EXISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODELS TRYGVE K. KARPER, ANTOINE MELLET, AND KONSTANTINA TRIVISA

Transcription

1 EXISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODELS TRYGVE K. KARPER, ANTOINE MELLET, AND KONSTANTINA TRIVISA Abstract. We establish the global existence of weak solutions to a class of kinetic flocking equations. The moels uner conieration inclue the kinetic Cucker-Smale equation [6, 7] with possibly non-symmetric flocking potential, the Cucker-Smale equation with aitional strong local alignment, an a newly propose moel by Motsch an Tamor [14]. The main tools employe in the analysis are the velocity averaging lemma an the Schauer fixe point theorem along with various integral bouns. Contents 1. Introuction an main results The kinetic Flocking moels 1.. Main results 4. A priori estimates an velocity averages 7.1. A priori estimates 7.. Velocity averaging an compactness Approximate solutions The operator T is well-efine The operator T is compact Proof of Proposition Existence of solutions (Proof of Theorem 1.1) Limit as λ Limit as δ 0 (an en of the proof of Theorem 1.1) Non-symmetric flocking an the Motsch-Tamor moel Other existence result 1 7. The entropy of flocking 3 References 6 References 6 1. Introuction an main results Moels escribing collective self-organization of biological agents are currently receiving consierable attention. In this paper, we will stuy a class of such moels. Date: February 0, Mathematics Subject Classification. Primary:35Q84; Seconary:35D30. Key wors an phrases. flocking, kinetic equations, existence, velocity averaging, Cucker- Smale, self-organize ynamics. The work of T.K was supporte by the Research Council of Norway through the project The work of A.M was supporte by the National Science Founation uner the Grant DMS The work of K.T. was supporte by the National Science Founation uner the Grant DMS

2 KARPER, MELLET, AND TRIVISA More precisely, we focus on kinetic type moels for the flocking behavior exhibite by certain species of birs, fish, an insects: Such moels are typically of the form f t + v x f + iv v (fl[f]) + β iv v (f(u v)) = 0, in (0, T ) (1.1) where f := f(t, x, v) is the scalar unknown, 1 is the spatial imension, an β 0 is a constant. The first two terms escribe the free transport of the iniviuals, an the last two terms take into account the interactions between iniviuals, who try to align with their neighbors. The alignment operator L has the form L[f] = K f (x, y)f(y, w)(w v) w y, (1.) where the kernel K f may epen on f an may not be symmetric in x an y (see (1.8) below). The last term in (1.1) escribes strong local alignment interactions (see below), where u enotes the average local velocity, efine by fv v R u(t, x) = f v. Equation (1.1) inclues the classical kinetic Cucker-Smale moel, which correspons to β = 0 an an alignment operator L[f] given by (1.) with a smooth kernel inepenent of f: K f (x, y) = K 0 (x, y) with K 0 symmetric (K 0 (x, y) = K 0 (y, x)). There is a consierable boy of literature concerning the kinetic Cucker-Smale equation an its variations (see [, 3, 4, 5, 6, 7, 8, 11, 1, 14]), but a general existence theory has thus far remaine absent. Notable exceptions are the stuies [1, 5] in which well-poseness for Cucker-Smaletype moels is establishe in the sense of measures an then extene to weak solutions ([1] as noise to the moel). The existence of classical solutions to the kinetic Cucker-Smale equation is establishe in [1], but the result oes not exten to (1.1). Compare to these results, the main contribution of this paper is the aition of the local alignment term (β > 0) an the fact that the function K f is allowe to be non-symmetric (such moels may not preserve the total momentum). More precisely, our analysis will inclue a moel recently propose by Motsch & Tamor [14], for which K f epens on f as follows: K f (x, y) = φ(x y) φ f v (1.3) with enoting the convolution prouct in space. In the remaining parts of this introuction, we first introuce the moels we consier in more etails an iscuss various variations propose in the literature. Following this we introuce the notion of weak solutions. We en the introuction by stating our main existence result an proviing a brief sketch of the main ieas use to prove it The kinetic Flocking moels. Our starting point is the pioneering moel introuce by Cucker an Smale [6]: Consier N iniviuals (birs, fish) each totally escribe by a position x i (t) an a velocity v i (t). The Cucker-Smale moel [6, 7] is given by the evolution ẋ i = v i, v i = 1 K 0 (x i, x j )(v j v i ). (1.4) N j i Roughly speaking, each particle attempts to align it s velocity with a local average velocity given by the form an support of K 0. In this paper, we focus not on the

3 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 3 particle moel but on the corresponing kinetic escription. This escription can be irectly erive for the empirical istribution function f(t, x, v) = 1 δ(x x i (t))δ(v v i (t)). N Specifically, by irect calculation one sees that f evolves accoring to where L is given by L[f] = i f t + v x f + iv v (fl[f]) = 0, (1.5) K 0 (x, y)f(y, w)(w v) w y, (1.6) The question we will aress in this paper is whether or not f is a function when the initial ata f 0 is a function. For this purpose, it is esirable that we o not have loss of mass at infinity. Note that there is no effect countering such loss in (1.5). Inee, (1.5) consists of only transport an alignment of the velocity to an average velocity. Hence, if the average velocity is non-zero, the equation will eventually transport all mass to infinity. To counter this, we a a confinement potential Φ to the equation. The only property we require of this potential is that Φ when x The Cucker-Smale: The simplest moel that we consier in this paper, is thus the Cucker-Smale moel [6, 7] with confinement potential: Cucker-Smale: f t + v x f iv v (f x Φ) + iv v (fl[f]) = 0. (1.7) where L is given by (1.6) with K 0 a smooth symmetric function The Motsch-Tamor correction. In the recent paper by Motsch an Tamor [14] it is argue that the normalization factor 1 N in (1.4) leas to some unesirable features. In particular, if a small group of iniviuals are locate far away from a much larger group of iniviuals, the internal ynamics in the small group is almost halte since the number of iniviuals is large. This is easily unerstoo if we also assume that K 0 has compact support an that the istance between the two groups are larger than the support of K 0. In that case, the ynamics of the two groups shoul be inepenent of each other, but the size of the alignment term in Cucker- Smale moel still epens on the total number of iniviuals. To remey this, Motsch an Tamor [14] propose a new moel, with normalize, non-symmetric alignment, given by φ(x y)f(y, w)(w v) w y R L[f] = R (1.8) φ(x y)f(y, w) w y R (which is the general operator (1.), with K f given by (1.3)). The resulting moel is the following: Motsch-Tamor: f t + v x f iv v (f x Φ) + iv v (f L[f] ) = 0, (1.9) with L given by (1.8). Note that unlike the Cucker-Smale moel [6, 7], the Motsch-Tamor moel [14] oes not preserve the total momentum vf v x.

