Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels

  • Received: 01 March 2022 Revised: 01 May 2022 Published: 16 June 2022
  • 34D23, 34E99, 82C22

  • This paper presents several sufficient frameworks for a collision avoidance and flocking dynamics of the Cucker–Smale (CS) model and thermodynamic CS (TCS) model with arbitrary dimensions and singular interaction kernels. In general, unlike regular kernels, singular kernels usually interfere with the global well-posedness of the targeted models from the perspective of the standard Cauchy–Lipschitz theory due to the possibility of a finite-in-time blow-up. Therefore, according to the intensity of the singularity of a kernel (strong or weak), we provide a detailed framework for the global well-posedness and emergent dynamics for each case. Finally, we provide an admissible set in terms of system parameters and initial data for the uniform stability of the d-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the d-dimensional CS with singular kernel if all agents have the same initial temperature.

    Citation: Hyunjin Ahn. Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels[J]. Networks and Heterogeneous Media, 2022, 17(5): 753-782. doi: 10.3934/nhm.2022025

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  • This paper presents several sufficient frameworks for a collision avoidance and flocking dynamics of the Cucker–Smale (CS) model and thermodynamic CS (TCS) model with arbitrary dimensions and singular interaction kernels. In general, unlike regular kernels, singular kernels usually interfere with the global well-posedness of the targeted models from the perspective of the standard Cauchy–Lipschitz theory due to the possibility of a finite-in-time blow-up. Therefore, according to the intensity of the singularity of a kernel (strong or weak), we provide a detailed framework for the global well-posedness and emergent dynamics for each case. Finally, we provide an admissible set in terms of system parameters and initial data for the uniform stability of the d-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the d-dimensional CS with singular kernel if all agents have the same initial temperature.



    The emergent dynamics of interacting many-body systems are often observed in complex ecosystems. Examples include the synchronization of fireflies and pacemaker cells [7,28,55], aggregation of bacteria [51], flocking of birds [26], and swarming of fish [27,50]. To briefly introduce them, we refer to [1,6,17,29,43,47,49,53,54]. We are interested in flocking dynamics in which each particle converges to a common velocity with an ordered formation by using limited information and simple laws. After the groundbreaking work [52] on the flocking model of birds proposed by Viscek et al., many mathematical models describing collective behavior have been widely investigated in the mathemathical community. Since [26], many mathematicians and physicists have been concerned with the Cucker–Smale (CS) type models derived from a Newtonian-like second-order model for position-velocity, governed by the following system in terms of (xi,vi):

    {dxidt=vi,t>0,i[N]:={1,,N},dvidt=κNjiψ(xixj)(vjvi),(xi(0),vi(0))=(x0i,v0i)Rd×Rd, (1)

    where N is the number of particles and κ is a strictly positive coupling strength. Many papers have studied on the CS model and its variants. This research comprises the following topics: the mean-field limit [4,5,20,36,38], kinetic model [10,40], hydrodynamic descriptions [30,32,41], particle analysis [10,17], time-delay effect [15,19], stochastic description [11], bi-cluster flocking [21,22], relativistic setting [3,5,8,35], unit-speed constraint [13,14,21,37,48], and collision avoidance [9,18,20,23,24,25,42,44,45,46].

    However, the above literature has only addressed the CS model without the temperature field. Therefore, the authors in [39] generalized the CS model to consider the temperature settings from the system of gas mixtures with rational reductions, called the thermodynamic CS (TCS) model. Afterward, in a follow-up paper [34], the authors derived an approximated TCS model by assuming that the diffusion velocities are sufficiently small, which is given by the following second-order system for position-velocity-temperature, (xi,vi,Ti):

    {dxidt=vi,t>0,i[N],dvidt=κ1Njiϕ(xixj)(vjTjviTi),dTidt=κ2Njiζ(xixj)(1Ti1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)R2d×R>0, (2)

    where N is the number of particles and κ1 and κ2 are strictly positive coupling strengths. For a brief introduction to the TCS type models, we refer to papers which are derivation of the TCS model [34,39], asymptotic behavior [34], uniform stability and uniform-in-time mean-field limit [33], hydrodynamic description [31], time-delay effect [12], unit-speed constraint [2] and collision avoidance [16].

    Throughout the paper, we choose the simplest singular communication weights (or singular interaction kernels) ψ, ϕ and ζ to predict collision avoidance between each pair of particles in (1) and (2):

    ψ(r):=1rα,ϕ(r):=1rβ,ζ(r):=1rγ,α,β,γ>0.

    We are only interested in the singularity when r=0; thus, it is not important to determine the explicit structures of the singular weights ϕ and ζ. Indeed, we aim to observe the flocking behavior of (1) and (2) when each of the communication weights has no regularity at zero. We must first consider the global well-posedness (i.e., the noncollisional phenomenon) problem of the targeted models because the singular kernels are not well defined at r=0. In addition to this motivation, collision avoidance between each pair of particles is an important issue in mechanical engineering and motion control engineering, to name a few for UAV (uncrewed aerial vehicle), drones and ACAS (airborne collision avoidance system), to name a few. Therefore, studying sufficient frameworks for collision avoidance in interacting many-body systems is of great significance. For more advanced works in this paper, we present several frameworks that differ from those in previous papers [9,16,18,20,23,24,25,42,44,45,46] related to collision avoidance, which can be summarized in the next paragraph. The main novelties of this paper are presented below. First of all, we prove the global well-posedness (i.e., collision avoidance) of the systems (1) and (2) in terms of quantities for L-diameters, independent of the number of particles N. To do this, we employ useful functionals to derive several dissipative structures with singular kernels. Second, we present the emergent dynamics in terms of L-diameters under sufficient frameworks in terms of the initial data and system parameters in (1) and (2). The flocking estimates are independent of N; therefore, it is natural to consider the uniform stability estimates of (1) and (2), which yield uniform-in-time mean-field limits from (1) and (2) to the corresponding Vlasov equation by taking N, respectively. Third, we rigorously demonstrate the uniform L2-stability result of (2) under the admissible initial data and system parameters, which can be trivially reduced to the uniform L2-stability result of (1) by removing (2)3 from Remark 1. Furthermore, we can derive the uniform stability estimate of (2) with a much simpler argument than the literature [5,33,36]. In summary, the ultimate goal of this paper is to extend the uniform stability independent of N of (1) on one-dimensional Euclidean space R1 studied in [20] to the CS and TCS models on Rd with arbitrary dimensions.

    This paper is organized as follows. In Section 2, we briefly revisit facts regarding the temperature field (2)3 and provide basic estimates for the global well-posedness of (1) and (2). In Section 3, we study several sufficient frameworks for collision avoidance, global well-posedness, and the emergent dynamics of (1) under strongly or weakly singular interaction kernels, respectively. In Section 4, we also present the global well-posedness of (2) on admissible data in terms of the initial data and system parameters by dividing the singularity of each communication weight in the targeted models into a weak case and strong case. However, unlike the case of (1) in Section 3, we provide sufficient frameworks for the emergent dynamics regardless of the intensity of the singularity. In Section 5, we provide a detailed proof for the uniform L2-stability of (2) under appropriate admissible data using the results from Section 3 and Section 4. Finally, Section 6 is devoted to summarizing the main results and discussing the remaining issues to be investigated in future work.

