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
{dxidt=vi,t>0,i∈[N]:={1,⋯,N},dvidt=κN∑j≠iψ(‖xi−xj‖)(vj−vi),(xi(0),vi(0))=(x0i,v0i)∈Rd×Rd, | (1) |
where
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,
{dxidt=vi,t>0,i∈[N],dvidt=κ1N∑j≠iϕ(‖xi−xj‖)(vjTj−viTi),dTidt=κ2N∑j≠iζ(‖xi−xj‖)(1Ti−1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)∈R2d×R>0, | (2) |
where
Throughout the paper, we choose the simplest singular communication weights (or singular interaction kernels)
ψ(r):=1rα,ϕ(r):=1rβ,ζ(r):=1rγ,α,β,γ>0. |
We are only interested in the singularity when
This paper is organized as follows. In Section 2, we briefly revisit facts regarding the temperature field
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]‖zi−zj‖,dX:=mini≠j,i,j∈[N]‖xi−xj‖forZ=(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
In this subsection, we briefly provide facts regarding the temperature system
Definition 2.1. [34,39] Let
S:=N∑i=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
1. (Conserved temperature sum) The total sum
N∑i=1Ti(t)=N∑i=1T0i:=NT∞,∀t∈[0,τ). |
2. (Entropy principle) The total entropy
dS(t)dt=12NN∑i,j=1ζ(‖xj−xi‖)|1Ti−1Tj|2≥0,∀t∈[0,τ). |
Due to the entropy principle and the simple structure of
Proposition 2. [12,34] (Monotonocity of max-min temperatures) Let
0<mini∈[N]T0i:=T∞m≤Ti(t)≤maxi∈[N]T0i:=T∞M,i∈[N],t∈[0,τ). |
Remark 1. By the standard Cauchy–Lipschitz theory, the TCSS model (2) for
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
maxi∈[N]‖vi‖≤maxi∈[N]‖v0i‖,t∈[0,τ). |
Proof. We choose an index
‖vMt‖:=maxi∈[N]‖vi(t)‖. |
Then, we take the inner product
12d‖vMt‖2dt=κNN∑j=1ψ(‖xMt−xj‖)⟨vj−vMt,vMt⟩≤0. |
Hence, we obtain
12d‖vMt‖2dt≤0,a.e.t∈(0,τ)⟹‖vMt‖≤‖vM0‖,t∈[0,τ), |
implying the desired result.
maxi∈[N]‖vi‖≤‖vM0‖≤maxi∈[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:
In this subsection, we study the global well-posedness of the CSS model (1) when
ψ(r)=1rα,r>0,α≥1. |
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
[l]:={i∈[N]|‖xl(t)−xi(t)‖→0ast→t0−}. |
Let
‖xl(t)−xi(t)‖≥δ>0,∀t∈[0,t0)and∀i∉[l]. |
Thus, we define the following
dX,[l]:=mini,j∈[l],i≠j‖xi−xj‖,DX,[l]:=maxi,j∈[l]‖xi−xj‖,DV,[l]:=maxi,j∈[l]‖vi−vj‖. |
For simplicity, we use the following notation:
ψij:=ψ(‖xi−xj‖),i,j∈[N],i≠j,andψij,[l]:=ψ(‖xi−xj‖),i,j∈[l],i≠j, |
where
Next, we employ the following functional
Ψij,[l](t):=ψ(‖xi−xj‖)|[l]|fori,j∈[l],i≠j,Ψii,[l](t):=ψ(dX,[l])−∑j≠i,j∈[l]ψ(‖xi−xj‖)|[l]|,fori∈[l]. |
Then, we observe that
1.
2.
3.
Theorem 3.1. Suppose that
α≥1andmini,j∈[N],i≠j‖x0i−x0j‖>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,i≠jand∀t∈[0,∞). |
Proof. First, we use the following relation
|d‖xi−xj‖2dt|=2|⟨xi−xj,vi−vj⟩|≤2‖xi−xj‖‖vi−vj‖ |
with the Cauchy–Schwarz inequality to have that for a.e.
