Citation: Irén Juhász Junger, Sarah Vanessa Homburg, Hubert Meissner, Thomas Grethe, Anne Schwarz Pfeiffer, Johannes Fiedler, Andreas Herrmann, Tomasz Blachowicz, Andrea Ehrmann. Influence of the pH value of anthocyanins on the electrical properties of dye-sensitized solar cells[J]. AIMS Energy, 2017, 5(2): 258-267. doi: 10.3934/energy.2017.2.258
[1] | Young-Pil Choi, Cristina Pignotti . Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14(4): 789-804. doi: 10.3934/nhm.2019032 |
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[6] | Hyunjin Ahn . Non-emergence of mono-cluster flocking and multi-cluster flocking of the thermodynamic Cucker–Smale model with a unit-speed constraint. Networks and Heterogeneous Media, 2023, 18(4): 1493-1527. doi: 10.3934/nhm.2023066 |
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[8] | Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang . Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013 |
[9] | Hyunjin Ahn . Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels. Networks and Heterogeneous Media, 2022, 17(5): 753-782. doi: 10.3934/nhm.2022025 |
[10] | Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim . Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media, 2018, 13(3): 379-407. doi: 10.3934/nhm.2018017 |
In the last years the study of collective behavior of multi-agent systems has attracted the interest of many researchers in different scientific fields, such as biology, physics, control theory, social sciences, economics. The celebrated Cucker-Smale model has been proposed and analyzed in [21,22] to describe situations in which different agents, e.g. animals groups, reach a consensus (flocking), namely they align and move as a flock, based on a simple rule: each individual adjusts its velocity taking into account other agents' velocities.
In the original papers a symmetric interaction potential is considered. Then, the case of non-symmetric interactions has been studied by Motsch and Tadmor [31]. Several generalizations and variants have been introduced to cover various applications' fields, e.g. more general interaction rates and singular potentials [8,10,19,27,30,32], cone-vision constraints [40], presence of leadership [17,37], noise terms [20,23,25], crowds dynamics [18,29], infinite-dimensional models [1,2,11,26,28,39], control problems [3,5,16,33]. We refer to [6,12] for recent surveys on the Cucker-Smale type flocking models and its variants.
It is natural to introduce a time delay in the model, as a reaction time or a time to receive environmental information. The presence of a time delay makes the problem more difficult to deal with. Indeed, the time delay destroys some symmetry features of the model which are crucial in the proof of convergence to consensus. For this reason, in spite of a great amount of literature on Cucker-Smale models, only a few papers are available concerning Cucker-Smale model with time delay [13,14,15,23,36]. Cucker-Smale models with delay effects are also studied in [34,35] when a hierarchical structure is present, namely the agents are ordered in a specific order depending on which other agents they are leaders of or led by.
Here we consider a distributed delay term, i.e. we assume that the agent
τ′(t)≤0andτ(t)≥τ∗,for t≥0, | (1) |
for some positive constant
τ∗≤τ(t)≤τ0for t≥0. | (2) |
It is clear that the constant time delay
Our main system is given by
xi(t)t=vi(t),i=1,⋯,N,t>0,vi(t)t=1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))(vk(s)−vi(t))ds, | (3) |
where
ϕ(xk(s),xi(t))={ψ(|xk(s)−xi(t)|)∑j≠iψ(|xj(s)−xi(t)|)if k≠i, 0if k=i, | (4) |
with the influence function
∫τ∗0α(s)ds>0, |
and
h(t):=∫τ(t)0α(s)ds,t≥0. |
We consider the system subject to the initial datum
xi(s)=:x0i(s),vi(s)=:v0i(s),i=1,⋯,N,s∈[−τ0,0], | (5) |
i.e., we prescribe the initial position and velocity trajectories
For the particle system (3), we will first discuss the asymptotic behavior of solutions in Section 2. Motivated from [13,26,31], we derive a system of dissipative differential inequalities, see Lemma 2.4, and construct a Lyapunov functional. This together with using Halanay inequality enables us to show the asymptotic velocity alignment behavior of solutions under suitable conditions on the initial data.
