PATTERN FORMATION OF THE CUCKER–SMALE TYPE KINETIC MODELS BASED ON GRADIENT FLOW

. In this paper, we study the pattern formation of the Cucker–Smale type kinetic models. Two distributed Cucker–Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results.


Introduction
Collective behavior is common in many biological systems composed of large numbers of individuals, such as the flocking of birds, the swarming of fish, and the herding of sheep.Many scholars have conducted modeling research on the above phenomena because of their potential applications in robot control, opinion formation of social networks, and other aspects.The Cucker-Smale model [11,12] is one of the well-known models to describe the flocking behavior of birds.The original Cucker-Smale model of  particles is given by: {︃ d d =   ,  = 1, 2, . . ., , Here (  (),   ()) ∈ R  × R  represent the position and velocity of the th particle, 1 ≤  ≤  .In this model, each particle adjusts its velocity by the weighted average of the differences between its velocity and those of the other particles.The original weighted communication kernel  0 : R + → R proposed by Cucker and Smale takes the form  0 () := 1 (1 +  2 )  ,  ≥ 0. (2) The above particle Cucker-Smale model has been extensively studied in the literature from different aspects, for example, collision avoiding [1,10], flocking with hierarchical leadership [22,31], rooted leadership flocking [23,24], bi-cluster flocking [17], discrete form flocking [28,29], infinite-particle [2,33,34] and so on.
The original Cucker-Smale model and its variants mentioned above, mainly focus on some asymptotic flocking behaviors.The terminology "flocking" means that all particles are moving with the same velocity and that the difference in position between particles is uniformly bounded.However, the flocking behavior can not provide a complete description of the spatial configuration, which limits its practical potential for applications such as robot control and pedestrian modeling.This is related to the pattern formation problems, which are very active in the field of multi-agent systems [18,26].In the recent work [9,13], the pattern formation of the particle Cucker-Smale model was studied under the regular and singular interactions, respectively.Nevertheless, the number of particles can be extremely large in real applications, which means that the numerical simulation for the ordinary differential equations dynamics is troublesome, even impossible.A common way to understand the large-scale particle system is to find its kinetic version through the mean-field limit.Unfortunately, no convergence rate results are obtained in [9,13], which implies that the distributed control laws of the above two particle models do not apply to kinetic models.This motivates us to consider the following two questions: -Question 1: What kind of distributed control law can induce the kinetic Cucker-Smale models to generate the pattern formation?-Question 2: Is it possible to establish the convergence rate results of the above pattern formation?
For Question 1, we design two distributed control laws based on gradient flow to enable the Cucker-Smaletype kinetic models to achieve the given spatial configuration with the same velocity.To this end, we fix the desired formation distribution  = (), target velocity ̃︀   , the "pattern" variable  ∈ R  , and take  as a measure with marginal .The first kinetic model is defined as follows: where  1 [] is the distributed control law given by Here  = (, , , ) is the distribution function at a phase point (, , ) ∈ R  × R  × R  at time .The second marginal of () is the desired formation distribution (, ) and (, ) = ().Since there is no differentiation with respect to the variable  in (5), it is a parameter to present the pattern.Pattern formation (see Thm. 2.6 below) of ( 5) and (6) means that the velocity support of  will concentrate on the target velocity ṽ , and the position support will evolve into the target formation distribution .
Next, we relax the distributed control term such that the kinetic model can produce pattern formation only with its own information.More precisely, we consider the following kinetic model:    + • ∇   + div  ( 2 []) = 0, (0, , , ) =  0 (, , ), (7) where  2 [] is the distributed control law given by Pattern formation (see Thm. 2.8 below) of ( 7) and (8) means that the velocity support of  will shrink to its initial average velocity, and the position support will evolve into the desired spatial configuration .The variable  in ( 5) and ( 7) is independent with the position variable  and velocity variable , then for any initial data  0 (, , ) we can rewrite it as  0 (, , ) =  0 (, ) ⊗ ().Moreover, the solutions (, , , ) corresponding to initial data  0 (, ) ⊗ () can also be rewritten as (, , , ) =  (, , ) ⊗ ().For simplicity of notation, we still denote the initial value and solutions by  0 (, , ) and (, , , ) in this paper.
For Question 2, we establish the convergence rate results (see Rem. 3.10 below) of the pattern formation by using gradient flow.To show our method directly, we present the natural relationship between the energy functional and its gradient through the corresponding particle models (see Eq. ( 21) below).And we obtain that the particle systems will exponentially generate the pattern formation.Unfortunately, the rate of convergence may depend on the number of particles.Therefore, we directly address the formation control problem of the kinetic models.A fundamental tool for our work is the notion of characteristic flow.As we will see, the solution () is exactly the push-forward of the initial measure under the characteristic particle flow.For model ( 5) and ( 6), we first use the characteristic flow to obtain the growing speed of the support of solution  (see Lem. 3.6 below).And then the decay rate of the communication kernel is estimated roughly.Furthermore, some differential inequalities are given and it is proved that the derivative of energy functional can be controlled by a part of itself.Finally, the corresponding Grönwall's inequality is obtained for the energy functional, which implies the pattern formation.By using the similar arguments used in the model ( 5) and ( 6), the pattern formation of the model ( 7) and ( 8) is established.
Modifications of the kinetic model ( 5) and ( 6) are similar to bonding forces proposed by Park et al. [27].We focus on pattern formation, whereas they focus on separation and cohesion.Moreover, the gradient flow structure on the vector field is essential in our model, and the origin of the gradient flow and Cucker-Smale model can be traced to [8,21,30].Jeongho Kim studied the equivalence relationship between the measure-valued solution to the first-and second-order kinetic Cucker-Smale equations through gradient flow [21].Choi and Zhang showed that the first-order kinetic Cucker-Smale mode has a gradient flow structure under the singular communication kernel [8].Peszek and Poyato [30] introduce a variant of the kinetic Cucker-Smale model with matrix-valued communication weights, and the uniform-in-time mean-field limit is established based on the fiber gradient flow.In this paper, pattern formation of the Cucker-Smale type kinetic models is obtained by constructing the energy functional through the gradient flow structure on the vector field.
The paper is organized as follows.Section 2 summarizes some preliminary knowledge for transport partial differential equations and presents our main results.In Section 3, we give a rigorous proof of the pattern formation of system (5), (6) and show the natural relationship between the kinetic model and the finite-dimensional particle model in terms of particle flow.In Section 4, we present the pattern formation of model ( 7) and (8) by using similar arguments in the proof of system ( 5) and (6).In Section 5, some numerical examples are provided to show the effectiveness of our results.Section 6 is devoted to concluding this paper.

