Emergent dynamics in the interactions of Cucker-Smale ensembles

Merging and separation of flocking groups are often observed in our natural complex systems. In this paper, we employ the Cucker-Smale particle model to model such merging and separation phenomena. For definiteness, we consider the interaction of two homogeneous Cucker-Smale ensembles and present several sufficient frameworks for mono-cluster flocking, bi-cluster flocking and partial flocking in terms of coupling strength, communication weight, and initial configurations.

1. Introduction. The terminology "flocking" employed in this paper has a universal meaning representing some collective phenomena, in which self-propelled individuals using only limited environmental information and simple rules organize into an ordered motion [35]. For example, the aggregation of bacteria, flocking of birds, swarming of fish and herding of sheep correspond to flocking phenomena [4,15,16,17,18,35,36,37]. It has been extensively studied in the literature [3,6,7,8,16,17,18] owing to possible applications to mobile and sensor networks, and in the control of robots and unmanned aerial vehicles [27,30,32]. After the pioneering works [26,38] of Winfree and Kuramoto several decades ago, many phenomenological agent-based models have been proposed and studied analytically and numerically. Among others, our main interest lies on the second-order particle system proposed by Cucker and Smale [14]. This model resembles Newton's equations for a many-body interaction system for point particles. Let where K is a positive coupling strength and ψ is the communication weight satisfying the positivity, boundedness, continuity, and monotonicity conditions: there exists a positive constant ψ ∞ such that ψ ∞ ≥ ψ(r) > 0, r ≥ 0, ψ(·) ∈ Lip(R + ) and (ψ(r 2 ) − ψ(r 1 ))(r 2 − r 1 ) ≤ 0, r 1 , r 2 ≥ 0. (1.2) In this paper, we are addressing the following situation. Suppose the situation where two homogeneous ensembles of C-S particles are interacting in the whole space. Then, what will happen asymptotically after they begin to interact? Do they form a single ensemble moving together? Or do they diverge as a separate flocking ensemble after the initial mixing? etc. We can think of several possible scenarios in this situation. Thus, our main interest is to provide several sufficient frameworks leading to the aforementioned possible scenarios. To fix the idea, let be two homogeneous C-S ensembles. In the sequel, we call them subsystem G 1 and subsystem G 2 , respectively. Here the adjective "homogeneous" means that each C-S particle in the same subsystem has the same mass, so that each particle is indistinguishable. Let x αi (t), v αi (t) ∈ R 2d be the phase-space coordinate of the ith Cucker-Smale flocking agent in group G α . Consider the interacting Cucker-Smale flocking system: x 1i = v 1i ,ẋ 2j = v 2j , t > 0, i = 1, 2, · · · , N 1 , j = 1, 2, · · · , N 2 , where K 1 , K 2 , and K d are nonnegative intra-system and inter-system coupling strengths, and the communication weight ψ α : R + → R is Lipschitz continuous and satisfies the following conditions: 0 < ψ α (s) ≤ ψ α (0) = 1 < +∞, ψ α (s) ∈ L 1 (R + ), α = 1, 2, d, (ψ α (s 2 ) − ψ α (s 1 ))(s 2 − s 1 ) ≤ 0, s 1 , s 2 ∈ R + . (1.4) Note that if we turn off inter-system coupling strength K d = 0, then system (1.3) becomes the collection of two C-S models. The well-posedness of system (1.3) -(1.4) is obvious owing to the standard Cauchy-Lipschitz theory of ordinary differential equations.
The main novelty of this paper is threefold. First, we present a sufficient framework for a mono-cluster flocking to the combined system (1.3)- (1.4). It turns out that the key factor for the emergence of mono-cluster flocking is basically dependent on the inter-system coupling strength K d . For a large inter-system coupling strength K d , the combined system leads to mono-cluster flocking for any nonnegative intra-system coupling strengths K 1 and K 2 (Theorem 3.3) for some admissible class of initial configurations. Second, we deal with a sufficient framework for the bi-cluster flocking of subsystems G 1 and G 2 . In this case, the inter-system coupling strength should be small, but the intra-system coupling strength should be large. We quantify this plausible guess by providing explicit lower and upper bounds for K α and K d in terms of initial configuration only. Third, we present a sufficient framework for a partial flocking. More precisely, we present the conditions for local flocking of subsystem G 1 .
The rest of this paper is organized as follows. In Section 2, we give the propagation of velocity moments and previous results on the flocking formations to the single ensemble of C-S particles, review some relevant results, and discuss the difference between our results presented in this paper. In Section 3, we study a sufficient framework leading to the formation of mono-cluster flocking of two ensembles. In Section 4, we present a sufficient framework leading to bi-cluster flocking, i.e., each ensemble flocks together, but the whole combined ensemble does not flock. In Section 5, we present a sufficient framework for the partial flocking. Under our framework, only one of the ensembles flocks, whereas the other ensemble does not flock. Finally, Section 7 is devoted to a brief summary of our paper.
Notation. Throughout the paper, we use a superscript to denote the component of a vector, e.g., x := (x 1 , · · · , x d ) ∈ R d . Subscripts are used to represent the ordering of particles. For vectors x, v ∈ R d , its 2 -norm and the inner product are defined as follows: where x i and v i are the ith components of x and v, respectively.

