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The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements

We present a coupled kinetic-macroscopic equation describing the dynamic behaviors of Cucker-Smale(in short C-S) ensemble undergoing velocity jumps and chemotactic movements. The proposed coupled model consists of a kinetic C-S equation supplemented with a turning operator for the kinetic density of C-S particles, and a reaction-diffusion equation for the chemotactic density. We study a global existence of strong solutions for the proposed model, when initial data is sufficiently regular, compactly supported in velocity and has finite mass and energy. The turning operator can screw up the velocity alignment, and result in a dispersed state. However, under suitable structural assumptions on the turning kernel and ansatz for the reaction term, the effects of the turning operator can vanish asymptotically due to the diffusion of chemical substances. In this situation, velocity alignment can emerge algebraically slow. We also present parabolic and hyperbolic Keller-Segel models with alignment dissipation in two scaling limits.


1.
Introduction. The emergence of collective behaviors such as velocity alignment is ubiquitous in an ensemble of self-propelled particles. Here the jargon "velocity alignment" represents some phenomenon in which the velocities of self-propelled particles tend to a common value asymptotically using only limited environmental information and simple rules [44]. In literature, the jargons such as swarming and herding are also used to represent similar phenomena. In recent years, several mathematical models have been proposed for the modeling of velocity alignment and coordinated control in literature [2,3,4,5,11,13,15,17,20,21,30,33,34,35,38,40,41,45], and they have been extensively investigated due to their potential Thus, our communication setting in (2) can be viewed as a perturbation of the all-to-all communication weight function, i.e., ψ = constant, and the condition (2) has been employed in literature [2,3,4,16,17,18] on the C-S flocking. In fact, the positive lower bound for ψ is crucial for the emergence of velocity alignment, when the model lacks a uniform compact support in position.
Note that the equation (1) can naturally arise from the kinetic description of the ensemble of mechanical C-S particles (see [20]), and (1), and its variants have been extensively studied in literature from diverse perspectives, e.g., global existence theories of classical solutions [5,21], measure-valued solutions [20], coupling with fluid equations [2,3,4], and the macroscopic C-S model and its asymptotic justification [20].
In this paper, we address two issues in relation with the kinetic model (1). First, we propose a new kinetic model for C-S particles with chemotactic movements with velocity jumps and attraction toward chemotactic substances. In chemotaxis literature [9,10,42], it is commonly reported that some bacteria exhibit an aggregation dynamics and chemotactic movements. For this, we employ the idea from the kinetic Keller-Segel model [36], i.e., we add a velocity jump process and chemotactic movements to R.H.S. of the kinetic C-S model (1) and a suitable field equation for chemotactic density via the turning operator. Second, we study a global existence of strong solution to the proposed model.
The main results of this paper is two-fold. First, we provide a coupled kineticparabolic PDE model describing the dynamics of C-S particles with chemotactic movements. For the dynamics of the chemotactic density, we use a reaction-diffusion equation. More precisely, to register the abrupt changes in velocities due to the attraction by the chemotactic substance, we introduce a nonlocal turning operator T [S](f ) whose kernel depends on the density of chemotactic substances (in short chemotactic density). Following the spirit of the Boltzmann equation, we supplement the turning operator T [S](f ) on the right-hand side of (1) to obtain To close the dynamics of f in (3), we need to add the spatio-temporal evolution of the chemotactic density S = S(x, t). To do this, we use the reaction-diffusion equation: for τ = 0, 1, Finally, we combine (3) and (4) to obtain a coupled PDE system with chemotactic movement: where ϕ = ϕ(S, ρ) is a reaction term representing chemical interactions between the C-S particle and chemical substances. In the case that the velocity alignment mechanism in (5) is turned off, i.e., κ 0 = 0, the coupled system (5) is reduced to the kinetic Keller-Segel model [6,8,13,24,23,25,26,32]. Second, we provide a global existence of strong solution to the coupled system (5) as follows. We first recall the concept of strong solution as follows.  R 2d When the initial data (f 0 , S 0 ) is sufficiently regular and has finite velocity moments, and the reaction term ϕ(S, ρ) has a specific ansatz, we show that there exists a global existence of strong solutions (see Theorem 4.1).
The rest of the paper is organized as follows. In Section 2, we briefly review a kinetic C-S model and discuss the our proposed model (5). We also list a sufficient assumption for the global existence of strong solutions to (5) with asymptotic velocity alignment property. In Section 3, we provide a local existence of a strong solution. To do this, we use a standard successive approximation and several a priori estimates. In Section 4, we discuss the global existence of strong solutions to the coupled model (5) by excluding the possibility of the finite-time blow up of f and S and their derivatives (see Theorem 4.1). Finally, Section 5 is devoted to summary of main results of this paper. In appendix, we present two macroscopic models using the scaling limits. In Appendix A, we present the drift-diffusion system using a formal parabolic limit. In Appendix B, we derive a hyperbolic model involving velocity alignment force and chemotactic movements using a hyperbolic scaling.
Notation: For any measurable functions f = f (x, v) and u = u(x) defined on R 2d and R d , we set