Preliminaries
In this section, we will provide some preliminary knowledge of transport partial differential equations and summarize our main results.
Remark 2.2.(i) Let  be a Borel measure on R  , and let  : R  → R  be a measurable map.The push-forward measure of  by  is the measure  # defined by  #() = ( −1 ()), for all Borel set  ⊂ R  .
(iii) Recall that supp  (the support of a measure ) is the closure of the set consisting of all points (, , ) in R 3 such that (  (, , )) > 0, ∀ > 0. If the measure  has a compact support, we can use  ∈  1 (R 3 ) as a test function in (10) and (11).
Now, we briefly recall some basic properties of 1-Wasserstein distance.For further details, we refer the reader to the book of Villani [32].Definition 2.3.Let (R  ) be the set of probability measure on R  .For any ,  ∈ (R  ), the 1-Wasserstein distance between  and  is defined by the formula where Π(, ) is the set of probability measure on R  × R  with marginals  and , respectively.More precisely, for any element  of Π(, ), for all integrable (resp.nonnegative) measurable functions ,  on R  , ∫︁ Moreover, the 1-Wasserstein distance  1 is equivalent to the bounded Lipschitz distance (, ): where

Statement of main results
In this section, we will present the pattern formation of the measure-valued solution.To achieve the desired formation distribution, we introduce some reasonable assumptions with the communication kernel  in this paper.

