0 χ(cos θ, r) e(cos θ, r, l(σ)) sin d−1 θ dθdr and the function χ solves 2 e(c, r, l(σ))∂ r χ where e(c, r, l) = exp − r 2 2σ + rcl σ − V (r) σ . Remark 1.2 Several considerations regarding the hydrodynamic equations (6)-(8) and the asymptotic limit to obtain them are needed: • The asymptotic limit in (5) is different from the one analysed in [13] where the friction term is penalized at higher order. The main technical difficulty in [13] compared to our present work is that to solve for the different orders on the expansion in [13] we had to deal with Fokker-Planck equations on the velocity sphere with speed α β .
• The hydrodynamic equations (6)- (8) in the particular case of the potential V (|v|) = β |v| 4 4 − α |v| 2 2 recover the ones obtained in [28,26,27,13] by taking the limit α → ∞ with β/α = O(1). In this limit, the particle density f is squeezed to a Dirac on the velocity sphere with speed α β . The constants can be computed exactly based on [36] and they converge towards the exact constants obtained in [28,27,13]. This is left to the reader for verification.
• The hydrodynamic equations (6)-(8) have the same structure as the equations derived in [28,27,13] just with different constants, and therefore they form a hyperbolic system as shown in [28,Subsection 4.4].
Theorem 1.2 Assume that the potential v → V (|v|) satisfies (4) and verifies the above hypothesis for some σ > 0. Let us consider (f ε ) ε>0 satisfying Therefore, at any (t, x) ∈ R + ×R d the dominant term in the Hilbert expansion The constants c ⊥ , c , c ′ are given by c ⊥ = R + r d+1 π 0 cos θ χ(cos θ, r) e(cos θ, r, |u|) sin d−1 θ dθdr |u| R + r d π 0 χ(cos θ, r) e(cos θ, r, |u|) sin d−1 θ dθdr c = R + r d+1 π 0 cos 2 θ χ Ω (cos θ, r) e(cos θ, r, |u|) sin d−2 θ dθdr 2|u| R + r d π 0 cos θχ Ω (cos θ, r) e(cos θ, r, |u|) sin d−2 θ dθdr c ′ = R + r d+1 π 0 χ Ω (cos θ, r) e(cos θ, r, |u|) sin d θ dθdr (d − 1)|u| R + r d π 0 cos θχ Ω (cos θ, r) e(cos θ, r, |u|) sin d−2 θ dθdr the function χ solves (9) and the function χ Ω solves Our paper is organized as follows. In Section 2 we investigate the function Z, whose variations will play a crucial role when determining the equilibria of the interaction mechanism Q. We identify a family of potentials such that a phase transition occurs for some critical diffusion coefficient σ 0 . Section 3 is devoted to the study of the linearization of Q and of its formal adjoint. We are led to study the spectral properties of the pressure tensor. The kernel of the adjoint of the linearization of Q is studied in Section 4. These elements will play the role of the collision invariants, when determining the macroscopic equations by the moment method. The main results, Theorem 1.1, 1.2, are proved in Section 5. Some examples are presented in Section 6.

Phase transitions and Potentials: Properties of Equilibria
For any u ∈ R d we denote by T u the family of orthogonal transformations of R d preserving u. Notice that T 0 is the family of all orthogonal transformations of R d .
Proof. For any O ∈ T u , we have For any ξ ∈ S d−1 ∩ (Ru) ⊥ , we consider O ξ = I d − 2ξ ⊗ ξ ∈ T u , and thus we obtain We assume that Observe that and therefore, under the hypothesis (12), it is easily seen that Z(σ, u) is finite for any σ > 0 and u ∈ R d . Similarly we check that for any σ > 0 and u ∈ R d , all the moments of M u are finite For further developments, we recall the formula for any non negative measurable function χ = χ(c, r) :] − 1, 1[×R ⋆ + → R, any Ω ∈ S d−1 and d ≥ 2. Here |S d−2 | is the surface of the unit sphere in R d−1 , for d ≥ 3, and |S 0 | = 2 for d = 2.
Proposition 2.1 Assume that the potential v → V (|v|) satisfies (12). Then the following statements hold true : 1. The function Z(σ, u) depends only on σ and |u|. We will simply write