4 4 KARPER, MELLET, AND TRIVISA Local alignment. It is also possible to combine the Cucker-Smale moel with the Motsch-Tamor moel letting the Cucker-Smale flocking term ominate the long-range interaction an the Motsch-Tamor term ominate short-range interactions. This will correct the aforementione eficiency of the kinetic Cucker-Smale moel. However, the large-range interactions is still close to that of the Cucker- Smale moel. In particular, we consier the singular limit where the Motsch- Tamor flocking kernel φ converges to a Dirac istribution. The Motsch-Tamor correction then converges to a local alignment term given by: L[f] = j ρv ρ = u v. where ρ(x, t) = f(x, v, t) v, j(x, t) = vf(x, v, t) v an u(x, t) is efine by the relation u = vf v R f v. (1.10) This leas to the following equation: Cucker-Smale with strong local alignment: f t + v x f iv v (f x Φ) + iv v (fl[f]) + β iv v (f(u v)) = 0 (1.11) with L given by (1.6) with K a given symmetric function. Note that though (1.11) is obtaine as singular limit of the non-symmetric Motsch-Tamor moel, it has more symmetry, an in particular preserves the total momentum (we will see later that it also has goo entropy inequality) Noise, self-propulsion, an friction. Finally, for the purpose of applications, there are many other aspects that are not inclue in the moels we have consiere so far. For instance, there might be unknown forces acting on the iniviuals such as win or water currents. Often these type of effects are simply moele as noise. If this noise is brownian it will lea to the aition of a Laplace term in the equations. We note that this term has a regularizing effect on the solutions, but that this regularizing effect is not require to prove the existence of solutions. In aition, we coul a self-propulsion an friction in the moels. This amounts to aing a term iv((a b v )vf) in the equation. The most general moel for which we will able to prove global existence of solutions is the following: Cucker-Smale with strong local alignment, noise, self-propulsion, an friction: f t + v x f iv v (f x Φ) + iv v (fl[f]) + β iv v (f(u v)) = σ v f iv((a b v )vf) with σ 0, a 0 an b 0. (1.1) 1.. Main results. We now list our main results. The existence of solutions for (1.7) when the alignment operator is given by (1.6) with smooth boune symmetric kernel K 0 presents no particular ifficulties. Our focus, instea, is on the local alignment moels (1.11) an (1.1) (with or without noise, friction an selfpropulsion). Throughout this paper, the potential Φ(x) is a smooth confinement potential, satisfying lim Φ(x) = +. x

5 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS Existence with local alignment an K symmetric. Our first result is: Theorem 1.1. Assume that f 0 0 satisfies f 0 L (R ) L 1 (R ), an ( v + Φ(x))f 0 L 1 (R ). Assume that L is the alignment operator given by (1.6) with K 0 symmetric (K 0 (x, y) = K 0 (y, x)) an boune. Then, for any σ 0, a 0 an b 0 there exists f such that f C(0, T ; L 1 (R )) L ((0, T ) R ), ( v + Φ(x))f L (0, ; L 1 (R )), an f is a solution of (1.1) in the following weak sense: for any ψ C c fψ t vf x ψ + f x Φ x ψ fl[f] v ψ vxt R +1 + σ v f v ψ βf(u v) v ψ vxt R +1 + (a b v )vf v ψ vxt = f 0 ψ(0, ) vx, R +1 R ([0, T ) R ), where u is such that j = ρu. (1.13) Remark 1.. Note that the efinition of u is ambiguous if ρ vanishes (vacuum). We thus efine u pointwise by j(x, t) if ρ(x, t) 0 u(x, t) = ρ(x, t) (1.14) 0 if ρ(x, t) = 0 Since ( j ) 1/ v f(x, v, t) v ρ 1/ we have j = 0 whenever ρ = 0 an so (1.14) implies j = ρu. We note also that u oes not belong to any L p space. However, we have uf x v f L (R ) v f(x, v, t) v x R R so that the term uf in the weak formulation (1.13) makes sense as a function in L. The proof of Theorem 1.1 is evelope in Sections 3 an 4. The main ifficulty is of course the nonlinear term fu (the other nonlinear term fl[f] is much more regular, since L[f] L ). Because u oes not belong to any L p space, we cannot o a fixe point argument irectly to prove the existence. Instea, we will introuce an approximate equation (see (3.1)), in which u is replace by a more regular quantity. Existence for this approximate equation will be prove via a classical Schauer fixe point argument (compactness will follow from velocity averaging lemma an energy estimates). The main ifficulty is then to pass to the limit in the regularization, which amounts to proving some stability property for (1.1). As pointe out above, since u cannot be expecte to converge in any L p space, we will pass to the limit in the whole term fu an then show that the limit has the esire form. We note that the friction an self-propulsion term introuces no aitional ifficulty. We will thus take a = b = 0 throughout the proof. The noise term has a regularizing effect, which will not be use in the proof. We thus assume that σ 0.

6 6 KARPER, MELLET, AND TRIVISA 1... Existence with local alignment an the Motsch-Tamor term. Our secon result concerns the Motsch-Tamor moel (1.9) with L given by (1.8). We can rewrite (1.9) as f t + v x f iv v (f x Φ) + iv v (f(ũ v)) = 0 (1.15) where φ(x y)f(y, w, t)w w y R ũ(x, t) = R R φ(x y)f(y, w, t) w y = φ(x y)j(y, t) y R φ(x y)ρ(y, t) y. Compare with (1.11), the term f(ũ v) is thus less singular than f(u v) (which we recover when φ(x y) = δ(x y)), an so the existence of solution for (1.9) can be prove following similar (or simpler) arguments, provie the necessary energy estimate hol. An this turns out to be quite elicate. Inee, the lack of symmetry of the alignment operator L implies that it oes not preserve momentum, an proving that the energy ( v + Φ(x))f(x, v, t) v x remains boune for all time proves elicate for general φ. We will prove an existence result when φ is compactly supporte. More precisely, we have: Theorem 1.3. Assume that f 0 0 satisfies f 0 L (R ) L 1 (R ), an ( v + Φ(x))f 0 L 1 (R ). Assume that L is the alignment operator (1.8) where φ is a smooth nonnegative function such that there exists r > 0 an R > 0 such that φ(x) > 0 for x r, φ(x) = 0 for x R. (1.16) Then there exists a weak solution of (1.15) in the same sense as in Theorem Entropy of flocking with symmetric kernel. To complete our stuy, restricting our attention to symmetric flocking, we will show that the moel (1.11) is enowe with a natural issipative structure. More precisely, we consier the usual entropy F(f) = R an the associate issipations D 1 (f) = 1 T 0 σ v f log f + f + fφ v x (1.17) β R β f σ β vf f(u v) v x (1.18) an D (f) = 1 K 0 (x, y)f(x, v)f(y, w) v w wyvx. (1.19) (Note that D 1 (f) is a local issipation ue to the noise an local alignment term while D (f) is the issipation ue to the non-local alignment term) We then have: Proposition 1.4. Assume that L is the alignment operator given by (1.6) with K 0 symmetric (K 0 (x, y) = K 0 (y, x)) an boune, an let β > 0 an σ 0. If f is a solution of (1.11) with sufficient integrability, then the following inequality hols: t F(f) + D 1 (f) + D (f) σ β K 0 (x, y)f(x, v)f(y, w) wyvx. (1.0) Furthermore, if the confinement potential Φ satisfies e Φ(x) x < + (1.1)