|dDX,[l](t)dt|≤DV,[l](t). | (3) |
Now, we take two indices
DV,[l](t):=‖vit(t)−vjt(t)‖,it,jt∈[l]. |
Then, it follows from
12ddt‖vit−vjt‖2=⟨vit−vjt,dvitdt−dvjtdt⟩=⟨vit−vjt,κNN∑k=1ψitk(vk−vit)−κNN∑k=1ψjtk(vk−vjt)⟩=⟨vit−vjt,κN∑k∉[l]ψitk(vk−vit)−κN∑k∉[l]ψjtk(vk−vjt)⟩+⟨vit−vjt,κN∑k∈[l]ψitk(vk−vit)−κN∑k∈[l]ψjtk(vk−vjt)⟩=:I1+I2. |
I1=⟨vit−vjt,κN∑k∉[l]ψitk(vk−vit)−κN∑k∉[l]ψjtk(vk−vjt)⟩≤DV,[l]‖κN∑k∉[l]ψitk(vk−vit)−κN∑k∉[l]ψjtk(vk−vjt)‖≤DV,[l](‖κN∑k∉[l]ψitk(vk−vit)‖+‖κN∑k∉[l]ψjtk(vk−vjt)‖)≤4κ(N−|[l]|)ψ(δ)maxi∈[N]‖v0i‖N⋅DV,[l]=:C(κ,[l],N,V(0),δ)DV,[l], | (4) |
where we used the definition of
●
I2=⟨vit−vjt,κN∑k∈[l]ψitk(vk−vit)−κN∑k∈[l]ψjtk(vk−vjt)⟩=⟨vit−vjt,κ|[l]|N∑k∈[l]Ψitk(vk−vit)−κ|[l]|N∑k∈[l]Ψjtk(vk−vjt)⟩=−κ|[l]|Nψ(dX,[l])⟨vit−vjt,vit−vjt⟩+κ|[l]|N⟨vit−vjt,∑k∈[l](Ψitk−Ψjtk)vk⟩=−κ|[l]|Nψ(dX,[l])⟨vit−vjt,vit−vjt⟩+κ|[l]|N⟨vit−vjt,∑k∈[l](Ψitk−min(Ψitk,Ψjtk)+min(Ψitk,Ψjtk)−Ψjtk)vk⟩. | (5) |
Here, since
⟨vit−vjt,vjt⟩≤⟨vit−vjt,vk⟩≤⟨vit−vjt,vit⟩, |
we can show that
I2≤−κ|[l]|Nψ(dX,[l])⟨vit−vjt,vit−vjt⟩+κ|[l]|N⟨vit−vjt,∑k∈[l](Ψitk−min(Ψitk,Ψjtk))vit⟩+κ|[l]|N⟨vit−vjt,∑k∈[l](min(Ψitk,Ψjtk)−Ψjtk)vjt⟩≤−κ|[l]|Nψ(dX,[l])⟨vit−vjt,vit−vjt⟩+κ|[l]|Nψ(dX,[l])⟨vit−vjt,vit−vjt⟩−κ|[l]|N∑k∈[l]min(Ψitk,Ψjtk)⟨vit−vjt,vit−vjt⟩=−κ|[l]|N∑k∈[l]min(Ψitk,Ψjtk)⟨vit−vjt,vit−vjt⟩≤−κ|[l]|Nψ(DX,[l])⟨vit−vjt,vit−vjt⟩=−κ|[l]|Nψ(DX,[l])D2V,[l], | (6) |
where we used the first property of
dDV,[l]dt≤−κ|[l]|Nψ(DX,[l])DV,[l]+C(κ,[l],N,V(0),δ). |
We then integrate to both sides of the above inequality from
∫tsψ(DX,[l])DV,[l]du≤N(DV,[l](s)+C(κ,[l],N,V(0),δ)(t−s))κ|[l]|. | (7) |
Moreover, let
Φ(t):=Φ(t;t1):=∫tt1ψ(u)du={logtt1,ifα=1,11−α(t1−α−t11−α),ifα>1. | (8) |
Therefore, it follows from (7) that for
|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)(t−s))κ|[l]|,0≤s≤t<t0. |
Now, we take
limt→t0−|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
limt→t0−|Φ(DX,[l](t))|=∞. |
Consequently, we have shown the noncollisional phenomenon of (1), i.e.,
xi(t)≠xj(t),∀(i,j)∈[N]2and∀t∈[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
0≤t1≤t<∞. |
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
In this subsubsection, from the proof of Theorem 3.1, we study the sufficient frameworks for the emergent dynamics of (1), assuming
Definition 3.2. Let
(i)(Group formation)⟺sup0≤t<∞maxi,j∈[N]‖xi(t)−xj(t)‖<∞,(ii)(Velocity alignment)⟺limt→∞maxi,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
α≥1andmini,j∈[N],i≠j‖x0i−x0j‖>0. |
Further assume that there exists a positive constant
DX(0)+DV(0)κψ(D∞X)≤D∞X. | (9) |
Then, we have the following asymptotic flocking result:
1. (Group formation)
2. (Velocity alignment)
Proof. If
S:={t>0|DX(s)<D∞X,∀s∈(0,t)} |
is nonempty. We define
DX(t∗)=D∞X. |
Now, we claim that
t∗=+∞. |
To prove the claim, we suppose that
|dDXdt|≤DV,dDVdt≤−κψ(DX)DV≤−κψ(D∞X)DV,a.e.t∈(0,t∗), |
where we used the definition of
