Research article Topical Sections

Influence of the pH value of anthocyanins on the electrical properties of dye-sensitized solar cells

  • Received: 31 December 2016 Accepted: 06 March 2017 Published: 13 March 2017
  • In recent years the harvesting of renewable energies became of great importance. This led to a rapid development of dye-sensitized solar cells which can be produced from low-purity materials. The best electrical properties are provided by cells prepared using synthetical, ruthenium based dyes. Unfortunately, most of them are toxic and expensive. The anthocyanins extracted for example from hibiscus flowers yield a more cost-effective and eco-friendly alternative to toxic dyes, however, with a loss of solar cell efficiency. In this article the possibility of improvement of the conversion efficiency by modification of the pH value of the dye is investigated. By decrease of the pH value, an increase of efficiency by a factor of two was achieved.

    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

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  • In recent years the harvesting of renewable energies became of great importance. This led to a rapid development of dye-sensitized solar cells which can be produced from low-purity materials. The best electrical properties are provided by cells prepared using synthetical, ruthenium based dyes. Unfortunately, most of them are toxic and expensive. The anthocyanins extracted for example from hibiscus flowers yield a more cost-effective and eco-friendly alternative to toxic dyes, however, with a loss of solar cell efficiency. In this article the possibility of improvement of the conversion efficiency by modification of the pH value of the dye is investigated. By decrease of the pH value, an increase of efficiency by a factor of two was achieved.


    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 i, i=1,,N, changes its velocity depending on the information received from other agents on a time interval [tτ(t),t]. Moreover, we assume normalized communication weights (cf [13]). Let us consider a finite number NN of autonomous individuals located in Rd, d1. Let xi(t) and vi(t) be the position and velocity of ith individual. Then, in the current work, we will deal with a Cucker-Smale model with distributed time delays. Namely, let τ:[0,+)(0,+), be the time delay function belonging to W1,(0,+). Throughout this paper, we assume that the time delay function τ satisfies

    τ(t)0andτ(t)τ,for t0, (1)

    for some positive constant τ. Then, denoting τ0:=τ(0), we have

    ττ(t)τ0for t0. (2)

    It is clear that the constant time delay τ(t)ˉτ>0 satisfies the conditions above.

    Our main system is given by

    xi(t)t=vi(t),i=1,,N,t>0,vi(t)t=1h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))(vk(s)vi(t))ds, (3)

    where ϕ(xk(s),xi(t)) are the normalized communication weights given by

    ϕ(xk(s),xi(t))={ψ(|xk(s)xi(t)|)jiψ(|xj(s)xi(t)|)if ki0if k=i (4)

    with the influence function ψ:[0,)(0,). Throughout this paper, we assume that the influence function ψ is bounded, positive, nonincreasing and Lipschitz continuous on [0,), with ψ(0)=1. Moreover, α:[0,τ0][0,) is a weight function satisfying

    τ0α(s)ds>0,

    and

    h(t):=τ(t)0α(s)ds,t0.

    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 x0i,v0iC([τ0,0];Rd).

    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 N to infinity:

    tft+vxft+v(1h(t)ttτ(t)α(ts)F[fs]dsft)=0, (6)

    where ft=ft(x,v) is the one-particle distribution function on the phase space Rd×Rd and the velocity alignment force F is given by

    F[fs](x,v):=Rd×Rdψ(|xy|)(wv)fs(y,w)dydwRd×Rdψ(|xy|)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):=max1i,jN|xi(t)xj(t)|anddV(t):=max1i,jN|vi(t)vj(t)|. (7)

    Definition 2.1. We say that the system with particle positions xi(t) and velocities vi(t), i=1,,N and t0, exhibits asymptotic flocking if the spatial and velocity diameters satisfy

    supt0dX(t)<andlimtdV(t)=0.

    We then state our main result in this section on the asymptotic flocking behavior of the system (3).

    Theorem 2.2. Assume N>2. Suppose that the initial data x0i,v0i are continuous on the time interval [τ0,0] and denote

    Rv:=maxs[τ0,0]max1iN|v0i(s)|. (8)

    Moreover, denoted βN=N2N1, assume that

    h(0)dV(0)+τ00α(s)(0sdV(z)dz)ds<βNτ0α(s)dX(s)+Rvτ0ψ(z)dzds, (9)

    where dX and dV denote, respectively, the spatial and velocity diameters defined in (7). Then the solution of the system (3)–(5) is global in time and satisfies

    supt0dX(t)<

    and

    dV(t)maxs[τ0,0]dV(s)eγtfor t0,

    for a suitable positive constant γ independent of t. The constant γ can be chosen also independent of N.

