GLOBAL CLASSICAL SOLUTIONS OF THE VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES

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1 GLOBAL CLASSICAL SOLUTIONS OF THE VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES YOUNG-PIL CHOI arxiv:164.86v1 [math.ap] 7 Apr 16 Abstract. In this paper, we are concerne with the global well-poseness an time-asymptotic ecay of the Vlasov-Fokker-Planck equation with local alignment forces. The equation can be formally erive from an agent-base moel for self-organize ynamics which is calle Motsch- Tamor moel with noises. We present the global existence an uniqueness of classical solutions to the equation aroun the global Maxwellian in the whole space. For the large-time behavior, we show the algebraic ecay rate of solutions towars the equilibrium uner suitable assumptions on the initial ata. We also remark that the rate of convergence is exponential when the spatial omain is perioic. The main methos use in this paper are the classical energy estimates combine with hyperbolic-parabolic issipation arguments. 1. Introuction We are concerne with a kinetic flocking equation in the presence of iffusion. More precisely, let F = F(x,v,t) be the ensity of particles which have position x R an velocity v R at time t. Then the evolution of the ensity F is governe by with the initial ata t F +v x F + v ((u F v)f) = v F, (1.1) where u F is the average local velocity given by vf(x,v,t)v R u F (x,t) := R F(x,v,t)v. F(x,v,) =: F (x,v), (1.) The purpose of this paper is to stuy the global existence an uniqueness of classical solutions to the equation (1.1) near the global Maxwellian an the large-time behavior of solutions. Recently, Motsch an Tamor introuce in [19] a new moel for self-organize ynamics in which the alignment force is normalize with a local average ensity. Compare to the celebrate Cucker- Smale flocking moel [8] where the alignment is scale with the total mass, the Motsch-Tamor moel takes into account not only the istance between agents but also their relative istance. More specifically, these two prototype moels are escribe by x i t = v i, v i t = 1 S i (x) N ψ(x i x j )(v j v i ), t >, i {1,...,N}, (1.3) j=1 where x i (t) = (x 1 i (t),,x i (t)) R an v i (t) = (v 1 i (t),,v i (t)) R are the position an velocity of i-th agent at time t, respectively. Here, the scaling function S i (x) an the communication Date: July 17, 18. Key wors an phrases. Global existence of classical solutions, large-time behavior, hypocoercivity, Vlasov equation, nonlinear Fokker-Planck equation. 1

2 CHOI weight ψ(x) are efine by N for Cucker-Smale moel, S i (x) := N ψ(x i x k ) for Motsch-Tamor moel k=1 an ψ(x) = 1/(1+ x ) β/ with β >, respectively. We refer to [5, 7, 14, 19] for the existence an large-time behavior of solutions to these two moels. Note that the motion of an agent governe by Cucker-Smale moel is moifie by the total number of agents. Thus it can not be aopte to escribe for far-from-equilibrium flocking ynamics. On the other han, the Motsch-Tamor moel oes not involve any explicit epenence on the number of agents, an this remeies several rawbacks of Cucker-Smale moel outline above. For a etaile escription of the moeling an relate literature, we refer reaers to [19] an the references therein. When the number of agents governe by Motsch-Tamor moel goes to infinity, i.e., N, we can formally erive a mesoscopic escription for the system (1.3) with a ensity function F = F(x,v,t) which is a solution to the following Vlasov-type equation: { t F +v x F + v ((ũ F v)f) =, (1.4) F(x,v,) =: F (x,v), where ũ F is given by ũ F (x,t) := (ψ bf )(x,t) (ψ a F (1.5) )(x,t) with a F (x,t) := F(x,v,t)v an b F (x,t) := vf(x,v,t)v. R R The equation (1.4) equippe with the noise effect is a non-local version of our main equation (1.1). We now consier the singular limit in which the communication weight ψ converges to a Dirac istribution, i.e., the communication rate is very concentrate aroun the closest neighbors of a given particle. In this framework, the Motsch-Tamor alignment force converges to the local one [16]: ũ F v u F v. From this observation, it is reasonable to expect our main equation (1.1) as the mean-fiel limit of the Motsch-Tamor moel with noise. It is also worth noticing that the equation (1.1) is of the classical form, usually calle the nonlinear Fokker-Planck equation[], which has foun applications in various fiels such as plasma physics, astrophysics, the physics of polymer fluis, population ynamics, an neurophysics [1]. We refer to [3, 15, 17] for the existence of weak solutions an hyroynamic limit of (1.1). In [, 6], the equation (1.1) couple to the incompressible flow are consiere, an global existence of weak solutions, hyroynamic limit, an large-time behavior are provie. As we mentione before, we are intereste in the stability of solutions near the global Maxwellian an the rate of convergence of these solutions towars it for the Cauchy problem (1.1)-(1.). For this, we efine the perturbation f = f(x,v,t) by where the global Maxwellian function M = M(v) = F = M + Mf, ) 1 exp ( v (π) / is normalize to have zero bulk velocity an unit ensity an temperature. The equation for the perturbation f satisfies t f +v x f +u F v f = Lf +Γ(f,f), (1.6)