DV(t)≤DV(0)exp(−κψ(D∞X)t),∀t∈[0,t∗]. |
Moreover, because
DX(t)=DX(0)+∫t0dDX(s)dsds≤DX(0)+∫t0DV(s)ds≤DX(0)+∫t0DV(0)exp(−κψ(D∞X)s)ds<DX(0)+DV(0)κψ(D∞X)≤D∞X,∀t∈[0,t∗], |
one can show that
DX(t)<D∞X,DV(t)≤DV(0)exp(−κψ(D∞X)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
α≥1,andmini,j∈[N],i≠j‖x0i−x0j‖>0, |
and further assume that
DV(0)≤κ∫∞DX(0)ψ(s)ds. | (10) |
Then, we obtain the following asymptotic flocking estimate: there exists a strictly positive number
1. (Group formation)
2. (Velocity alignment)
Proof. First, we use the same arguments employed in Theorem 3.1 and Theorem 3.3 to deduce that
|dDXdt|≤DV,dDVdt≤−κψ(DX)DV. | (11) |
Now, we consider the following Lyapunov functional:
L±(DX,DV):=DV±κΨ(DX), |
where
ddtL±(DX,DV)=dDVdt±κdDXdtψ(DX)≤−κψ(DX)DV±κdDXdtψ(DX)=κψ(DX)(−DV±dDXdt)≤0. | (12) |
Next, we utilize (12) to induce
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)≤κ∫∞DX(0)ψ(s)ds. |
Hence, there is a smallest positive real number
DV(0)=κ∫D∞XDX(0)ψ(s)ds; |
thus
DX(t)≤D∞X,∀t∈[0,∞). |
Due to the above estimate and (11), from Grönwall's lemma, it follows that
DV(t)≤DV(0)exp(−κψ(D∞X)t),t∈[0,∞). |
Finally, we prove the desired asymptotic flocking results.
Remark 3. If
∫∞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
∫∞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
α≥1andmini,j∈[N],i≠j‖x0i−x0j‖>0. |
Further assume that there are two strictly positive constants
DV(0)=κ∫D∞X1DX(0)ψ(s)ds≤max(κ∫∞DX(0)ψ(s)ds,κψ(D∞X2)(D∞X2−DX(0))). |
Then, it follows that the following asymptotic flocking holds.
1. (Group formation)
2. (Velocity alignment)
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
Proposition 3.[8,16,42] Let
0<α<1,d=1,x01≠x02. |
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
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
0<α<1andmini,j∈[N],i≠j‖x0i−x0j‖>0. |
Further suppose that there exists a strictly positive number
DV(0)=κ∫D∞X1DX(0)ψ(s)ds<κψ(D∞X1)mini,j∈[N],i≠j‖x0i−x0j‖. | (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
inf0≤t<∞mini,j∈[N],i≠j‖xi−xj‖≥mini,j∈[N],i≠j‖x0i−x0j‖−DV(0)κψ(D∞X1)>0. |
Furthermore, we gain the following asymptotic flocking estimate, as follows:
1. (Group formation)
2. (Velocity alignment)
Proof. Suppose that the local well-posedness of (1) with
‖xi(t)−xj(t)‖≥‖x0i−x0j‖−∫t0‖vi(s)−vj(s)‖ds≥‖x0i−x0j‖−∫t0DV(s)ds≥‖x0i−x0j‖−∫∞0DV(s)ds≥‖x0i−x0j‖−DV(0)κψ(D∞X1)≥mini,j∈[N],i≠j‖x0i−x0j‖−DV(0)κψ(D∞X1)>0. |
Therefore, by the standard Cauchy–Lipschitz theory and Lemma 2.2, we demonstrate that there exists a positive
In this section, we recall the previous results for the global well-posedness of (2) with
ϕ(r)=1rβ,β≥1andr>0. |
We describe the basic dissipative structures in terms of
Before we describe the main results, we revisit the basic notion for the asymptotic flocking for the TCSS model (2).