7 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 7 then there exists C epening only on K 0, Φ an f 0 (x, v) x v such that t F(f) + 1 D 1(f) + 1 K 0 (x, y)ϱ(x)ϱ(y) u(x) u(y) yx C R β F(f(t)). (1.) The first inequality (1.0) shows that the nonlocal alignment term is responsible for some creation of entropy. The secon inequality (1.) shows that this term can be controlle by D 1 (f) an the entropy itself. This last inequality is particularly useful in the stuy of singular limits of (1.11) with ominant local alignment (β ). Such limits will be investigate in [13]. Finally, we can now prove the existence of weak solutions satisfying the entropy inequality: Theorem 1.5. Assume that L is the alignment operator given by (1.6) with K symmetric (K 0 (x, y) = K(y, x)) an boune, an let β > 0 an σ 0. Assume furthermore that f 0 satisfies f 0 L (R ) L 1 (R ), an ( v + Φ(x))f 0 L 1 (R ). then there exist a weak solution of (1.11) (in the sense of Theorem 1.1) satisfying F(f(t)) + an, if Φ satisfies (1.1), F(f(t)) + 1 for all t > t 0 t 0 t 0 D 1 (f) s D 1 (f) + D (f) s e σ β K M t F(f 0 ) (1.3) K 0 (x, y)ϱ(x)ϱ(y) u(x) u(y). A priori estimates an velocity averages yxs e C β t F(f 0 ) In this section, we collect some results that we will nee for the proof of Theorem 1.1. Since the only interesting case is when β > 0, we will take β = 1 throughout the proof. First, we erive some priori estimates satisfie by solutions of (1.11) (when introucing the regularize equation in Section 3, we will make sure that these estimates still hol). Then, we recall the classical averaging lemma which will play a crucial role in the proof..1. A priori estimates. Any smooth solution of (1.11) satisfies the following conservation of mass: f(x, v, t) x v = f 0 (x, v, t) x v =: M. R R Since f 0 0, we have f 0, an so the conservation of mass implies a a priori boun in L (0, T ; L 1 (R )). It is well known that solutions of (1.11) also satisfy a a priori estimates in L (0, T ; L p (R )) (cf. [1]) for all p [1, ]. More precisely: Lemma.1. Let f be a smooth solution of (1.11), then f L (0,T ;L p (R )) + σ v f p p L ((0,T ) ) ect/p f 0 L p (R ) (.1) with C = [1 + K 0 L M] an p = p p 1, for all p [1, ].

8 8 KARPER, MELLET, AND TRIVISA Proof. For p <, a simple computation yiels f p 4(p 1) x v = σ (f p/ ) x v t p (p 1) f p iv v L[f] x v + (p 1) f p x v Since iv v L[f] = K 0 (x, y)f(y, w) y w we also have We euce: t iv v L[f] K 0 L M. f p 4(p 1) x v = σ (f p/ ) x v p +(p 1)[1 + K 0 L M] f p x v which implies (.1) by a Gronwall argument. Next, we will nee better integrability of f for large v an x. Let us enote v E(f) = f + Φ(x)f x v. We then have the following important a priori estimate: Lemma.. Let f be a smooth solution of (1.11), then t E(f) + u v f x v R + 1 K 0 (x, y)f(x, v)f(y, w) v w wyvx R R = σ f x v = σm. (.) In particular, E(f(t)) σmt + E(f 0 ). (.3) Remark.3. Inequality (.) will play a funamental role in this paper. It implies that the iniviuals remain somewhat localize in space an velocity for all time. We note that it is not quite the stanar entropy inequality when σ > 0 (the natural entropy inequality for (1.11) when σ > 0, which involves a term of the form f log f, will be etaile in Section 7. It is not neee for the proof of Theorem 1.1). Our existence result can be generalize to other moels, provie we can still establish (.3). For instance, self-propulsion an friction can be taken into account, via a term of the form iv((a b v )vf) in the right han sie of (1.11). It is easy to check that (.3) then becomes E(f) [ E(f 0 ) + σm a ] e at. Another important generalization is to consier non-symmetric flocking interactions (i.e. when K 0 (x, y) K 0 (y, x)). The erivation of a boun on E(f) requires stronger assumptions on the convolution kernel, an these assumptions are ifficult to check in the case of Motsch-Tamor moel (because of the normalization of the convolution kernel). This will be iscusse in Section 5.

9 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 9 Proof of Lemma.. Using Equation (1.11), we compute ) (Φ(x) t E(f) = + v t f vx R = fl[f]v + f(u v) v vx + σ f x v R = 1 (.4) K 0 (x, y)f(x, v)f(y, w) v w wyvx R 4 f u v vx + σ f x v R where we use the symmetry K 0 (x, y) = K 0 (y, x) an the fact that f(u v) v = 0. The lemma follows. Finally, we recall the following lemma which will be proven to be very useful in the upcoming analysis: Lemma.4. Assume that f satisfies f L ([0,T ] R ) M, v f vx M. R Then there exists a constant C = C(M) such that ρ L (0,T ;L p ( )) C, for every p [1, + ), j L (0,T ;L p ( )) C, for every p [1, + +1 ), (.5) where ρ = f v an j = vf v. Proof. Let p (1, ) an let q be such that 1/p + 1/q = 1. Then we have: ρ(x, t) = (1 + v ) /p f 1/p f 1/q (v) v (1 + v ) /p ( ) 1/p ( ) 1/q (1 + v ) f(v) f(v) v v. (1 + v ) q/p In particular, if q/p >, we euce an so ρ(x, t) C f(t) 1/q L ( (1 + v ) f(v) v ρ(t) p L = p ) 1/p ρ(x, t) p x C (1 + v ) f(v) v x. Noting that the conition q/p > is equivalent to p < +, this implies the first inequality in (.5). A similar argument hols for j: j(x, t) (1 + v ) /p f 1/p f 1/q (v) v (1 + v ) /p 1 ( ) 1/p ( ) 1/q (1 + v ) f(v) f(v) v v. (1 + v ) q/p q In particular, if q/p q >, we euce an so j(x, t) C f(t) 1/q L ( (1 + v ) f(v) v) 1/p (.6) j(t) p L = p n(x, t) p x C (1 + v ) f(v) v x,

10 10 KARPER, MELLET, AND TRIVISA where the conition q/p q > is equivalent to p < Velocity averaging an compactness. In the proof of Theorem 1.1, we will nee some compactness results for the ensity ϱ = f v an the first moment j = fv v of sequences of approximate solutions. Such compactness will be obtaine by using the celebrate velocity averaging lemma for quantities of the form ϱ ψ = fψ(v) v, R where ψ is a locally supporte function together with the boun (.3). We first recall the following result (see Perthame an Souganiis [15]): Proposition.5. Let {f n } n be boune in L p loc (R+1 ) with 1 < p <, an {G n } n be boune in L p loc (R+1 ). If f n an G n satisfy f n t + v x f n = k vg n, f n t=0 = f 0 L p (R ), for some multi-inex k an ψ C c k (R ), then {ϱ n ψ } is relatively compact in L p loc (R+1 ). The velocity averaging lemma cannot be irectly applie to conclue compactness of ϱ an j since the function ψ is require to be compactly supporte. Also, we woul like to get compactness in L p (+1 ) instea of L p loc (R+1 ). We will thus nee the following lemma, consequence of the ecay of f for large x, v (provie by Lemma.): Lemma.6. Let {f n } n an {G n } n be as in Proposition.5 an assume that f n is boune in L (R +1 ), ( v + Φ)f n is boune in L (0, T ; L 1 (R +1 )). Then, for any ψ(v) such that ψ(v) c v an q < + +1, the sequence { } f n ψ(v) v, (.7) n is relatively compact in L q ((0, T ) ). Proof. 1. We first prove compactness of the sequence { f n ψ(v) v } n in Lq loc. Since ψ is not compactly supporte, we consier ϕ k (v), a sequence of smooth functions satisfying ϕ k (v) = 1 for v k an ϕ k (v) = 0 for v k + 1. Proposition.5 then implies that for all k N, the sequence m k,n ψ = ϕ k f n ψ(v) v converges strongly in L q loc ((0, T ) R ) (up to a subsequence) to some m k ψ. Now, for k 1 > k an any 0 < α < 1, we have that m k1,n ψ m k,n ψ C f n ψ(v) v C v k +1 (k ) α f n v 1+α v C ( ) (1+α) k α (ϱ n ) (1 α) f n v v. By integrating this inequality over space an applying of the Höler inequality, m k1,n ψ m k,n ψ q L q (R ) C ( ) q(1+α) ( (k ) qα (ϱ n ) q(1 α) q(1+α) x f n v vx R ) q(1+α).