    Notation. Throughout the paper, we employ the following notation and abbreviations:

    :=l2-norm,,:=standard inner product,(T)CS with singular kernel: = (T)CSS,X:=(x1,,xN),V:=(v1,,vN),T:=(T1,,TN),[N]:={1,,N},DZ:=maxi,j[N]zizj,dX:=minij,i,j[N]xixjforZ=(z1,,zN){X,V,T}.

    In this section, we provide basic materials to guarantee the global well-posedness of the CSS model (1) and TCSS model (2). For this, we revisit previous results for the temperature field (2)3 in Section 2.1 to be used throughout the paper. In Section 2.2, we provide the uniform boundedness of speed for each particle in (1) and (2).

    In this subsection, we briefly provide facts regarding the temperature system (2)3 for global well-posedness. In the literature [39], the authors proved that the total temperature sum is conserved and that the entropy principle holds, stated as follows.

    Definition 2.1. [34,39] Let τ(0,] and (X,V,T) be a solution to the singular system (2) for t(0,τ). Then, the total entropy is defined as

    S:=Ni=1ln(Ti).

    We now present the previous results on the conservation of the temperature sum and the monotonicity of the total entropy as follows.

    Proposition 1. [34,39] For a fixed τ(0,], suppose that (X,V,T) is a solution to the singular system (2) for t(0,τ). Then, the following assertions hold.

    1. (Conserved temperature sum) The total sum Ni=1Ti is conserved:

    Ni=1Ti(t)=Ni=1T0i:=NT,t[0,τ).

    2. (Entropy principle) The total entropy S is monotonically increasing:

    dS(t)dt=12NNi,j=1ζ(xjxi)|1Ti1Tj|20,t[0,τ).

    Due to the entropy principle and the simple structure of (2)3, the authors in [12,34] proved that the temperature for each particle of (2) on t[0,τ) is uniformly bounded.

    Proposition 2. [12,34] (Monotonocity of max-min temperatures) Let τ(0,]. Assume that (X,V,T) is a solution to the singular system (2) for t(0,τ). Then, min1iNTi(t) is monotonically increasing and max1iNTi(t) is monotonically decreasing in t[0,τ). Hence, we have the uniform boundedness of temperature as below.

    0<mini[N]T0i:=TmTi(t)maxi[N]T0i:=TM,i[N],t[0,τ).

    Remark 1. By the standard Cauchy–Lipschitz theory, the TCSS model (2) for [0,τ) can be reduced to the CSS model (1) for [0,τ) if the initial temperature data for (2) have the same positive constant, T0>0, that is,

    T01=T0N=T0>0.

    In this subsection, we derive that the maximum speed is uniformly bounded by physical constraints in terms of the initial data in (1) to verify the global well-posedness. More concretely, we show that the maximum speed is monotonically decreasing in (1).

    Lemma 2.2. Let τ(0,] and (X,V) be a solution to the singular system (1) for t(0,τ). Then, it follows that

    maxi[N]vimaxi[N]v0i,t[0,τ).

    Proof. We choose an index Mt[N] dependent on time t[0,τ) such that

    vMt:=maxi[N]vi(t).

    Then, we take the inner product vMt with ˙vMt in (1)2 to obtain the following for a.e. t(0,τ),

    12dvMt2dt=κNNj=1ψ(xMtxj)vjvMt,vMt0.

    Hence, we obtain

    12dvMt2dt0,a.e.t(0,τ)vMtvM0,t[0,τ),

    implying the desired result.

    maxi[N]vivM0maxi[N]v0i,t[0,τ).

    Therefore, if we prove that (1) has a noncollisional phenomenon at any time, then we have global well-posedness with Lemma 2.2 and the Cauchy–Lipschitz theory. For detailed descriptions, we refer to Section 3. In the case of the system (2), immediately determining information about the maximum speed is challenging, so we use another method to guarantee the uniformly boundedness of the maximum speed in Section 4.

    In this section, we establish sufficient frameworks in terms of the initial data and system parameters for the global well-posedness, collision avoidance, and emergent dynamics of (1), by dividing them into two cases: α1 and 0<α<1, according to the singularity of ψ. To achieve this, we will employ useful functionals to derive the dissipative differential inequalities for position-velocity diameters DX and DV.

    In this subsection, we study the global well-posedness of the CSS model (1) when ψ is a strongly singular kernel.

    ψ(r)=1rα,r>0,α1.

    Next, we rigorously verify the global well-posedness of (1) under a strongly singular interaction kernel. It suffices to demonstrate the noncollisional state between each pair of particles on any finite time. We assume that t0 is the first collision time of the singular system (1), and [l] denotes the set of all particles that collide with the l-th particle at time t0, i.e.,

    [l]:={i[N]|xl(t)xi(t)0astt0}.

    Let δ be a strictly positive real number satisfying

    xl(t)xi(t)δ>0,t[0,t0)andi[l].

    Thus, we define the following L-diameters in terms of position-velocity from the perspective of [l] for t[0,t0):

    dX,[l]:=mini,j[l],ijxixj,DX,[l]:=maxi,j[l]xixj,DV,[l]:=maxi,j[l]vivj.

    For simplicity, we use the following notation:

    ψij:=ψ(xixj),i,j[N],ij,andψij,[l]:=ψ(xixj),i,j[l],ij,

    where |[l]| is a cardinal number of the set [l].

    Next, we employ the following functional Ψij,[l] for (i,j)[l]2:

    Ψij,[l](t):=ψ(xixj)|[l]|fori,j[l],ij,Ψii,[l](t):=ψ(dX,[l])ji,j[l]ψ(xixj)|[l]|,fori[l].

    Then, we observe that Ψij satisfies the following three properties:

    1. Ψij,[l]ψ(DX,[l])|[l]|,i,j[l], ij,

    2. j[l]Ψij,[l]=ψ(dX,[l]),i[l],

    3. j[l]Ψij,[l](vjvi)=ji,j[l]ψij|[l]|(vjvi),i[l].

    Theorem 3.1. Suppose that (X,V) is a solution to (1) with a strongly singular kernel and noncollisional position initial data, that is,

    α1andmini,j[N],ijx0ix0j>0.

    Then, we can obtain the global well-posedness of (1), or, equivalently, we have the global-in-time collisionless state:

    xi(t)xj(t),(i,j)[N]2,ijandt[0,).

    Proof. First, we use the following relation

    |dxixj2dt|=2|xixj,vivj|2xixjvivj

    with the Cauchy–Schwarz inequality to have that for a.e. t(0,t0),

    |dDX,[l](t)dt|DV,[l](t). (3)

    Now, we take two indices it,jt[l] dependent on time t(0,t0) such that

    DV,[l](t):=vit(t)vjt(t),it,jt[l].

    Then, it follows from (1)2 that for a.e. t(0,t0),

    12ddtvitvjt2=vitvjt,dvitdtdvjtdt=vitvjt,κNNk=1ψitk(vkvit)κNNk=1ψjtk(vkvjt)=vitvjt,κNk[l]ψitk(vkvit)κNk[l]ψjtk(vkvjt)+vitvjt,κNk[l]ψitk(vkvit)κNk[l]ψjtk(vkvjt)=:I1+I2.