    Remark 1. If the influence function ψ is not integrable, i.e., it has a heavy tail, then the right hand side of (9) becomes infinite, and thus the assumption (9) is satisfied for all initial data and time delay τ(t) satisfying (1). This is reminiscent of the unconditional flocking condition for the Cucker-Smale type models, see [11,12,13]. On the other hand, if the influence function ψ is integrable and the weight function α(s) is given by α(s)=δτ(s), then the system (3) with a constant time delay τ(t)τ becomes the one with discrete time delay:

    xi(t)t=vi(t),i=1,,N,t>0,vi(t)t=Nk=1ϕ(xk(tτ),xi(t))(vk(tτ)vi(t)).

    Note the above system is studied in [13]. For this system, the assumption (9) reduces to

    dV(0)+0τdV(z)dz<βNdX(τ)+Rvτψ(z)dz.

    Since dV(z)0, the left hand side of the above inequality increases as the size of time delay τ>0 increases. On the other hand, the integrand on the right hand side decreases as τ increases. This asserts that small size of the time delay provides a larger set of initial data.

    Remark 2. Observe that our theorem above gives a flocking result when the number of agents N is greater than two. The result for N=2 is trivial for our normalized model.

    Remark 3. Note that 0<βN1 as N. Then, this implies that for any β(0,1) there exists NN such that βNβ for NN.

    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 Rv>0 be given by (8) and let (x,v) be a local-in-time C1-solution of the system (3)–(5). Then the solution is global in time and satisfies

    max1iN|vi(t)|Rvfor tτ0.

    Proof. Let us fix ϵ>0 and set

    Sϵ:={t>0:max1iN|vi(s)|<Rv+ϵ s[0,t)}.

    By continuity, Sϵ. Let us denote Tϵ:=supSϵ>0. We want to show that Tϵ=+. Arguing by contradiction, let us suppose Tϵ finite. This gives, by continuity,

    max1iN|vi(Tϵ)|=Rv+ϵ. (10)

    From (3)–(5), for t<Tϵ and i=1,,N, we have

    |vi(t)|2t2h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))(|vk(s)||vi(t)||vi(t)|2)ds2h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))max1kN|vk(s)||vi(t)|ds2|vi(t)|2.

    Note that

    max1kN|vk(s)|Rv+ϵandNk=1ϕ(xk(s),xi(t))=1,

    for s[tτ(t),t] with t<Tϵ. Thus we obtain

    |vi(t)|2t2[(Rv+ϵ)|vi(t)||vi(t)|2],

    which gives

    |vi(t)|t(Rv+ϵ)|vi(t)|. (11)

    From (11) we obtain

    limtTϵ max1iN|vi(t)|eTϵ(max1iN|vi(0)|Rvϵ)+Rv+ϵ<Rv+ϵ.

    This is in contradiction with (10). Therefore, Tϵ=+. Being ϵ>0 arbitrary, the claim is proved.

    In the lemma below, motivated from [13,38] we derive the differential inequalities for dX and dV. We notice that the diameter functions dX and dV are not C1 in general. Thus we introduce the upper Dini derivative to consider the time derivative of these functions: for a given function F=F(t), the upper Dini derivative of F at t is defined by

    D+F(t):=lim suph0+F(t+h)F(t)h.

    Note that the Dini derivative coincides with the usual derivative when the function is differentiable at t.

    Lemma 2.4. Let Rv>0 be given by (8) and let (x,v) be the global C1-solution of the system (3)–(5) constructed in Lemma 2.3. Then, for almost all t>0, we have

    |D+dX(t)|dV(t),D+dV(t)1h(t)ttτ(t)α(ts)(1βNψ(dX(s)+Rvτ0))dV(s)dsdV(t).

    Proof. The first inequality is by now standard, Then, we concentrate on the second one. Due to the continuity of the velocity trajectories vi(t), there is an at most countable system of open, mutually disjoint intervals {Iσ}σN such that

    σN¯Iσ=[0,)

    and thus for each σN there exist indices i(σ), j(σ) such that

    dV(t)=|vi(σ)(t)vj(σ)(t)|for tIσ.