We next derive, analogously to [13] where the case of a single pointwise time delay is considered, a delayed Vlasov alignment equation from the particle system (3) by sending the number of particles
∂tft+v⋅∇xft+∇v⋅(1h(t)∫tt−τ(t)α(t−s)F[fs]dsft)=0, | (6) |
where
F[fs](x,v):=∫Rd×Rdψ(|x−y|)(w−v)fs(y,w)dydw∫Rd×Rdψ(|x−y|)fs(y,w)dydw. |
We show the global-in-time existence and stability of measure-valued solutions to (6) by employing the Monge-Kantorowich-Rubinstein distance. As a consequence of the stability estimate, we discuss a mean-field limit providing a quantitative error estimate between the empirical measure associated to the particle system (3) and the measure-valued solution to (6). We then extend the estimate of large behavior of solutions for the particle system (3) to the one for the delayed Vlasov alignment equation (6). For this, we use the fact that the estimate of large-time behavior of solutions to the particle system (3) is independent of the number of particles. By combining this and the mean-field limit estimate, we show that the diameter of velocity-support of solutions of (6) converges to zero as time goes to infinity. Those results will be proved in Section 3.
We start with presenting a notion of flocking behavior for the system (3), and for this we introduce the spatial and, respectively, velocity diameters as follows:
dX(t):=max1≤i,j≤N|xi(t)−xj(t)|anddV(t):=max1≤i,j≤N|vi(t)−vj(t)|. | (7) |
Definition 2.1. We say that the system with particle positions
supt≥0dX(t)<∞andlimt→∞dV(t)=0. |
We then state our main result in this section on the asymptotic flocking behavior of the system (3).
Theorem 2.2. Assume
Rv:=maxs∈[−τ0,0]max1≤i≤N|v0i(s)|. | (8) |
Moreover, denoted
h(0)dV(0)+∫τ00α(s)(∫0−sdV(z)dz)ds<βN∫τ∗0α(s)∫∞dX(−s)+Rvτ0ψ(z)dzds, | (9) |
where
supt≥0dX(t)<∞ |
and
dV(t)≤maxs∈[−τ0,0]dV(s)e−γtfor t≥0, |
for a suitable positive constant
Remark 1. If the influence function
xi(t)t=vi(t),i=1,⋯,N,t>0,vi(t)t=N∑k=1ϕ(xk(t−τ),xi(t))(vk(t−τ)−vi(t)). |
Note the above system is studied in [13]. For this system, the assumption (9) reduces to
dV(0)+∫0−τdV(z)dz<βN∫∞dX(−τ)+Rvτψ(z)dz. |
Since
Remark 2. Observe that our theorem above gives a flocking result when the number of agents
Remark 3. Note that
For the proof of Theorem 2.2, we will need several auxiliary results. Inspired by [13], we first show the uniform-in-time bound estimate of the maximum speed of the system (3).