    (Estimate of I1): If we apply Lemma 2.2 and the Cauchy–Schwarz inequality to I1, then there exists a nonnegative constant C(κ,[l],N,V(0),δ) satisfying

    I1=vitvjt,κNk[l]ψitk(vkvit)κNk[l]ψjtk(vkvjt)DV,[l]κNk[l]ψitk(vkvit)κNk[l]ψjtk(vkvjt)DV,[l](κNk[l]ψitk(vkvit)+κNk[l]ψjtk(vkvjt))4κ(N|[l]|)ψ(δ)maxi[N]v0iNDV,[l]=:C(κ,[l],N,V(0),δ)DV,[l], (4)

    where we used the definition of δ and monotonocity of ψ.

    (The estimate of I2): For this, we employ the properties of Ψ to get that

    I2=vitvjt,κNk[l]ψitk(vkvit)κNk[l]ψjtk(vkvjt)=vitvjt,κ|[l]|Nk[l]Ψitk(vkvit)κ|[l]|Nk[l]Ψjtk(vkvjt)=κ|[l]|Nψ(dX,[l])vitvjt,vitvjt+κ|[l]|Nvitvjt,k[l](ΨitkΨjtk)vk=κ|[l]|Nψ(dX,[l])vitvjt,vitvjt+κ|[l]|Nvitvjt,k[l](Ψitkmin(Ψitk,Ψjtk)+min(Ψitk,Ψjtk)Ψjtk)vk. (5)

    Here, since

    vitvjt,vjtvitvjt,vkvitvjt,vit,

    we can show that

    I2κ|[l]|Nψ(dX,[l])vitvjt,vitvjt+κ|[l]|Nvitvjt,k[l](Ψitkmin(Ψitk,Ψjtk))vit+κ|[l]|Nvitvjt,k[l](min(Ψitk,Ψjtk)Ψjtk)vjtκ|[l]|Nψ(dX,[l])vitvjt,vitvjt+κ|[l]|Nψ(dX,[l])vitvjt,vitvjtκ|[l]|Nk[l]min(Ψitk,Ψjtk)vitvjt,vitvjt=κ|[l]|Nk[l]min(Ψitk,Ψjtk)vitvjt,vitvjtκ|[l]|Nψ(DX,[l])vitvjt,vitvjt=κ|[l]|Nψ(DX,[l])D2V,[l], (6)

    where we used the first property of Ψ. Hence, we combine (4) with (6) to obtain for a.e. t(0,t0),

    dDV,[l]dtκ|[l]|Nψ(DX,[l])DV,[l]+C(κ,[l],N,V(0),δ).

    We then integrate to both sides of the above inequality from s to t for 0st<t0 to yield

    tsψ(DX,[l])DV,[l]duN(DV,[l](s)+C(κ,[l],N,V(0),δ)(ts))κ|[l]|. (7)

    Moreover, let Φ be a primitive of a strongly singular kernel ψ with α1. Then, for fixed t1>0,

    Φ(t):=Φ(t;t1):=tt1ψ(u)du={logtt1,ifα=1,11α(t1αt11α),ifα>1. (8)

    Therefore, it follows from (7) that for 0st<t0,

    |Φ(DX,[l](t))||tsdduΦ(DX,[l](u))du|+|Φ(DX,[l](s))|=|tsψ(DX,[l](u))(dduDX,[l](u))du|+|Φ(DX,[l](s))|tsψ(DX,[l](u))|dduDX,[l](u)|du+|Φ(DX,[l](s))|tsψ(DX,[l](u))DV,[l](u)du+|Φ(DX,[l](s))|,

    which implies from (7) that

    |Φ(DX,[l](t))||Φ(DX,[l](s))|+N(DV,[l](s)+C(κ,[l],N,V(0),δ)(ts))κ|[l]|,0st<t0.

    Now, we take s=0 and tt0 for the above inequality to obtain

    limtt0|Φ(DX,[l](t))||Φ(DX,[l](0))|+N(DV,[l](0)+C(κ,[l],N,V(0),δ)t0)κ|[l]|<,

    which yields a contradiction to the definition of t0 and (8) because

    limtt0|Φ(DX,[l](t))|=.

    Consequently, we have shown the noncollisional phenomenon of (1), i.e.,

    xi(t)xj(t),(i,j)[N]2andt[0,).

    Finally, one combines this with Lemma 2.2 to have the desired global well-posedness of (1) under the strongly singular kernel from the standard Cauchy–Lipschitz theory.

    Remark 2. Note that Φ(t) with 0<α<1 in (8) is always finite if

    0t1t<.

    Therefore, we may take a different strategy from the proof of Theorem 3.1 to guarantee the global well-posedness of (1) under a weakly singular kernel (0<α<1).

    In this subsubsection, from the proof of Theorem 3.1, we study the sufficient frameworks for the emergent dynamics of (1), assuming α1. Before we continue, we revisit the definition of the emergent dynamics of (1).

    Definition 3.2. Let Z:=(X,V) be a global-in-time solution to the singular system (1). The configuration Z exhibits asymptotic flocking if

    (i)(Group formation)sup0t<maxi,j[N]xi(t)xj(t)<,(ii)(Velocity alignment)limtmaxi,j[N]vj(t)vi(t)=0.

    We present sufficient frameworks for the asymptotic flocking of (1) with a strongly singular kernel using two approaches, the bootstrapping argument (or continuous argument) and the Lyapunov functional.

    Theorem 3.3. (Asymptotic flocking via bootstrapping) Let (X,V) be a global-in-time solution to (1) with

    α1andmini,j[N],ijx0ix0j>0.

    Further assume that there exists a positive constant DX satisfying

    DX(0)+DV(0)κψ(DX)DX. (9)

    Then, we have the following asymptotic flocking result:

    1. (Group formation) DX(t)<DX,

    2. (Velocity alignment) DV(t)DV(0)exp(κψ(DX)t).

    Proof. If DV(0)=0, then we have nothing to prove. Therefore, we can assume DV(0)>0. We observe from the condition (9) that the set

    S:={t>0|DX(s)<DX,s(0,t)}

    is nonempty. We define t:=supS>0. Hence,

    DX(t)=DX.

    Now, we claim that

    t=+.

    To prove the claim, we suppose that t<+ for the proof by contradiction. Then, by replacing [l] with [N] in the proof of Theorem 3.1, we use the arguments of Theorem 3.1 to establish the following:

    |dDXdt|DV,dDVdtκψ(DX)DVκψ(DX)DV,a.e.t(0,t),

    where we used the definition of S. Then, Grönwall's lemma implies that

    DV(t)DV(0)exp(κψ(DX)t),t[0,t].

    Moreover, because

    DX(t)=DX(0)+t0dDX(s)dsdsDX(0)+t0DV(s)dsDX(0)+t0DV(0)exp(κψ(DX)s)ds<DX(0)+DV(0)κψ(DX)DX,t[0,t],

    one can show that DX(t)<DX, resulting in a contradiction, i.e., t=. Thus, one has

    DX(t)<DX,DV(t)DV(0)exp(κψ(DX)t),t[0,).

    Therefore, we achieve the desired results.

    Next, we provide a different approach to achieve the asymptotic flocking of (1) via the Lyapunov functional method introduced in [38].

    Theorem 3.4. (Asymptotic flocking via the Lyapunov functional) Let (X,V) be a global-in-time solution to (1) with

    α1,andmini,j[N],ijx0ix0j>0,

    and further assume that

    DV(0)κDX(0)ψ(s)ds. (10)

    Then, we obtain the following asymptotic flocking estimate: there exists a strictly positive number DX>0 such that

    1. (Group formation) DX(t)DX,

    2. (Velocity alignment) DV(t)DV(0)exp(κψ(DX)t), t[0,).