11 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 11 Lemma.4 implies that ϱ n is boune in L (0, T ; L p ( )) for p [1, + ), an so the right han sie above is boune if q is such that q(1 α) q(1 + α) < + q < α. For any q (1, + +1 ), we can thus choose α small enough so that m k1,n ψ m k,n ψ q L q (R ) C 1 k qα Passing to the limit n, we conclue that the sequence {m k ψ } is Cauchy in L q loc ((0, T ) R ) an thus converges to m ψ. By a iagonal extraction process, we euce the existence of a subsequence along which m n,n ψ = f n ψ(v)ϕ n (v) v m ψ in L q loc ((0, T ) R ) (for any q < ( + )/( + 1)). Finally, we have f n ψ(v) v m ψ (1 ϕ n )ψ(v)f n v + mn,n ψ m ψ ψ(v) f n v v n+1 + mn,n ψ m ψ (.8) 1 n α v 1+α f n v + mn,n ψ m ψ, which converges to zero in L q loc ((0, T ) R ) for q < ( + )/( + 1) (in particular, the first term in the right han sie is boune in L q for the same reason as above)... We have thus establishe the compactness in L q loc ((0, T ) R ). To prove compactness in L q ((0, T ) ), we argue as above, but instea of using the fact that R v f n x v is boune, we nee to show that ψ(v)f n v ecay for x. To prove this, we procee as in the proof of Lemma.4 (see in particular (.6)), to show that for l < an p < +l ψ(v)f n v p +1, By integrating for x k, we then see that p ψ(v)f n v x x k C ( ϱ n C Φ x + Φ(k) x k Φ(k) l ( ) 1 C Φ(k) + 1. Φ(k) l ) f n p q L (1 + v (R l )f n v ( ( C ϱ n + (ϱ n ) l v f n v ϱ n Φ x x k ) l ( x k ) l ). f n v vx ) l

12 1 KARPER, MELLET, AND TRIVISA In particular, for any q < + +1, we can choose l < such that the inequality above hols with p = q, an so q ψ(v)f n v x 0 as k uniformly w.r.t. n. x k We can now procee as in the first part of the proof to show that the sequence ψ(v)f n v converges in L q ((0, T ) ). 3. Approximate solutions In this section, we prove the existence of solutions for an approximate equation (by a fixe point argument). In the next section, we pass to the limit in the approximation to obtain a solution of (1.11) an prove Theorem 1.1. As pointe out in the introuction, the main ifficulty in (1.11) is the lack of R vf fv estimates on the velocity u = R f v. We thus consier the following equation, in which the velocity term u has been regularize: { t f + v x f iv v (f x Φ) + iv v (fl[f]) = σ v f iv v (f(χ λ (u δ ) v)) where f(x, v, 0) = f 0 (x, v), the function χ λ is the truncation function u δ is efine by: u δ = χ λ (u) = u 1 u λ. (3.1) vf v δ + f v = ρ u. (3.) δ + ρ Formally, we see that we recover (1.11) in the limit δ 0 an λ. The rigorous arguments for taking these limits are given in the ensuing section. In this section, we prove the following existence result for fixe δ an λ. More precisely, we prove: Proposition 3.1. Let f 0 0 satisfy the conition of Theorem 1.1. Then, for any δ > 0, λ > 0 there exists a solution f C(0, T ; L 1 (R )) of (3.1) satisfying f L (0,T ;L p (R )) + σ v f p p L ((0,T ) ) ect/p f 0 L p (R ) (3.3) for all p [1, ], an sup E(f) E(f 0 ) + σmt. (3.4) t [0,T ) The proof of Proposition 3.1 relies on a fixe point argument: Fix ( p 0 1, + ) + 1 an for a given ū L p0 (0, T ; L p0 ( )) let f be the solution of { t f + v x f iv v (f x Φ) + iv v (fl[f]) = σ v f iv v (f(χ λ (ū) v)) f(x, v, 0) = f 0 (x, v). We then consier the mapping ū T (ū) := u δ = (3.5) vf v δ + f v = ρu δ + ρ. (3.6)

13 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 13 In the remaining parts of this section, we prove the existence of a fixe point for the mapping T : L p0 (0, T ; L p0 ( )) L p0 (0, T ; L p0 ( )) an thereby prove Proposition The operator T is well-efine. First, we nee to check that the operator T is well-efine, an to erive some bouns on f. We start with the following result: Lemma 3.. For all ū L p (0, T ; L p ( )), there exists a unique f C(0, T ; L 1 ( )) solution of (3.5). Furthermore, f is non-negative an satisfies f L (0,T ;L p (R )) + σ v f p p L ((0,T ) ) ect/p f 0 Lp (R ), (3.7) p p 1 with C = [1 + K 0 L M] an p = sup E(f(t)) + 1 T fv x v t 1 t [0,T ] 0 R, for all p [1, ] an T 0 λ MT R f χ λ (ū) vxt + σmt + σmt. (3.8) Proof. Since χ λ (ū) L ((0, T ) ), the existence of a solution to (3.5) is classical (we prove it in Section 6 for the sake of completeness, see Theorem 6.3). Moreover, since χ λ (ū) is inepenent of v, we have that fχ λ (ū) v f p 1 v = 1 f p iv v χ λ (ū) v = 0. R p Hence, we can perform the same computations as in Lemma.1 to obtain (3.7). Next, using the equation (3.5), we write t E(f) = ) (Φ + v f vx t R = fl[f]v + fχ λ (ū) f v vx + σ f v x R = 1 K 0 (x, y)f(x, v)f(y, w) v w wyvx f v vx R 4 R + fχ λ (ū)v vx + σm. R Finally, writing fχ λ (ū)v vx 1 f χ λ (ū) vx + 1 f v vx, R R R we euce (3.8). The following lemma establishes the continuity of the operator T. Lemma 3.3. The operator T is continuous an there exists C(δ, λ) such that for all ū L p0 (0, T ; L p0 ( )), T (ū) L p 0 (0,T ;L p 0 (R )) C(δ, λ) Proof. We have T (ū) 1 δ j, an we recall that p 0 < So lemma.4 together with (3.8) implies that there exists a constant C such that j L (0,T ;L p 0 ( )) C, where C only epens on f 0 L an v f 0 x v.

14 14 KARPER, MELLET, AND TRIVISA 3.. The operator T is compact. Compactness of the operator T follows from the following lemma: Lemma 3.4. Let {ū n } n N be a boune sequence in L p0 (0, T ; L p0 ( )). Then up to a subsequence, T (ū n ) converges strongly in L p0 (0, T ; L p0 ( )). Proof. By efinition of T, we have that T (ū n ) = jn δ + ϱ n. So in orer to prove that the sequence T (ū n ) is relatively compact in L p0 (0, T ; L p0 ( )), we have to show that ϱ n converges a.e an that j n is relatively compact in L p0 (0, T ; L p0 ( )). This follows from Lemma.6. Inee, let G n = f n x Φ + σ v f n f n (χ λ (ū n ) v) f n L[f n ] then (3.1) can be rewritten as f n t + v v f n = iv v G n. Furthermore, we have the following lemma (whose proof is postpone to the en of this section): Lemma 3.5. For any q, there exists a constant C inepenent of n such that G n L (0,T ;L q ( )) C for all n 0. (3.9) In particular, Sobolev embeings imply that G n is relatively compact in W 1,r loc ((0, T ) R ), for r < (+1) 1,p0 1, an hence also in Wloc ((0, T ) R ). In view of (3.3) an (3.8), we can apply Lemma.6 (with ψ(v) = 1 an ψ(v) = v) to conclue the existence of functions ϱ an j such that (up to a subsequence) ϱ n = f n v n ϱ in L p0 ((0, T ) ) an a.e., R j n = vf n v n j in L p0 ((0, T ) ). Proof of Lemma 3.5. By repeate applications of the Höler inequality (we rop the n epenence for the sake of clarity), G(t) L q (R ) Φ L (R ) f L q (R ) + v f L q (R ) + Cλ f L q (R ) + K 0 L (R )( f L 1 (R ) vf L q (R ) + f L q (R ) vf L 1 (R )), where we have that ( ) q/ vf L q v f v x f q/ an vf L f L L q q v f v x, for q [1, ), which are both boune by Lemma 3.. It remains to boun the term involving v f. For this purpose, we first observe that (.1) with p = 1 provies the boun T 0 Q 1 f vf vxt CM. (3.10)