The von Mises-Fisher distribution M u is an equilibrium if and only if
Proof.
1. Applying formula (13) with Ω = u/|u|, if u = 0, and any Ω ∈ S d−1 if u = 0, we obtain and therefore Z depends only on σ and |u|. 2. We consider the integrable vector field a(v) = M u (v)v, v ∈ R d . It is easily seen that for any O ∈ T u , we have and therefore the vector field a is left invariant by T u . Our conclusion follows by Lemma 2.1.
and we are done observing that for any v such that v · u > 0 we have

The von Mises-Fisher distribution M u is an equilibrium if and only if
By the previous statement we know that R d M u (v)v dv ∈ Ru and therefore M u is an where Ω = u |u| if u = 0 and Ω is any vector in S d−1 if u = 0. But we have and therefore M u is an equilibrium if and only if l = |u| is a critical point of Z(σ, ·).

Remark 2.2
As Z depends only on σ, |u|, we can write for any Ω ∈ S d−1 and u ∈ R d . We deduce that for any Ω ∈ S d−1 and u ∈ R d , we have where At this point, we know that for any σ > 0, the equilibria are related to the critical points of Z(σ, ·). In order to find possible bifurcation points of the disordered state u = 0, let us analyze the variations of Z(σ, ·) for small σ. We assume the following hypothesis on the potential For such a potential, we can minimize Φ u (v) with respect to v ∈ R d , for any u ∈ R d . Indeed, the function Φ u is convex, continuous on R d and By (12) we deduce that lim |v|→+∞ Φ u (v) = +∞ and therefore Φ u has a minimum point v ∈ R d . This minimum point is unique ). We intend to analyze the sign of ∂ l Z(σ, |u|) for small σ.
We need to determine the sign of ( We assume that V (·) possesses another critical point r 0 > 0 and V ′ (r) < 0 for any 0 < r < r 0 and V ′ (r) > 0 for any r > r 0 .

We have
Proof. 1. By (15) we know that Φ 0 is strictly convex on R d and we deduce that r → r 2 2 + V (r) is strictly convex on R + . Therefore the function r → r + V ′ (r) is strictly increasing on R + and maps [0, r 0 ] to [0, r 0 ]. It remains to check that it is unbounded when r → +∞. Suppose that there is a constant C such that r + V ′ (r) ≤ C, r ∈ R + . After integration with respect to r, one gets implying that which contradicts (12).
2. Let us consider 0 < |u| < r 0 . Therefore, v = 0 and , and thus Clearly, for any 0 < δ < r 0 /2, we have Similarly, for any |u| > r 0 , we have |v| > r 0 and As before, for any δ > 0, we obtain The previous arguments allow us to complete the analysis of the variations of Z(σ, |u|), when σ is small. The convergence when σ ց 0 in (16) can be handled by dominated convergence, The Notice that (18) guarantees (12) and (15). Indeed, the function v → V λ (|v|) being convex, it is bounded from below by a linear function and therefore Obviously, Φ 0 is strictly convex, as sum between the strictly convex function v → (1 − λ) |v| 2 2 and the convex function v → V λ (|v|).
In order to conclude the study of the variations of Z for small σ > 0, we consider potentials V satisfying V (| · |) ∈ C 2 (R d ), (17) and (18). We come back to (16). Notice that As we know, cf.
Motivated by the above behavior of the function Z, we assume that the potential v → V (|v|) satisfies (12) (such that Z is well defined) and belongs to the family V defined by: there exists σ 0 > 0 verifying and strictly decreasing on [l(σ), +∞[; In fact, the critical diffusion coefficient σ 0 vanishes the second order derivative of Z with respect to l, at l = 0, as shown next.
After passing to the limit when n → +∞, we obtain a contradiction and therefore lim σրσ 0 l(σ) = 0. We have proved that σ → l(σ) is continuous.
3 Given a potential V (| · |) ∈ V, then the unique bifurcation point from the disordered state happens at σ 0 . In fact, if we define the function as in [3]. Then by (14), we get σ∂ l Z(σ, l) = Z(σ, l)H(σ, l). By taking the derivative with respect to l, we obtain Therefore, for the curve l(σ) such that H(σ, l(σ)) = 0, we get ∂ l H(σ 0 , 0) = 0. Using implicit differentiation and the continuity of the curves and the functions involved, it is also easy to check that ∂ σ H(σ 0 , 0) = 0. Therefore, to clarify the behavior of the two curves at σ 0 , one needs to work more to compute the lim σրσ 0 l ′ (σ). In any case, this shows that σ 0 is the only bifurcation point from the manifold of disorder states u = 0 for potentials V (| · |) ∈ V without the need of applying the Crandall-Rabinowitz bifurcation theorem. It would be interesting to use Crandall-Rabinowitz for general potentials to identify more general conditions for bifurcations.
In the last part of this section, we explore some properties of the potentials V in the class V. We show that under the hypothesis (18), we retrieve a weaker version of (17). (18) and there is the limit lim σց0 l(σ) = r 0 > 0, then V ′ (r) ≤ 0 for any 0 < r ≤ r 0 and V ′ (r) ≥ 0 for any r ≥ r 0 .
It remains to determine the fluid equations satisfied by the macroscopic quantities ρ, Ω. When |u| = 0, the continuity equation leads to ∂ t ρ = 0. In the sequel we concentrate on the case |u| = l(σ), 0 < σ < σ 0 (that is, the modulus of the mean velocity is given, as a function of σ). We follow the strategy in [13,1]. We consider We introduce the usual scalar products Mu and we denote by |·| Mu , · Mu the associated norms. Moreover we need a Poincaré inequality. This comes from the equivalence between the Fokker-Planck and Schrödinger operators. As described in [8], we can write it as We have a spectral decomposition of the operator H u under suitable confining assumptions (cf. Theorem XIII.67 in [39]).
, is bounded from below and is coercive i.e. |∇ Then H −1 u is a self adjoint compact operator in L 2 (R d ) and H u admits a spectral decomposition, that is, a nondecreasing sequence of real numbers (λ n u ) n∈N , lim n→+∞ λ n u = +∞, and a L 2 (R d )-orthonormal basis (ψ n u ) n∈N such that H u ψ n u = λ n u ψ n u , n ∈ N, λ 0 u = 0, λ 1 u > 0. Therefore, under the hypotheses in Lemma 3.1, for any u ∈ R d there is λ u > 0 such that for any χ ∈ H 1 Mu we have The fluid equations are obtained by taking the scalar product of (22) with elements in the kernel of the (formal) adjoint of L f , that is with functions ψ = ψ(v) such that see also [4,5,14,15,34,35]. For example, ψ = 1 belongs to the kernel of and we obtain the continuity equation (7) ∂ t In the sequel we determine the formal adjoint of the linearization of the collision operator Q around its equilibria. 1. The linearization L f = dQ f is given by