    Proof. First, we use the same arguments employed in Theorem 3.1 and Theorem 3.3 to deduce that

    |dDXdt|DV,dDVdtκψ(DX)DV. (11)

    Now, we consider the following Lyapunov functional:

    L±(DX,DV):=DV±κΨ(DX),

    where Ψ(t):=t0ψ(s)ds. Then, we apply (11) to obtain

    ddtL±(DX,DV)=dDVdt±κdDXdtψ(DX)κψ(DX)DV±κdDXdtψ(DX)=κψ(DX)(DV±dDXdt)0. (12)

    Next, we utilize (12) to induce L±(DX(t),DV(t))L±(DX(0),DV(0)) and moreover,

    κ|DX(t)DX(0)ψ(s)ds|DV(t)+κ|DX(t)DX(0)ψ(s)ds|DV(0).

    Therefore, we combine this with the condition (10) to yield

    κ|DX(t)DX(0)ψ(s)ds|DV(0)κDX(0)ψ(s)ds.

    Hence, there is a smallest positive real number DX satisfying

    DV(0)=κDXDX(0)ψ(s)ds;

    thus

    DX(t)DX,t[0,).

    Due to the above estimate and (11), from Grönwall's lemma, it follows that

    DV(t)DV(0)exp(κψ(DX)t),t[0,).

    Finally, we prove the desired asymptotic flocking results.

    Remark 3. If α>1, then it follows from a direct calculation that

    DX(0)ψ(s)ds<.

    Therefore, we know that the assumption (10) can not be rejected. However, we can remove (10) in the case of 0<α1 because

    DX(0)ψ(s)ds=.

    Ultimately, we combine the sufficient frameworks of Theorem 3.3 with Theorem 3.4 to conclude the following.

    Corollary 1. (Refined asymptotic flocking) Let (X,V) be a global-in-time solution to (1) with

    α1andmini,j[N],ijx0ix0j>0.

    Further assume that there are two strictly positive constants DX1 and DX2 such that

    DV(0)=κDX1DX(0)ψ(s)dsmax(κDX(0)ψ(s)ds,κψ(DX2)(DX2DX(0))).

    Then, it follows that the following asymptotic flocking holds.

    1. (Group formation) DX(t)max(DX1,DX2),

    2. (Velocity alignment) DV(t)DV(0)exp(κψ(max(DX1,DX2))t), t[0,).

    Thus, we attain a larger admissible set in terms of the initial data and system parameters where asymptotic flocking occurs in (1) with a strongly singular kernel.

    In this subsubsection, we present the global well-posedness of (1) with the weakly singular interaction kernel

    ψ(r)=1rα,0<α<1andr>0.

    For this, we first provide the following proposition regarding the existence of a collisional phenomenon of the two-particle system (1) on R1.

    Proposition 3.[8,16,42] Let (X,V) be a solution to the two-particle system (1) such that

    0<α<1,d=1,x01x02.

    Then, there exist sufficient conditions only in terms of the initial data and system parameters satisfying a finite-in-time collision. That is, there is a strictly positive time t0(0,) such that

    x1(t0)=x2(t0).

    Thus, due to Proposition 3, we easily observe that the global well-posedness does not hold for arbitrary noncollisional position data. Hence, we deduce that the global well-posedness may be guaranteed under more restrictive sufficient conditions than those of Theorem 3.1 in the CSS model (1) with a weakly singular interaction kernel.

    Theorem 3.5. (Global well-posedness and asymptotic flocking) Let (X,V) be a solution to (1) with

    0<α<1andmini,j[N],ijx0ix0j>0.

    Further suppose that there exists a strictly positive number DX1 satisfying

    DV(0)=κDX1DX(0)ψ(s)ds<κψ(DX1)mini,j[N],ijx0ix0j. (13)

    Then, we have the global well-posedness of (1) with a weakly singular kernel.

    More precisely, we attain the strict positivity of the relative distance between the pairwise particles along (1) with 0<α<1:

    inf0t<mini,j[N],ijxixjmini,j[N],ijx0ix0jDV(0)κψ(DX1)>0.

    Furthermore, we gain the following asymptotic flocking estimate, as follows:

    1. (Group formation) DX(t)DX1,t[0,),

    2. (Velocity alignment) DV(t)DV(0)exp(κψ(DX1)t).

    Proof. Suppose that the local well-posedness of (1) with 0<α<1 holds on t(0,t) for t(0,). Then, it follows from Theorem 3.4 and the condition (13) that

    xi(t)xj(t)x0ix0jt0vi(s)vj(s)dsx0ix0jt0DV(s)dsx0ix0j0DV(s)dsx0ix0jDV(0)κψ(DX1)mini,j[N],ijx0ix0jDV(0)κψ(DX1)>0.

    Therefore, by the standard Cauchy–Lipschitz theory and Lemma 2.2, we demonstrate that there exists a positive ϵ such that the uniqueness and existence of the solution to (1) with 0<α<1 on (0,t+ϵ) can be guaranteed. Hence, we obtain the global well-posedness and strict positivity of the relative distance between each pair of particles. Thus, the same arguments employed in Theorem 3.4 with Remark 3 can be employed to obtain the asymptotic flocking of (1) with 0<α<1. Consequently, we reach the desired assertions.

    In this section, we recall the previous results for the global well-posedness of (2) with β1, equivalently,

    ϕ(r)=1rβ,β1andr>0.

    We describe the basic dissipative structures in terms of DX, DV, and DT under 0<β<. For this, we employ some useful functionals, as in Section 3.1, to derive several differential inequalities with respect to position-velocity-temperature, which are independent of the number of particles N. As the main results of this section, we present the global well-posedness and emergent dynamics of (2).

    Before we describe the main results, we revisit the basic notion for the asymptotic flocking for the TCSS model (2).

    Definition 4.1. Let Z:=(X,V,T) be a global-in-time solution to (2). The configuration Z exhibits the asymptotic flocking if

    (i)(Group formation)sup0t<maxi,j[N]xi(t)xj(t)<,(ii)(Velocity alignment)limtmaxi,j[N]vj(t)vi(t)=0,(iii)(Temperature equilibrium)limtmaxi,j[N]|Tj(t)Ti(t)|=0.

    To analyze the asymptotic flocking phenomenon of (2), the existence and uniqueness of the solution to (2) on a global time interval are essential. Therefore, a noncollision result is crucial to obtain the global well-posedenss of (2).

    In this subsection, we briefly introduce the previous result related to the global well-posedness of (2) with a strongly singular kernel studied in [16].

    Proposition 4. [16] (Global well-posedness of TCSS with β1) Suppose that (X,V,T) is a solution to (2) satisfying

    1βγ2andmini,j[N],ijx0ix0j>0.

    Then, we have the global well-posedness (i.e., global collisionless state) of (2) in the sense that

    xi(t)xj(t),(i,j)[N]2,ijandt[0,).