Definition 4.1. Let
(i)(Group formation)⟺sup0≤t<∞maxi,j∈[N]‖xi(t)−xj(t)‖<∞,(ii)(Velocity alignment)⟺limt→∞maxi,j∈[N]‖vj(t)−vi(t)‖=0,(iii)(Temperature equilibrium)⟺limt→∞maxi,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
1≤β≤γ2andmini,j∈[N],i≠j‖x0i−x0j‖>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,i≠jand∀t∈[0,∞). |
However, in this paper, we propose reasonable sufficient frameworks for the emergent dynamics in Section 4 and uniform stability independent of
Next, we consider the following functional
{Φij(t):=ϕ(‖xi−xj‖)Nfori,j∈[N],i≠j,Φii(t):=ϕ(dX)−∑j≠iϕ(‖xi−xj‖)N. |
We can easily verify that
1.
2.
Moreover, we also employ the functional
{Ψij(t):=ζ(‖xi−xj‖)Nfori,j∈[N],i≠j,Ψii(t):=ζ(dX)−∑Nj=1,j≠iζ(‖xi−xj‖)N. |
Then, we observe that
1.
2.
We can now derive several dissipative differential inequalities of (1.2) with
Lemma 4.2. Let
0<β<∞andmini,j∈[N],i≠j‖x0i−x0j‖>0. |
Then, for a.e.
1.
2.
3.
Proof. For the first assertion, we note that
|d‖xi−xj‖2dt|=2|⟨xi−xj,vi−vj⟩|≤2‖xi−xj‖‖vi−vj‖, |
which combines with the Cauchy–Schwarz inequality, for a.e.
|dDX(t)dt|≤DV(t). |
Next, we prove the third assertion with respect to the
DT(t)=TMt(t)−Tmt(t),mt,Mt∈[N]. |
Then, it follows from the properties of
dDTdt=˙TMt−˙Tmt=κ2NN∑k=1ζMtk(1TMt−1Tk)−κ2NN∑k=1ζmtk(1Tmt−1Tk)=κ2N∑k=1ΨMtk(1TMt−1Tk)−κ2N∑k=1Ψmtk(1Tmt−1Tk)=κ2ζ(min1≤i,j≤N‖xi−xj‖)(1TMt−1Tmt)−κ2N∑k=11Tk(ΨMtk−Ψmtk)=κ2ζ(min1≤i,j≤N‖xi−xj‖)(1TMt−1Tmt)−κ2N∑k=11Tk(ΨMtk−min(ΨMtk,Ψmtk)+min(ΨMtk,Ψmtk)−Ψmtk)≤κ2ζ(min1≤i,j≤N‖xi−xj‖)(1TMt−1Tmt)+κ2TmtN∑k=1(Ψmtk−min(ΨMtk,Ψmtk))−κ2TMtN∑k=1(ΨMtk−min(ΨMtk,Ψmtk))=−κ2(1Tmt−1TMt)N∑k=1(min(ΨMtk,Ψmtk))≤−κ2DT(T∞M)2N∑k=1(min(ΨMtk,Ψmtk))≤−κ2ζ(DX)(T∞M)2DT. |
For the second assertion, we take two indices
DV(t):=‖vit(t)−vjt(t)‖,it,jt∈[N], |
and assume that
12ddt‖vit−vjt‖2=⟨vit−vjt,dvitdt−dvjtdt⟩=⟨vit−vjt,κ1NN∑k=1ϕitk(vkTk−vitTit)−κ1NN∑k=1ϕjtk(vkTk−vjtTjt)⟩=⟨vit−vjt,κ1N∑k=1Φitk(vkTk−vitTit)−κ1N∑k=1Φjtk(vkTk−vjtTjt)⟩=−κ1ϕ(dX)⟨vit−vjt,vitTit−vjtTjt⟩+κ1⟨vit−vjt,N∑k=1(Φitk−Φjtk)vkTk⟩. |
We apply the properties of
Φitk−Φjtk=Φitk−min(Φitk,Φjtk)+min(Φitk,Φjtk)−Φjtk |
to have
12ddt‖vit−vjt‖2=−κ1ϕ(dX)⟨vit−vjt,vitTit−vjtTjt⟩+κ1⟨vit−vjt,N∑k=1(Φitk−min(Φitk,Ψjtk)+min(Φitk,Φjtk)−Φjtk)vkTk⟩:=I1+I2. |
⟨vit−vjt,vjt⟩≤⟨vit−vjt,vk⟩≤⟨vit−vjt,vit⟩ |
to
I2≤κ1⟨vit−vjt,N∑k=1(Φitk−min(Φitk,Φjtk))vitTk⟩+κ1⟨vit−vjt,N∑k=1(min(Φitk,Φjtk)−Φjtk)vjtTk⟩=−κ1N∑k=1min(Φitk,Φjtk)‖vit−vjt‖2Tk+κ1⟨vit−vjt,N∑k=1ΦitkvitTk⟩−κ1⟨vit−vjt,N∑k=1ΦjtkvjtTk⟩. | (14) |
where we used the nonnegativity of
I1+I2≤−κ1N∑k=1min(Φitk,Φjtk)‖vit−vjt‖2Tk+κ1⟨vit−vjt,N∑k=1Φitkvit(1Tk−1Tit)⟩−κ1⟨vit−vjt,N∑k=1Φjtkvjt(1Tk−1Tjt)⟩≤−κ1ϕ(DX)‖vit−vjt‖2T∞M+κ1‖vit−vjt‖‖vit‖N∑k=1Φitk|1Tk−1Tit|+κ1‖vit−vjt‖‖vjt‖N∑k=1Φjtk|1Tk−1Tjt|. |
Therefore, using Proposition 2, the property of
|1Ti−1Tj|≤DT(T∞m)2,‖vi‖=|vi−∑Nj=1vjN|≤DV,i,j∈[N], |
one can show that
12dD2Vdt=I1+I2≤−κ1ϕ(DX)T∞MD2V+2κ1ϕ(dX)(T∞m)2DTD2V, |
which implies that for a.e.