2.
Preliminaries. In this section, we briefly review the flocking theorems for the C-S model (1.1) and present estimates on the propagation of velocity moments.
2.1. The Cucker-Smale model. In this subsection, we briefly review available flocking estimates for the emergence of mono-cluster flocking (global flocking) for the C-S model in (1.1)-(1.2). Mono-cluster flocking formation for system (1.1) was first studied by Cucker and Smale [14] for a special ansatz of ψ cs (s) = (1 + s 2 ) − β 2 , β ≥ 0. They showed that mono-cluster flocking occurs for any initial data for the case of long-range communication β ∈ [0, 1), whereas for the short-range case, they showed that mono-cluster flocking is possible for initial configurations close to the flocking state using the self-bounding argument. Later, Cucker and Smale's results were further refined and generalized to several physical settings, e.g., stochastic perturbations [2,13,19,22], bonding force and formation control [31], collision avoidance [1,11], effect of informed agent [12], nonlinear friction [21], relation with mechanical model [24], network effects [28,29,34], kinetic and fluid description [20,23,25], etc. So far, the most refined flocking estimates on the mono-cluster formation for (1.1) can be summarized as follows. We first define a mixed norm: where || · || is the standard 2 -norm in R d .

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S.-Y. HA, D. KO, Y. ZHANG AND X. ZHANG Theorem 2.1. [1,23] Let (x, v) be a solution to (1.1)-(1.2) with initial data (x 0 , v 0 ) satisfying the following conditions: Then there exists a positive number x M such that  [9,10], it has been shown that local flocking, in particular bicluster flocking, can emerge from some well-prepared configurations close to bicluster configurations.