    Then, by using the simplified notation i:=i(σ), j:=j(σ) (we may assume that ij), we have for every tIσ,

    12D+d2V(t)=(vi(t)vj(t))(vi(t)tvj(t)t)=(vi(t)vj(t))(1h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))vk(s)ds1h(t)Nk=1ttτ(t)α(ts)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):=Nk=1ϕkij(s,t).

    Note that, from the definition (4) of ϕ(xk(s),xi(t)), it results ˉϕij(s,t)<1 for all ij, 1i,jN and for all t>0, s[tτ(t),t]. Then, we get

    1h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))vk(s)ds1h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xj(t))vk(s)ds=1h(t)Nk=1ttτ(t)α(ts)(ϕ(xk(s),xi(t))ϕkij(s,t))vk(s)ds1h(t)Nk=1ttτ(t)α(ts)(ϕ(xk(s),xj(t))ϕkij(s,t))vk(s)ds=1h(t)ttτ(t)α(ts)(1ˉϕij(s,t))Nk=1(akij(s,t)akji(s,t))vk(s)ds, (13)

    where

    akij(s,t)=ϕ(xk(s),xi(t))ϕkij(s,t)1ˉϕij(s,t),ij, 1i,j,kN.

    Observe that akij(s,t)0 and Nk=1akij(s,t)=1. Thus, if we consider the convex hull of a finite velocity point set {v1(t),,vN(t)} and denote it by Ω(t), then

    Nk=1akij(s,t)vk(s)Ω(s)for all 1ijN.

    This gives

    |Nk=1(akij(s,t)akji(s,t))vk(s)|dV(s),

    which, used in (13), implies

    1h(t)|Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))vk(s)dsNk=1ttτ(t)α(ts)ϕ(xk(s),xj(t))vk(s)ds|1h(t)ttτ(t)α(ts)(1ˉϕij(s,t))dV(s)ds. (14)

    Now, by using the first equation in (3), we estimate for any 1i,kN and s[tτ(t),t],

    |xk(s)xi(t)|=|xk(s)xi(s)tsddtxi(z)dz||xk(s)xi(s)|+τ0supz[tτ(t),t]|vi(z)|.

    Then, Lemma 2.3 gives

    |xk(s)xi(t)|dX(s)+Rvτ0,fors[tτ(t),t],

    and due to the monotonicity property of the influence function ψ, for ki, we deduce

    ϕ(xk(s),xi(t))ψ(dX(s)+Rvτ0)N1. (15)

    On the other hand, we find

    ˉϕij=Nk=1ϕkij=ki,jϕkij+ϕiij+ϕjij=ki,jϕkij.

    Then, from (15), we obtain

    ˉϕij(s,t)N2N1ψ(dX(s)+Rvτ0)=βNψ(dX(s)+Rvτ0).

    Using the last estimate in (14), we have

    1h(t)|Nk=1ttτ(t)α(ts)ϕ(xk(s),xi(t))vk(s)ds1h(t)Nk=1ttτ(t)α(ts)ϕ(xk(s),xj(t))vk(s)ds|1h(t)ttτ(t)α(ts)(1βNψ(dX(s)+Rvτ0))dV(s)ds,

    that, used in (12), concludes the proof.

    Lemma 2.5. Let u be a nonnegative, continuous and piecewise C1-function satisfying, for some constant 0<a<1, the differential inequality

    ddtu(t)ah(t)ttτ(t)α(ts)u(s)dsu(t)for almost all t>0. (16)

    Then we have

    u(t)supt[τ0,0]u(s)eγtfor all t0,

    with γ=1τ10H(aτ0exp(τ0)), where H is the product logarithm function, i.e., H satisfies z=H(z)exp(H(z)) for any z0.

    Proof. Note that the differential inequality (16) implies

    ddtu(t)asups[tτ0,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 t>0, we introduce the following Lyapunov functional for the system (3)–(5):

    L(t):=h(t)dV(t)+βNτ(t)0α(s)|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|ds+τ(t)0α(s)(0sdV(t+z)dz)ds=:h(t)dV(t)+L1(t)+L2(t),

    where Rv is given by (8) and the diameters dX(t), dV(t) are defined in (7). We first estimate L1 as