15 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 15 Using this together with the Höler inequality an q = p p+1, we get that T T v f q vxt = f q f q v f q vxt 0 Q = 0 T 0 T 0 Q f q L q q ( ( p p+1 f L p So using (3.7) an (3.8), we conclue the proof. Q Q ) q 1 f vf vx t 1 f vf vx ) p p+1 t C, (3.11) 3.3. Proof of Proposition 3.1. Lemma 3.3 an 3.4, together with Schauer fixe point theorem imply the existence of a fixe point u L p0 ((0, T ) ) of T. The corresponing solution of (3.5) solves (3.1). Furthermore, it is reaily seen that (3.7) implies (3.3). Finally, since we have an so (3.8) yiels (3.4). u δ = vf v δ + f v u = f χ λ (u δ ) x v ρ u x f v xv vf v f v, (3.1) 4. Existence of solutions (Proof of Theorem 1.1) For δ, λ > 0, we enote by f δ,λ the solution of (3.1) given by Proposition 3.1. We also enote ρ δ,λ = f δ,λ v an j δ,λ = vf δ,λ v an u δ δ,λ = j δ,λ. δ + ρ δ,λ In this section we show how to pass to the limit λ an δ 0, thereby proving Theorem 1.1. First, We recall that Proposition 3.1 implies the following bouns (which are uniform with respect to δ an λ): Corollary 4.1. There exists C inepenent of δ an λ such that an f δ,λ Lp (0,T ;L p (R )) e CT/p f 0 Lp (R ), (4.1) f δ,λ In particular, Lemma.4 implies ( ) v + Φ vx C. (4.) ρ δ,λ L (0,T ;L p ( )) C for all p < +. j δ,λ L (0,T ;L p ( )) C for all p < Finally, in orer to use the averaging Lemma.6, we rewrite (3.1) as t f δ,λ + v x f δ,λ = iv v G δ,λ (4.3) with G δ,λ = f δ,λ Φ + σ v f δ,λ f δ,λ (χ λ (u δ δ,λ) v) f δ,λ L[f δ,λ ], an we will nee the following result:

16 16 KARPER, MELLET, AND TRIVISA Lemma 4.. For any q, there exists a constant C inepenent of δ an λ such that G δ,λ L (0,T ;L q ( )) C. (4.4) Proof. This lemma is similar to Lemma 3.5. The only aitional ifficulty is to boun the term f δ,λ χ λ (u δ δ,λ ) uniformly with respect to λ. But we note that f δ,λ χ λ (u δ δ,λ ) f δ,λu δ,λ an f δ,λ u δ,λ L (0,T ;L (R )) f δ,λ L ((0,T ) R ) f δ,λ v x v an so (4.1) an (4.) imply the existence of a constant C > 0, inepenent of λ an δ, such that f δ,λ u δ,λ L (0,T ;L (R )) C which conclues the proof Limit as λ. We now fix δ > 0 an consier a sequence λ n. We enote by f n = f δ,λn the corresponing solution of (3.1) an vf u n n v δ = δ + f n v = jn δ + ρ n. We then have: Lemma 4.3. Up to a subsequence, f n converges weakly in L (0, T ; L 1 (R ) L (R )), to some function f, an u n δ converges strongly to u δ = R fv v δ+ R f v in L p ((0, T ) R ), p < Furthermore, f is a weak solution of (3.1) with λ =, an it satisfies the a priori estimates of Corollary 4.1 an Lemma 4.. Proof. By virtue of (4.1), there exists a function f L (0, T ; L 1 (R ) L (R )) such that, up to a subsequence, f n f in L (0, T ; L 1 (R ) L (R )). Furthermore, using (4.3) an Lemma 4., we can reprouce the arguments of Lemma 3.4 to show that ϱ n an j n converge strongly an almost everywhere to ϱ an j in L p ((0, T ) R ) for p < We euce: u n δ u δ := j δ + ϱ in Lp ((0, T ) R ), p < We can now pass to the limit in the equation f n t + v x f n iv(f n x Φ) + iv v (f n L[f n ]) = σ v f n iv v (f n (χ λ (u n δ ) v)). The only elicate term is the nonlinear term f n χ λ (u n δ ) (the other terms are either linear or involve quantities that are more regular than f n χ λn (u n )). We write f n χ λ (u n δ ) = f n u n δ + f n (χ λ (u n δ ) u n δ ). The strong convergence of u n δ implies that the first term converges to fu δ in D. An using the fact that u n δ 1 δ jn, Lemma 4.1 implies that u n is boune in L (0, T ; L p ( )) for p < + +1, an so u n δ χ λn (u n δ ) f n x v C f n L u n δ x v u n δ >λn converges to zero uniformly w.r.t t as λ n goes to infinity. We euce that which conclues the proof. f n χ λ (u n δ ) fu δ in D.

17 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS Limit as δ 0 (an en of the proof of Theorem 1.1). For all δ > 0, we have shown that there exists a weak solution f to the equation f t + v x f iv(f x Φ) + iv v (fl[f]) = σ v f iv v (f(u δ v)). (4.5) where u δ = j δ+ϱ. We now consier a sequence δ n 0 an enote by f n = f δn the corresponing weak solution of (4.5). Proceeing as before, (using the bouns (4.1), Lemma 4. an the velocity averaging Lemma.6) we can show that there exists a function f such that up to a subsequence, the following convergences hol: f n f in L ((0, T ) L 1 (R ) L (R )), ϱ n ϱ j n j in L p ((0, T ) R )-strong an a.e., in L p ((0, T ) R )-storng an a.e., (4.6) for any p < It remains to show that f is a weak solution of (1.11). More precisely, we have to pass to the limit in the following weak formulation of (4.5): f n ψ t vf n x ψ + f n x Φ x ψ f n L[f n ] v ψ vxt R +1 (4.7) + σ v f n v ψ f n (u n δ v) v ψ vxt = f 0 ψ(0, ) vx. R +1 R where ψ Cc ([0, ) R ). Since f n converges in the weak-star topology of L, the most elicate term is the term involving u n δ (un δ is not boune in any space). The key lemma is thus the following: Lemma 4.4. Let ϕ Cc ( ) an enote ϱ ϕ (x, t) = f(x, v, t)ϕ(v) v. Then, (up to another subsequence), Furthermore, ϱ n ϕ ϱ ϕ in L p ((0, T ) R ) as n. ρ n ϕu n δ ρ ϕ u in D ((0, T ) ) as n, where u is such that j(x, t) = ρ(x, t)u(x, t) a.e. Lemma 4.4 implies that for all ϕ Cc ( ) an φ Cc ([0, T ) ) there exists a subsequence such that T lim f n (x, v, t)u n δ (x, t)ϕ(v)φ(x, t) v xt n 0 R T = lim ρ n ϕ(x, t)u n δ (x, t)φ(x, t) xt n 0 R T T = ρ ϕ (x, t)u(x, t)φ(x, t) xt = f(t, x)u(t, x)ϕ(v)φ(x, t) xt. 0 0 R We can thus pass to the limit in (4.7) an show that the function f (which oes not epen on the subsequence) satisfies (1.13) for all test function ψ(x, v, t) = ϕ(v)φ(x, t) in Cc ([0, T ) ). We conclue the proof using the ensity of the sums an proucts of functions of the form ϕ(v)φ(x, t) in Cc ([0, T ) ) (recall that fu is in L ). Proof of Lemma 4.4. For a given ϕ Cc ( ), the same reasoning use to show the convergence of ρ n implies that, up to a subsequence, ϱ n ϕ ϱ ϕ in L p ((0, T ) R ) as n. (4.8)