Assumptions
(A1) () ∈  1 (R + ) is non-increasing and bounded: 0 < () ≤ 1; (A2) For any fixed constant  > 0, there exists some constant (A3) For any fixed constant  > 0, (  ) is not integrable at ∞. Remark 2.4.Assumption (A1) is a general condition to study the complex system.Assumptions (A1) and (A2) can guarantee that the function  is global Lipschitz with respect to  and we denote   as its Lipschitz constant.Assumption (A3) indicates that the force brought by long-term interaction is large enough.The classical communication kernel  0 () = 1 (1+ 2 )  does not satisfy the above assumptions, and we provide another communication kernel as below: • The new communication kernel meets all the above assumptions (A1)-(A3), since We stress that the new communication kernel captures the main feature of the classical one, that is the communication influence will reduce to zero when the distance between particles tends to infinity.Now, we can present the pattern formation of the system (5), (6) (kinetic model I) and the system (7), (8) (kinetic model II), respectively.Kinetic model I Theorem 2.5.Under the assumptions (A1)-(A3), if the initial probability measure  0 has compact support in  0 ×  0 ×  0 , there exists a unique global in time measure-valued solution  to the kinetic model I.Moreover, the following two assertions hold (i) The velocity support of  will shrink to the target velocity: (ii) The position support of  will evolve into the target formation distribution (): The above conclusions can be rephrased in terms of the density with respect to the solutions.For this purpose, we introduce a bit of notation: given a measure  ∈ (R 3 ), we define its translate The velocity support of  will concentrate on its initial average velocity: (ii) The position support of  will evolve into the relative formation distribution: | −  − ( − )| 2 (, d, d, d)(, d, d, d) = 0. ( 16)

Kinetic model I
In this section, we provide a rigorous proof of the pattern formation of the solution  to the kinetic model I.We first show our arguments directly through the corresponding particle model (see equality (21) and Thm.3.3 below).Unfortunately, the rate of convergence depends on the number of particles.Therefore, we directly address the formation control problem of the kinetic model I by using the modified method in the proof of particle model.For this purpose, some basic properties of the kinetic model I are given first.Next, we provide an estimate of the growth rate of the compact support of , which is important in the proof of Theorem 2.5.Moreover, the stability results of the solution  and the mean-filed limit are provided.Finally, the pattern formation of the kinetic model I is presented.