dDVdt≤−κ1ϕ(DX)T∞MDV+2κ1ϕ(dX)(T∞m)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
Theorem 4.3. (Global well-posedness and asymptotic flocking) Suppose that the initial data and system parameters satisfy
0<β<∞andmini,j∈[N],i≠j‖x0i−x0j‖>0. |
Further assume that there exist two positive constants
dX(0)−exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)T∞Mκ1ϕ(D∞X)≥d∞X,DX(0)+exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)T∞Mκ1ϕ(D∞X)≤D∞X. | (15) |
Then, we have the following global well-posedness and asymptotic flocking results on
1. (Collision avoidance and Group formation)
DX(t)<D∞XanddX(t)>d∞X, |
2. (Velocity alignment)
DV(t)≤exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)exp(−κ1ϕ(D∞X)T∞Mt), |
3. (Temperature equilibrium)
DT(t)≤DT(0)exp(−κ2ζ(D∞X(T∞M)2)t). |
Proof. First, suppose that
S:={t>0|dX(s)>d∞X,DX(s)<D∞X,∀s∈(0,t)andt≤τ} |
is nonempty. Here, we define
DX(t∗)=D∞XordX(t∗)=d∞X. |
Thus, it suffices to demonstrate that
t∗=τ. |
Suppose that
|dDXdt|≤DV,dDTdt≤−κ2ζ(DX)(T∞M)2DT≤−κ2ζ(D∞X)(T∞M)2DT,a.e.t∈(0,t∗), |
where, Grönwall's lemma yields
DT(t)≤DT(0)exp(−κ2ζ(D∞X)(T∞M)2t),∀t∈[0,t∗]. | (16) |
Next, we use the second assertion of Lemma
dDVdt≤−κ1ϕ(DX)T∞MDV+2κ1ϕ(dX)(T∞m)2DTDV≤−κ1ϕ(D∞X)T∞MDV+2κ1ϕ(d∞X)(T∞m)2DTDV≤−κ1ϕ(D∞X)T∞MDV+2κ1ϕ(d∞X)DT(0)(T∞m)2exp(−κ2ζ(D∞X)(T∞M)2t)DV. | (17) |
Hence, we apply the comparison principle for ODE to (17) to obtain
DV(t)≤exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)exp(−κ1ϕ(D∞X)T∞Mt). | (18) |
Therefore, it follows from (18) and the first assertion of Lemma
DX(t)=DX(0)+∫t0dDX(s)dsds≤DX(0)+∫t0DV(s)ds≤DX(0)+∫t0exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)exp(−κ1ϕ(D∞X)T∞Ms)ds<DX(0)+exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)T∞Mκ1ϕ(D∞X)≤D∞X,∀t∈[0,t∗], | (19) |
which means that
dX(t)=dX(0)−∫t0d(dX(s))dsds≤dX(0)−∫t0DV(s)ds≥dX(0)−∫t0exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)exp(−κ1ϕ(D∞X)T∞Ms)ds>dX(0)−exp(2κ1DT(0)ϕ(d∞X)(T∞M)2κ2ζ(D∞X)(T∞m)2)DV(0)T∞Mκ1ϕ(D∞X)≥d∞X,∀t∈[0,t∗], | (20) |
which leads to
In this section, we define a sufficient framework for the uniform