18 18 KARPER, MELLET, AND TRIVISA We now introuce the function m n = ρ n ϕu n δ. We have the following pointwise boun: m n ϕ an proceeing as in Lemma.4, we euce: Hence, up to a subsequence, v f n v, m n L (0,T ;L p ( )) C for all p < ( + )/( + 1). an it only remains to show that m n m in L (0, T ; L p ( )), m = ϱ ψ u, where u is such that j = ϱu. First, we check that such a function u exists: Consier the set A R = {(x, t) B R (0, T ) ; ρ = 0}. By irect calculation, we see that ( ) 1/ ( j n x t ρ n u n x t ρ n x t A R A R A R ( ) 1/ C ρ n x t 0, A R an hence j = 0 a.e. in A R. Consequently, we can efine the function j(x, t) if ρ(x, t) 0 u(x, t) = ρ(x, t) 0 if ρ(x, t) = 0 ) 1/ an we then have j = ρu. It only remains to prove that m = ρ ϕ u. First, we observe that a similar argument implies that m = 0 whenever ρ ϕ = 0, so we only have to check that m(x, t) = ρ ϕ (x, t)u(x, t) whenever ρ ϕ (x, t) 0. For this purpose, let us consier the set B ɛ R = {(x, t) B R (0, T ) ; ρ > ɛ}. Egorov s theorem together with (4.6) asserts the existence of a set C η BR ɛ with BR ɛ \ Cɛ R η on which ρn ϕ an ρ n converge uniformly on C η to ρ ϕ an ρ. We then have (for n large enough) ρ n ɛ/ in C η, an since m n = jn δ n + ρ n ρn ϕ, we can pass to the limit in C η (pointwise) to euce m = j ρ ρ ϕ = uρ ϕ in C η. Since this hols for all η > 0, we have m = uρ ϕ in BR, ɛ for every R an ɛ. We conclue that, m = uρ ϕ in {ρ > 0}.

19 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS Non-symmetric flocking an the Motsch-Tamor moel So far we have limite our attention to the case of symmetric flocking kernel. In this section, we will exten our existence result to inclue some non-symmetric kernel K(x, y) K(y, x). The critical step is to erive the appropriate energy boun on the solution. As we will see in Proposition 5.1 below, this is rather straight forwar provie the flocking kernel satisfies a conition of the form K(x, y)ϱ(x) x C, ϱ L 1 +( ). In particular, if K is boune the result follows reaily. For this reason we will focus on the Motsch-Tamor moel for which K epens on f an is singular (see (1.3)). To the authors knowlege this is the most ifficult case currently foun in the mathematical literature. Let us recall the Motsch-Tamor moel for flocking: f t + v x f iv v (f x Φ) + iv v (f L[f] ) = 0, (5.1) with normalize, non-symmetric alignment, given by φ(x y)f(y, w)(w v) w y R L[f] = R = ũ v (5.) φ(x y)f(y, w) w y R where ũ is efine by the equalities ρ = φ(x y)f(y, w) w y, ρũ = φ(x y)wf(y, w) w y. R R This moel only iffers from (1.11) by the fact that u is replace by ũ. Because the function ũ involves the convolution with a smooth kernel φ, we expect this function to be smoother than u, an the proof of the existence of a solution for (5.1) is actually simpler, provie that we can erive the necessary a priori estimates. This is our goal in the remaining parts of this section. More precisely, we are going to prove: Proposition 5.1. Assume that φ is a smooth nonnegative function an that there exists r > 0 an R > 0 such that φ(x) > 0 for x r, φ(x) = 0 for x R. (5.3) Let f be a smooth solution of (5.1)-(5.) an efine ( ) v E(t) = + Φ(x) f(x, v, t) x v. R Then, there exists a constant C sup B R (0) φ inf Br (0) φ ( R r ) epening only on φ such that E(t) E(0)e Ct. (5.4) We note that the constant C in (5.4) is invariant if we replace φ(x) by φ(λx) (in which case both R an r are scale by a factor λ) or by λφ(x) (in which case the sup an inf are both scale by a factor λ). In particular, if we take a sequence φ ɛ = ɛ φ(x/ɛ), which converges to δ 0 as ɛ 0 (so that L converges with the local alignment term consiere in the previous section), the estimate (5.4) hols uniformly with respect to ɛ. Proof of Proposition 5.1. Multiplying (5.1) by v + Φ(x) an integrating with respect to x an v, we get: t E(t) = f L[f] v x v R

20 0 KARPER, MELLET, AND TRIVISA = f(ũ v) v x v R = f(ũ v) x v + f(ũ v) ũ x v R R 1 f(ũ v) x v + ρũ x. (5.5) R It remains to see that ρũ x can be controlle by E(t). First, we notice that ρũ φ(x y)w f(y, w, t) w y an so ρũ x In orer to conclue, we thus nee the following lemma. ρ(x, t) R ρ(x, t) φ(x y)w f(y, w, t) w y x. (5.6) Lemma 5.. Uner the assumptions of Proposition 5.1, there exists a constant C sup φ ( R ) inf Br (0) φ r such that φ(x y) ρ(x) R ρ(x) x C y R for all nonnegative functions ρ L 1 ( ). Equation (5.6) together Lemma 5. now imply ρũ x C w f(y, w, t) w y CE(t) R an so (5.5) yiels which gives the lemma. E (t) CE(t) Proof of Lemma 5.. First, we note that φ(x y) ρ(x) x sup φ R ρ(x) B R (y) ρ(x) ρ(x) x. Next, we cover B R (y) with balls of raius r/ (with r as in (5.3)): We have B R (y) with N (R/r). We can thus write where We euce N B r/ (x i ) i=1 φ(x y) ρ(x) R ρ(x) x sup φ N ρ(x) = φ(x z)ρ(z) z φ(x y) ρ(x) ρ(x) x sup φ N i=1 i=1 B r/ (x i) B r/ (x i) B r/ (x i) B r/ (x i) ρ(x) ρ(x) x φ(x z)ρ(z) z. ρ(x) x. φ(x z)ρ(z) z