Particle model I
In order to show our method directly, we introduce the following particle model I: where  = ( 1 , . . .,   ) ∈ R   is the target formation vector.After doing the following translation we can rewrite (17) as below: Note that the vector field of the system (18) satisfies the Lipschitz condition.By the standard Cauchy-Lipschitz theory, the existence and uniqueness of the solutions of system ( 18) can be verified.Let (x(), v()) be the corresponding solution of system ( 18) and set Then the system (18) can be reformulated as a gradient-like system of the form: Now, we define the energy functional: The relationship between  (x) and ∇   (x) is important: Moreover, we can obtain that  1 (x, v) is nonincreasing along the trajectory of the system (19).
Lemma 3.1.The energy functional is nonincreasing with respect to time  and Furthermore, there exists some constant  > 0 such that Proof.Combining Furthermore, there exists some constant  > 0 such that max Now, we introduce a modified version of the energy functional: where  > 0 will be defined later.Then, by the Cauchy inequality The above two inequalities mean that ) be the solution to system (18), then we have Proof.We first note that d d Here we used the Cauchy inequality and equality (21).Combining the above results we obtain Now, we can provide the pattern formation of the particle model ( 18).
Theorem 3.3.Let (x(), v()) be the solution to system (18).Then, all particles will achieve the desired spatial configuration and move with the target velocity ṽ .That is, for all 1 ≤  ≤  , Proof.When  ≤ 1 2 , we only need to prove that lim Combining (23) with the nonincreasing property of function , there exists some constant  > 0 such that )︃ .
Then, equation (26) gives By Grönwall's inequality Combining the above inequality with ( 27), the proof is completed.
Remark 3.4.(i) The assumptions (A2) and (A3) are not necessary for the pattern formation of the particle model I, here the communicate kernel  only needs to be assumed that is Lipschitz continuous, and satisfy the assumption (A1).
(ii) The gradient flow structure is crucial in the above proof.The third term of the energy functional (25) controls the energy of the position variable, which can be observed from the equality (21).
With the above preparations, the following estimates are provided.
)) be the measure-valued solution to the kinetic model I that has compact support.Then for any  ≥ 0 we have: where Proof.Note that Then we have Here we used the change of variable (x, , v) ↔ (ȳ, , ω) and Fubini's theorem.Combining the above two equalities, equation ( 31) is obtained.Now, we estimate the growing speed of compact support and its characteristics.
Proof.According to Lemma 3.5, there exists some constant  > 0 depending only on  0 such that Then where  > 0 depends only on  0 .Therefore, the following estimates hold By Grönwall's inequality, there exists some positive constant  > 0 depending only on  0 such that Then, combining Remark 2.2 with the definition of the characteristic system (30), the proof is completed.
Next, we show the stability results and explain that the kinetic model I is the mean-field limit of the particle model I. Lemma 3.7.Assume that the initial probability densities  0 ,  0 satisfy For any fixed  > 0, there exists some positive constant   > 0 depending only on   ,  ,  0 such that where ,  are solutions of system (5) and (6) with initial data  0 ,  0 , respectively.
Proof.Combining Lemma 3.6 with supp  0 ⋃︀ supp  0 ⊂  0 ×  0 ×  0 , we deduce that there exists some constant   > 0 depending only on  0 and  such that supp () Next, we define the forward characteristic flows of  and  with the initial data (, , ), (, , ) ∈ R 3 as follows: The forward characteristic flows of  and  are defined by ((), ,  ()) and ( (), ,  ()).We are devoted to proving that for any where the constant   > 0 depends only upon   ,  and  0 .
From the definition of the characteristic flows and (12) we get and We complete the proof with the arbitrariness of  0 .
The following corollary is presented.
Proof.We first recall the definition of measure-valued solutions of system ( 5) and ( 6).For any  ∈  with Combining the arbitrariness of the test function  with the Lemma 3.7 (consider functions  localized around any given particle (  (),   ,   ())), the proof is completed.
Then, combining the fact 0 <  ≤ 1 with the Cauchy inequality |ω|(, d, d, d) |ω|(, d, d, d) Here we used the fact that |ω|(, d, d, d) and the Cauchy inequality For simplicity of notation, we denote Then, we can reformulate the energy functional as below: With the above preparations, we give a rigorous proof of the pattern formation of the kinetic model I below.
Proof of Theorem 2.5.We split the proof into two parts.
Clearly, a fixed point of the map Γ is a solution of the kinetic model I in [0,  ].Note that the function  is globally Lipschitz, which implies that the nonlinear velocity field  1 [] is locally Lipschitz with respect to (, , ) and grows linearly.Combining the theory developed in [3] with the above facts, we can obtain the local existence and uniqueness of the measure-valued solution of the kinetic model I by Banach fixed point theorem; see Theorem 3.10 of [3] for details.Finally, Lemma 3.6 yields that we can extend this local solution by standard argument.Since in our case, the growth of characteristics is bounded.
Pattern formation.Now, we are devoted to proving the pattern formation of the kinetic model I. From Lemma 3.6, there exists some positive constant  1 > 0 such that sup Now, the constant  1 in the energy functional (39) is fixed.Then, from Lemma 3.9 we get Furthermore, combining Lemma 3.6 with the assumption (A2), there exists some constant  1 > 0 such that By Grönwall's inequality Since for the fixed constant  1 > 0, ( 1  1 ) is not integrable at ∞. Finally, we have The proof is completed.
Remark 3.10.If we choose the communicate rate kernel as , the convergence rate of the above pattern formation will be )︂ .
Remark 3.11.Combining Corollary 3.8 with Lemma 3.7, we can use the stability result and Dorbrushin's mean-field limit argument to prove the wellposedness results.For Dorbrushin's mean-field limit argument, we first construct the empirical measure sequence   0 = 1  ∑︀  =1  ((0),,(0)) to approximate the initial measure  0 (, , ).Then we know the corresponding measure-valued solution sequence   () = 1  ∑︀  =1  ((),,()) is a Cauchy sequence in  1 through the stability results.Moreover, the solution sequence   () converges to some limit (, , , ), and we can verify (, , , ) is the measure-valued solution through the test function.The uniqueness of the solutions follows from the stability results.
Remark 3.12.The mean-field approximation can be used to prove the pattern formation.First, using the similar argument in Lemma 3.6, we estimate the growing speed of the  -particle flow and obtain the bounded results depending only on the initial diameter of the velocity , position , and pattern .Then we use ( 1  1 ) instead of  0 in energy  2 .Of course, we need to rescale the energy  2 by dividing the particle number  (Ensure energy  2 is the discrete form of energy  1 ).Moreover, the energy  1 is continuous with respect to convergence in  1 , so inequality (45) follows for general measures by approximation.Finally, we can pass the limit and obtain the pattern formation.Now, we show that the measure-valued solution of the kinetic model I will converge to the target formation distribution under bounded Lipschitz distance.