2.
Preliminaries. In this section, we briefly discuss a kinetic C-S model and a generalized kinetic C-S model incorporating velocity jumps and chemotactic movement, and present several a priori estimates for the proposed model (5).

2.1.
A kinetic C-S model. Let x i and v i be the position and velocity of the i-th particle unit mass, respectively. Then, the dynamics of mechanical state (x i , v i ) is governed by the following N -body system: When the number of particles is sufficiently large, i.e., N 1, the numerical integration of (6) is almost impossible to implement. Thus, we are forced to approximate the system (6) using a corresponding mean-field kinetic model (see [21,20,31] for formal and rigorous derivations). Let f = f (x, v, t) be a one-particle probability density function of C-S particles (or cells) at position x ∈ R d and time t ∈ R + with velocity v ∈ R d . Then, the spatio-temporal evolution of f is described by the Vlasov-Mckean type equation: For notational simplicity, we suppress t-dependence in f and use a handy notation for velocity moments of f : for Next, we discuss the basic properties of the velocity alignment term F a [f ]f in (7) in the following lemma.
be a strong solution to (7) which decays to zero sufficiently fast at infinity: Then, we have Proof. (i) We used the relation (ii) Again, we use the relation Next, we set the first three velocity moments M 0 , M 1 and M 2 which represent the total mass, momentum, and twice the value of energy, respectively: Based on the estimates in Lemma 2.1, we obtain the conservation of momentum and dissipation of energy for (7) in the following proposition.
Proposition 1. Let f = f (x, v) be a strong solution to (7) which decays to zero sufficiently fast at infinity: Then, we obtain the conservation of momentum and dissipation of energy: for a.e. t > 0, Proof. (i) The conservation of mass follows from the far-field decay condition of f and divergence form of (7). For the conservation of momentum, we multiply (7) by v, and we integrate the resulting equation over R 2d and use Lemma 2.1(i) to find Therefore, the right-hand side of (9) becomes zero. This implies the conservation of momentum.
(ii) We multiply (7) by |v| 2 and integrate the resulting equation over R 2d to obtain the dissipation of the energy.
Next, we discuss a priori velocity alignment estimate following the idea in [21]. For this, we introduce a Lyapunov functional L 0 [f ] measuring the velocity dispersion of the kinetic density f : where v c is the average velocity defined by Then, it follows from the conservations of mass and momentum in Proposition As discussed in [16], the functional L 0 measures the variance of the velocity around the average velocity v c . More precisely, zero convergence of L[f (t)] as t → ∞ implies the formation of velocity alignment in probability. This can be seen as follows. From Chebyshev inequality, for any ε > 0, we have This implies Then, for any strong solution f = f (x, v) to (7) decaying to zero at infinity sufficiently fast: we have an exponential velocity alignment: Proof. Because (7) conserves the total momentum, we may assume that v c (0) = 0, and the functional L 0 (t) becomes twice that of the total energy: From the estimate (ii) in Lemma 2.1, where we have used a lower bound of ψ and zero momentum. Then, Gronwall's inequality yields the desired estimate.