3 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 3 where the linear part Lf an the nonlinear part Γ(f,f) are given by Lf := 1 ( ( )) f v M v an Γ(f,f) = u F M M ( ) 1 vf +v M, (1.7) respectively. Note that Lf can be written as Lf = v f + 1 ( v ) f. 4 Before we state our main result, we introuce several simplifie notations. For functions f(x, v), g(x), h(v), we enote by f L p, g L p, h L p v the usual L p (R R ), L p x(r ), L p v(r )-norms, respectively. We also introuce norms µ an µ as follows. f µ := R ( v f(v) +µ(v) f(v) ) v, µ(v) := 1+ v, an f µ := R R ( v f(v) +µ(v) f(v) ) vx. f g represents that there exists a positive constant C > such that f Cg. We also enote by C a generic positive constant epening only on the norms of the ata, but inepenent of T, an rop x-epenence of ifferential operators x, that is, i f := xi f for 1 i an f := x f. For any nonnegative integer s, H s enotes the s-th orer L Sobolev space. C s ([,T];E) is the set of s-times continuously ifferentiable functions from an interval [,T] R into a Banach space E, an L p (,T;E) is the set of the L functions from an interval (,T) to a Banach space E. s enotes any partial erivative α with multi-inex α, α = s. Theorem 1.1. Let 3 an s [/]+. Suppose F M + Mf an f H s ǫ 1. Then we have the global existence of the unique classical solution f(x, v, t) to the equation (1.6) satisfying f C([, );H s (R R )), F M + Mf, an f(t) H s +c 1 t (a,b) H s 1s+c 1 t k+l s k l v {I P}f µ s c f H s, for some positive constants c 1,c >. Furthermore, if f L v (L 1 ) is boune, we have f(t) H s C ( f H s + f L v (L 1 )) (1+t) 4, t, where C is a positive constant inepenent of t. Remark 1.1. In the case of the perioic spatial omain, i.e., the particles governe by the equation (1.1) are in the torus T, we have the exponential ecay of f(t) H s uner aitional assumptions on the initial ata. We give the etails of that in Remark 5.1. Remark 1.. Our strategy can also be extene to the non-local version of (1.1), i.e., the equation (1.1) with ũ F efine in (1.5), when the communication weight ψ satisfies ψ L 1 (R ) an ψ L 1 = 1. For the proof of Theorem 1.1, we employ a similar strategy as in [11] where the global existence an large-time behavior of classical solutions to the kinetic Cucker-Smale equation with the ensityepenent iffusion are investigate. As we briefly mentione above, the alignment of Cucker-Smale equation is normalize with the total mass, an this makes the equation consiere in [11] has weak nonlinearity compare to our equation (1.6). To be more precise, if we similarly formulate the equation (1.6) with the alignment of Cucker-Smale equation, the nonlinear part Γ(f, f) of this equation becomes bilinear. However, in our situation the nonlinear operator Γ(f, f) efine in (1.7) is not bilinear ue to the ifferent normalization. Thus, a more elicate analysis is require for the energy estimates. Our careful analysis of the nonlinear issipation term enables to obtain the

4 4 CHOI uniform boun of a total energy functional E(f) (see (5.1)), by which we can show that the existence of global classical solutions an its large-time behavior of the equation (1.6). The rest of this paper is organize as follows. In Section, we recall some basic properties of the linear operator L an hypocoercivity for a linearize Cauchy problem with a non-homogeneous microscopic source. The proof of hypocoercivity property showing the algebraic time ecay rate is postpone to Appenix A for the sake of a simpler presentation. We also provie several useful lemmas. Section 3 is evote to present the local existence an uniqueness of the equation (1.6). In Section 4, we show a priori estimates of solutions which consist of the classical energy estimates an the macro-micro ecomposition argument. Finally, in Section 5 we provie etails of the proof of Theorem Preliminaries.1. Coercivity an hypocoercivity. In this part, we recall some properties of the linear Fokker- Planck operator L an provie the hypocoercivity for a type of linear Fokker-Planck equation with other sources. For this, we first ecompose the Hilbert space L v as L v = N N, N = span{ M,v M}. Note that M,v 1 M,,v M form an orthonormal basis of N. We now efine the projector P by P : L v N, f Pf = {af +b f v} M, then it is clear to get a f = M,f an b f = v M,f, where, enotes the inner prouct in the Hilbert space L v (R ). We also introuce projectors P an P 1 as follows. P f := a f M an P 1 f := b f v M. Then P can be written as P = P P 1. In the following proposition, we provie some issipative properties of the linear Fokker-Planck operator L. For the proof, we refer to [4, 11]. Proposition.1. The linear operator L has the following properties. (i) ( ) Lf,f = f v M v, R M Ker L = Span{ M}, an Range L = Span{ M}. (ii) There exists a positive constant λ > such that the coercivity estimate hols: Lf,f λ {I P}f µ + bf. (iii) There exists a positive constant λ > such that Lf,f λ {I P }f µ. Forthe sakeofsimplicity, weropthe superscriptf, anenotea f anb f byaanb, respectively. We next show the hypocoercivity property for the Caucy problem with a non-homogeneous source. Consier the following linear Cauchy problem: with the initial ata: t f = Bf +b v M +h, x R, t >, (.1) f(x,v,) =: f (x,v). (.)

5 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 5 Here h = h(x,v,t) an f = f (x,v) are given, an the linear operator B is efine by B = v +L, Lf = v f + 1 ( v ) f. 4 We efine e Bt as the solution operator to the Cauchy problem (.1) with h, i.e., the solution to the equations (.1)-(.) can be formally written as f(t) = e Bt f + t e B(t s) h(s)s. Before we state the estimate for algebraic ecay of e Bt, we set the inex of rate σ,q,m by σ,q,m = ( 1 q 1 ) + m. Proposition.. Let 1 q an 1. (i) For any k,l with l k, an f satisfies k f L (R R ) an l f L v(r ;L q x(r )), we have k e Bt f L C(1+t) σ (,q,m l f L v (Lq ) + k f L ), t >, where m = k l an C is a positive constant epening only on,m, an q. (ii) For any k,l with l k. Suppose the non-homogeneous source h satisfies µ(v) 1/ k h(t) L (R R ) an µ(v) 1/ l h(t) Z q for t. Furthermore, h satisfies then we have t k e B(t s) h(s)s C t h, M = an h,v M = for (x,t) R R +, (.3) L ) (1+t s) σ,q,m ( µ 1/ l h(s) L v (Lq ) + µ 1/ k h(s) L s, t, where m = k l an C is a positive constant epening only on,m, an q. Proof. Although the proof is similar as in [1, 11], we provie the etaile proof in Appenix A for the reaer s convenience... Technical lemmas. We first recall several Sobolev inequalities which will be use in the rest of this paper. Lemma.1. (i) For any pair of functions f,g (H k L )(R ), we obtain Furthermore if f L (R ), we have (ii) For f H [/]+1 (R ), we have k (fg) L f L k g L + k f L g L. k (fg) f k g L f L k 1 g L + k f L g L. f L f H [/]. (iii) For f (H k L )(R ), let k N,p [1, ], an h C k (B(, f L )) where B(,R) enotes the ball of raius R > centre at the origin in R, i.e., B(,R) := {x R : x R}. Then there exists a positive constant c = c(k,p,h) such that k h(f) L p c f k 1 L k f L p. In the lemma below, we provie some relations between the macro components (a,b) an f.