    However, in this paper, we propose reasonable sufficient frameworks for the emergent dynamics in Section 4 and uniform stability independent of N in Section 5, going beyond the previous paper [16]. This independence leads to deriving a kinetic Vlasov equation corresponding to the particle model (2) via the uniform-in-time mean-field limit. This kinetic equation represents the dynamics of an infinite number of particles based on the standard BBGKY hierarchy. (For the related papers on the uniform-in-time mean-field limit and the BBGKY hierarchy, we refer to [4,5,10,20,36,38,40].) Furthermore, the authors of the previous literature [16] provided an inaccurate framework for the emergent dynamics of (2) when β1. Moreover, they did not provide a sufficient framework for the global well-posedness and emergent dynamics of (2) when 0<β<1. To remedy these issues, we must first provide the following lemma concerning the dissipative differential inequalities of (2) to establish sufficient frameworks for global well-posedness when 0<β<1 and the emergent dynamics for 0<β<.

    Next, we consider the following functional Φij defined by

    {Φij(t):=ϕ(xixj)Nfori,j[N],ij,Φii(t):=ϕ(dX)jiϕ(xixj)N.

    We can easily verify that Φij satisfies the following properties:

    1. ΦijϕijNfori,j[N],ij,andNj=1Φij=ϕ(dX),

    2. Nj=1Φij(vjTjviTi)=jiϕijN(vjTjviTi),whereϕij:=ϕ(xixj).

    Moreover, we also employ the functional Ψ defined by

    {Ψij(t):=ζ(xixj)Nfori,j[N],ij,Ψii(t):=ζ(dX)Nj=1,jiζ(xixj)N.

    Then, we observe that Ψij satisfies the following properties:

    1. ΨijζijNfori,j[N],ij,andNj=1Ψij=ζ(dX),

    2. Nj=1Ψij(1Ti1Tj)=Nj=1ζijN(1Ti1Tj),whereζij:=ζ(xixj).

    We can now derive several dissipative differential inequalities of (1.2) with 0<β< using Φ and Ψ.

    Lemma 4.2. Let (X,V,T) be a solution to (2) on [0,τ) for τ(0,] such that

    0<β<andmini,j[N],ijx0ix0j>0.

    Then, for a.e. t(0,τ),

    1. |dDXdt|DV,

    2. dDVdtκ1ϕ(DX)TMDV+2κ1ϕ(dX)(Tm)2DTDV,

    3. dDTdtκ2ζ(DX)(TM)2DT.

    Proof. For the first assertion, we note that

    |dxixj2dt|=2|xixj,vivj|2xixjvivj,

    which combines with the Cauchy–Schwarz inequality, for a.e. t(0,τ), to yield

    |dDX(t)dt|DV(t).

    Next, we prove the third assertion with respect to the L-diameter for temperature. For this, we select two indices Mt and mt, depending on time t, satisfying

    DT(t)=TMt(t)Tmt(t),mt,Mt[N].

    Then, it follows from the properties of Ψij and (2)3 that, for a.e. t(0,τ),

    dDTdt=˙TMt˙Tmt=κ2NNk=1ζMtk(1TMt1Tk)κ2NNk=1ζmtk(1Tmt1Tk)=κ2Nk=1ΨMtk(1TMt1Tk)κ2Nk=1Ψmtk(1Tmt1Tk)=κ2ζ(min1i,jNxixj)(1TMt1Tmt)κ2Nk=11Tk(ΨMtkΨmtk)=κ2ζ(min1i,jNxixj)(1TMt1Tmt)κ2Nk=11Tk(ΨMtkmin(ΨMtk,Ψmtk)+min(ΨMtk,Ψmtk)Ψmtk)κ2ζ(min1i,jNxixj)(1TMt1Tmt)+κ2TmtNk=1(Ψmtkmin(ΨMtk,Ψmtk))κ2TMtNk=1(ΨMtkmin(ΨMtk,Ψmtk))=κ2(1Tmt1TMt)Nk=1(min(ΨMtk,Ψmtk))κ2DT(TM)2Nk=1(min(ΨMtk,Ψmtk))κ2ζ(DX)(TM)2DT.

    For the second assertion, we take two indices it and jt, depending on time t(0,τ), satisfying

    DV(t):=vit(t)vjt(t),it,jt[N],

    and assume that Ni=1vi=0 without loss of generality. Then, we use (2)2 to obtain that for a.e. t(0,τ),

    12ddtvitvjt2=vitvjt,dvitdtdvjtdt=vitvjt,κ1NNk=1ϕitk(vkTkvitTit)κ1NNk=1ϕjtk(vkTkvjtTjt)=vitvjt,κ1Nk=1Φitk(vkTkvitTit)κ1Nk=1Φjtk(vkTkvjtTjt)=κ1ϕ(dX)vitvjt,vitTitvjtTjt+κ1vitvjt,Nk=1(ΦitkΦjtk)vkTk.

    We apply the properties of Φ and the following relation

    ΦitkΦjtk=Φitkmin(Φitk,Φjtk)+min(Φitk,Φjtk)Φjtk

    to have

    12ddtvitvjt2=κ1ϕ(dX)vitvjt,vitTitvjtTjt+κ1vitvjt,Nk=1(Φitkmin(Φitk,Ψjtk)+min(Φitk,Φjtk)Φjtk)vkTk:=I1+I2.

    (Estimate of I2) For this, we apply the following inequality

    vitvjt,vjtvitvjt,vkvitvjt,vit

    to I2 to show that

    I2κ1vitvjt,Nk=1(Φitkmin(Φitk,Φjtk))vitTk+κ1vitvjt,Nk=1(min(Φitk,Φjtk)Φjtk)vjtTk=κ1Nk=1min(Φitk,Φjtk)vitvjt2Tk+κ1vitvjt,Nk=1ΦitkvitTkκ1vitvjt,Nk=1ΦjtkvjtTk. (14)

    where we used the nonnegativity of Φ. Then, it follows from (14), Proposition 2 and the properties of Φ that for a.e. t(0,τ),

    I1+I2κ1Nk=1min(Φitk,Φjtk)vitvjt2Tk+κ1vitvjt,Nk=1Φitkvit(1Tk1Tit)κ1vitvjt,Nk=1Φjtkvjt(1Tk1Tjt)κ1ϕ(DX)vitvjt2TM+κ1vitvjtvitNk=1Φitk|1Tk1Tit|+κ1vitvjtvjtNk=1Φjtk|1Tk1Tjt|.

    Therefore, using Proposition 2, the property of Φ and the following relations:

    |1Ti1Tj|DT(Tm)2,vi=|viNj=1vjN|DV,i,j[N],

    one can show that

    12dD2Vdt=I1+I2κ1ϕ(DX)TMD2V+2κ1ϕ(dX)(Tm)2DTD2V,

    which implies that for a.e. t(0,τ),

    dDVdtκ1ϕ(DX)TMDV+2κ1ϕ(dX)(Tm)2DTDV.

    Finally, we get the desired second assertion.

    In this subsection, we study the sufficient framework for the global well-posedness and emergent dynamics of (2) under 0<β< using Lemma 4.2 and the bootstrapping argument. More concretely, we present a sufficient framework for the strict positivity of each relative distance for all particles in (2), assuming 0<β<.

    Theorem 4.3. (Global well-posedness and asymptotic flocking) Suppose that the initial data and system parameters satisfy

    0<β<andmini,j[N],ijx0ix0j>0.