2.2.
Modeling of chemotactic movements. In this subsection, we introduce a generalized kinetic C-S model with chemotactic movements. As reported in [22,29,46], collective behaviors in animal groups often emerge via individual interactions (communications). In 2007, Cucker-Smale [11] introduced a simple analytical model for flocking behavior of animal groups, which generalizes Vicsek's model [45]. After the Cucker-Smale's seminal work, most works on the flocking adopted their simple velocity alignment mechanism, which results in the smooth change of individual velocities. However, as we often see in the school of fish in aquarium, fish change their velocity abruptly, i.e., they have several components such as free swimming, tumbling and swarming etc in their motion. To model abrupt change of velocity, we adopt turning operator which is commonly used in kinetic Keller-Segel models [6,7,9,25,26,36,37]. As far as the authors know, there is only one work [19] to incorporating the velocity alignment and turning operator to model the collective behaviors in phototactic Cyanobacteria. The authors presented a coupled model without any theoretical studies on the coupled system. Next, we describe the velocity jump process via the turning operator: for a given (x, t) ∈ R d × R + , we set T =⇒ v.
Let S = S(x, t) be the concentration of the chemotactic substance. Then, the contribution of the rate of change in f along the particle trajectory due to chemotactic movements will be registered by the turning operator T [S](f ) (see [1,12,13,27,28]): where the quantity λ[S] denotes the turning frequency. Now, we combine (7) and (11) to obtain a kinetic chemotaxis-Cucker-Smale (CCS) model (5).
2.2.1. Propagation of velocity moments. Due to the effects of chemotactic attraction, the propagation of the moments of (5) will be different from the kinetic Cucker-Smale model (7), which has been discussed in Proposition 1. For the simplicity of notation, we set be a strong solution to (5) which decays to zero at infinity sufficiently fast: Then, for t > 0, we have Proof. (i) For the estimates of M 0 , we use similar calculation in Lemma 2.1 to obtain On the other hand, we use the identity and definition of turning kernel in (11) to see We combine (12) and (13) to derive a conservation of mass dM0 dt = 0. For the estimate on momentum M 1 , we again use Proposition 1 and the fact that the convection and the alignment forcing terms do not affect the total momentum. Therefore, we have (ii) We multiply (5) by |v| 2 and integrate the resulting relation with respect to (x, v) to obtain Remark 1. In general, the coupled system (5) does not conserve total momentum, and the energy itself may not be monotone in time unlike the kinetic C-S model.
3. Local existence of strong solutions. In this section, we first briefly describe an assumption, based on which we present the local existence of strong solutions.
3.1. Assumptions. In this section, we first give structural ansatz for the turning kernel T [S] and reaction term ϕ(S, ρ) for the local existence theory to (5).
• (A1): The dimension d of the spatial domain is any positive integer and (5) 2 is a reaction-diffusion equation.
• (A2): The turning kernel T [S](x, t, v, v ) is smooth with respect to v and v , and has compact supports in v and v . Moreover, if S = S(x, t) is smooth with respect to x, then T [S] is also smooth to x. More precisely, T [S] has the following properties: Here we use B Rv (0) to denote the ball centered at 0 with radius R v . • (A3) The reaction term ϕ(S, ρ) in (5) 2 takes the following ansatz [39,43]: For the structural assumption (A2), the turning kernel has compact support in the velocity domain. In [36,37], the authors studied a Boltzmann type equation where the velocity alignment force is absent. Therefore, micro-velocity becomes a parameter and the support of velocity will not change. However, in our model, the velocity is a variable and the support of this variable may change during the evolution of solutions.
Under the assumption (A1) -(A3), system (5) becomes Our first main result is concerned with the local existence of strong solutions.
Then, there exists a local-in-time strong solution (f, S) to (5) and a positive constant T * such that where K is a compact set containing B Rv (0) and K is another compact set containing K.
Proof. Since the proof is rather lengthy, we will present its proof in the following subsection.
3.2. Proof of Theorem 3.1. In this subsection, we present the local existence of strong solutions to (5) by performing the following four steps. • Step A: Construction of approximate solutions using the standard successive approximations. • Step B: Derivation of a priori estimates for approximate solutions.
• Step C: Establishment of the convergence of approximate solutions • Step D: Verification that the limit function is in fact our desired strong solution.