6 6 CHOI Lemma.. Let T >, 3, an s [/]+. For k s, we have k (a,b) L C k f L, where C is a positive constant. Furthermore, if we assume sup t T f(t) H s ε 1, then we have ( ( b 1+a) k C k (a,b) L an b b k Cε L 1+a) k (a,b) L, L for some positive constant C >. Proof. By the efinition of a an b, it is clear to get k (a,b) L C k f L, (.4) for some positive constant C >. Then we use (.4) an the assumption on f to fin sup (a(t),b(t)) H s ε 1. t T In particular, we obtain (a,b) L C (a,b) L (,T;H s ) ε 1, an this yiels from Lemma.1 that ( 1 1+a) k k a L. L This an together with the Sobolev inequality in Lemma.1, we euce ( ( b 1+a) k b L 1 k + L 1+a) k b L 1 L 1+a L b H [/] k a L + k b L k (a,b) L. Similarly, we also have ( ( b b 1+a) k b L b k L 1+a) L + b 1+a L k b L b L k (a,b) L Cε k (a,b) L. Before we complete this section, we give a type of Gronwall s inequality an an elementary integral calculus whose proofs can be foun in [1, 4]. Lemma.3. For a positive constant T >, let {a n } n N be a sequence of nonnegative continuous functions efine on [, T] satisfying t t a n+1 (t) C 1 +C a n (s)s+c 3 a n+1 (s)s, t T, where C i,i = 1,,3 are nonnegative constants. Then there exists a positive constant K such that for all n N K n t n if C 1 =, a n (t) n! Ke Kt if C 1 >. Lemma.4. For any < α 1 an β > 1, t (1+t τ) α (1+τ) β τ C(1+t) min{α,β} for t.

7 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 7 3. Local existence an uniqueness In this section, we investigate the local existence an uniqueness of classical solutions to the Cauchy problem (1.6). For this, we first present a linearize equation of (1.6) where the average local velocity u F is given an satisfies certain smallness an regularity assumptions. For the linearize equation, we show the global existence an uniqueness of classical solutions. Then we construct approximate solutions {f n } n N, an show that they are convergent in L -space. Finally, we conclue that the limit function is our esire solution using the Sobolev embeing theorem an the lower semicontinuity of the H s -norm. The theorem below is our main result in this section. Theorem 3.1. Let 3 an s [/]+. There exist positive constants ǫ, T, an N > such that if f H s ǫ an M + Mf, then there exists the unique classical solution f to the Cauchy problem (1.6) such that M + Mf an f C([,T ];H s (R R )). In particular, we have sup f(t) H s N. t T 3.1. Local well-poseness of a linearize system. In this part, we linearize the system (1.6) an show the invariance property of the solutions in an appropriate norm. For a given f C([,T];H s (R R )), we consier t f +v f +u F v f = Lf +Γ( f,f), (3.1) with the initial ata f(x,v,) =: f (x,v), where u F = v f M v R 1+ f M v R ( ) 1 an Γ( f,f) = u F vf +v M. By the stanar linear solvability theory, the unique solution f C([,T];H s (R R )) to (3.1) is well-efine for any positive time T >. For notational simplicity, we set ā := f M v an b := v f M v. R R Lemma 3.1. Let 3 an s [/]+. There exist constants ǫ, T, an N > such that if M + Mf, f H s ǫ, an sup f(t) H s N, t T we have M + Mf an f C([,T ];H s (R R )). Furthermore, we have sup f(t) H s N. t T Proof. We first notice that the equation (3.1) can be rewritten in terms of F(x,v,t): t F +v F +u F v F = v ( v F +vf). Then by the maximum principle for the linearize equation (3.1), we fin F M + Mf.

8 8 CHOI We next show the boun on f in the rest of the proof. It follows from (3.1) that for k s 1 t k f L +λ {I P } k f µ k Γ( f,f), k f x+ ( ) k k l u F v l f, k f x R l l<k R 1 ( ) k k l u F v l f, k f x+ k u F v M, k f x l l k R R + ( ) k k l u F v l f, k f x l R =: l<k 3 I i. i=1 First, we can easily fin I = k u F k b x k u F L k b L k (ā, b) L k b L k f L k f L, R ue to Lemma.. For the estimate of I 1, we obtain I 1 ( ) 1/ k l u F v l f k f xv k l u F l f L x k f R R R v µ. l k On the other han, the last term of the above inequality is boune by ( ) { 1/ k l k l u F l f L x u F L l f L for k l [/], R v k l u F L l f L v (L ) for k l [/]+1. This implies Similarly, we also euce I 1 f L v (H s ) I 3 f L v (H s ) α+β s α+β s α β vf µ. α β v f µ. Thus we have 1 t k f L +λ {I P } k f µ k f L k f L + f L v (H s ) Note that P k f µ can be estimate as an this yiels α+β s P k f µ = k a M µ = v k a M +(1+ v ) k a M vx R R k a x (1+ v )M v R R C k a L C k f L, (3.) α β v f µ. (3.3) k f µ {I P } k f µ + P k f µ {I P } k f µ +C k f L. (3.4)

9 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 9 Hence, by combining (3.3) an (3.4) we have 1 t k f L +λ k f µ In a similar way, we also fin k f L k f L + f L v (H s ) t f H s +λ α+β s C f L v (H s ) Since f(t) H s N for t T, t f H s +(λ CN) α β v f µ α+β s α+β s an by integrating it over [,t](t T ) we get f(t) H s +(λ CN) t α+β s Finally, we choose ǫ, T >, an N > such that α+β s α β vf µ +C k f L. α β vf µ +C f H s +C f H s. α β v f µ CN +C f H s, α β v f(τ) µ τ ǫ +CN T +C sup τ T f(τ) H st. to conclue < ǫ 1, λ C > N, an CT 1 4, sup f(t) H s ǫ +CN T N. t T 3.. Proof of Theorem 3.1. We construct the approximate solutions f m for the system (1.6) as follows. t f m+1 +v x f m+1 +u F m f m+1 = Lf m+1 +Γ(f m,f m+1 ), (3.5) with initial ata an first iteration step: an f m (x,v) t= = f for all m 1, (x,v) R R, f (x,v,t) = for (x,v,t) R R R +. Here u F m an Γ(f m,f m+1 ) are given by vf m M v R u F m = 1+ R f m an Γ(f m,f m+1 ) = u F m M v ( 1 vfm+1 +v ) M, respectively. Set a m := f m M v an b m := vf m M v. R R Let I(s,T;N) be the solution space for f efine by { I(s,T;N) := f C([,T];H s (R R )) : M + } Mf an sup f(t) H s N. t T Then as a irect consequence of Lemma 3.1, we have the uniform boun estimate of f in the proposition below.