    Further assume that there exist two positive constants dX and DX such that

    dX(0)exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)TMκ1ϕ(DX)dX,DX(0)+exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)TMκ1ϕ(DX)DX. (15)

    Then, we have the following global well-posedness and asymptotic flocking results on (0,):

    1. (Collision avoidance and Group formation)

    DX(t)<DXanddX(t)>dX,

    2. (Velocity alignment)

    DV(t)exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)exp(κ1ϕ(DX)TMt),

    3. (Temperature equilibrium)

    DT(t)DT(0)exp(κ2ζ(DX(TM)2)t).

    Proof. First, suppose that [0,τ) is a maximal interval for which an unique solution of (2) under 0<β< exists and τ< for the proof by contradiction. When DV(0)=0, then we have nothing to prove. Therefore, we now assume DV(0)>0. We note from (15) that the following set

    S:={t>0|dX(s)>dX,DX(s)<DX,s(0,t)andtτ}

    is nonempty. Here, we define t:=supS>0, which implies that

    DX(t)=DXordX(t)=dX.

    Thus, it suffices to demonstrate that

    t=τ.

    Suppose that t<τ for the proof by contradiction. Then, we utilize the third assertion of Lemma 4.2 with S to attain that

    |dDXdt|DV,dDTdtκ2ζ(DX)(TM)2DTκ2ζ(DX)(TM)2DT,a.e.t(0,t),

    where, Grönwall's lemma yields

    DT(t)DT(0)exp(κ2ζ(DX)(TM)2t),t[0,t]. (16)

    Next, we use the second assertion of Lemma 4.2 together with S and (16) to get that for a.e. t(0,t),

    dDVdtκ1ϕ(DX)TMDV+2κ1ϕ(dX)(Tm)2DTDVκ1ϕ(DX)TMDV+2κ1ϕ(dX)(Tm)2DTDVκ1ϕ(DX)TMDV+2κ1ϕ(dX)DT(0)(Tm)2exp(κ2ζ(DX)(TM)2t)DV. (17)

    Hence, we apply the comparison principle for ODE to (17) to obtain

    DV(t)exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)exp(κ1ϕ(DX)TMt). (18)

    Therefore, it follows from (18) and the first assertion of Lemma 4.2 that

    DX(t)=DX(0)+t0dDX(s)dsdsDX(0)+t0DV(s)dsDX(0)+t0exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)exp(κ1ϕ(DX)TMs)ds<DX(0)+exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)TMκ1ϕ(DX)DX,t[0,t], (19)

    which means that DX(t)<DX. Moreover, we again employ the first assertion of Lemma 4.2 to deduce that

    dX(t)=dX(0)t0d(dX(s))dsdsdX(0)t0DV(s)dsdX(0)t0exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)exp(κ1ϕ(DX)TMs)ds>dX(0)exp(2κ1DT(0)ϕ(dX)(TM)2κ2ζ(DX)(Tm)2)DV(0)TMκ1ϕ(DX)dX,t[0,t], (20)

    which leads to dX(t)>dX. Thus, we have t=τ. However, from the standard Cauchy–Lipschitz theory with Proposition 2, (18), (20), and the conservation of momentum (i.e., Nk=1vi=constant), there exists a positive ϵ such that the uniqueness and existence of solution to (2) with 0<β< on (0,τ+ϵ) can be guaranteed. In conclusion, we obtain the global well-posedness (i.e.,τ=). Consequently, it follows from the set S with t=, (16), (18), (19), and (20) that the desired results hold.

    In this section, we define a sufficient framework for the uniform L2-stability estimate of (2) with an arbitrary singular interaction kernel on 0<β< and dimension dN. With Remark 1, if we set T01==T0N=T>0, it follows that the TCSS model (2) can be reduced to the CSS model (1). Therefore, it suffices to construct a sufficient framework for the L2-uniform stability of (2). To do this, we revisit the TCSS model (2) which is governed by the following system in terms of (X,V,T):

    {dxidt=vi,t>0,i[N],dvidt=κ1NNj=1ϕ(xixj)(vjTjviTi),dTidt=κ2NNj=1ζ(xixj)(1Ti1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)R2d×R>0,Ni=1v0i=Nv=0,Ni=1T0i=NT. (21)

    The definition of the uniform L2-stability estimate of (2) is as follows.

    Definition 5.1. (Uniform L2-stability): For the two solutions (X,V,T) and (ˉX,ˉV,ˉT) to (2) with the initial data (X0,V0,T0) and (ˉX0,ˉV0,ˉT0), respectively, if there exists a positive constant G independent of t such that

    sup0t<(X(t)ˉX(t)+V(t)ˉV(t))G(X0ˉX0+V0ˉV0),

    then we say that the equation (2) satisfies the uniform L2-stability.

    In particular, we estimate G defined in Definition 5.1 so that this is independent of the number of particle N in (2) because the independence of N is very important for deriving uniform-in-time mean-field limits based on the standard BBGKY method. Assume that there exist two global-in-time solutions (X,V,T) and (ˉX,ˉV,ˉT) of (2) with the initial data (X0,V0,T0) and (ˉX0,ˉV0,ˉT0), respectively. Then, without loss of generality, we further suppose that

    v=ˉv=0,

    where the assumption v=ˉv for two average momentums v and ˉv is very crucial to derive the uniform L2-stability of (2). Indeed, under appropriate conditions for asymptotic flocking, the uniform stability between two clusters with different average velocities cannot be established. However, we verify the uniform L2-stability estimate when the averages of the sum of the initial temperatures are different from each other, that is,

    T=ˉTorTˉT.

    We use the following simple notation:

    ϕ(xixj):=ϕij,ζ(xixj):=ζij,ϕ(ˉxiˉxj):=ˉϕij,ζ(ˉxiˉxj):=ˉζijDˉX,DˉV,DˉT,ˉTm,ˉTMis defined similarly as before.

    We study the system of differential inequalities in terms of XˉX, VˉV, and TˉT to deduce a sufficient framework for the uniform L2-stability of (21).

    Lemma 5.2. Suppose that (X,V,T) and (ˉX,ˉV,ˉT) are two global-in-time solutions to (21) such that

    0<β<,min(mini,j[N],ijˉx0iˉx0j,mini,j[N],ijx0ix0j)>0.

    and (15) hold, respectively. Then, it follows that for a.e. t(0,),

    1. (Differentiation of the difference between X and ˉX)

    |dXˉXdt|VˉV,

    2. (Differentiation of the difference between V and ˉV)

    dVˉVdt(κ1ϕ(DX)TMκ1ϕ(dX)DT(Tm)2)VˉV+2κ1ϕ(dX)DˉVTmˉTmTˉT+2βκ1min(dX,dˉX)β+1(DˉVˉTm+DˉVDˉT(ˉTm)2)XˉX,

    3. (Differentiation of the difference between T and ˉT)

    dTˉTdt2γκ2DˉT(ˉTm)2(min(dX,dˉX))β+1XˉX+κ2ζ(dX)(DˉTTm(ˉTm)2+DTˉTm(Tm)2)TˉT.

    Proof. (Proof of (1)) For the first assertion, we apply the Cauchy–Schwarz inequality to show that

    12|dXˉX2dt|=|XˉX,VˉV|XˉXVˉV,

    which implies that for a.e. t(0,),

    |dXˉXdt|VˉV.