Propagation of velocity moments.
In this subsection, we study the temporal evolution of the normalized first and second velocity moments. For this, we set Then, m i,j satisfies the following estimates.
Lemma 2.3. Let x, v be a global solution of the coupled system (1.3). Then, we have Proof. In the sequel, we only prove the assertions (i) and (iii). The estimates for (ii) and (iv) can be treated similarly.
(i) We use the symmetry of ψ l , l = 1, 2 in the transformation (iii) We take an inner product (1.3) 2 with 2v 1i to find Remark 2.4. Recall that the total momentum P and energy E satisfy the following estimates: As a direct corollary of Lemma 2.3, we have the estimates for M 1 and M 2 as follows.
Corollary 2.5. Let x 1 , v 1 and x 2 , v 2 be a global solution of the coupled system (1.3). Then, we have 3. Emergence of mono-cluster flocking. In this section, we study a sufficient condition for the mono-cluster flocking in the interaction of two homogeneous C-S ensembles. We will see that the inter-ensemble coupling strength K d will play a key role in the mono-cluster flocking estimates as long as the intra-ensemble coupling strengths K 1 and K 2 are nonnegative.
3.1. Lyapunov functionals. In this subsection, we introduce nonlinear functionals measuring the formation of mono-cluster flocking for system (1.3). In order to study the global flocking, we introduce the global averages and fluctuations around them: Then (x c , v c ) and (x α ,v α ) satisfẏ Note that the dynamics of (x c , v c ) and (x α ,v α ) are coupled except for N 1 = N 2 . We now define Lyapunov functionals X and V as the weighted l 2 -norms: Note that X and V measure the deviations from the global averages, and it is easy to see that the functional X and V are Lipschitz continuous in t, so it is differentiable for almost all t ∈ (0, ∞). Before we proceed to the flocking estimate, we recall the definition of mono-cluster flocking as follows. 3.2. Temporal evolution of the Lyapunov functionals. In this subsection, we derive a system of dissipative differential inequalities (SDDI) for (X , V) in (3.2). Note that, in the following sections, we let x := (x 1 , x 2 ), v := (v 1 , v 2 ), and N 1 + N 2 := N .
Then, the Lyapunov functionals defined in (3.2) satisfy Proof. (i) (Derivation of the first inequality): By the definition of X and the Cauchy inequality, we have This yields the desired first inequality.
(ii) (Derivation of the second inequality): We multiply (3.1) 2 by 2v 1i and (3.1) 3 by 2v 2j , and sum the results together, and then using similar calculations to the proof of Lemma 2.3 (ii), we have where we used On the other hand, note that Then, we substitute the relation 1 This yields the desired differential inequality for V.

3.3.
Emergence of a mono-cluster flocking. In this subsection, we provide the proof of the emergence of mono-cluster flocking using the SDDI in Proposition 3.2. We now present our first main result.
Theorem 3.3. Suppose that initial data (x 0 , v 0 ) are given and the intra-and interensemble coupling strengths K α and K d satisfy the following conditions:

Proof. •
Step A (Existence of x 1M ). It follows from Proposition 3.2 that we have We now define a Lyapunov functional L 0 following [23]: Then, we use (3.5) and (3.6) to obtain or equivalently In particular, this yields We set Then, F(β) is a continuous and increasing function of β, and by assumption (3.4), Hence, by the intermediate value theorem, we can choose the largest value of x 1M such that Then, we claim sup 0≤t<∞ Proof of claim (3.8). Suppose not, i.e., there exists t * ∈ (0, ∞) such that Then, for such X (t * ), we have which is contradictory to (3.7). • Step B (Exponential decay of V). We use (3.8) and the non-increasing property of ψ d to obtain This yields the desired result. Thus, in principle our flocking estimates can be done for the coupled particle system (1.3) and its kinetic counterpart with singular communication weights [5,23,33] in a priori settings. However, we leave this issue for future work.
2. The condition (3.4) on the lower bound for K d implies that, as V 0 increases or X 0 increases, the lower bound for K d increases. This is what we can expect to happen.
3. Consider the system with a bi-partite interaction, i.e., there is no intraensemble interaction, i.e., K 1 = K 2 = 0: for i = 1, 2, · · · , N 1 , j = 1, 2, · · · , N 2 , Then, the result of Theorem 3.3 yields that, as long as the inter-ensemble coupling strength K d is sufficiently large, we still have mono-cluster flocking for the initial configuration. This is a rather counterintuitive result.
In the following two sections, we study the formation of bi-cluster and multicluster flocking. 4. Emergence of bi-cluster flocking. In this section, we study the dynamics of system (1.3) in a small inter-coupling regime K d 1.
In this regime, we present sufficient conditions where each sub-ensemble G 1 and G 2 flock by themselves, but there is no mono-cluster flocking. Note that, for a large inter-ensemble we have a mono-cluster flocking wherein two sub-ensembles flock together independent of the detailed geometry of the initial configurations.