The formal adjoint of L f is
3. We have the identity Proof.

We have
Therefore we obtain 3. For any i ∈ {1, ..., d} we have and therefore We identify now the kernel of L ⋆ f .

The function
for some vector W ∈ ker(M u − σI d ).
Moreover, the linear map W : As ρ > 0 we deduce that W [ψ] ∈ ker(M u − σI d ) and by the second statement in Proposition 3.1 it comes that 2. =⇒ 1. Let ψ be a function satisfying (24) for some vector W ∈ ker(M u −σI d ). Multiplying by M u (v)(v − u) and integrating with respect to v yields As we know that W ∈ ker(M u − σI d ), we deduce that W = W [ψ], implying that ψ belongs to ker We focus on the eigenspace ker(M u − σI d ).

comes by the change of variable
For the formula (26) and therefore, We claim that R d We deduce that and by taking into account that |u| By Remark 2.2, we know that and finally we have As l(σ) is a maximum point of Z(σ, ·), we have ∂ 2 ll Z(σ, l(σ)) ≤ 0 and therefore M u ≤ σI d . 4 The kernel of L ⋆ f By Lemmas 3.2, 3.3, any solution of (24) with W ∈ (Ru) ⊥ belongs to the kernel of the formal adjoint L ⋆ f . Generally we will solve the elliptic problem for any W ∈ R d . We consider the continuous bilinear symmetric form a u : Mu and the linear form L : We are looking for variational solutions of (27) i.e., Mu and a u (ψ, θ) = L(θ) for any θ ∈ H 1 Mu .
When taking θ = 1 ∈ H 1 Mu , we obtain the following necessary condition for the solvability of (27) which is satisfied for any W ∈ R d , because M u has mean velocity u. It happens that (29) also guarantees the solvability of (27). For that, it is enough to observe that the bilinear form a u is coercive on the Hilbert spaceH 1 Mu := {θ ∈ H 1 Mu : ((θ, 1)) Mu = 0}. Indeed, for any θ ∈ H 1 Mu such that ((θ, 1)) Mu = 0, we have thanks to the Poincaré inequality (23) and therefore Thanks to Lax-Milgram lemma on the Hilbert spaceH 1 Mu , there is a unique function ψ ∈H 1 Mu such that a u (ψ,θ) = L(θ) for anyθ ∈H 1 Mu .
From now on, for any W ∈ R d , we denote by ψ W the unique solution of (28), verifying Let us introduce the Hilbert spaces endowed with the scalar product We denote the induced norms by |ξ| Mu = (ξ, ξ)