10 1 CHOI Proposition 3.1. Let 3 an s [/]+. There exist positive constants ǫ, T, an N > such that if f H s (R R ) with f H s ǫ an M + Mf, then for each m, f m is well-efine an f m I(s,T ;N). We next show that the approximations f m are Cauchy sequences in C([,T ];L (R R )). Lemma 3.. Let f m be a sequence of approximate solutions with the initial ata f satisfying f H s ǫ. Then f m is Cauchy sequence in C([,T ];L (R R )). Proof. It follows from (3.5) that t (f m+1 f m )+v (f m+1 f m )+u F m v (f m+1 f m ) = (u F m u F m 1) v f m +L(f m+1 f m )+ 1 u F m 1 v( f m+1 f m) ( 1 +(u F m u F m 1) vfm +v ) M. Then by using similar estimates as in Lemma 3.1 with k = we get 1 t fm+1 f m L +λ {I P }(f m+1 f m ) µ 1 u F m 1 v(f m+1 f m ),f m+1 f m x R 1 + (u F m u F m 1) R vfm +v M v f m,f m+1 f m x where we use We now use the estimate C f m+1 f m L f m+1 f m µ +C f m f m 1 L f m+1 f m µ, u F m u F m 1 L a m a m 1 L + b m b m 1 L f m f m 1 L. f m+1 f m µ {I P }(f m+1 f m ) µ +C f m+1 f m L to obtain 1 t fm+1 f m L +λ f m+1 f m µ C f m+1 f m L +C fm f m 1 L + λ (3.6) fm+1 f m µ. Applying the Gronwall s inequality for (3.6), we euce Finally, we use Lemma.3 to have This conclues the esire result. t (f m+1 f m )(t) L (f m f m 1 )(s) L s. (f m+1 f m )(t) L Tm+1 (m+1)! for t [,T ]. Proof of Theorem 3.1. Note that f m is boune in C([,T ];H s (R R )). Thus we use the Gagliaro-Nirenberg interpolation inequality together with the convergence estimate in Lemma 3. to euce f m f in C([,T ];H s 1 (R R )), as m. Furthermore, it follows from the lower semicontinuity of the norm that f m I(s,T ;N) implies M + Mf an sup f(t) H s N. t T

11 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 11 Finally, if we let f,g be the classical solutions obtaine from the above with the same initial ata. Then we have t f(t) g(t) L C f(τ) g(τ) L τ, for some positive constant C >, an the stanar argument yiels the uniqueness of the classical solutions. 4. A priori estimates In this section, we provie a priori estimates for the global existence of classical solutions to the equation (1.6). By using the classical energy metho together with the careful analysis of the local average velocity, we obtain several uniform a priori estimates of energy inequalities. These energy estimates play a crucial role in obtaining the global well-poseness of solutions with the help of the local existence as well as the continuum argument Energy estimates. We first present a priori estimate of f L v (H s ) for the equation (1.6). Lemma4.1. Let 3 an s [/]+, an let T > be given. Suppose that f C([,T];H s (R R )) is the solution to the equation (1.6) satisfying Then we have t f L v (Hs ) +C 1 for some positive constants C 1,C >. sup f(t) H s ǫ 1 1. t T k s k {I P}f µ Cǫ 1 (a,b) H s 1, Proof. L -estimate: We first easily fin 1 ( ( f xv = f v x f +u F v M + v ) ) t R R R R f vf +Lf xv = u F v Mf xv + u F v R R R R f xv + flf xv R R 3 =: I i. i=1 Here, I 1 an I 3 are estimate by R b I 1 = u F bx = R 1+a x, I 3 = f L{I P}f xv + f LPf xv = L{I P}f,f x b x. R R R R R R Thus, we obtain I 1 +I 3 = L{I P}f,f x R For the estimate of I, we notice that R a b 1+a x. vf,f = v, Pf + vpf,{i P}f + v, {I P}f = ab+ vpf,{i P}f + v, {I P}f.

12 1 CHOI This euces 1 f xv = L{I P}f,f x+ u F vpf,{i P}f x t R R R R + 1 u F v {I P}f x R L{I P}f,f x+ u F L Pf L {I P}f µ R where we use + u F L {I P}f µ L{I P}f,f x+cǫ 1 (a,b) H +Cǫ 1 {I P}f µ, s 1 R u F L C u F H [/] C (a,b) H s 1 an Pf L C (a,b) L Cǫ 1, ue to Lemma.. Thus we have t f L +λ {I P}f µ Cǫ 1 (a,b) H s 1. H s x-estimate: For 1 k s, we take k to (1.6) to get where Then we fin t k f +v k+1 f + k (u F v f) = L k f + k Γ(f,f), Γ(f,f) = u F v M + u F v f. 1 t k f L = L k f, k f x+ k Γ(f,f), k f x R R ( ) k k l u F v l f, k f x l R =: l<k 3 J i, i=1 where J 1 is easily estimate by J 1 = L k {I P}f, k f x+ L k Pf, k f x R R = L k {I P}f, k f x k b x. R R For the estimate of J, we ecompose it into two terms: J = k (u F v M), k f x+ 1 k (u F vf), k f x R R =: J 1 +J. Here, J 1 is estimate as J 1 = k u F k bx R = R k b 1+a x+ R k b l<k 1+a x+cǫ 1 (a,b) H s 1, ( ) ( ) k 1 l b k l k bx l R 1+a (4.1)

13 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 13 where we use for k l [/]+1 R l b k l an for k l [/] Similarly, we obtain R l b k l ( ) 1 k bx l b L 1+a ( ) 1 1+a ( 1 1+a) k l L k b L l+1 b H [/] k l a L k b L Cǫ 1 (a,b) H s 1, ( k bx l b L 1 1+a) k l k b L L l b L k l+1 a H [/] k b L Cǫ 1 (a,b) H s 1. J = 1 ( ) k k l u F v ( {I P} l f +P l f ),{I P} k f +P k f x l l k R Cǫ 1 k {I P}f µ + (a,b) H. s 1 1 k s This yiels R k b J 1+a x+cǫ 1 1 k s k {I P}f µ + (a,b) H. (4.) s 1 Finally, we again use similar arguments as the above to fin J 3 Cǫ 1 1 k s Hence by combining (4.1)-(4.3) we have t f L v (Hs 1 ) +(λ Cǫ 1) 1 k s k {I P}f µ + (a,b) H. (4.3) s 1 k {I P}f µ 1 k s R Cǫ 1 (a,b) H s 1. a 1+a k b x+cε (a,b) H s 1 We next provie the mixe space-velocity erivative of {I P}f in the following lemma. Lemma4.. Let 3 an s [/]+, an let T > be given. Suppose that f C([,T];H s (R R )) is the solution to the equation (1.6) satisfying sup f(t) H s ǫ 1 1. t T