Lemma 2.3. Let
max1≤i≤N|vi(t)|≤Rvfor t≥−τ0. |
Proof. Let us fix
Sϵ:={t>0:max1≤i≤N|vi(s)|<Rv+ϵ∀ s∈[0,t)}. |
By continuity,
max1≤i≤N|vi(Tϵ)|=Rv+ϵ. | (10) |
From (3)–(5), for
|vi(t)|2t≤2h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))(|vk(s)||vi(t)|−|vi(t)|2)ds≤2h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))max1≤k≤N|vk(s)||vi(t)|ds−2|vi(t)|2. |
Note that
max1≤k≤N|vk(s)|≤Rv+ϵandN∑k=1ϕ(xk(s),xi(t))=1, |
for
|vi(t)|2t≤2[(Rv+ϵ)|vi(t)|−|vi(t)|2], |
which gives
|vi(t)|t≤(Rv+ϵ)−|vi(t)|. | (11) |
From (11) we obtain
limt→Tϵ− max1≤i≤N|vi(t)|≤e−Tϵ(max1≤i≤N|vi(0)|−Rv−ϵ)+Rv+ϵ<Rv+ϵ. |
This is in contradiction with (10). Therefore,
In the lemma below, motivated from [13,38] we derive the differential inequalities for
D+F(t):=lim suph→0+F(t+h)−F(t)h. |
Note that the Dini derivative coincides with the usual derivative when the function is differentiable at
Lemma 2.4. Let
|D+dX(t)|≤dV(t),D+dV(t)≤1h(t)∫tt−τ(t)α(t−s)(1−βNψ(dX(s)+Rvτ0))dV(s)ds−dV(t). |
Proof. The first inequality is by now standard, Then, we concentrate on the second one. Due to the continuity of the velocity trajectories
⋃σ∈N¯Iσ=[0,∞) |
and thus for each
dV(t)=|vi(σ)(t)−vj(σ)(t)|for t∈Iσ. |
Then, by using the simplified notation
12D+d2V(t)=(vi(t)−vj(t))⋅(vi(t)t−vj(t)t)=(vi(t)−vj(t))⋅(1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))vk(s)ds−1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xj(t))vk(s)ds)−|vi(t)−vj(t)|2. | (12) |
Set
ϕkij(s,t):=min {ϕ(xk(s),xi(t)),ϕ(xk(s),xj(t))}andˉϕij(s,t):=N∑k=1ϕkij(s,t). |
Note that, from the definition (4) of
1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))vk(s)ds−1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xj(t))vk(s)ds=1h(t)N∑k=1∫tt−τ(t)α(t−s)(ϕ(xk(s),xi(t))−ϕkij(s,t))vk(s)ds−1h(t)N∑k=1∫tt−τ(t)α(t−s)(ϕ(xk(s),xj(t))−ϕkij(s,t))vk(s)ds=1h(t)∫tt−τ(t)α(t−s)(1−ˉϕij(s,t))N∑k=1(akij(s,t)−akji(s,t))vk(s)ds, | (13) |
where
akij(s,t)=ϕ(xk(s),xi(t))−ϕkij(s,t)1−ˉϕij(s,t),i≠j, 1≤i,j,k≤N. |
Observe that
N∑k=1akij(s,t)vk(s)∈Ω(s)for all 1≤i≠j≤N. |
This gives
|N∑k=1(akij(s,t)−akji(s,t))vk(s)|≤dV(s), |
which, used in (13), implies
1h(t)|N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))vk(s)ds−N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xj(t))vk(s)ds|≤1h(t)∫tt−τ(t)α(t−s)(1−ˉϕij(s,t))dV(s)ds. | (14) |
Now, by using the first equation in (3), we estimate for any
|xk(s)−xi(t)|=|xk(s)−xi(s)−∫tsddtxi(z)dz|≤|xk(s)−xi(s)|+τ0supz∈[t−τ(t),t]|vi(z)|. |
Then, Lemma 2.3 gives
|xk(s)−xi(t)|≤dX(s)+Rvτ0,fors∈[t−τ(t),t], |
and due to the monotonicity property of the influence function
ϕ(xk(s),xi(t))≥ψ(dX(s)+Rvτ0)N−1. | (15) |
On the other hand, we find
ˉϕij=N∑k=1ϕkij=∑k≠i,jϕkij+ϕiij+ϕjij=∑k≠i,jϕkij. |
Then, from (15), we obtain
ˉϕij(s,t)≥N−2N−1ψ(dX(s)+Rvτ0)=βNψ(dX(s)+Rvτ0). |
Using the last estimate in (14), we have
1h(t)|N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xi(t))vk(s)ds−1h(t)N∑k=1∫tt−τ(t)α(t−s)ϕ(xk(s),xj(t))vk(s)ds|≤1h(t)∫tt−τ(t)α(t−s)(1−βNψ(dX(s)+Rvτ0))dV(s)ds, |
that, used in (12), concludes the proof.