Construction of approximate solutions.
In this part, we present a sequence of approximate solutions {(f n , S n )} ∞ n=0 for system (16) as follows. Initial step (n = 0): We set where (f 0 , S 0 ) represents the initial data.
Inductive step (n ≥ 1): Suppose that the (n − 1)-th iterate (f n−1 , S n−1 ) has been constructed. Then, with this (n − 1)-th iterate (f n−1 , S n−1 ), we define the n-th iterate (f n , S n ) as a solution of the Cauchy problem to the following linear system: Note that (17) 1 can be rewritten in the following quasi-linear form: From the method of characteristics for (18), it follows that for a given x, v ∈ R d at time t, consider the the forward characteristics: which are given by the solution of the system: Then, it is easy to see that equation (19) 3 can be rewritten as Note that the approximation scheme (17) guarantees the positivity of f n . Hence, if the limit function for the sequence {f n } exists, then it will be nonnegative as well. Moreover, the non-negativity of S n follows from the maximum principle of elliptic equation.

3.2.2.
A priori estimates for approximate solutions. Suppose that the initial data satisfy Then, it is reasonable to expect that the approximate solution (f n , S n ) satisfies Before we begin technical estimates, we briefly discuss how to obtain a local solution: Suppose that f n−1 is in the desired function space in (22), and that it has finite moments and compact support with respect to v. We solve the linear system (17) using the method of characteristics, and we further show the regularity of the solution. For this, we show the following statements: • For small T > 0, we show that both M n 1 and M n 2 are bounded in the timeinterval [0, T ).
• We show that f n and S n are locally bounded in the function space (22), and thus we can extract a weak* limit in their space. • We show that f n and S n form Cauchy sequences in L ∞ space, and we construct a weak solution of the nonlinear system. As the weak* limit is unique, this constructed solution and the weak* limit coincide and thus have regularity.
Lemma 3.2. For T ∈ (0, ∞), suppose that the initial data (f 0 , S 0 ) satisfy (21), and the (n − 1)-th iterate (f n−1 , S n−1 ) satisfies Then, we have Proof. We split the proof into several pieces. (i) (Regularity of f n ): We integrate the first equation in (17) over R 2d to obtain: Thus, we have f n ∈ L 1 (R 2d ). Consider a backward equation of the characteristic curve: Where F := (F 1 , F 2 ) is the vector field Then, it is easy to verify that F is C 1 with respect to x n and v n . Hence, we can conclude that ( x, v) and consider the equation Here we use h n (τ ) to represent Therefore, we can write f n (x, v, t) in terms of an integral form as following Thus f (x, v, t) is Lipschitz continuous with respect to t. On the other hand, it follows from the regularity of S n−1 , T ± [S n ](f n−1 ), x n (τ ; x, v, t), ψ and v n (τ ; x, v, t) that we have where we have used the facts: Now, we apply Gronwall's lemma to (26) to determine the boundedness of the velocity support: |v n j (t)| ≤ C(T ). Next, we consider the second moments M n 2 . From the definition of M n 2 , we have Then, we use the assumptions (23) for S n−1 , f n−1 , M n−1