{dxidt=vi,t>0,i∈[N],dvidt=κ1NN∑j=1ϕ(‖xi−xj‖)(vjTj−viTi),dTidt=κ2NN∑j=1ζ(‖xi−xj‖)(1Ti−1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)∈R2d×R>0,N∑i=1v0i=Nv∞=0,N∑i=1T0i=NT∞. | (21) |
The definition of the uniform
Definition 5.1. (Uniform
sup0≤t<∞(‖X(t)−ˉX(t)‖+‖V(t)−ˉV(t)‖)≤G∞(‖X0−ˉX0‖+‖V0−ˉV0‖), |
then we say that the equation (2) satisfies the uniform
In particular, we estimate
v∞=ˉv∞=0, |
where the assumption
T∞=ˉT∞orT∞≠ˉT∞. |
We use the following simple notation:
∙ϕ(‖xi−xj‖):=ϕij,ζ(‖xi−xj‖):=ζij,ϕ(‖ˉxi−ˉxj‖):=ˉϕij,ζ(‖ˉxi−ˉxj‖):=ˉζij∙DˉX,DˉV,DˉT,ˉT∞m,ˉT∞Mis defined similarly as before. |
We study the system of differential inequalities in terms of
Lemma 5.2. Suppose that
0<β<∞,min(mini,j∈[N],i≠j‖ˉx0i−ˉx0j‖,mini,j∈[N],i≠j‖x0i−x0j‖)>0. |
and (15) hold, respectively. Then, it follows that for a.e.
1. (Differentiation of the difference between
|d‖X−ˉX‖dt|≤‖V−ˉV‖, |
2. (Differentiation of the difference between
d‖V−ˉV‖dt≤−(κ1ϕ(D∞X)T∞M−κ1ϕ(d∞X)DT(T∞m)2)‖V−ˉV‖+2κ1ϕ(d∞X)DˉVT∞mˉT∞m‖T−ˉT‖+2βκ1min(dX,dˉX)β+1(DˉVˉT∞m+DˉVDˉT(ˉT∞m)2)‖X−ˉX‖, |
3. (Differentiation of the difference between
d‖T−ˉT‖dt≤2γκ2DˉT(ˉT∞m)2(min(d∞X,d∞ˉX))β+1‖X−ˉX‖+κ2ζ(d∞X)(DˉTT∞m(ˉT∞m)2+DTˉT∞m(T∞m)2)‖T−ˉT‖. |
Proof.
12|d‖X−ˉX‖2dt|=|⟨X−ˉX,V−ˉV⟩|≤‖X−ˉX‖‖V−ˉV‖, |
which implies that for a.e.
|d‖X−ˉX‖dt|≤‖V−ˉV‖. |
12d‖T−ˉT‖2dt=κ2NN∑i=1(Ti−ˉTi)(N∑j=1ζij(1Ti−1Tj)−N∑j=1ˉζij(1ˉTi−1ˉTj))=κ2NN∑i,j=1(ζij−ˉζij)(Ti−ˉTi)(1ˉTi−1ˉTj)+κ2NN∑i,j=1ζij(Ti−ˉTi)(1Ti−1Tj−1ˉTi+1ˉTj):=I1+I2. |
I1=κ2NN∑i,j=1(ζij−ˉζij)(Ti−ˉTi)(1ˉTi−1ˉTj)≤κ2NN∑i,j=1|ζij−ˉζij||Ti−ˉTi||1ˉTi−1ˉTj|≤κ2DˉTN(ˉT∞m)2N∑i,j=1|ζij−ˉζij||Ti−ˉTi|≤γκ2DˉTN(ˉT∞m)2(min(d∞X,d∞ˉX))γ+1N∑i,j=1(‖xi−ˉxi‖+‖xj−ˉxj‖)|Ti−ˉTi|≤2γκ2DˉT(ˉT∞m)2(min(d∞X,d∞ˉX))γ+1‖X−ˉX‖‖T−ˉT‖, |
where we used 1) the uniform boundedness of the Lipschitz norm of the
I2=κ2NN∑i,j=1ζij(Ti−ˉTi)(1Ti−1Tj−1ˉTi+1ˉTj)=κ22NN∑i,j=1ζij((Ti−ˉTi)−(Tj−ˉTj))(1Ti−1Tj−1ˉTi+1ˉTj)=κ22NN∑i,j=1ζij((Ti−ˉTi)−(Tj−ˉTj))(1Ti−1Tj−1ˉTi+1ˉTj)=κ22NN∑i,j=1ζij((Ti−ˉTi)−(Tj−ˉTj))(−((Ti−ˉTi)−(Tj−ˉTj))TiˉTi)+κ22NN∑i,j=1ζij((Ti−ˉTi)−(Tj−ˉTj))(Tj−ˉTj)(1TjˉTj−1TiˉTi)≤κ22NN∑i,j=1ζij((Ti−ˉTi)−(Tj−ˉTj))(Tj−ˉTj)(1TjˉTj−1TiˉTi)≤κ22NN∑i,j=1ζij(|Ti−ˉTi|+|Tj−ˉTj|)|Tj−ˉTj||1TjˉTj−1TiˉTi|≤κ2ζ(d∞X)2N(DˉTT∞m(ˉT∞m)2+DTˉT∞m(T∞m)2)N∑i,j=1(|Ti−ˉTi|+|Tj−ˉTj|)|Tj−ˉTj|≤κ2ζ(d∞X)(DˉTT∞m(ˉT∞m)2+DTˉT∞m(T∞m)2)‖T−ˉT‖2, |