Kinetic model II
In this section, we are devoted to presenting the pattern formation of the kinetic model II.The proof arguments are roughly similar to the kinetic model I.One difference is that the average velocity of kinetic model II is conservative.

A prior estimates
To establish the pattern formation of the solution to the kinetic model II, we first provide the following lemma.
For simplicity, we denote || 2 (, d, d, d); || 2 (, d, d, d); Then, we can derive the following inequality from Lemma 4.1: Moreover In fact, equation ( 56) can be obtained by the following differential inequality: With the above preparations, we can estimate the characteristics and the growing speed of compact support of the solution.
Then, equations (57) and (58) give where  depends only on  0 .Furthermore, the following estimates hold for the characteristic flow: where  depends only on  0 .Therefore, By Grönwall's inequality, there exists some positive constant  > 0 depending only on  0 such that Then, by Remark 2.2 the proof is completed.

Remarks on the finite-dimensional Cucker-Smale model
In this section, we first provide the stability result of the solution of system ( 7) and (8).Then, we show that the kinetic equations ( 7) and ( 8) is the mean-field limit of the finite-dimensional Cucker-Smale model.The method is similar to what we used in Lemma 3.7, we omit it here.Lemma 4.3.Assume that the initial probability densities  0 ,  0 satisfy Then for any fixed  > 0, there exists some positive constant   > 0 depending only on   ,  ,  0 such that where ,  are solutions to model (7) and (8) with initial data  0 ,  0 , respectively.Now, we present the relationship between the finite-dimensional Cucker-Smale model with the kinetic model II.For this purpose, we introduce the following dynamics: which is a simplified version of the model studied in [13].For  particles moving in R  , we design the target formation vector  = ( 1 , . . .,   ) ∈ R   .It was shown in [13] that all particles will achieve the relative target position asymptotically: However, the convergence rate result of the above pattern formation has not been obtained.Using the method in this paper, we can establish the convergence rate result of the pattern formation of the above particle model.In fact, the target formation vector  = ( 1 , . . .,   ) ∈ R   can be viewed as the discretization of the target formation distribution.Considering the agents as Dirac masses, system (60) can be viewed as the discretization of ( 7) and (8).We complete the proof with the arbitrariness of the test function  and the Lemma 4.3.
Furthermore, the following corollary can be derived from the Theorem 2.8.
Corollary 4.5.Fix the formation vector  = ( 1 , . . .,   ) ∈ R   .For any given initial data {( 0 ,  0 )}  =1 , the solutions of system (60) will achieve the relative target position asymptotically: The convergence rate of the above pattern formation is exponential.Let (x(), v()) be the corresponding solution to system (60) and set Then we have With the above facts, we can establish the exponential convergence rate of pattern formation of system (60) by gradually using similar arguments in the system (17).