Lemma 2.5. Let
ddtu(t)≤ah(t)∫tt−τ(t)α(t−s)u(s)ds−u(t)for almost all t>0. | (16) |
Then we have
u(t)≤supt∈[−τ0,0]u(s)e−γtfor all t≥0, |
with
Proof. Note that the differential inequality (16) implies
ddtu(t)≤asups∈[t−τ0,t]u(s)−u(t). |
Then, the result follows from Halanay inequality (see e.g. [24,p. 378]).
We are now ready to proceed with the proof of Theorem 2.2.
Proof of Theorem 2.2. For
L(t):=h(t)dV(t)+βN∫τ(t)0α(s)|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|ds+∫τ(t)0α(s)(∫0−sdV(t+z)dz)ds=:h(t)dV(t)+L1(t)+L2(t), |
where
D+L1(t)=βN{τ′(t)α(τ(t))|∫dX(t−τ(t))+Rvτ0dX(−τ(t))+Rvτ0ψ(z)dz|+∫τ(t)0α(s) sign (∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz)ψ(dX(t−s)+Rvτ0)D+dX(t−s)ds}≤βN∫τ(t)0α(s)ψ(dX(t−s)+Rvτ0)dV(t−s)ds, | (17) |
for almost all
sgn(x):={−1if x<0, 0if x=0, 1if x>0. |
Analogously, we also get
D+L2(t)≤∫τ(t)0α(s)(dV(t)−dV(t−s))ds=h(t)dV(t)−∫τ(t)0α(s)dV(t−s)ds, | (18) |
for almost all
D+L(t)≤h′(t)dV(t)+∫τ(t)0α(s)(1−βNψ(dX(t−s)+Rvτ0))dV(t−s)ds−h(t)dV(t)+βN∫τ(t)0α(s)ψ(dX(t−s)+Rvτ0)dV(t−s)ds+h(t)dV(t)−∫τ(t)0α(s)dV(t−s)ds=h′(t)dV(t). |
On the other hand, since
h(t)dV(t)+βN∫τ(t)0α(s)|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|ds+∫τ(t)0α(s)(∫0−sdV(t+z)dz)ds≤h(0)dV(0)+∫τ00α(s)(∫0−sdV(z)dz)ds. | (19) |
Moreover, it follows from the assumption (9) that there exists a positive constant
h(0)dV(0)+∫τ00α(s)(∫0−sdV(z)dz)ds≤βN∫τ∗0α(s)∫d∗dX(−s)+Rvτ0ψ(z)dzds. |
This, together with (19) and (2), implies
h(t)dV(t)+∫τ(t)0α(s)(∫0−sdV(t+z)dz)ds≤βN{∫τ∗0α(s)∫d∗dX(−s)+Rvτ0ψ(z)dzds−∫τ∗0α(s)|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|ds}=βN∫τ∗0α(s){∫d∗dX(−s)+Rvτ0ψ(z)dz−|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|}ds. | (20) |
Now, observe that, if
∫τ∗0α(s){∫d∗dX(−s)+Rvτ0ψ(z)dz−|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|}ds=∫τ∗0α(s)∫d∗dX(t−s)+Rvτ0ψ(z)dzds. | (21) |
Similarly, when
∫τ∗0α(s){∫d∗dX(−s)+Rvτ0ψ(z)dz−|∫dX(t−s)+Rvτ0dX(−s)+Rvτ0ψ(z)dz|}ds≤∫τ∗0α(s)∫d∗dX(−s)+Rvτ0ψ(z)dzds≤∫τ∗0α(s)∫d∗dX(t−s)+Rvτ0ψ(z)dzds. | (22) |
Thus, from (20), (21) and (22) we deduce that
h(t)dV(t)+∫τ(t)0α(s)(∫0−sdV(t+z)dz)ds≤βN∫τ∗0α(s)∫d∗dX(t−s)+Rvτ0ψ(z)dzds. |
Note that, for
dX(t−s)=dX(t)+∫t−stD+dX(z)dz≤dX(t)+2Rvτ0, |
due to Lemma 2.3 and the first inequality of Lemma 2.4. Analogously, we also find for
dX(t)=dX(t−s)+∫tt−sD+dX(z)dz≤dX(t−s)+2Rvτ0. |
This gives
dX(t−s)−2Rvτ0≤dX(t)≤dX(t−s)+2Rvτ0,fors∈[0,τ(t)]. | (23) |