21 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 1 an using the fact that when x, z B r/ (x i ) we have x z r, we euce φ(x y) ρ(x) ρ(x) x sup φ inf Br(0) φ an the proof is complete. N i=1 sup φ inf Br(0) φ N C sup φ inf Br(0) φ B r/ (x i) B r/ (x i) ( ) R r ρ(x) x ρ(z) z 6. Other existence result In Section 3, we constructe a sequence of approximate solutions of our kinetic flocking equation (1.11). In our iscussion therein the existence of these approximate solutions relie on the existence of solutions to equations of the form f t + v x f iv v (f x Φ) + iv v (L[f]f) = σ v f + iv v (F vf Ef), where E an F are given functions. The purpose of this section is to prove this result. The precise statement is given in Theorem 6.3 below. We commence by recalling the following result ue to Degon [10, 9]: Proposition 6.1. Let a(v) be a boune function, σ > 0, then for any E [L (0, T ; L ( ))] an F L (0, T ; L ( )), there exists a unique weak solution f C 0 (0, T ; L 1 ( )) of f t + v x f iv v (f x Φ) = σ v f + iv v (F a(v)f Ef). (6.1) Furthermore, f satisfies f L (0,T ;L p (R )) + σ v f p p L ((0,T ) ) e F L T q f 0 Lp (R ) (6.) with q = p p 1, for any p [1, ], an v f(x, v, t) x v C R v f 0 (x, v) x v R (6.3) (1 + Φ(x))f(x, v, t) x v C R (1 + Φ(x))f 0 (x, v) x v, R (6.4) with C = C( x Φ L, E L, F L, T ). By passing to the limit (weakly) in (6.1), we euce the following corollary. Corollary 6.. For any E [L (0, T ; L ( ))] an F L (0, T ; L ( )) an for any σ 0, there exists a unique weak solution f C 0 (0, T ; L 1 ( )) of f t + v x f iv v (f x Φ) = σ v f + iv v (F vf Ef) (6.5) Furthermore, f satisfies (6.), (6.3) an (6.4). The main result of this section is the following: Theorem 6.3. For any E [L (0, T ; L ( ))] an F L (0, T ; L ( )) an for any σ 0, there exists a unique weak solution f C 0 (0, T ; L 1 ( )) of f t + v x f iv v (f x Φ) + iv v (L[f]f) = σ v f + iv v (F vf Ef). (6.6) Furthermore, f satisfies (6.), (6.3) an (6.4).

22 KARPER, MELLET, AND TRIVISA Proof. Let us enote by Ẽ an F the functions E an F in (6.6) (which are given). We will argue the existence of a fixe point in the following sense: For E an F given, let f solve (6.5) an efine the operators T 1 (E, F ) = R Ẽ + K 0 (x, y)f(y, w)w wy, T (E, F ) = F + K 0 (x, y)f(y, w) wy, T (E, F ) = [T 1 (E, F ), T (E, F )]. (6.7) Then, any fixe point T (E, F ) = [E, F ] is a solution to (6.6). We will prove the existence of such a fixe point by verifying the postulates of Schaefer theorem. However, to facilitate this we will work with the space of continuous functions C 0 ((0, T ) ) instea of L ((0, T ) ). We temporarily assume that Ẽ, F are in C 0 ((0, T ) ). The result follows by passing to the limit. Let us first verify that the operator T is compact. For this purpose, let {[E n, F n ]} n be a uniformly boune sequence in [C 0 ([0, T ) )] an {f n } n be the corresponing sequence solutions to (6.5). By virtue of (6.3) - (6.4), it is clear that T (E n, F n ) is uniformly (in n) boune in [L ((0, T ) )]. Since K 0 is Lipschitz, we have that K 0 (x + ɛ, y)f n (y, w)w wy K 0 (x, y)f n (y, w)w wy ɛc K 0 W 1,, where C is inepenent of n. In particular, { } K 0 (x, y)f n (y, w)w wy is both uniformly boune an equicontinuous. Clearly, it follows that T 1 is compact in C 0 ((0, T ) ). A similar argument hols for T an hence T is compact in C 0 ((0, T ) ). Next, let us verify that the operator T is continuous. Observe that this actually follows from compactness provie that T (E n, F n ) T (E, F ), whenever [E n, F n ] [E, F ] in C 0 ((0, T ) ). In turn, this is immeiate if f n f, where f is a weak solution of (6.5) with E an F being the above escribe limits. Consequently, we can conclue continuity of T if we can pass to the limit in f n t + v x f n iv v (f n x Φ) = σ v f n + iv v (F n vf n E n f n ) (6.8) Note that the bouns (6.) - (6.4), together with the assumption that F n an E n are uniformly boune, provies the existence of a constant C, inepenent of n, sup (1 + Φ(x) + v )f n (x, v, t) x v + f n L (0,T ;L ( )) C, t (0,T ) R for any given finite total time T. Since F n an E n converge strongly there is no problems with passing to the limit in (6.8) to conclue that the limit f solves (6.5). Hence, T (E n, F n ) T (E, F ) in C 0 ((0, T ) ) an consequently is also continuous. To conclue the existence of a fixe point, it remains to verify that, n {[E, F ] = λt (E, F ) for some λ [0, 1]}, is boune. For this purpose we take [E, F ] in this set an f a weak solution of ( f t + v x f iv v (f x Φ) + iv v (L[f]f) = σ v f + iv v λ F ) vf λẽf. (6.9)

23 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 3 The corresponing energy estimate becomes E(t) := ( 1 R v + Φ)f vx + λ t K 0 (x, y)f(x, v)f(y, w) w v wyvx 0 R 4 λ F L t 0 E(s) s + λ Ẽ L + E(0). Since λ 1, an application of the Gronwall inequality yiels sup E(t) C ( F ) L, Ẽ L, T (E(0) + 1). t (0,T ) Equippe with this boun, we euce from (6.7) that E L = λ T 1 (E, F ) L ( Ẽ L + C F ) ( ) L, Ẽ L K 0 L M 1 v f 0 vx R F L = λ T (E, F ) L F L + K 0 L M. To summarize, the operator T o satisfy the postulates of the Schaefer fixe point theorem an hence we conclue the existence of a fixe point. This fixe point is a solution of (6.6) an hence our proof is complete. 7. The entropy of flocking Equations of the form (1.11) are expecte to possess a natural issipative structure often expresse through the notion of entropy. In our existence analysis, we have not relie on such inequalities because the energy inequality (.) was enough. This inequality is however of limite use for the analysis of asymptotic behavior involving singular noise. In this section, we prove that our weak solutions satisfy bouns akin to classical entropy inequalities when β > 0 an σ > 0 (in the case σ = 0, the computation below reuces to (.)). We will restrict to consiering symmetric flocking as the non-symmetric case oes not seem to posses any nice issipative structure (in fact, the non-symmetric case oes not even conserve momentum). We recall that the entropy is given by F(f) = R an the associate issipations by D 1 (f) = 1 T 0 σ v f log f + f + fφ v x β R β f σ β vf f(u v) v x an D (f) = 1 K 0 (x, y)f(x, v)f(y, w) v w wyvx The first part of Proposition 1.4 follows from the following lemma: Lemma 7.1. Let f be a sufficiently integrable solution of (1.11), then t F(f) + D 1 (f) + D (f) = σ β K 0 (x, y)f(x, v)f(y, w) wyvx. (7.1)