2
, and the compact support of turning operator T with respect to v and v to obtain Therefore, we have Again it follows from the relation M n 1 ≤ M 0 M n 2 that we also have sup (iii) (Regularity of S n ): We use the defining equation for v n (t) in (19) to show that f n has compact support with respect to v. Then, there exists a positive constant C such that Thus, we have ). Finally, we prove the regularity of S n by showing the following a priori estimates. First, since S n satisfies a linear parabolic equation (17) 2 , we can use the Green's function of heat equation and the non-negative property of ρ n−1 and S n to obtain . For the first order derivative, we can use Green's function of heat equation to obtain We can do second order derivative estimate similarly to obtain Therefore, we finally obtain the desire regularity of S n as following 3.2.3. Convergence of approximate solutions. In this part, we show that the limit of f n and S n exists, when n tends to infinity. First, we show that f n and S n are uniformly bounded in a short-time interval. Let L be a positive constant such that Lemma 3.3. There exists a positive constant T * such that if Proof. We will show that the lower-order estimate and the higher-order estimate can be obtained with the same method.
• Step A (zeroth order estimate of f n ): It follows from (20) that Note that for T * = 0, it follows from (27) that From the continuity of f n with respect to time t, we can find T * independent of n such that ||f n (t)|| L ∞ < L, 0 ≤ t ≤ T * . • Step B (uniform bound of M n i ): Directly calculation shows that Since the turning effect exists only for low-velocity particles due to (14) 1 and the turning kernel has the estimate (14) 2 , we have Similarly, we have Therefore, we combine (28), (29) and (30) to obtain a differential inequality: Then, we immediately obtain the estimate of M n 0 and M n 2 as below: Hence, we can apply the similar method as that used for f n to imply that for proper T * independent of n, we have Step C (uniform compact velocity support): Now, we can investigate the equation of v n (t) in (19): We multiply v n to (31) and denote the 2 norm of v n by |v n |. Then we obtain Therefore, we obtain the estimate of |v n | as below: This shows that for sufficiently small T * Therefore, we can find a large enough compact set K such that if v n (t) / ∈ K , then Since the initial datum for f n is f 0 which has a compact support. Thus, |v n (0)| has a common upper bound for any n. Moreover, due to the choice of K, the support of turning operator is contained in K. Therefore, for any v n (t) / ∈ K , we have v n (τ ) / ∈ K along the characteristic curve from 0 to t. Thus according to (25), we have d dt f n (τ ) = 0, 0 ≤ τ ≤ t ≤ T * .
This shows that the support of f n with respect to v is contained in K and thus uniformly bounded for 0 ≤ t ≤ T * . Due to this uniformly compact support, we have for 0 ≤ t ≤ T * , • Step D (lower-order estimate of S n ): First, we check the zeroth-order estimate of S n , and we find that S 0 (y)dy.
The right hand side is the convolution of heat kernel and S 0 . Therefore, we apply the maximum principle to show that Next, we check the first-order estimate of S n . For each i, we have for any 0 ≤ t ≤ T * , Therefore, we can choose T * sufficiently small to guarantee • Step F (higher-order estimate): We can take the derivative of the equation of f n and S n to obtain the partial differential equation of ∂ xi f n , ∂ vi f n and ∇ 2 xixj S n (x, t). Then we apply the same method from step A to step E and obtain the estimates of ||∂ xi f n || L ∞ , ||∂ vi f n || L ∞ and ||∇ 2 xixj S n (x, t)||.
Next, we will show that the sequences {f n } and {S n } are Cauchy in the space L ∞ (R 2d ). We use (17) to derive the equation for f n+1 − f n : Then, we can rewrite the relation (32) as Because we have proved the uniform compact support of f n with respect to v, we have |x n | ≤ (|v n (0)| + ψ M L 2 T * )T * by finite speed of propagation. Thus, we can assume T * sufficiently small so that 0 ≤ t ≤ T * < L and Lemma 3.4. Let (f n , S n ) be a sequence of approximate solutions defined by (17). Then, the sequence (f n , S n ) is Cauchy for sufficiently small T * in the space: Proof. We split the estimate into two parts.
(Estimate of I 11 ): We apply the compact support (34) of the velocity, position, and the higher order estimate in Lemma 3.3 to obtain where we used Lemma 3.3 and the relation (34) to find (Estimate of I 12 ): By directly calculation, we have , and this yields (Estimate of I 13 ): It follows from (11) that we have Therefore, we have Therefore, we have (Estimate of I 15 ): For this part, we apply the property (11) and use the similar argument as in I 13 to have (Estimate of I 16 ): For the last part, we apply the same argument as in I 14 to obtain Finally, we integrate (33) along the characteristic curve and combine estimates (35), (36), (38), (39), (40), and (41) to obtain Then, Gronwall's lemma and f n+1 (x, v, 0) = f n (x, v, 0) imply We next return to the estimate of ||S n+1 − S n || L ∞ .
• Case B (Estimate of S n+1 − S n ): We consider the representation formula for S n to obtain This yields Finally, we combine (43) and (44), and apply Lemma 3.3 to obtain The term on the right hand side makes up a convergent series, from which we conclude that the sequences f n and S n are Cauchy sequences.
We next return to the proof of Theorem 3.1. Proof of Theorem 3.1: Now, we can construct a limit function This gives a solution to (5) in the distribution sense. Moreover, we have uniform bounds for f n and S n in the spaces: ).

C. CHEN, S. HA AND X. ZHANG
Thus, there exists a weak* limit (f ∞ , S ∞ ) such that ). Because f n and S n are Cauchy, the limit functions (f, S) coincide with the weak* limit (f ∞ , S ∞ ). Thus, we have . This finish the proof of Theorem 3.1.
4. Global existence of strong solutions. In this section, we show that the local strong solution will never blow up in any finite-time interval; thus the local strong solution can be extended to the global strong solution by the continuous induction argument. More precisely, the main result of this paper is given as follows.
. Proof. Since the proof is rather lengthy, we postpone the proof to the end of this section.
In the following subsections, we present a series of a priori estimates.