14 14 CHOI Then for fixe 1 l s we have t k s l k l v {I P}f L +C Cǫ 1 k s l +C k s+1 l +Cχ { l s} 1 β l 1 α+β s k s l k l v {I P}f µ k l v{i P}f µ + (a,b) H s 1 k {I P}f µ + (a,b) H s l α β v{i P}f µ, (4.4) for some positive constants C,C >. Here χ A enotes the characteristic function of a set A. Proof. By applying {I P} to the equation (1.6), we fin t {I P}f +{I P}(v f +u F v f) = {I P}Lf +{I P}Γ(f,f). Since {I P}(v M) =, we get {I P}Γ(f,f) = 1 u F v{i P}f + 1 u F [{I P},v]f = 1 u F v{i P}f + 1 u F [v,p]f. This euces t {I P}f +v {I P}f +u F v {I P}f = L{I P}f + 1 u F v{i P}f + 1 u F [v,p]f +P(v {I P}f +u F v {I P}f) {I P}(v Pf +u F v Pf), where we use {I P}L = L{I P}. Then we have for k +l s with fixe 1 l s 1 t k l v {I P}f L = k l v(v {I P}f)+ k l v(u F {I P}f), k l v{i P}f x R + k l v(l{i P}f)+ 1 R k l v(u F v{i P}f)+ 1 k l v(u F [v,p]f), k l v{i P}f x + k l v (P(v {I P}f +u F v {I P}f)), k l v {I P}f x R + k l v ({I P}(v Pf +u F v Pf)), k l v {I P}f x R 9 =: K i. i=1

15 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 15 Estimate of K 1 : A straightforwar computation yiels K 1 = k [ l v,v ]{I P}f, k l v {I P}f x R C [ l v,v ] k {I P}f L +δ k l v{i P}f L C α+1 {I P}f L +Cχ { l s} α β v {I P}f L α s l +δ k l v{i P}f L, 1 β l 1 α+β s where δ > is a positive constant which will be etermine later, an [, ] enotes the commutator operator, i.e., [A,B] := AB BA. Estimate of K : Similar to (3.), we fin K = ( ) k k α u F v α l v α {I P}f, k l v {I P}f x α<k R C k α u F α l+1 v {I P}f L v k l v {I P}f L x α<k R v C (a,b) H s 1 k l v {I P}f L Cǫ 1 Estimate of K 3 : Since k+l s k+l s k l v{i P}f L. [ l v,l] = [ l v, v ]+[ l v, 1 4 ( v )] = [ l v, v ], we obtain K 3 = k [ l v,l]{i P}f, k l v {I P}f x L k l v {I P}f, k l v {I P}f x R R = k [ l v, v ]{I P}f, k l v {I P}f x L k l v {I P}f, k l v {I P}f x. R R On the other han, the first term of the above equality can be estimate as k [ l v, v ]{I P}f, k l v {I P}f x R C [ l v, v ] k {I P}f L +δ k l v{i P}f L C α+1 {I P}f µ +Cχ { l s} α β v {I P}f µ α s l +δ k l v{i P}f L. 1 β l 1 α+β s

16 16 CHOI Estimates of K 4 an K 5 : Similar to the estimate of K, we get K 4 k α u F α l v(v{i P}f) L v k l v{i P}f L v x α k R (a,b) H s α β v{i P}f µ Cǫ 1 K 5 α+β s α+β s α β v{i P}f µ, k α u F l v ([v,p] α f) L v k l v {I P}f L x α k R v ( (a,b) H s (a,b) H s 1 + k l ) v {I P}f L Cǫ 1 ( (a,b) H s 1 + k l v {I P}f L ). Estimates of K i,i = 6,,9: Similarly, we have K 6 +K 7 C α+1 {I P}f L +C (a,b) H s α s l +δ k l v {I P}f L C α+1 {I P}f L +Cǫ 1 α s l α+β s α+β s α β v{i P}f L α β v{i P}f L +δ k l v{i P}f L, K 8 +K 9 C (a,b) H s l +C (a,b) H s ( (a,b) H s 1 + k l v{i P}f L ) +δ k l v{i P}f L C (a,b) H s l +Cǫ 1( (a,b) H s 1 + k l v {I P}f L ) +δ k l v {I P}f L. 4.. Macro-micro ecomposition. In this part, we erive the hyperbolic-parabolic system which the macro components a an b satisfy. For this, we use the local conservation law of mass an the local balance law of momentum. More precisely, we multiply (1.6) by M an v i M for 1 i an take the velocity integration over R to get t a+ b =, (4.5) t b i + i a+ j A ij ({I P}f) =. Furthermore, we rewrite (1.6) as where j=1 t Pf +v Pf +u F v Pf 1 u F vpf (u F b) v M = t {I P}f +l+r, (4.6) l = v {I P}f +L{I P}f, r = b 1+a b v{i P}f + (1+a) {I P}f. Then we now apply the moment functional A ij (g) := (v i v j 1) M,g for 1 i,j to (4.6) to euce t A ij ({I P}f)+ i b j + j b i b ib j 1+a = A ij(l+r). (4.7) The similar erivation of the system (4.5)-(4.7) is use for the stuy of collisional kinetic equations [9, 13]. Using the system (4.5)-(4.7), we fin a temporal energy functional which has the issipation