    D+L1(t)=βN{τ(t)α(τ(t))|dX(tτ(t))+Rvτ0dX(τ(t))+Rvτ0ψ(z)dz|+τ(t)0α(s) sign (dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz)ψ(dX(ts)+Rvτ0)D+dX(ts)ds}βNτ(t)0α(s)ψ(dX(ts)+Rvτ0)dV(ts)ds, (17)

    for almost all t0, due to (1) and the first inequality in Lemma 2.4. Here sgn() is the signum function defined by

    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(ts))ds=h(t)dV(t)τ(t)0α(s)dV(ts)ds, (18)

    for almost all t0. Then, from Lemma 2.4, (17) and (18), we have, for almost all t>0,

    D+L(t)h(t)dV(t)+τ(t)0α(s)(1βNψ(dX(ts)+Rvτ0))dV(ts)dsh(t)dV(t)+βNτ(t)0α(s)ψ(dX(ts)+Rvτ0)dV(ts)ds+h(t)dV(t)τ(t)0α(s)dV(ts)ds=h(t)dV(t).

    On the other hand, since h(t)=α(τ(t))τ(t)0, we have D+(h(t)L(t))0, namely

    h(t)dV(t)+βNτ(t)0α(s)|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|ds+τ(t)0α(s)(0sdV(t+z)dz)dsh(0)dV(0)+τ00α(s)(0sdV(z)dz)ds. (19)

    Moreover, it follows from the assumption (9) that there exists a positive constant d>dX(s)+Rvτ0 such that

    h(0)dV(0)+τ00α(s)(0sdV(z)dz)dsβNτ0α(s)ddX(s)+Rvτ0ψ(z)dzds.

    This, together with (19) and (2), implies

    h(t)dV(t)+τ(t)0α(s)(0sdV(t+z)dz)dsβN{τ0α(s)ddX(s)+Rvτ0ψ(z)dzdsτ0α(s)|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|ds}=βNτ0α(s){ddX(s)+Rvτ0ψ(z)dz|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|}ds. (20)

    Now, observe that, if dX(ts)dX(s), then

    τ0α(s){ddX(s)+Rvτ0ψ(z)dz|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|}ds=τ0α(s)ddX(ts)+Rvτ0ψ(z)dzds. (21)

    Similarly, when dX(ts)<dX(s), then

    τ0α(s){ddX(s)+Rvτ0ψ(z)dz|dX(ts)+Rvτ0dX(s)+Rvτ0ψ(z)dz|}dsτ0α(s)ddX(s)+Rvτ0ψ(z)dzdsτ0α(s)ddX(ts)+Rvτ0ψ(z)dzds. (22)

    Thus, from (20), (21) and (22) we deduce that

    h(t)dV(t)+τ(t)0α(s)(0sdV(t+z)dz)dsβNτ0α(s)ddX(ts)+Rvτ0ψ(z)dzds.

    Note that, for s[0,τ(t)],

    dX(ts)=dX(t)+tstD+dX(z)dzdX(t)+2Rvτ0,

    due to Lemma 2.3 and the first inequality of Lemma 2.4. Analogously, we also find for s[0,τ(t)] that

    dX(t)=dX(ts)+ttsD+dX(z)dzdX(ts)+2Rvτ0.

    This gives

    dX(ts)2Rvτ0dX(t)dX(ts)+2Rvτ0,fors[0,τ(t)]. (23)

    Thus we get

    τ0α(s)ddX(ts)+Rvτ0ψ(z)dzdsτ0α(s)dmax{dX(t)Rvτ0,0}ψ(z)dzdsh(0)dmax{dX(t)Rvτ0,0}ψ(z)dz.

    Combining this and (20), we obtain

    h(t)dV(t)+τ(t)0α(s)(0sdV(t+z)dz)dsh(0)βNdmax{dX(t)Rvτ0,0}ψ(z)dz.

    Since the left hand side of the above inequality is positive, we have

    dX(t)d+Rvτ0fort0.

    We then again use (23) to find

    dX(ts)+Rvτ0dX(t)+3Rvτ0d+4Rvτ0,

    for s[0,τ(t)] and t0. Hence, by Lemma 2.4 together with the monotonicity of ψ we have

    D+dV(t)(1ψβN)h(t)ttτ(t)α(ts)dV(s)dsdV(t),

    for almost all t>0, where ψ=ψ(d+4Rvτ0). We finally apply Lemma 2.5 to complete the proof. Note that, in order to have an exponential decay rate of dV independent of N, it is sufficient to observe that βN1/2 for N3.