    (Proof of (3)) For the third assertion, we use (21)3 for a.e. t(0,) to obtain the following:

    12dTˉT2dt=κ2NNi=1(TiˉTi)(Nj=1ζij(1Ti1Tj)Nj=1ˉζij(1ˉTi1ˉTj))=κ2NNi,j=1(ζijˉζij)(TiˉTi)(1ˉTi1ˉTj)+κ2NNi,j=1ζij(TiˉTi)(1Ti1Tj1ˉTi+1ˉTj):=I1+I2.

    (Estimate of I1) To estimate I1, we first observe from Proposition 2 that for a.e. t(0,),

    I1=κ2NNi,j=1(ζijˉζij)(TiˉTi)(1ˉTi1ˉTj)κ2NNi,j=1|ζijˉζij||TiˉTi||1ˉTi1ˉTj|κ2DˉTN(ˉTm)2Ni,j=1|ζijˉζij||TiˉTi|γκ2DˉTN(ˉTm)2(min(dX,dˉX))γ+1Ni,j=1(xiˉxi+xjˉxj)|TiˉTi|2γκ2DˉT(ˉTm)2(min(dX,dˉX))γ+1XˉXTˉT,

    where we used 1) the uniform boundedness of the Lipschitz norm of the ζ with triangle inequality and Theorem 4.1 to estimate the third inequality, and 2) the Cauchy–Schwarz inequality to estimate the last inequality.

    (The estimate of I2) (Estimate of I2) We apply the standard technique of interchanging i and j and dividing by 2 to yield

    I2=κ2NNi,j=1ζij(TiˉTi)(1Ti1Tj1ˉTi+1ˉTj)=κ22NNi,j=1ζij((TiˉTi)(TjˉTj))(1Ti1Tj1ˉTi+1ˉTj)=κ22NNi,j=1ζij((TiˉTi)(TjˉTj))(1Ti1Tj1ˉTi+1ˉTj)=κ22NNi,j=1ζij((TiˉTi)(TjˉTj))(((TiˉTi)(TjˉTj))TiˉTi)+κ22NNi,j=1ζij((TiˉTi)(TjˉTj))(TjˉTj)(1TjˉTj1TiˉTi)κ22NNi,j=1ζij((TiˉTi)(TjˉTj))(TjˉTj)(1TjˉTj1TiˉTi)κ22NNi,j=1ζij(|TiˉTi|+|TjˉTj|)|TjˉTj||1TjˉTj1TiˉTi|κ2ζ(dX)2N(DˉTTm(ˉTm)2+DTˉTm(Tm)2)Ni,j=1(|TiˉTi|+|TjˉTj|)|TjˉTj|κ2ζ(dX)(DˉTTm(ˉTm)2+DTˉTm(Tm)2)TˉT2,

    since Theorem 4.3 holds and

    |1TjˉTj1TiˉTi|1Ti|1ˉTi1ˉTj|+1ˉTj|1Ti1Tj|DˉTTm(ˉTm)2+DTˉTm(Tm)2.

    Therefore, we combine I1 with I2 for a.e. t(0,) to attain the following:

    dTˉTdt2γκ2DˉT(ˉTm)2(min(dX,dˉX))β+1XˉX+κ2ζ(dX)(DˉTTm(ˉTm)2+DTˉTm(Tm)2)TˉT,

    which gives the desired second assertion of the lemma.

    (Proof of (3)) For this, by using the second equation (21)2, one can easily check that

    12dVˉV2dt=κ1NNi=1viˉvi,Nj=1ϕij(vjTjviTi)Nj=1ˉϕij(ˉvjˉTjˉviˉTi)=κ1NNi,j=1(ϕijˉϕij)viˉvi,ˉvjˉTjˉviˉTi+κ1NNi,j=1ϕijviˉvi,vjTjviTiˉvjˉTj+ˉviˉTi:=J1+J2.

    (Estimate of J1) We employ Theorem 4.1 and the following relation

    ˉvi=ˉviNi=1ˉviNN1NDˉVDˉV

    to deduce that

    J1=κ1NNi,j=1(ϕijˉϕij)viˉvi,ˉvjˉTjˉviˉTiκ1NNi,j=1|ϕijˉϕij|viˉviˉvjˉTjˉviˉTiβκ1Nmin(dX,dˉX)β+1Ni,j=1(xiˉxi+xjˉxj)viˉviˉvjˉTjˉviˉTiβκ1Nmin(dX,dˉX)β+1×Ni,j=1(xiˉxi+xjˉxj)viˉvi(ˉvjˉviˉTj+ˉvi|1ˉTj1ˉTi|)βκ1Nmin(dX,dˉX)β+1Ni,j=1(xiˉxi+xjˉxj)viˉvi(DˉVˉTm+DˉVDˉT(ˉTm)2)2βκ1min(dX,dˉX)β+1(DˉVˉTm+DˉVDˉT(ˉTm)2)XˉXVˉV,

    where we used the Lipschitz norm of f(r)=1rβ,r>min(dX,dˉX) in the second inequality and utilized Theorem 4.3 with Proposition 2 in the fourth inequality and used the Cauchy–Schwarz inequality to estimate the last inequality.

    (Estimate of J2) Again, from the standard technique of interchanging i and j and dividing by 2,

    J2=κ1NNi,j=1ϕijviˉvi,vjTjviTiˉvjˉTj+ˉviˉTi=κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjviTiˉvjˉTj+ˉviˉTi=κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTiviTiˉvjTi+ˉviTi+κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjvjTi+ˉvjTiˉviTiˉvjˉTj+ˉviˉTi:=J21+J22.

    (Estimate of J21) We note from Proposition 2 and Theorem 4.3 that

    J21=κ12NNi,j=1ϕijviˉvivj+ˉvj2Tiκ12NNi,j=1ϕ(DX)viˉvivj+ˉvj2TM=κ12NNi,j=1ϕ(DX)(viˉvi2+vjˉvj2TM)=κ1ϕ(DX)TMVˉV2,

    because we assumed that v=ˉv=0.

    (Estimate of J22) Moreover, due to Theorem 4.3,

    J22=κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjvjTi+ˉvjTiˉviTiˉvjˉTj+ˉviˉTi=κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjˉvjTj+ˉvjTjvjTi+ˉvjTiˉviTiˉvjˉTj+ˉviˉTi=κ12NNi,j=1ϕijviˉvivj+ˉvj,ˉvjTjˉviTiˉvjˉTj+ˉviˉTi+κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjˉvjTjvjTi+ˉvjTi:=J221+J222.

    (Estimate of J221) We employ the Cauchy–Schwarz inequality, Proposition 2, Theorem 4.3, and v=ˉv=0 to deduce that

    J221=κ12NNi,j=1ϕijviˉvivj+ˉvj,ˉvjTjˉviTiˉvjˉTj+ˉviˉTiκ12NNi,j=1ϕij(viˉvi+vjˉvj)(ˉvj|1Tj1ˉTj|+ˉvi|1Ti1ˉTi|)κ1ϕ(dX)DˉV2NTmˉTmNi,j=1(viˉvi+vjˉvj)(|TiˉTi|+|TjˉTj|)2κ1ϕ(dX)DˉVTmˉTmVˉVTˉT.

    (The estimate of J222) Likewise, it is easy to verify that

    J222=κ12NNi,j=1ϕijviˉvivj+ˉvj,vjTjˉvjTjvjTi+ˉvjTiκ12NNi,j=1ϕij(viˉvi+vjˉvj)(vjˉvj|1Tj1Ti|)κ1ϕ(dX)DT2N(Tm)2Ni,j=1(viˉvi+vjˉvj)vjˉvjκ1ϕ(dX)DT(Tm)2VˉV2.