since Theorem 4.3 holds and
|1TjˉTj−1TiˉTi|≤1Ti|1ˉTi−1ˉTj|+1ˉTj|1Ti−1Tj|≤DˉTT∞m(ˉT∞m)2+DTˉT∞m(T∞m)2. |
Therefore, we combine
d‖T−ˉT‖dt≤2γκ2DˉT(ˉT∞m)2(min(d∞X,d∞ˉX))β+1‖X−ˉX‖+κ2ζ(d∞X)(DˉTT∞m(ˉT∞m)2+DTˉT∞m(T∞m)2)‖T−ˉT‖, |
which gives the desired second assertion of the lemma.
12d‖V−ˉV‖2dt=κ1NN∑i=1⟨vi−ˉvi,N∑j=1ϕij(vjTj−viTi)−N∑j=1ˉϕij(ˉvjˉTj−ˉviˉTi)⟩=κ1NN∑i,j=1(ϕij−ˉϕij)⟨vi−ˉvi,ˉvjˉTj−ˉviˉTi⟩+κ1NN∑i,j=1ϕij⟨vi−ˉvi,vjTj−viTi−ˉvjˉTj+ˉviˉTi⟩:=J1+J2. |
‖ˉvi‖=‖ˉvi−∑Ni=1ˉviN‖≤N−1N⋅DˉV≤DˉV |
to deduce that
J1=κ1NN∑i,j=1(ϕij−ˉϕij)⟨vi−ˉvi,ˉvjˉTj−ˉviˉTi⟩≤κ1NN∑i,j=1|ϕij−ˉϕij|‖vi−ˉvi‖‖ˉvjˉTj−ˉviˉTi‖≤βκ1Nmin(dX,dˉX)β+1N∑i,j=1(‖xi−ˉxi‖+‖xj−ˉxj‖)‖vi−ˉvi‖‖ˉvjˉTj−ˉviˉTi‖≤βκ1Nmin(dX,dˉX)β+1×N∑i,j=1(‖xi−ˉxi‖+‖xj−ˉxj‖)‖vi−ˉvi‖(‖ˉvj−ˉviˉTj‖+‖ˉvi‖|1ˉTj−1ˉTi|)≤βκ1Nmin(dX,dˉX)β+1N∑i,j=1(‖xi−ˉxi‖+‖xj−ˉxj‖)‖vi−ˉvi‖(DˉVˉT∞m+DˉVDˉT(ˉT∞m)2)≤2βκ1min(dX,dˉX)β+1⋅(DˉVˉT∞m+DˉVDˉT(ˉT∞m)2)‖X−ˉX‖‖V−ˉV‖, |
where we used the Lipschitz norm of
J2=κ1NN∑i,j=1ϕij⟨vi−ˉvi,vjTj−viTi−ˉvjˉTj+ˉviˉTi⟩=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−viTi−ˉvjˉTj+ˉviˉTi⟩=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTi−viTi−ˉvjTi+ˉviTi⟩+κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−vjTi+ˉvjTi−ˉviTi−ˉvjˉTj+ˉviˉTi⟩:=J21+J22. |
J21=−κ12NN∑i,j=1ϕij‖vi−ˉvi−vj+ˉvj‖2Ti≤−κ12NN∑i,j=1ϕ(D∞X)‖vi−ˉvi−vj+ˉvj‖2T∞M=−κ12NN∑i,j=1ϕ(D∞X)(‖vi−ˉvi‖2+‖vj−ˉvj‖2T∞M)=−κ1ϕ(D∞X)T∞M‖V−ˉV‖2, |
because we assumed that
J22=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−vjTi+ˉvjTi−ˉviTi−ˉvjˉTj+ˉviˉTi⟩=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−ˉvjTj+ˉvjTj−vjTi+ˉvjTi−ˉviTi−ˉvjˉTj+ˉviˉTi⟩=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,ˉvjTj−ˉviTi−ˉvjˉTj+ˉviˉTi⟩+κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−ˉvjTj−vjTi+ˉvjTi⟩:=J221+J222. |
J221=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,ˉvjTj−ˉviTi−ˉvjˉTj+ˉviˉTi⟩≤κ12NN∑i,j=1ϕij(‖vi−ˉvi‖+‖vj−ˉvj‖)(‖ˉvj‖|1Tj−1ˉTj|+‖ˉvi‖|1Ti−1ˉTi|)≤κ1ϕ(d∞X)DˉV2NT∞mˉT∞mN∑i,j=1(‖vi−ˉvi‖+‖vj−ˉvj‖)(|Ti−ˉTi|+|Tj−ˉTj|)≤2κ1ϕ(d∞X)DˉVT∞mˉT∞m‖V−ˉV‖‖T−ˉT‖. |