Pattern formation of the kinetic model II
From the microscopic point of view, pattern formation of model ( 7) and ( 8) means that all particles reach the relative target position with the initial average velocity over a long time.We use  2 () and ̃︀  2 () to describe this phenomenon.Similarly, there exists a corresponding energy functional for model (7) and ( 8): where  > 0 and  2 > 0 will be chosen later.The following lemma is provided to estimate the derivative of the third term of the energy functional with respect to time .As we showed in Lemma 4.2, the forward characteristic flow is bounded.Therefore, we can deduce that there exists some continuous monotone increasing integrable function  1 () such that (() −  () − ( − )) We used the Lebesgue dominated theorem in the second equality with  1 being the dominating function and || 2 (, d, d, d).
We first use the Cauchy inequality to derive the second inequality.Then, following from the fact that 0 <  ≤ 1 and Hölder's inequality, the third inequality is proved.Estimate of ℐ 2 .Similarly, we have Combining the above estimates, the proof is completed.Then, we can prove Theorem 2.8 as below: Proof of Theorem 2.8.As we stated in the proof of Theorem 2.5, combining the Lemma 4.2 with the method used in [3], we can get the wellposedness results of model ( 7) and ( 8).Now, we are devoted to demonstrating the pattern formation of system ( 7) and (8).We first recall that By Cauchy inequality With the help of (68), the proof is completed.

Simulations
In this section, some numerical simulations are presented to illustrate our theoretical results, Theorems 2.5, 2.6, and 2.8.Throughout this section, we choose the communicate kernel as () = 1 1+ln(1+ 2 ) .Model I. We consider a "Square evolved into a circle" pattern (see Fig. 1) in the 2-dimensional space.The initial position distribution is chosen as uniform distribution on a square and the target position distribution is set to the uniform distribution on a circle (see Fig. 2) }︂ .
Figure 1 performs random uniform  -particle approximation ( = 450) for the solutions of the kinetic model I.We observe that position support of the solution will evolve into formation distribution.Moreover, Figure 3 shows that the velocity support will concentrate on target velocity ṽ = 0.As the number of particles increases, the target pattern becomes more intuitive, as shown in Figure 4.When particle number is finite, the convergence rate of pattern formation is exponential.We confirm it from the numerical results, as shown in Figure 5. Furthermore, when the number of particles increases, it can be seen that the convergence rate is constantly slowing down, as shown in Figure 6.
Model II.We provide a "Circle evolved into five rings" pattern (see Fig. 7) in the 2-dimensional space.The initial position distribution is chosen as uniform distribution on a circle and we approximate it with random uniform  -particle approximation ( = 300).The target pattern is chosen as the five rings (see Fig. 8).We can see that the position support of the solution will evolve into the target pattern, and the velocity support of solutions will concentrate on the initial average velocity   = 0 (see Fig. 9).The convergence rate of pattern formation also presented in Figure 10.The key difference between model II and model I is the conservation of the average velocity.
It can be noted that even if the initial velocity support of the solutions is concentrated at a single point, model I and model II still need a long time to reach the target pattern, as shown in Figures 3 and 9.

Conclusion
In this paper, we studied the pattern formation in two kinetic models of the Cucker-Smale type, i.e., the velocity support of the solution will concentrate on one single point, and the position support of the solution will evolve into the target formation distribution.For kinetic mode I, we first gave some estimates to demonstrate the basic properties of the kinetic model.Then, we provided the growing speed of the support of the solutions.Furthermore, we presented the stability results and showed the relationship between the corresponding finitedimensional Cucker-Smale model and the kinetic model I. Finally, based on gradient flow, we provided a rigorous proof of the asymptotic pattern formation.In particular, we obtained the convergence rate of the   So far, nearly all the pattern formation analyses for the Cucker-Smale model are focused on the particle version, while in this paper we contribute a theoretical analysis for the kinetic version.The difficulty mainly arises from the loss of the nonexpansive property of velocity support, which prevents us from using the typical cyclic iteration method proposed in [5].Our approach is based on the gradient flow structure on the velocity field, and the new communication kernel.
We guess pattern formation will appear for the kinetic Cucker-Smale model with the original communication kernel.This will be left for our future work.

Figure 5 .
Figure 5. Convergence rate of pattern formation for model I,  = 450.

Figure 10 .
Figure 10.Convergence rate of pattern formation for model II,  = 300.