4.1.
Description of the main results. In this subsection, we briefly discuss our main results on the formation of bi-cluster flocking. Since we have bi-cluster flocking asymptotics in mind, we introduce local ensemble averages and local fluctuations around them: for α = 1, 2, we set Here we use the same notation for the local fluctuations as for the global fluctuations in Section 3 for notational simplicity. Then, it is easy to see that And then (x αc , v αc ) and (x α ,v α ) satisfẏ 2. The whole system (G 1 , G 2 ) exhibits a time-asymptotic bi-cluster flocking if and only if both subsystems G 1 and G 2 exhibit a time-asymptotic flocking, but the whole system does not exhibit a time-asymptotic mono-cluster flocking.
3. The whole system (G 1 , G 2 ) exhibits a time-asymptotic partial flocking if and only if only one of G 1 and G 2 exhibits a time-asymptotic flocking, but the other does not.
Our main results on the emergence of bi-cluster flocking can be summarized as follows.
Theorem 4.2. Suppose that the following framework (F A ) holds for the initial data (x 0 , v 0 ) to system (1.3).
• (F A 1): (Restriction on initial configurations) • (F A 2): (Restriction on coupling strengths): for α = 1, 2, Then, the whole system (G 1 , G 2 ) exhibits a time-asymptotic bi-cluster flocking. More precisely, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), there exist positive constants x ∞ α and C α , α = 1, 2 that depend only on the initial data and ψ such that  (0)) ≥ 0 means that the particles in different groups depart each other initially. Actually, this geometric condition is not that crucial for the validity of Theorem 4.1 as can be seen in Corollary 4.4. This condition will be attained in a finite time for proper coupling strengths, even if we begin with initial data that do not satisfy this condition.
2. The smallness condition on K d is needed to prevent mono-cluster flocking, whereas the largeness condition on K α is needed to enable flocking of each subsystem.
In fact, we can get rid of the condition min 1≤i≤N1 1≤k≤N2 For this, we set Corollary 4.4. Suppose that the following framework (F A ) holds for the initial data (x 0 , v 0 ) to system (1.3).
• (F A 1): (Restriction on initial configurations) • (F A 2): (Restriction on coupling strengths): for α = 1, 2, where we have used some quantities that only depend on the initial data: Then, the whole system (G 1 , G 2 ) exhibits a time-asymptotic bi-cluster flocking. More precisely, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), there exist x ∞ α and C α , α = 1, 2, that only depend on the initial data and ψ such that Remark 4.5. Note that the conditions imposed on the coupling strengths are explicitly computable from initial data.

Proof of Theorem 4.2.
In this subsection, we present a proof of Theorem 4.2 on the formation of bi-cluster flockings resulting from the interaction of two C-S ensembles in the low inter-coupling regime K d 1.
Proposition 4.6. Suppose that the coupling strengths satisfy and let (x, v) be a global solution to (1.3). Then, for α = 1, 2, we have where ψ dM is the time-dependent maximal communication weight between distinct ensembles: Proof. Since the estimates for subsystem G 2 are the same as for subsystem G 1 , we only treat estimates for α = 1.

Thus we can obtain
We use the above relation and (4.1) to obtain (ii) Since the first inequality can be proved similarly with Proposition 3.2, here we only prove the second one. We multiply (4.1) 4 by 2v 1i and sum the results with respect to i to obtain On the other hand, note that We now combine all estimates in (4.3) and (4.4) to obtain Similarly, we have Proposition 4.7. Suppose that the coupling strengths satisfy Proof. We only prove α = 1 and the case α = 2 can be treated similarly. We set We claim that for any Proof of claim (4.5). Now we take an arbitrary t 1 ∈ [0, ∞). Set For any j ∈ I t1 , we have We use the continuity to get that, there exists t > 0 such that for any t ∈ [t 1 , t 1 + t], any j ∈ I t1 and any i ∈ {1, · · · , N }\I t1 , it holds that Thus, for any Thus, the claim (4.5) holds. Now we set

Now we claim
Otherwise, we assume T < ∞. Then We use claim (4.5) to know there exists t > 0 such that Thus, we can have This contradicts the definition of T . Thus we obtain T = ∞. The conclusion follows.
In the following two subsections, we proceed to prove Theorem 4.2 as follows. • Step A (Local-in-time estimate). We will show that each sub-ensemble G α satisfies the flocking estimates for some finite time T : Step B (Continuation to the whole time interval). We will show that time T in Step A can be chosen to be infinity.