24 4 KARPER, MELLET, AND TRIVISA Proof. Using the equation (1.11), we calculate ( ) σ t F(f) = f t R β log f + v + Φ vx σ = R β L[f] vf 1 f (σ f βf(u v)) σ β vf vx + fl[f]v (σ f βf(u v)v) vx R + vf x Φ + vf x Φ vx R = σ R β f iv v L[f] + vfl[f] vx ( ) β σ f f(u v) ( σ f β β vf + vf) vx := I + II. By efinition of L[f], we euce I := σ R β f iv v L[f] + vfl[f] vx σ = R β K 0(x, y)f(x, v)f(y, w) iv v v wyvx + K 0 (x, y)f(x, v)f(y, w)(w v)v wyvx R σ = β K 0(x, y)f(x, v)f(y, w) wyvx D (f), (7.) (7.3) where we have use the symmetry of K 0 (x, y)f(x)f(y) to conclue the last equality. By aing an subtracting u, we rewrite II as follows: ( ) β σ II : = R f β vf f(u v) ( σ β vf + vf) vx. = D 1 (f) + uσ v f + fu(u v) vx (7.4) R = D 1 (f) + ϱu ϱu x = D 1 (f). We conclue by setting (7.3) an (7.4) in (7.). To prove the secon inequality (1.) in Proposition 1.4, we must prove that the right-han sie in (7.1) can be controlle by the issipation an the entropy. We will nee the following classical lemma: Lemma 7.. Let ϱ L 1 +( ) be a given ensity an let Φ be a confinement potential satisfying (1.1). Then, the negative part of ϱ log ϱ is boune as follows ϱ log ϱ x 1 ϱφ x + 1 e Φ x. (7.5) R R e In particular, we have R f log + f + fv Inequality (1.) now follows from the following lemma: + fφ vx CF(f) (7.6)

25 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 5 Lemma 7.3. Let g L p ( ) be given an let Φ be a confinement potential satisfying (1.1). There is a constant C > 0, epening only on K 0, Φ an the total mass M, such that 1 K 0 (x, y)ϱ(x)ϱ(y) u(x) u(y) D (f) σ β yx 1 D(f) CF(f(t)) K 0 (x, y)f(x, v)f(y, w) wyvx. Proof of Lemma 7.3. By symmetry of K 0 (x, y), we have 1 = = = K 0 (x, y)ϱ(x)ϱ(y) u(x) u(y) yx R K 0 (x, y)ϱ(x)ϱ(y) (u(x) u(y)) u(x) yx R K 0 (x, y)f(x, v)f(y, w) (v w) u(x) wyvx R K 0 (x, y)f(y, w) (v w) ( f(x, v)(u(x) v) σ ) β vf(x, v) wyvx + K 0 (x, y)f(y, w)f(x, v) (v w) v wyvx + σ K 0 (x, y)f(y, w) (v w) v f(x, v) wyvx β = I + II + III. (7.7) Let us first consier the last term. Integration by parts provies the ientity III = σ β = σ β K 0 (x, y)f(y, w) (v w) v f(x, v) wyvx R K 0 (x, y)f(y, w)f(x, v) wyvx. (7.8) By symmetry of the kernel K 0 (x, y), we have that II = = K 0 (x, y)f(y, w)f(x, v) (v w) v wyvx R v w K 0 (x, y)f(y, w)f(x, v) wyvx. R (7.9) It remains to boun I. For simplicity, let us introuce the notation V (x, v) = 1 (f(x, v)(u(x) v) σ ) f(x, v) β vf(x, v).

26 6 KARPER, MELLET, AND TRIVISA Using this notation, some straight forwar manipulations, an the Höler inequality, we obtain using Lemma 7., I = K 0 (x, y) f(x, v)f(y, w) (v w) V (x, v) wyvx R = K 0 (x, y) f(x, v)ϱ(y)(v u(y))v (x, v) yvx R ( ) = K 0 (x, y)ϱ(y) y v f(x, v)v (x, v) vx R ( ) K 0 (x, y)ϱ(y)u(y) y f(x, v)v (x, v) vx (7.10) ( ) 1 ( ) 1 K L M v f(x, v) vx V (x, v) vx R ( ) ( ) 1 + K L M 1 fv x V (x, v) vx C(K, M) F(f) + 1 β D 1(f). We conclue the result by setting (7.8) - (7.10) in (7.7). Proof. Let f be the solution of (3.1) given by Proposition 3.1. similar to the proof of Lemma 7.1 yiels A computation t F(f) + D 1 (f) + D (f) = σ β K 0 (x, y)f(x, v)f(y, w) wyvx + fv [χ λ (u δ ) u] vx. where fv [χ λ (u δ ) u] vx = ρu [χ λ (u δ ) u] x 1 ρu x + 1 ρχ λ (u δ ) x 0 ρu x since χ λ (u) u. We euce that the solution of the approximate equation (3.1) satisfy the entropy inequality t F(f) + D 1 (f) + D (f) σ β K 0 (x, y)f(x, v)f(y, w) wyvx. Integrating in time an passing to the limit (using the convexity of the entropy), we euce (1.3). References [1] F. Bolley, J.A. Cañizo, J.A. Carrillo, Stochastic Mean-Fiel Limit: Non-Lipschitz Forces & Swarming, Math. Mo. Meth. Appl. Sci. 1: , 011. [] J.A. Carrillo an J. Rosao. Uniqueness of boune solutions to aggregation equations by optimal transport methos. European Congress of Mathematics, no. 3-16, Eur. Math. Soc., Zrich, 010. [3] J.A. Carrillo, M. Fornasier, J. Rosao an G. Toscani. Asymptotic flocking ynamics for the kinetic Cucker-Smale moel. SIAM J. Math. Anal. 4 no. 1, (010).

27 EXISTENCE OF SOLUTIONS TO KINETIC FLOCKING MODELS 7 [4] J.A. Carrillo, M. Fornasier, G. Toscani, an V. Francesco. Particle, kinetic, an hyroynamic moels of swarming. Mathematical moeling of collective behavior in socio-economic an life sciences 97336, Moel. Simul. Sci. Eng. Technol., Birkhuser Boston, Inc., Boston, MA, 010. [5] J.A. Cañizo, J.A. Carrillo an J. Rosao. A well-poseness theory in measures for some kinetic moels of collective motion. Mathematical Moels an Methos in Applie Sciences Vol. 1 No. 3: , 011. [6] F. Cucker an S. Smale. Emergent behavior in flocks. IEEE Transactions on automatic control, 5 no. 5: 85-86, 007. [7] F. Cucker an S. Smale. On the mathematics of emergence. Japanese Journal of Mathematics, no. (1):197-7, 007. [8] R. Duan, M. Fornasier, an G. Toscani. A Kinetic Flocking Moel with Diffusion. Comm. Math. Phys , 010. [9] P. Degon. Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 an space imensions. Ann. Sci. École Norm. Sup. (4) 19 no. 4: , [10] P. Degon. Existence globale e solutions e l équation e Vlasov-Fokker-Planck, en imension 1 et. (French) [Global existence of solutions of the Vlasov-Fokker-Planck equation in imension 1 an.] C. R. Aca. Sci. Paris Sér. I Math. 301 no. 3:7376, [11] S.-Y. Ha, an J.-G. Liu. A simple proof of the Cucker-Smale flocking ynamics an mean-fiel limit. Commun. Math. Sci. Vol. 7, No..: [1] S.-Y. Ha, an E. Tamor. From particle to kinetic an hyroynamic escriptions of flocking. Kinet. Relat. Moels 1 no. 3: , 008. [13] Hyroynamic limit of the kinetic Cucker-Smale flocking with strong local alignment. Preprint 01. [14] S. Motsch an E. Tamor. A new moel for self-organize ynamics an its flocking behavior. Journal of Statistical Physics, Springer, 141 (5): , 011. [15] B. Perthame an P.E. Souganiis. A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4) 31 no. 4: , (Karper) Center for Scientific Computation an Mathematical Moeling, University of Marylan, College Park, MD 074 aress: karper@gmail.com URL: folk.uio.no/~trygvekk (Mellet) Department of Mathematics, University of Marylan, College Park, MD 074 aress: mellet@math.um.eu URL: math.um.eu/~mellet (Trivisa) Department of Mathematics, University of Marylan, College Park, MD 074 aress: trivisa@math.um.eu URL: math.um.eu/~trivisa