    In this section, we are interested in the behavior of solutions to the particle system (3) as the number of particles N goes to infinity. At the formal level, we can derive the following delayed Vlasov alignment equation from (3) as N:

    tft+vxft+v(1h(t)ttτ(t)α(ts)F[fs]dsft)=0, (24)

    for (x,v)Rd×Rd, t>0, with the initial data:

    fs(x,v)=gs(x,v),(x,v)Rd×Rd,s[τ0,0],

    where ft=ft(x,v) is the one-particle distribution function on the phase space Rd×Rd. Here the interaction term F[fs] is given by

    F[fs](x,v):=Rd×Rdψ(|xy|)(wv)fs(y,w)dydwRd×Rdψ(|xy|)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 T>0, we call ftC([0,T);P1(Rd×Rd)) a measure-valued solution of the equation (24) on the time interval [0,T), subject to the initial datum gsC([τ0,0];P1(Rd×Rd)), if for all compactly supported test functions ξCc(Rd×Rd×[0,T)),

    T0Rd×Rdft(tξ+vxξ+1h(t)ttτ(t)α(ts)F[fs]dsvξ)dxdvdt+Rd×Rdg0(x,v)ξ(x,v,0)dxdv=0,

    where P1(Rd×Rd) denotes the set of probability measures on the phase space Rd×Rd with bounded first-order moment and we adopt the notation ftτ0:≡gtτ0 for t[0,τ0].

    We next introduce the 1-Wasserstein distance.

    Definition 3.2. Let ρ1,ρ2P1(Rd) be two probability measures on Rd. Then 1-Waserstein distance between ρ1 and ρ2 is defined as

    W1(ρ1,ρ2):=infπΠ(ρ1,ρ2)Rd×Rd|xy|dπ(x,y),

    where Π(ρ1,ρ2) represents the collection of all probability measures on Rd×Rd with marginals ρ1 and ρ2 on the first and second factors, respectively.

    Theorem 3.3. Let the initial datum gtC([τ0,0];P1(Rd×Rd)) and assume that there exists a constant R>0 such that

    supp gtB2d(0,R)for all t[τ0,0],

    where B2d(0,R) denotes the ball of radius R in Rd×Rd, centered at the origin.

    Then for any T>0, the delayed Vlasov alignment equation (24) admits a unique measure-valued solution ftC([0,T);P1(Rd×Rd)) in the sense of Definition 3.1.

    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 ftC([0,T];P1(Rd×Rd)) be such that

    supp ftB2d(0,R)for all t[0,T],

    for some positive constant R>0. Then we use the similar estimate as in [13,Lemma 3.1] to have that there exists a constant C>0 such that

    |F[ft](x,v)|Cand|F[ft](x,v)F[ft](˜x,˜v)|C(|x˜x|+|v˜v|),

    for (x,v)B2d(0,R),t[0,T]. Subsequently, this gives

    |1h(t)ttτ(t)α(ts)F[fs]ds|C

    and

    |1h(t)ttτ(t)α(ts)(F[fs](x,v)F[fs](˜x,˜v))ds|C

    for (x,v)B2d(0,R),t[0,T]. This, together with an application of [4,Theorem 3.10], provides the local-in-time existence and uniqueness of measure-valued solutions to the equation (24) in the sense of Definition 3.1. On the other hand, once we obtain the growth estimates of support of ft in both position and velocity, these local-in-time solutions can be extended to the global-in-time ones. Thus in the rest of this proof, we provide the support estimate of ft in both position and velocity. We set

    RX[ft]:=maxx¯suppxft|x|,RV[ft]:=maxv¯suppvft|v|,

    for t[0,T], where suppxft and suppvft represent x- and v-projections of suppft, respectively. We also set

    RtX:=maxτ0stRX[fs],RtV:=maxτ0stRV[fs]. (25)

    We first construct the system of characteristics

    Z(t;x,v):=(X(t;x,v),V(t;x,v)):[0,τ0]×Rd×RdRd×Rd

    associated with (24),

    X(t;x,v)t=V(t;x,v),V(t;x,v)t=1h(t)ttτ(t)α(ts)F[fs](Z(t;x,v))ds, (26)

    where we again adopt the notation ftτ0:≡gtτ0 for t[0,τ0]. The system (26) is considered subject to the initial conditions

    X(0;x,v)=x,V(0;x,v)=v, (27)

    for all (x,v)Rd×Rd. Note that the characteristic system (26)–(27) is well-defined since the F[fs] is locally both bounded and Lipschitz. We can rewrite the equation for V as

    dV(t;x,v)dt =1h(t)ttτ(t)α(ts)(Rd×Rdψ(|X(t;x,v)y|)wdfs(y,w)Rd×Rdψ(|X(t;x,v)y|)s(y,w))dsV(t;x,v)=1h(t)τ(t)0α(s)(Rd×Rdψ(|X(t;x,v)y|)wdfts(y,w)Rd×Rdψ(|X(t;x,v)y|)dfts(y,w))dsV(t;x,v).