(4.6)
We first show that T * 1 exists and is positive.
Proof. (i) We first show that T * > 0.
(v 2k (0) − v 1i (0)) · e 0 1,2 We now take a minimum over i and k to obtain Then, by the continuity, there exists δ > 0 such that Thus, by the non-increasing property of ψ d , we have ψ dM (t) ≤ ψ d λ 0 t , for all t ∈ [0, T * 1 ). Lemma 4.9. (Flocking estimate in [0, T * 1 )) Suppose that the initial data (x 0 , v 0 ) satisfy (F A 1) and the coupling strengths satisfy Then, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), there exist positive constants x ∞ α and C α independent of time t such that Proof. (i) (Existence of an upper bound x ∞ α ): We fix α ∈ {1, 2} and define a Lyapunov functional L 1α : It follows from Proposition 4.6 and Lemma 4.8 that . We integrate the aforementioned relation to obtain In particular, this yields (4.7) On the other hand, the assumption on K α implies We use (4.7) and (4.8) to see the existence of a solution to the following equation: We set x ∞ α to be the largest positive value. Then, by the same argument employed in Theorem 3.3, we have We now apply Lemma A.1 in Appendix A with to find the desired flocking estimate: As a final step, we are now ready to complete the proof of Theorem 4.2 by showing that T * 1 = ∞.

4.2.2.
Step B: T * 1 = ∞. : Suppose that initial data (x 0 , v 0 ) and coupling strengths satisfy the framework (F A ). We claim Proof of claim (4.9). Suppose not, i.e., 0 < T * 1 < ∞. Then, it follows from the definition of T * in (4.6) that On the other hand, we use Proposition 4.6 and Proposition 4.7 to obtain: for (4.11) We use assumptions on the initial data K d and (4.11) to derive the following relation: This contradicts relation (4.10). Thus T * 1 = ∞. 4.3. Proof of Corollary 4.4. In this subsection, we provide the proof of Corollary 4.4 under the following condition: The main idea is to show that there exists a positive finite time such that all the position difference along the average velocity difference will be positive, i.e., for some finite time T 0 ∈ (0, ∞), This is plausible because for K d = 0, two C-S subsystems are independent so that the quantity (x 2k (t)−x 1i (t))·(v 2c (t)−v 1c (t)) will grow in time. Then, after t = T 0 , we can restart the C-S flow (1.3) with a new initial data (x α (T 0 ), v α (T 0 )) and apply the result of Theorem 4.2 to derive the desired bi-cluster estimates. For notational simplicity, we set

4.3.1.
From mixing to segregation. In the sequel, for the moment, we will assume that there exists a time T 0 such that min 1≤i≤N1 1≤k≤N2 Then, we can apply the first assertion in Theorem 4.2 for the solution starting from t = T 0 . To verify (4.12), we need several a priori estimates. By posterior calculation, we set In the sequel, we will show that T 0 defined in (4.13) satisfies the estimate (4.12).
Lemma 4.10. Suppose that the initial data (x 0 , v 0 ) satisfy (F A 1) and that the inter-subsystem coupling strength K d is sufficiently small in the sense that Proof. (i) It follows from Proposition 4.6 (i) that we have This and the assumption on (ii) It follows from Proposition 4.7 that we have for: Remark 4.11. By Proposition 4.6(iii), we can see that Thus, we have Proposition 4.12. Suppose that the initial data satisfy (F A 1) and that the intergroup coupling strength K d is sufficiently small: where D(x 1 (0), x 2 (0)) is defined in (4.2). Then, we have Proof. We first claim that (4.14) Proof of claim (4.14). We use the relation to obtain Then, we use the condition on K d to obtain We now integrate the aforementioned relation in t from t = 0 to t = T 0 to obtain ∆ 1i,2k (t) ≥ ∆ 1i,2k (0) + λ 2 0 t, t ∈ [0, T 0 ]. This and the definition of T 0 in (4.13) imply ∆ 1i,2k (T 0 ) ≥ ∆ 1i,2k (0) + λ 2 0 T 0 ≥ 0. We now take an infimum over i, k to obtain the desired result.
Remark 4.13. Note that the intra-coupling strengths K α , α = 1, 2, do not play any role in the mixing phase.