4.1.
A priori estimates. In this subsection, we provide several a priori estimates.
Proof. (i) Integrating the equation of S with respect to x, we obtain d dt S(t) L 1 ≤ 0.
(ii) Note that S satisfies By Duhamel's principle for the inhomogeneous heat equation, we have We next use S 0 (y)dy ≤ ||S 0 || L ∞ , x ∈ R d , t > 0.

A KINETIC CUCKER-SAMEL MODEL WITH CHEMOTACTIC MOVEMENTS 525
This yields the desired estimate.
(iii) For the second estimate, we again use the relation (46) to see , let (f, S) be a strong solution to (16). Then, we have the following estimates: where C is a positive constant that is independent of t.
Proof. (i) (The estimate of M 1 ): We multiply v by the first equation of (16) and integrate the resulting equation with respect to x and v to obtain: where we used the estimate in Lemma 4.2, and V = B Rv is defined in (A2) in Section 3.1. This yields the desired result.
(i) (The estimate of M 2 ): We multiply |v| 2 by the first equation of (16) and integrate the resulting relation to obtain: The second term in (47) can be estimated as follows.
Next, we estimate the size of the velocity support of f . For this, we consider the following backward characteristic equation: for x, v ∈ R d and 0 ≤ τ ≤ t < T , Proof. We set a sequence of increasing balls: Suppose that the initial configuration has compact support in the velocity domain We claim: f (x, v, t) ≡ 0, for v / ∈ D n2 and n 2 ≥ n 1 + 1.
This immediately show that supp v (f (x, ·, t)) ⊂ D. Next, we prove the claim by contradiction. Suppose we have Then we notice from the definition of D n that, for any v / ∈ D n1 , v is out of V which is the support of the turning kernel. Thus, only alignment force affect the dynamic and we have On the other hand, due to the continuity of v(τ ; t, x, v) and the fact However, the facts v ∈ D c n2 and v(τ 0 ; t, x, v) ∈ D c n1 ∩ D n2 imply that |v| ≥ r n2 > |v(τ 0 ; t, x, v)|, which is contradictory to (53). Therefore, assumption (51) is not true and we finish the proof.
Remark 2. For D obtained in the above lemma, we actually have V ⊂ D, where V is the turning kernel effect region in assumption (A2), which was discussed in Section 3.1.

4.2.
Velocity alignment estimate. In this subsection, we present the a priori velocity alignment estimate. For the velocity alignment estimate of model (5), we cannot use the Lyapunov functional (10) as we do not know a priori the asymptotic velocity of the particles. This is due to the lack of the conservation of momentum. Hence, we instead use a new functional L 1 : Then, it is easy to see that the functional L 1 can be rewritten as follows.
Lemma 4.5. Let (f, S) be a strong solution to (5). Then, the Lyapunov functional L 1 (t) decays to zero in that Proof. We use Lemma 2.2 and Lemma 4.2 to see the time-variation of L 1 (t): The last inequality follows the decay rate of ||S|| L ∞ in Lemma 4.2. From Duhammel's principle, the above relation yields

4.3.
Proof of Theorem 4.1. In this subsection, we provide the proof of Theorem 4.1 with respect to the global existence of strong solutions. In Theorem 3.1, we have already seen that the coupled system (5) yields the local existence of strong solutions. Thus, for the global existence, we will extend these local solutions to global ones by showing the finiteness of the following quantity in any finite-time interval: for any T ∈ (0, ∞), Lemma 4.6. Assume that the assumptions (A1) -(A3) hold, and the initial data Then, for any T ∈ (0, ∞), there exists a positive constant C and time dependent constant C(T ) such that Proof. (i) (Estimate of ||S|| L p ): For p = 1 and ∞, the desired estimates follow from Lemma 4.2. On the other hand, for p ∈ (1, ∞), we use the interpolation lemma to find ||S(t)|| L p ≤ C. (ii) (Estimate of ||f || L ∞ ): We first multiply p|f | p−1 by the kinetic equation (5), and integrate with respect to x and v to obtain (54) • (Estimate of I 21 ): We use • (Estimate of I 22 ): We use the assumption (A2) in Section 3.1 to obtain (56) Finally, in (54), we combine (55) and (56) to obtain Then, it follows from Gronwall's lemma that we have We let p → ∞ to obtain Lemma 4.7. For i = 1, · · · , d, we have Proof. Denote the diameter of the velocity support by V(t). Then, it follows from Lemma 4.4 that V(t) ≤ CV(0).