    Then, arguing as in the proof of Lemma 2.3, we get

    d|V(t)|dtRtV|V(t)|,

    due to (25). Using again a similar argument as in the proof of Lemma 2.3 and the comparison lemma, we obtain

    RtVR0Vfort0,

    which further implies RtXR0X+tR0V for t0. This completes the proof.

    In this subsection, we discuss the rigorous derivation of the delayed Vlasov alignment equation (24) from the particle system (3) as N. For this, we first provide the stability of measure-valued solutions of (24).

    Theorem 3.4. Let fitC([0,T);P1(Rd×Rd)), i=1,2, be two measure-valued solutions of (24) on the time interval [0,T], subject to the compactly supported initial data gisC([τ0,0];P1(Rd×Rd)). Then there exists a constant C=C(T) such that

    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)ttτ(t)α(ts)W1(f1s,f2s)ds).

    Then we have

    W1(f1t,f2t)e2CTmaxs[τ0,0]W1(g1s,g2s),

    for t[0,T).

    Remark 4. Since the empirical measure

    fNt(x,v):=1NNi=1δ(xNi(t),vNi(t))(x,v),

    associated to the N-particle system (3) with the initial data

    gNs(x,v):=1NNi=1δ(x0i(s),v0i(s))(x,v)

    for s[τ0,0], is a solution to the equation (24), we can use Theorem 3.4 to have the following mean-field limit estimate:

    supt[0,T)W1(ft,fNt)Cmaxs[τ0,0]W1(gs,gNs),

    where C is a positive constant independent of N.

    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 gP1(Rd×Rd),

    dX[g]:=diam(suppxg),dV[g]:=diam(suppvg),

    where suppxf denotes the x-projection of suppf and similarly for suppvf.

    Theorem 3.5. Let T>0 and ftC([0,T);P1(Rd×Rd)) be a weak solution of (24) on the time interval [0,T) with compactly supported initial data gsC([τ0,0];P1(Rd×Rd)). Furthermore, we assume that

    h(0)dV[g0]+τ00α(s)(0sdV[gz]dz)ds<τ0α(s)(dX[gs]+R0Vτ0ψ(z)dz)ds. (28)

    Then the weak solution ft satisfies

    dV[ft](maxs[τ0,0]dV[gs])eCtfor t0,supt0dX[ft]<,

    where C is a positive constant independent of t.

    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}NN, which is a family of N-particle approximations of gs, i.e.,

    gNs(x,v):=1NNi=1δ(x0i(s),v0i(s))(x,v)for s[τ0,0],

    where the x0i,v0iC([τ0,0];Rd) are chosen such that

    maxs[τ0,0]W1(gNs,gs)0asN.

    Note that we can choose (x0i,v0i)i=1,,N such that the condition (9) is satisfied uniformly in NN due to the assumption (28). Let us denote by (xNi,vNi)i=1,,N the solution of the N-particle system (3)–(5) subject to the initial data (x0i,v0i)i=1,,N constructed above. Then it follows from Theorem 2.2 that there exists a positive constant C1>0 such that

    dV(t)(maxs[τ0,0]dV(s))eC1tfor t0,

    with the diameters dV, dX defined in (7), where C1>0 is independent of t and N. Note that the empirical measure

    fNt(x,v):=1NNi=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 T>0, we have the following stability estimate

    W1(ft,fNt)C2maxs[τ0,0]W1(gs,gNs)fort[0,T),

    where the constant C2>0 is independent of N. This yields that sending N implies dV[ft]=dV(t) on [0,T) for any fixed T>0. Thus we have

    dV[ft](maxs[τ0,0]dV[gs])eC1tfort[0,T).

    Since the uniform-in-t boundedness of dX[ft] just follows from the above exponential decay estimate, we conclude the desired result.

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