4.3.2.
Emergence of bi-cluster flocking. In this part, we finally provide the proof of Corollary 4.4. We set for all t ∈ [T 0 , T ) .
Proof. It follows from Lemma 4.10 that This yields Thus we haveT * 1 > T 0 . For the second estimate of ψ dM (t), we use the same argument as in Lemma 4.8. ) Suppose that the initial data (x 0 , v 0 ) satisfy (F A 1) and the coupling strengths satisfy let (x, v) be the solution to the coupled system (1.3) with initial data (x 0 , v 0 ). Then, there exist positive constants x ∞ α and C α,T0 independent of time t such that sup Proof. Since the proof is almost the same as in Lemma 4.9, we omit it here.
We next provide the proof of Corollary 4.4 by showing thatT * 1 = ∞. Suppose that initial data (x 0 , v 0 ) and the coupling strengths satisfy the framework (F A ).
Procedure A: (The conditions on K α in Corollary 4.4 imply the conditions on K α in Lemma 4.15). Suppose that K α satisfies On the other hand, we use Corollary 2.5, Proposition 4.6, and Remark 4.11 to see that Thus, we have We combine (4.16) and (4.17) to recover the condition in Lemma 4.15: Procedure B: We claim thatT * 1 = ∞. By Lemma 4.14, we haveT * 1 > T 0 . Suppose thatT * 1 < ∞. Then, by the definition ofT * 1 in (4.15), we have By the same arguments in Section 4.2.2, we have: Then, straightforward calculations using Lemma 4.10 and the assumption of K d imply Thus, we prove thatT * 1 = ∞. We now apply Lemma 4.15 withT * 1 = ∞ to obtain the desired flocking estimates. This completes the proof of Corollary 4.4.

5.
Emergence of partial flocking. In this section, we continue to study the dynamics of system (1.3) in a small inter-system coupling. In the previous section, we see that if the intra-subsystem coupling strengths K α are sufficiently large and the inter-subsystem coupling strengths K d is small, then each subsystem evolves to the flocking state so that the total system exhibits a bi-cluster flocking. In this section, we consider the case in which exactly one of the intra-system coupling strengths is sufficiently large, but the other is not. To fix the idea, we assume that K 1 1 and K 2 1.

Statement of the main results.
In this subsection, we briefly discuss the main results for the emergence of partial flocking (see Definition 4.1) for some class of initial configurations under the following situation: In this case, subsystem G 1 flocks, but the other subsystem, G 2 , does not. More precisely, our result is as follows.
Theorem 5.1. Suppose that the following framework (F B ) holds for the initial data (x 0 , v 0 ) to system (1.3).
• (F B 1): (Restriction on initial configurations) • (F B 2): (Restriction on coupling strengths): where positive constants Λ 0 and µ 0 are given by the following relations: Then, the subsystem G 1 and G 2 exhibit a time-asymptotic partial flocking. More precisely, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), there exist x ∞ 1 and C 1 that only depend on the initial data and ψ 1 such that As a corollary of Theorem 5.1, we get rid of the assumptions min 1≤i≤N1 1≤k≤N2 For this, we define Corollary 5.2. Suppose that the following framework (F B ) holds for the initial data (x 0 , v 0 ) to system (1.3).
• (F B 2): (Restriction on coupling strengths): where P 1 (0) and R 1 (0) are defined in (4.2). Then, the subsystems G 1 and G 2 exhibit a time-asymptotic partial flocking. More precisely, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), the following estimates hold: there exist x ∞ 1 > 0 and T 0 such that, for t ∈ [T 0 , ∞), In the following two subsections, we present the proof of the aforementioned two results.