Then, we have
Thus, ||∂ xi S|| L ∞ will not blow up in finite time.
Proof. We estimate the terms one-by-one: Then, we integrate the equation (59) along the particle trajectory to obtain Thus, in (60), we combine (61), (62), and (63) to find This yields From the above estimates, we apply Gronwall's inequality to conclude that ||∇ i x ∇ j v f (t)|| ∞ will not blow up in finite-time, where i + j = 1. Finally, we need to consider the second-order regularity of the chemical concentration S(x, t). Proof. We use the relation (58) and Lemma 4.8 to rewrite ∆ x S as follows.
Now, we are ready to present the proof of Theorem 4.1 Proof of Theorem 4.1: We combine Lemmata 4.6, 4.7, 4.8, and 4.9 to obtain a strong solution in any time period [0, T ): . if the assumptions of the initial data and the turning kernel in Theorem 4.1 hold. Moreover, by applying Lemma 4.5, we can obtain asymptotic behavior of the strong solution. More precisely, flocking emerges asymptotically: Remark 3. The chemical diffusion may attract particles to positions far away from the initial position, although the possibility of particles moving towards infinity is very small. Thus, the support of the kinetic density f = f (x, v, t) with respect to x is unbounded.

5.
Conclusion. In this paper, we presented a new mathematical modeling of the spatio-temporal dynamics of C-S flocking particles with chemotactic movements. Our combined model consists of two coupled equation. For the evolution of the kinetic density, we employed the kinetic C-S model which is a Vlasov-McKean type equation, whereas for the chemical density, we used the standard reaction-diffusion equation. These are coupled through the turning operator which represent the abrupt change of velocities of flocking particles. For this coupled system, we provided a global well-posedness of strong solution and presented a velocity alignment estimate for the special choice of reaction term. In particular, for velocity alignment estimate, we employed a robust Lyapunov functional approach measuring the velocity variation, and we showed that under suitable setting on the communication function and reaction term, we show that this functional tends to zero algebraically fast. This zero convergence implies the velocity alignment in probability. Finally, we also provide two macroscopic models using two scaling limits, parabolic limit and hyperbolic limit from the proposed coupled model, respectively and provide analytical forms for transport coefficients. There are still lots of interesting problems to be explored in future. For example, we assume that the positive lower bound for the communication weight function to show that the constructed strong solution satisfies velocity alignment. Moreover, to figure out the dynamic effects of singular communication weight functions will be also an interesting problem. We will leave this issues for a future work.
On the other hand, we substitute the ansatz (69) into (68) to obtain By comparing the terms in O(1) and O(ε), we can see that Thus, once we have an expression for f 1 in terms of ρ 0 and S 0 , we will obtain the desired leading-order dynamics for ρ 0 and S 0 . To derive such a represention for f 1 , we return to (67), and substitute the ansatzes (69) and (70) into (67) to obtain where T 0S [S 0 , S 1 ] is a turning operator whose kernel is the Fréchet derivative of T 0 with respect to S, and it is evaluated at S 0 in the direction of S 1 . We now compare terms in O(1), O(ε), O(ε 2 ), · · · .
At this stage, we assume that there exists a bounded equilibrium velocity distribution F = F (v) > 0 that is independent of x, t, and S satisfying the detailed balance principle and suitable normalization conditions: Then, it follows from the entropy inequality [7] that the kernel of T 0 [S 0 ] is spanned by F . Thus, we can set f 0 (x, v, t) = ρ 0 (x, t)F (v).
• (Derivation of the momentum equation): We multiply v by the first equation in (82) to get We now use the ansatz for f as in [14]: We substitute this ansatz (84) into the following terms in (83) to obtain where P is the stress tensor defined by Then, we combine (84) and (85) to obtain the momentum balance equations after ε → 0: ψ(|x − y|)(u(x, t) − u(y, t))ρ(x, t)ρ(y, t)dy.