Thus we get
∫τ∗0α(s)∫d∗dX(t−s)+Rvτ0ψ(z)dzds≤∫τ∗0α(s)∫d∗max{dX(t)−Rvτ0,0}ψ(z)dzds≤h(0)∫d∗max{dX(t)−Rvτ0,0}ψ(z)dz. |
Combining this and (20), we obtain
h(t)dV(t)+∫τ(t)0α(s)(∫0−sdV(t+z)dz)ds≤h(0)βN∫d∗max{dX(t)−Rvτ0,0}ψ(z)dz. |
Since the left hand side of the above inequality is positive, we have
dX(t)≤d∗+Rvτ0fort≥0. |
We then again use (23) to find
dX(t−s)+Rvτ0≤dX(t)+3Rvτ0≤d∗+4Rvτ0, |
for
D+dV(t)≤(1−ψ∗βN)h(t)∫tt−τ(t)α(t−s)dV(s)ds−dV(t), |
for almost all
In this section, we are interested in the behavior of solutions to the particle system (3) as the number of particles
∂tft+v⋅∇xft+∇v⋅(1h(t)∫tt−τ(t)α(t−s)F[fs]dsft)=0, | (24) |
for
fs(x,v)=gs(x,v),(x,v)∈Rd×Rd,s∈[−τ0,0], |
where
F[fs](x,v):=∫Rd×Rdψ(|x−y|)(w−v)fs(y,w)dydw∫Rd×Rdψ(|x−y|)fs(y,w)dydw. |
For the equation (24), we provide the global-in-time existence and uniqueness of measure-valued solutions and mean-field limits from (3) based on the stability estimate. We also establish the large-time behavior of measure-valued solutions showing the velocity alignment.
In this part, we discuss the global existence and uniqueness of measure-valued solutions to the equation (24). For this, we first define a notion of weak solutions in the definition below.
Definition 3.1. For a given
∫T0∫Rd×Rdft(∂tξ+v⋅∇xξ+1h(t)∫tt−τ(t)α(t−s)F[fs]ds⋅∇vξ)dxdvdt+∫Rd×Rdg0(x,v)ξ(x,v,0)dxdv=0, |
where
We next introduce the
Definition 3.2. Let
W1(ρ1,ρ2):=infπ∈Π(ρ1,ρ2)∫Rd×Rd|x−y|dπ(x,y), |
where
Theorem 3.3. Let the initial datum
supp gt⊂B2d(0,R)for all t∈[−τ0,0], |
where
Then for any
Proof. The proof can be done by using a similar argument as in [13,Theorem 3.1], thus we shall give it rather concisely. Let
supp ft⊂B2d(0,R)for all t∈[0,T], |
for some positive constant
|F[ft](x,v)|≤Cand|F[ft](x,v)−F[ft](˜x,˜v)|≤C(|x−˜x|+|v−˜v|), |
for
|1h(t)∫tt−τ(t)α(t−s)F[fs]ds|≤C |
and
|1h(t)∫tt−τ(t)α(t−s)(F[fs](x,v)−F[fs](˜x,˜v))ds|≤C |
for
RX[ft]:=maxx∈¯suppxft|x|,RV[ft]:=maxv∈¯suppvft|v|, |
for
RtX:=max−τ0≤s≤tRX[fs],RtV:=max−τ0≤s≤tRV[fs]. | (25) |
We first construct the system of characteristics
Z(t;x,v):=(X(t;x,v),V(t;x,v)):[0,τ0]×Rd×Rd→Rd×Rd |
associated with (24),
X(t;x,v)t=V(t;x,v),V(t;x,v)t=1h(t)∫tt−τ(t)α(t−s)F[fs](Z(t;x,v))ds, | (26) |
where we again adopt the notation
X(0;x,v)=x,V(0;x,v)=v, | (27) |
for all