J222=κ12NN∑i,j=1ϕij⟨vi−ˉvi−vj+ˉvj,vjTj−ˉvjTj−vjTi+ˉvjTi⟩≤κ12NN∑i,j=1ϕij(‖vi−ˉvi‖+‖vj−ˉvj‖)(‖vj−ˉvj‖|1Tj−1Ti|)≤κ1ϕ(d∞X)DT2N(T∞m)2N∑i,j=1(‖vi−ˉvi‖+‖vj−ˉvj‖)‖vj−ˉvj‖≤κ1ϕ(d∞X)DT(T∞m)2‖V−ˉV‖2. |
Finally, we combine
d‖V−ˉV‖dt≤−(κ1ϕ(D∞X)T∞M−κ1ϕ(d∞X)DT(T∞m)2)‖V−ˉV‖+2κ1ϕ(d∞X)DˉVT∞mˉT∞m‖T−ˉT‖+2βκ1min(dX,dˉX)β+1⋅(DˉVˉT∞m+DˉVDˉT(ˉT∞m)2)‖X−ˉX‖, |
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
Next, we are ready to prove the uniform
Next, we set the following simple notation to simply express the three differential inequalities of Lemma 5.2.
‖X−ˉX‖=:X,‖V−ˉV‖=:V,‖T−ˉT‖=:T. |
Theorem 5.3. Let
0<β<∞,min(mini,j∈[N],i≠j‖x0i−x0j‖,mini,j∈[N],i≠j‖ˉx0i−ˉx0j‖)>0 |
and (15) hold, respectively. Then, the uniform
1. (Uniform stability for
X(t)≤G∞(X(0)+V(0)+T(0)), |
2. (Uniform stability for
V(t)≤G∞(X(0)+V(0)+T(0))exp(−(C1−ϵ)t), |
3. (Uniform stability for
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
∙|dXdt|≤V,a.e.t∈(0,∞),∙dVdt≤−C1V+˜C1exp(−C1t)V+˜C2exp(−C1t)X+˜C3exp(−C1t)T,∙dTdt≤˜C4exp(−C1t)X+˜C5exp(−C1t)T. | (22) |
Now, for an arbitrarily given
W(t):=V(t)exp((C1−ϵ)t). |
Then, (22) can be converted to the following inequalities:
∙|dXdt|≤Wexp(−(C1−ϵ)t),a.e.t∈(0,∞),∙dWdt≤−ϵC1W+˜C1exp(−ϵt)W+˜C2exp(−ϵt)X+˜C3exp(−ϵt)T,∙dTdt≤˜C4exp(−C1t)X+˜C5exp(−C1t)T. | (23) |
By defining
∙|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
ddt(X+W+T)≤Cexp(−ˉC1t)(X+W+T). |
Therefore, we can conclude that
X+V+T≤X+W+T≤exp(C¯C1)(X(0)+W(0)+T(0))=exp(C¯C1)(X(0)+V(0)+T(0)), |
yielding the uniform
Remark 5. Further,
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)=max0≤s≤t‖Z(s)−ˉZ(s)‖ |
with arguments that were too technical and lengthy. However, we used the following substitution
u(t)=‖V(t)−ˉV(t)‖exp((C1−ϵ)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
● (Question 1): Can we enlarge the sufficient framework for the emergent dynamics independent of
● (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.
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 |