1/2
Mu , ξ ∈ L 2 Mu and ξ Mu = ((ξ, ξ)) Mu . Obviously, a vector field ξ = ξ(v) belongs to H 1 Mu iff ξ i ∈ H 1 Mu for any i ∈ {1, ..., d} and we have Let us consider the closed subspacẽ Thanks to (23), for any ξ ∈ H 1 Mu we have the inequality We introduce the continuous bilinear symmetric form a u : Mu and the linear form L : Mu . Under the hypothesis (12), it is easily seen that L is bounded on H 1 Mu .

Proposition 4.1
There is a unique solution F of the variational problem F ∈H 1 Mu and a u (F, η) = L(η), for any η ∈ H 1 Mu .

For any
The vector field F is left invariant by the family T u .
By Lax-Milgram lemma, applied on the Hilbert spaceH 1 Mu , there is a unique vector field F ∈H 1 Mu such that a u (F, η) = L(η), for any η ∈H 1 Mu .
Actually, the above equality holds true for any η ∈ H 1

It remains to check that for any
Mu . Notice also that for any θ ∈ H 1 Mu we have θW ∈ H 1 Mu and Thank to the uniqueness we obtain We are done if we prove that v → OF ( t Ov) solves the same problem as F . Clearly we have The vector field F expresses in terms of two functions which are left invariant by the family T u .

Proposition 4.2
There is a function ψ, which is left invariant by the family T u , such that We claim that F ′ (v) is parallel to the orthogonal projection of v over (RΩ) ⊥ . Indeed, for any v ∈ R d \ (RΩ), let us consider When d = 2, since E(v) and F ′ (v) are both orthogonal to Ω, there exists a function ψ = ψ(v) such that If d ≥ 3, let us denote by ⊥ E, any unitary vector orthogonal to E and Ω. Introducing the from which it follows that ⊥ E · F ′ (v) = 0, for any vector ⊥ E orthogonal to E and Ω. Hence, there exists a function ψ(v) such that It is easily seen that the function ψ is left invariant by the family T u . Indeed, for any O ∈ T u we have Similarly, ψ Ω is left invariant by the family T u The functions ψ, ψ Ω will enter the fluid model satisfied by the macroscopic quantities ρ, Ω, |u|.
It is convenient to determine the elliptic partial differential equations satisfied by them.

The fluid model
The balances for the macroscopic quantities ρ, u follow by using the elements in the kernel of L ⋆ f . Proof. (of Theorem 1.1) The use of ψ = 1 ∈ ker L ⋆ f leads to (7). By Lemma 3.3, we know that (Ru) ⊥ ⊂ ker(M u − σI d ) and thus, for any (t, We It is easily seen (use the change of variable v = ( Therefore one gets with cos θχ(cos θ, r)e(cos θ, r, l(σ)) sin d−1 θ dθdr l(σ) R + r d π 0 χ(cos θ, r)e(cos θ, r, l(σ)) sin d−1 θ dθdr .
As before, using also R d ψ Ω (v)M u (v) dv = 0, we write Similarly, observe that is F (v) = v − u, v ∈ R d , which belongs toH 1 Mu , and therefore the functions ψ, ψ Ω such that are given by By straightforward computations we obtain In this case (10), (11) are the Euler equations, as expected when taking the limit ε ց 0 in the Fokker-Planck equations
One can check that these potentials belong to the family V, see [36]. We include an example V 1,1 (|v|) = |v| 4 4 − |v| 2 2 for the sake of completeness. In this case the critical diffusion can be computed explicitly. In particular, for d = 2 we have σ 0 = 1/π.
Taking the second derivative with respect to l one gets cf. Remark 2.2 rl cos θ σ (r cos θ − l) 2 − σ σ 2 sin d−2 θ dθdr and therefore It is easily seen that