    Finally, we combine J1 with J21, J221 and J222 to concldue that for a.e. t(0,),

    dVˉVdt(κ1ϕ(DX)TMκ1ϕ(dX)DT(Tm)2)VˉV+2κ1ϕ(dX)DˉVTmˉTmTˉT+2βκ1min(dX,dˉX)β+1(DˉVˉTm+DˉVDˉT(ˉTm)2)XˉX,

    which yields the desired second assertion.

    Remark 4. The results in Lemma 5.2 are similar to those in the previous paper [33]; however, we proved Lemma 5.2 in a much more concise way. The authors of [33] verified Lemma 5.22 and the result is similar to Lemma 5.23 by dividing N particles into two sets for technical calculations, and too long estimates were made for each of them.

    Next, we are ready to prove the uniform L2-stability of (21) under a sufficient framework in terms of the initial data and system parameters, which can be used to derive the uniform-in-time mean-field limit from (21) to the corresponding kinetic Vlasov equation. Thus, we must estimate G defined in Definition 5.1 so that it is independent of the initial data (X0,V0,T0) as well as the number of particle N for the sake of an uniqueness of measure-valued solution to the kinetic equation. For papers related to the uniform-in-time mean-field limit and the measure-valued solution framework, refer to [4,5,20,33,36,38].

    Next, we set the following simple notation to simply express the three differential inequalities of Lemma 5.2.

    XˉX=:X,VˉV=:V,TˉT=:T.

    Theorem 5.3. Let (X,V,T) and (ˉX,ˉV,ˉT) be two global-in-time solutions of (21) such that

    0<β<,min(mini,j[N],ijx0ix0j,mini,j[N],ijˉx0iˉx0j)>0

    and (15) hold, respectively. Then, the uniform L2-stability estimate holds for (21). More precisely, there exist G>0 and C1>0 such that for ϵ(0,C1),

    1. (Uniform stability for X)

    X(t)G(X(0)+V(0)+T(0)),

    2. (Uniform stability for V)

    V(t)G(X(0)+V(0)+T(0))exp((C1ϵ)t),

    3. (Uniform stability for T)

    T(t)G(X(0)+V(0)+T(0)),t(0,).

    Proof. First, it follows from Theorem 4.3 and Lemma 5.2 that there exist a strictly positive constant C1 and nonnegative constants {˜Ck}5k=1 independent of t, N and the initial data such that

    |dXdt|V,a.e.t(0,),dVdtC1V+˜C1exp(C1t)V+˜C2exp(C1t)X+˜C3exp(C1t)T,dTdt˜C4exp(C1t)X+˜C5exp(C1t)T. (22)

    Now, for an arbitrarily given ϵ(0,C1), we set

    W(t):=V(t)exp((C1ϵ)t).

    Then, (22) can be converted to the following inequalities:

    |dXdt|Wexp((C1ϵ)t),a.e.t(0,),dWdtϵC1W+˜C1exp(ϵt)W+˜C2exp(ϵt)X+˜C3exp(ϵt)T,dTdt˜C4exp(C1t)X+˜C5exp(C1t)T. (23)

    By defining ˉC1=min(ϵ,C1ϵ), we estimate (23) as follows:

    |dXdt|Wexp(ˉC1t),a.e.t(0,),dWdt˜C1exp(ˉC1t)W+˜C2exp(ˉC1t)X+˜C3exp(ˉC1t)T,dTdt˜C4exp(ˉC1t)X+˜C5exp(ˉC1t)T. (24)

    Now, we sum up from (24)1 to (24)3 to guarantee that there exists C0, independent of t, N and the initial data, such that

    ddt(X+W+T)Cexp(ˉC1t)(X+W+T).

    Therefore, we can conclude that

    X+V+TX+W+Texp(C¯C1)(X(0)+W(0)+T(0))=exp(C¯C1)(X(0)+V(0)+T(0)),

    yielding the uniform L2-stability of (21). Moreover, we get the desired results (1), (2) and (3).

    Remark 5. Further, G, estimated in the proof of Theorem 5.3, is independent of the number of particles N, time t, and the given initial data (X(0),V(0),T(0)), (ˉX(0),ˉV(0),ˉT(0)).

    Remark 6. In previous articles [5,33,36], the authors verified the uniform stability estimates of the targeted models with regular kernels that are monotonically decreasing, nonnegative, bounded, and Lipschitz continuous. Their verifications were conducted by employing functionals, such as

    MZ(t)=max0stZ(s)ˉZ(s)

    with arguments that were too technical and lengthy. However, we used the following substitution

    u(t)=V(t)ˉV(t)exp((C1ϵ)t)

    to obtain a much more improved proof in Theorem 5.3 than those in previous papers.

    In this paper, we provided several sufficient frameworks, independent of the number of particles N, for collision avoidance (that is, global well-posedness) and the emergent dynamics of the CS and TCS models under strongly and weakly singular kernels, respectively. We first derived the dissipative structures with the L-diameters DX, DV (and DT) and then used the Lyapunov functional approach and appropriate bootstrapping arguments with technical estimates to obtain the collision avoidance and asymptotic flocking results. In particular, a collisional phenomenon of the two-particle CSS and TCSS models on R1 under the weakly singular kernel; therefore we adopted a sufficient framework for the global well-posedness and emergent behavior so that the two models have strictly positive lower bounds on all pairwise distances. Furthermore, to construct an admissible set for the emergent dynamics of the TCSS model, we introduced sufficient frameworks. These were introduced to ensure that the distance between each pair of particles to have a positive lower bound regardless of having weakly or strongly singular kernels, due to the dissipative velocity structure with ϕ(dX) and finite-in-time blow-up when dX=0, unlike the CSS model. Finally, we described the sufficient frameworks for the L2-uniform stability results of the TCSS model, which can be used to derive uniform-in-time mean-field limits of the CSS and TCSS models. In summary, this work is meaningful in that it provides sufficient frameworks to enable deriving uniform-in-time mean-field limits from the CS and TCS models with singular kernels to the corresponding kinetic Vlasov equation, respectively. However, several remaining questions require study in future work:

    ● (Question 1): Can we enlarge the sufficient framework for the emergent dynamics independent of N of the TCSS model with a strongly singular kernel without the strict positivity of each relative distance?

    ● (Question 2): Can we improve the sufficient frameworks for the uniform stability of the CSS and TCSS models in Section 5 when each nonzero relative distance converges to zero?

    ● (Question 3): Can we also prove the noncollisional phenomena of the CS and TCS models with singular kernels on complete Riemannian manifolds?

    This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2022R1C12007321). The author would like to thank Dr. Woojoo Shim for the suggestion of the main idea in the proof of Theorem 5.3.



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  • This article has been cited by:

    1. Hyunjin Ahn, Junhyeok Byeon, Seung-Yeal Ha, Interplay of unit-speed constraint and singular communication in the thermodynamic Cucker–Smale model, 2023, 33, 1054-1500, 10.1063/5.0165245
    2. Hyunjin Ahn, Se Eun Noh, Finite-in-time flocking of the thermodynamic Cucker–Smale model, 2024, 19, 1556-1801, 526, 10.3934/nhm.2024023
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