5.2.
Proof of Theorem 5.1. In this subsection, we present a proof of Theorem 5.1. For this, we first remind about a similar result on the SDDI. Proposition 5.3. Suppose that the coupling strengths satisfy and let (x, v) be a global solution to (1.3). Then, for α = 1, 2, we have where ψ dM and ψ 2M are given by the following relations: Proof. It is an analogue of the proof of Proposition 4.6 and Proposition 4.7.
In the following two subsections, we proceed to prove Theorem 5.1 as follows. • Step A (Local-in-time estimate). We will show that, for some finite time T , subsystem G 1 satisfies the flocking estimate, but subsystem G 2 does not: Ct, for some positive constant C. • Step B (Continuation to the whole time interval). We will show that the time T in Step A can be chosen to be infinity.

5.2.1.
Step A (Local-in-time flocking estimates). In this part, we show that the flocking estimates hold at least locally in time. For this, we set

Similarly, we have
x 2k (t) − x 1i (t) ≥ Λ 0 t. Thus, by the non-increasing property of ψ 2 and ψ d , we have the desired estimates.
Lemma 5.6. Suppose that the initial data (x 0 , v 0 ) satisfy (F B 1), then the following estimate holds: Proof. It follows from Proposition 5.4 that, for i ∈ {1, · · · , N 2 }, Lemma 5.7. Suppose that the initial data (x 0 , v 0 ) satisfy (F B 1) and the coupling strengths K 2 and K d satisfy Then, we haveT * 0 = T * 0 . Proof. It follows from Lemma 5.4 that we haveT * 0 > 0. We now assume that On the other hand, by the initial assumption and Lemma 5.6, for t ∈ [0,T * 0 ] and i = k The last inequality is from the assumptions of K 2 and K d . This gives a contradiction to (5.3). Hence we obtainT * 0 = T * 0 . Lemma 5.8. Keep the assumption of Lemma 5.7, we have Proof. The estimates follow directly from the Proposition 5.4.
Lemma 5.9. (Local-in-time flocking estimate) Suppose that the initial data (x 0 , v 0 ) satisfy (F B 1) and the coupling strengths satisfy Then, for the solution (x, v) to system (1.3) with initial data (x 0 , v 0 ), there exist positive constantsx ∞ 1 and C 1 independent of time t such that Proof. (i) (Existence ofx ∞ 1 ): Define a Lyapunov functional L 2 : It follows from Proposition 5.3 and Lemma 5.5 that we have . We integrate the aforementioned relation to obtain In particular, this yields On the other hand, the condition on K 1 implies We setx ∞ 1 to be a positive number satisfying the following relation: Then, by using (5.4) and (5.5), we have (ii) (Decay estimate of V 1 ): It follows from Proposition 5.4 and the result of (i) that we have This yields the desired decay estimate of V 1 . The two remaining estimates are direct results of Lemma 5.5 and Lemma 5.7.

5.2.2.
Step B: T * 0 = ∞. In this part, we complete the proof of Theorem 5.1. Suppose that the initial data and coupling strength satisfy the framework (F B ) in Theorem 5.1. Then, it follows from Lemma 5.4 that we have T * 0 > 0. Suppose that T * 0 < ∞. Then, by the definition of T * 0 in (5.2), we have (v 2k (T * 0 ) − v 1i (T * 0 )) · e 0 2k,1 = Λ 0 . (5.6) It follows from Lemma 5.5-Lemma 5.8 that we have where we have used the initial assumption and the assumptions of K 2 , K d to get the last inequality. This gives a contradiction to (5.6). Thus T * 0 = ∞. Now we apply Lemma 5.9 with T * 0 = ∞ to get the desired estimates and complete the proof of Theorem 5.1.

5.3.
Proof of Corollary 5.2. In this subsection, we provide the proof of Corollary 5.2. As in Section 4.2, we will show that there exists a positive time T 0 such that min 1≤i≤N1 1≤k≤N2 For this, we introduce the following auxiliary functions: and we set Then, we have, for t ∈ [0, , for all i = k.
Proof. (i) It follows from Proposition 5.3 that where we have used the assumption of K d .
(ii) It follows from Proposition 5.3 that Thus, we have where we have used the assumptions of K 2 and K d .
(iii) By (ii), we have Proposition 5.11. Suppose that the initial data (x 0 , v 0 ) satisfy (F B 1) and the coupling strengths K 2 and K d are sufficiently small: to obtain Then, we have where we have used the assumptions of K 2 and K d .