17 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 17 rate (a,b) H s 1. Before we show it, we first provie the estimates of the term in the right han sie of the equation (4.7) in the lemma below. Lemma4.3. Let 3 an s [/]+, an let T > be given. Suppose that f C([,T];H s (R R )) is the solution to the equation (1.6) satisfying sup f(t) H s ǫ 1 1. t T Then there exists a positive constant C > such that k A ij (l) L C an k s 1 k s 1 k s k A ij (r) L Cǫ 1 k s k {I P}f L, (4.8) k {I P}f L. (4.9) Proof. The estimate (4.8) can be obtaine by using similar arguments as in [4]. For the estimate (4.9), we obtain This yiels k A ij (r) = A ij ( k r) = (v i v j 1) ( b v M, k (1+a) {I P}f b = ( ) k (v i v j 1) M, 1 ( b l k l 1+a l k ( ) k (v i v j 1) ( b M, k l l 1+a k s 1 l k l k ( b 1+a) k l L l {I P}f L. ( k A ij (r) L C b 1+a) Cǫ 1 where we use the estimate in Lemma.. k s H s 1 k s k {I P}f L, ) 1+a v{i P}f ) v l {I P}f ) v l {I P}f k {I P}f L Inspire by Kawashima s hyperbolic-parabolic issipation arguments [18], we introuce the temporal energy functional E (f) by E (f) := k ( i b j + j b i ) k A ij ({I P}f)x k a k bx. R R k s 1 1 i,j k s 1 Lemma4.4. Let 3 an s [/]+, an let T > be given. Suppose that f C([,T];H s (R R )) is the solution to the equation (1.6) satisfying sup f(t) H s ǫ 1 1. t T Then there exist positive constants C 3,C > such that t E (f(t))+c 3 (a,b) H s 1 C {I P}f L v (Hs ).

18 18 CHOI Proof. We first notice that 1 i,j k ( i b j + j b i ) L = k+1 b L + k b L. On the other han, it also follows from (4.7) that k ( i b j + j b i ) L = k ( i b j + j b i ) k A ij ({I P}f)x t 1 i,j 1 i,j R + k ( i t b j + j t b i ) k A ij ({I P}f)x R =: 1 i,j + 1 i,j 3 J i, i=1 R k ( i b j + j b i ) k ( ) bi b j 1+a +A ij(l+r) x where J i,i =,3 are estimate as follows. J = ( i a+ ) l A il ({I P}f) k j A ij ({I P}f)x 1 i,j R k 1 l ǫ 1 a H s 1 +C {I P}f L v (Hs 1 ), J i,j 1 i,j k ( i b j + j b i ) L +C 1 i,j b i b j 1+a +C {I P}f L H s 1 v (Hs ) k ( i b j + j b i ) L +Cǫ 1 b H s 1 +C {I P}f L v (Hs ). Here, we use the equation (4.5) for J an the estimates in Lemma 4.3 for J 3. Thus, we obtain k ( i b j + j b i ) k A ij ({I P}f)x+ b H t s 1 + b H s 1 R k s 1 1 i,j ǫ 1 a H s 1 +C {I P}f L v (Hs ) +Cǫ 1 b H s 1. We again use (4.5) to get the issipation rate of a H s 1. By taking k to (4.5) an multiplying it by k i a, we fin k+1 a L = k a k bx+ k i t a k b i x t R 1 i R k i a j k A ij ({I P}f)x 1 i,j R k a k bx+ 1 t R k+1 a L + k b L +C {I P}f L v (Hs 1 ), ue to the local conservation law of mass (4.5) 1. This implies k a k bx+ 1 t R k+1 a L k b L +C {I P}f L v (Hs 1 ). Now we combine all the estimates above to have ( ) 1 t E (f)+ b H + s 1 ǫ 1 a H Cǫ 1 b s 1 H s 1 +C {I P}f L v (Hs ).

19 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 19 Remark 4.1. Using the efinition of E (f), we fin E (f) ( k {I P}f L + k (a,b) ) L k s k s f L v (Hs ), ( k {I P}f L + k Pf L ) i.e., the temporal energy functional E (f(t)) is boune by f(t) L v (Hs ) for all t [,T]. 5. Proof of Theorem Existence an uniqueness. In this subsection, we provie the etails of proof for the part of global existence an uniqueness of classical solutions in Theorem 1.1. We first combine the estimates in Lemmas 4.1 an 4.4 to get ( ν 1 f L t v (Hs ) +E (f) )+(C 1 ν 1 C) k {I P}f µ+(c 3 Cǫ 1 ν 1 ) (a,b) Hs 1, k s for some large positive constant ν 1 1. Note that for some ν 1 > large enough it is clear to get ν 1 f L v (Hs ) +E (f) f L v (Hs ) in the sense that there exists a positive constant C > such that 1 C f L v (Hs ) ν 1 f L v (Hs ) +E (f) C f L v (Hs ), ue to Remark 4.1. It also follows from the linear combination of (4.4) over 1 l s that t 1 l s C l k s l Cǫ 1 +C k+l s k s k l v {I P}f L +C 4 1 l s k s l k l v {I P}f µ + (a,b) H s 1 k {I P}f µ +C (a,b) H s 1, k l v {I P}f µ for some positive constants C,C 4,C l >, l = 1,,s. We now set a total energy functional E(f) an a issipation rate D(f): E(f) := ν (ν 1 f L v (Hs ) +E (f) )+ C l k l v{i P}f L, D(f) := k+l s 1 l s k s l k l v {I P}f µ + (a,b) H s 1, (5.1) where ν 1 is a sufficiently large positive constant. Then we obtain t E(f(t))+C 5D(f(t)), where C 5 > is a positive constant inepenent of T an f, an this yiels t } {E(f(t))+C 5 D(f(τ))τ E(f ). sup t T

20 CHOI On the other han, since k l vpf L k f L, we fin f H s = k l v (Pf +{I P}f) L k+l s f L + k f L + 1 k s 1 l s k s l k l v{i P}f L, (5.) an this implies E(f) f Hs. Hence we have t { f(t) Hs +C sup t T } D(f(τ))τ C f Hs, (5.3) where C,C > are positive constants inepenent of T an f. We now choose a positive constant L = min{ǫ,ǫ 1 }, where ǫ an ǫ 1 are appeare in Theorem 3.1 an in Section 4, respectively. We also choose the initial ata f satisfying M + Mf an f H s Define the lifespan of solutions to the system (1.6) by { } T := sup t : sup f(τ) H. s τ t Since f H s L 1+C L ǫ, L 1+C. it follows from Theorem 3.1 an the continuation argument that the lifespan T > is positive. If T < +, we can euce from the efinition of T that On the other han, it follows from (5.3) that sup f(τ) H s = L. (5.4) τ T sup f(τ) H s C f H s L C τ T L 1+C, an this is a contraction to (5.4). Therefore T = +, an this conclues that the local solution f obtaine in Theorem 3.1 can be extene to the infinite time. 5.. Large-time behavior. In this part, we show the time-ecay estimate of solutions obtaine in Section 5.1, an complete the proof of Theorem 1.1. We also remark the exponential ecay rate of convergence when the spatial perioic omain is consiere. We first show the estimate of time ecay for the solutions f L in terms of the total energy function E(f) given in (5.1). Lemma 5.1. If f L v (L 1 ) is boune, we have f(t) L C( ) E(f )+ f L v (L 1 ) (1+t) / t ( t +C (1+t τ) / E(f(τ)) τ +C (1+t τ) E(f(τ))τ) /4, for t. (5.5)