dV(t;x,v)dt =1h(t)∫tt−τ(t)α(t−s)(∫Rd×Rdψ(|X(t;x,v)−y|)wdfs(y,w)∫Rd×Rdψ(|X(t;x,v)−y|)s(y,w))ds−V(t;x,v)=1h(t)∫τ(t)0α(s)(∫Rd×Rdψ(|X(t;x,v)−y|)wdft−s(y,w)∫Rd×Rdψ(|X(t;x,v)−y|)dft−s(y,w))ds−V(t;x,v). |
Then, arguing as in the proof of Lemma 2.3, we get
d|V(t)|dt≤RtV−|V(t)|, |
due to (25). Using again a similar argument as in the proof of Lemma 2.3 and the comparison lemma, we obtain
RtV≤R0Vfort≥0, |
which further implies
In this subsection, we discuss the rigorous derivation of the delayed Vlasov alignment equation (24) from the particle system (3) as
Theorem 3.4. Let
W1(f1t,f2t)≤Cmaxs∈[−τ0,0]W1(g1s,g2s)for t∈[0,T). |
Proof. Again, the proof is very similar to [13,Theorem 3.2], see also [7,9]. Indeed, we can obtain
ddtW1(f1t,f2t)≤C(W1(f1t,f2t)+1h(t)∫tt−τ(t)α(t−s)W1(f1s,f2s)ds). |
Then we have
W1(f1t,f2t)≤e2CTmaxs∈[−τ0,0]W1(g1s,g2s), |
for
Remark 4. Since the empirical measure
fNt(x,v):=1NN∑i=1δ(xNi(t),vNi(t))(x,v), |
associated to the
gNs(x,v):=1NN∑i=1δ(x0i(s),v0i(s))(x,v) |
for
supt∈[0,T)W1(ft,fNt)≤Cmaxs∈[−τ0,0]W1(gs,gNs), |
where
In this part, we provide the asymptotic behavior of solutions to the equation (24) showing the velocity alignment under suitable assumptions on the initial data. For this, we first define the position- and velocity-diameters for a compactly supported measure
dX[g]:=diam(suppxg),dV[g]:=diam(suppvg), |
where supp
Theorem 3.5. Let
h(0)dV[g0]+∫τ00α(s)(∫0−sdV[gz]dz)ds<∫τ∗0α(s)(∫∞dX[g−s]+R0Vτ0ψ(z)dz)ds. | (28) |
Then the weak solution
dV[ft]≤(maxs∈[−τ0,0]dV[gs])e−Ctfor t≥0,supt≥0dX[ft]<∞, |
where
Let us point out that the flocking estimate at the particle level, see Section 2 and Remark 3, is independent of the number of particles, thus we can directly use the same strategy as in [11,13,26]. However, we provide the details of the proof for the completeness.
Proof of Theorem 3.5. We consider an empirical measure
gNs(x,v):=1NN∑i=1δ(x0i(s),v0i(s))(x,v)for s∈[−τ0,0], |
where the
maxs∈[−τ0,0]W1(gNs,gs)→0asN→∞. |
Note that we can choose
dV(t)≤(maxs∈[−τ0,0]dV(s))e−C1tfor t≥0, |
with the diameters
fNt(x,v):=1NN∑i=1δ(xNi(t),vNi(t))(x,v) |
is a measure-valued solution of the delayed Vlasov alignment equation (24) in the sense of Definition 3.1. On the other hand, by Theorem 3.4, for any fixed
W1(ft,fNt)≤C2maxs∈[−τ0,0]W1(gs,gNs)fort∈[0,T), |
where the constant
dV[ft]≤(maxs∈[−τ0,0]dV[gs])e−C1tfort∈[0,T). |
Since the uniform-in-
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