21 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 1 Proof. We rewrite the equation (1.6) in the mil form: f(t) = e Bt f + where t e B(t τ) (H 1 (f(τ))+h (f(τ))) τ, H 1 (f) := 1 u F v{i P}f u F v {I P}f, H (f) := (u F b) v M + 1 u F vpf u F v Pf. Then it follows from Proposition. that ) f(t) L (E(f C )+ f L v (L1 ) (1+t) / +C t (1+t τ) /( µ 1/ H 1 (f(τ)) L v (L1 ) + µ 1/ H 1 (f(τ)) L )τ ( t +C (1+t τ) /4( ) H (f(τ)) L v (L 1 ) + H (f(τ)) L ) τ, since H 1 satisfies the conition (.3). Note that (5.6) (u F b) v M L v (L 1 ) + (u F b) v M L C u F b L 1 +C u F b L C(1+ a L ) b L CE(f). This an together with similar estimates as in [11], we obtain µ 1/ H 1 (f(τ)) L v (L1 ) + µ 1/ H 1 (f(τ)) L CE(f(τ)), H (f(τ)) L v (L 1 ) + H (f(τ)) L CE(f(τ)). Finally, we combine (5.6) an (5.7) to conclue the esire result. We set E (t) := sup (1+τ) / E(f(τ)). τ t Then it follows from (5.5) that t f(t) L (E(f C )+ f L v (L1 ) )(1+t) / +C f H (1+t τ) / E(f(τ))τ s ( t +C f 1/3 H (1+t τ) /4 E(f(τ)) 11/1 τ s =: 3 I i, i=1 ) ue to E(f) E(f ) C f Hs. We also use Lemma.4 to euce I C f H se (t) t (1+t τ) / (1+τ) / τ C f H se (t)(1+t) /. Similarly, we use the fact 11/4> max{1,/4} to obtain ( t ) I 3 C f 1/3 H se (t) 11/6 (1+t τ) /4 (1+τ) 11/4 τ C f 1/3 H se (t) 11/6 (1+t) /. Thus we have f(t) L C ( N + f H se (t)+ f 1/3 H se (t) 11/6) (1+t) /, (5.7)

22 CHOI where N := f H + f s L v (L1 ) >, an this yiels ( E (t) C N + f H se (t)+ f 1/6 H se (t) 11/6). Since f H s 1, this conclues the uniform bouneness of E (t). Hence, we have the algebraic ecay of the solutions to the equation (1.6). Remark 5.1. If we consier the perioic spatial omain T, we fin the following ifferential inequality using similar arguments as in Section 4. E(f(t))+CD(f(t)), t where E(f) an D(f) are given in (5.1). It also follows from (5.) that E(f) f Hs. We now further assume a (x)x = an T b (x)x =, T then this implies a(x,t)x = an T b(x,t)x =, for t, T ue to the conservations of mass an momentum. Thus, by using the Poincaré inequality, we get This euces k+l s Hence we have k l vpf L f H s k+l s D(f), i.e., E(f) D(f). This conclues an k s (a,b) L C (a,b) L. k (a,b) L k l v Pf L + 1 k s k+l s E(f(t))+CE(f(t)), t k (a,b) L (a,b) H s 1. k l v {I P}f L f(t) H s CE(f(t)) CE(f )e Ct C f H se Ct for t. We consier the following Cauchy problem: Appenix A. Proof of Proposition. t f +v f b v M = Lf +h, (A.1) where h is a given function satisfying the conitions in Proposition.. For an integrable function g : R R, we efine its Fourier transform ĝ := Fg by ĝ(k) = Fg(k) := e πix k g(x)x, R where k R an i 1 C is the imaginary unit. We enote the ot prouct a b for a,b C by (a b), where b is the complex conjugate of b. Then we first show the L -estimate of f in the following lemma. Lemma A.1. Let f be a solution of the equation (A.1) satisfying the conitions in Proposition.. Then we have t f L v +λ {I P} f µ C µ 1/ ĥ L v.

23 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 3 Proof. The Fourier transform of (A.1) gives t f +iv k f b v M = L f + ĥ = L{I P} f b v M +ĥ. Then we use {I P }{I P} = {I P} to get 1 t f L +λ {I P} f v µ (ĥ f). On the other han, we obtain (ĥ f) = (ĥ {I P} f) δ µ 1/ {I P} f L v + 1 4δ µ 1/ f L v Cδ {I P} f µ + 1 4δ µ 1/ f L v, for any positive constant δ >. We now choose the δ > such that Cδ < λ/ to complete the proof. Similar as in Section 4., we erive the macroscopic balance laws: t a+ b =, t b i + i a+ j A ij ({I P}f) =, j=1 t A ij ({I P}f)+ i b j + j b i = A ij (l+h), where l = v {I P}f +L{I P}f. Then it follows from (A.) that for 1 m b m t j A jm ({I P}f) 1 m A jj ({I P}f) = 1 1 j 1 j m A jj (l+h) 1 j By taking the Fourier transform to (A.3), we euce k bm t ik j A jm ({I P} f) 1 = 1 1 j 1 j 1 j ik m A jj ( l+ĥ) 1 j 1 j m j A jm (l+h). 1 j m ik j A jm ( l+ĥ). ik m A jj ({I P} f) This yiels t ik j A jm ({I P} f) 1 ik m A jj ({I P} f) b m + k b m 1 j 1 j m = 1 ik m A jj ( l+ĥ) ik j A jm ( l+ĥ) bm 1 j 1 j + ik j A jm ({I P} f) 1 ik m A jj ({I P} f) t bm =: I 1 +I, 1 j m (A.) (A.3)

24 4 CHOI where I 1 is estimate as follows. I 1 1 k b m +C 1 i,j ( A ij ( l) + A ij (ĥ) ) 1 k b m +C(1+ k ) {I P} f L v +C µ 1/ ĥ L v. For the estimate of I, we use the following Fourier transform of (A.) : t bm +ik m â+ ik j A mj ({I P} f) =. Then we euce I = ik j A jm ({I P} f) 1 1 j 1 j m δ k â +C(1+ k ) {I P} f L v. Thus we have t 1 j ik j A jm ({I P} f) 1 1 j ik m A jj ({I P} f) ik m â 1 j m 1 j ik m A jj ({I P} f) b m + k b m δ k â +C(1+ k ) {I P} f L v +C µ 1/ ĥ L v. We next obtain the issipation of k â. For this, we fin from (A.4) that ( t ik b ) â + k â + k i k j A ij ({I P} f) â =. 1 i,j Moreover, we get ( t ik b ) ( â = t ik b ) ( â + ik b ) ( t â = t ik b ) â k b. (A.4) ik j A mj ({I P} f) (A.5) Using the above equation, we have ( t Re ik b ) â + 1 k â k b +C k {I P} f L. (A.6) v Set E( f) := 3 1 m,j ik 1+ k ik ( j 1+ k A jm ({I P} f) ) bm 3 ( b â). Combining (A.5) an (A.6), we fin that E( f) satisfies We next set t ReE( f)+ k 4(1+ k ) 1 m,j ik ( m 1+ k A jj ({I P} f) ) bm ( â + b ) C {I P} f L v +C µ 1/ ĥ L v. Ẽ( f) := f L v +κree( f), where κ > is a positive constant which will etermine later. Then we obtain t Ẽ( f)+(λ Cκ) {I P} f L + k ( v 4(1+ k â + b ) C µ 1/ ĥ L ). v

25 VLASOV-FOKKER-PLANCK EQUATION WITH LOCAL ALIGNMENT FORCES 5 Since an {I P} f L v + k 1+ k ( â + b ) k 1+ k ( {I P} f L v + â + b ) ( Ẽ(ˆf) C {I P} f L + â + b ), v we have tẽ( f)+ C k 1+ k Ẽ( f) C µ 1/ ĥ L, v for some positive constant C >. This implies Ẽ( f(k,t)) Ẽ( f(,k))e C k 1+ k t +C t Let h =, then f(t) = e Bt f is the solution to (A.1) an α e Bt f L = k R α Ẽ( f(k,t)) k k α e C k 1+ k t f (k) L k R v k (α β) e C k 1+ k t k β f (k) L k + v k 1 e C k 1+ k (t τ) µ 1/ {I P} f(τ) L τ. v k 1 C(1+t) q + α β β f L v (Lq ) +Ce C t k f L, e C t k α f (k) L v k where k α = k α1 1 kα kα. Here Höler an Hausorff-Young inequalities are use as in [18]. We conclue the esire result by using the similar techniques as the above. Acknowlegments The author was supporte by Engineering an Physical Sciences Research Council(EP/K844/1). The author also acknowleges the support of the ERC-Starting Grant HDSPCONTR High-Dimensional Sparse Optimal Control. References [1] L. Bouin, L. Desvillettes, C. Granmont, an A. Moussa, Global existence of solutions for the couple Vlasov an Navier-Stokes equations, Diff. Int. Eqns,, (9), [] J. A. Carrillo, Y.-P. Choi, an T. Karper, On the analysis of a couple kinetic-flui moel with local alignment forces, Ann. I. H. Poincaré - AN, 33, (16), [3] J. A. Carrillo, Y.-P. Choi, E. Tamor, an C. Tan, Critical threshols in 1D Euler equations with nonlocal forces, Math. Mo. Meth. Appl. Sci., 6, (16), [4] J. A. Carrillo, R. Duan, an A. Moussa, Global classical solutions close to the equilibrium to the Vlasov-Fokker- Planck-Euler system, Kinetic an Relate Moels, 4, (11), [5] J. A. Carrillo, M. Fornasier, J. Rosao, an G. Toscani, Asymptotic flocking ynamics for the kinetic Cucker- Smale moel, SIAM J. Math. Anal., 4, (1), [6] Y.-P. Choi, Compressible Euler equations interacting with incompressible flow, Kinetic an Relate Moels, 8, (15), [7] Y.-P. Choi, S.-Y. Ha, an Z. Li, Emergent ynamics of the Cucker-Smale flocking moel an its variants, preprint (16). [8] F. Cucker an S. Smale, Emergent behavior in flocks, IEEE Trans. Auto. Control, 5, (7), [9] R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence an uniform stability in L ξ (HN x ), J. Diff. Eqns., 44, (7), [1] R. Duan, Hypocoercivity of linear egenerately issipative kinetic equations, Nonlinearity, 4, (11), [11] R. Duan, M. Fornasier, an G. Toscani, A kinetic flocking moel with iffusion, Comm. Math. Phys., 3, (1), [1] T. D. Frank, Nonlinear Fokker-Planck Equations: Funamentals an Applications, Springer-Verlag Berlin Heielberg, (5). [13] Y. Guo, The Boltzmann equation in the whole space, Iniana Univ. Math. J., 53, (4),

26 6 CHOI [14] S.-Y. Ha an E. Tamor, From particle to kinetic an hyroynamic escriptions of flocking, Kinetic an Relate Moels, 1, (8), [15] T. K. Karper, A. Mellet, an K. Trivisa, Existence of weak solutions to kinetic flocking moels, SIAM J. Math. Anal., 45, (13), [16] T. K. Karper, A. Mellet, an K. Trivisa, On strong local alignment in the kinetic Cucker-Smale moel, Hyperbolic Conservation Laws an Relate Analysis with Applications Springer Proceeings in Math. Stat. 49, (14), 7 4. [17] T. Karper, A. Mellet, an K. Trivisa, Hyroynamic limit of the kinetic Cucker-Smale flocking moel, Math. Mo. Meth. Appl. Sci., 5, (15), [18] S. Kawashima, Systems of a hyperbolic-parabolic composite type with applications to the equations of magnetohyroynamics, Thesis Kyoto University, [19] S. Motsch an E. Tamor, A new moel for self-organize ynamics an its flocking behavior, J. Stat. Phys., 144, (11), [] C. Villani, A review of mathematical topics in collisional kinetic theory In:Hanbook of mathematical flui ynamics, Vol. I, Amsteram: North-Hollan, (), (Young-Pil Choi) Fakultät für Mathematik Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany aress: ychoi@ma.tum.e