Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Special Issues

Global well-posedness of strong solution to the kinetic Cucker-Smale model with external potential field

  • In this paper, we studied the initial problem of the kinetic Cucker-Smale model with noise, driven by pairwise alignment interactions under the influence of external potential field. Without the general compact support or smallness assumption on the initial datum, we established the global existence of strong solution. The proof was based on weighted energy estimates.

    Citation: Linglong Du, Anqi Du, Zhengyan Luo. Global well-posedness of strong solution to the kinetic Cucker-Smale model with external potential field[J]. Networks and Heterogeneous Media, 2025, 20(2): 460-481. doi: 10.3934/nhm.2025021

    Related Papers:

    [1] 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
    [2] 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
    [3] Shenglun Yan, Wanqian Zhang, Weiyuan Zou . Multi-cluster flocking of the thermodynamic Cucker-Smale model with a unit-speed constraint under a singular kernel. Networks and Heterogeneous Media, 2024, 19(2): 547-568. doi: 10.3934/nhm.2024024
    [4] Hyunjin Ahn, Woojoo Shim . Interplay of a unit-speed constraint and time-delay in the flocking model with internal variables. Networks and Heterogeneous Media, 2024, 19(3): 1182-1230. doi: 10.3934/nhm.2024052
    [5] 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
    [6] Hyunjin Ahn, Se Eun Noh . Finite-in-time flocking of the thermodynamic Cucker–Smale model. Networks and Heterogeneous Media, 2024, 19(2): 526-546. doi: 10.3934/nhm.2024023
    [7] 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
    [8] Hyunjin Ahn . Asymptotic flocking of the relativistic Cucker–Smale model with time delay. Networks and Heterogeneous Media, 2023, 18(1): 29-47. doi: 10.3934/nhm.2023002
    [9] Hyeong-Ohk Bae, Seung Yeon Cho, Jane Yoo, Seok-Bae Yun . Effect of time delay on flocking dynamics. Networks and Heterogeneous Media, 2022, 17(5): 803-825. doi: 10.3934/nhm.2022027
    [10] 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
  • In this paper, we studied the initial problem of the kinetic Cucker-Smale model with noise, driven by pairwise alignment interactions under the influence of external potential field. Without the general compact support or smallness assumption on the initial datum, we established the global existence of strong solution. The proof was based on weighted energy estimates.



    This paper is concerned with the following kinetic Cucker-Smale model with external potential field:

    {tf+vxfxU(x)vf+v(L[f]f)=σΔvf,f(t,x,v)|t=0=f0(x,v). (1.1)

    Here, f=f(t,x,v) is the particle distribution function in space (x,v)Ω=Rd×Rd, at time t0. The function U(x)=12|x|2 represents the harmonic potential field. The constant σ>0 represents the noise strength. The alignment force L[f] is expressed in the form:

    L[f](t,x,v)=Ωφ(|xy|)f(t,y,v)(vv)dydv.

    The interaction kernel function φ(|xy|) is a positive nonincreasing C2 function. Without loss of generality, we assume that

    max{|φ|,|φ|,|φ|}1.

    The system (1.1) arises as a mean-field kinetic description of the following stochastic Cucker-Smale model with external potential field:

    {dxi=vidt,dvi=1NNj=1φ(|xixj|)(vjvi)dtxU(xi)dt+σdW(i)t,i=1,,N, (1.2)

    where the deterministic system was studied by [1]. Here, (xi(t),vi(t)) are the position and velocity pair of ith-particle, W(i)t denote independent Wiener processes, and σ is the magnitude of the noise. The communication weight function φ:RdR+ satisfies some symmetry conditions.

    The particle Cucker-Smale model was originally proposed to understand the flocking phenomena in bird populations by Cucker and Smale [2,3]. Under the "molecular chaos" assumption, Ha and Tadmor [4] derived the kinetic Cucker-Smale model formally from the particle Cucker-Smale model using the BBGKY hierarchy method, e.g., [5,6,7,8]. For large-scale particle systems, Ha and Liu [9] rigorously justified the mean-field limit from the multi-particle Cucker-Smale model to the kinetic Cucker-Smale model, utilizing tools such as measure-valued solutions and the Kantorovich-Rubinstein distance. Furthermore, Carrillo et al. [10] proved that the solutions approached exponentially fast in velocity to the mean velocity of the initial condition, while in space they converged to a translational flocking solution.

    The Cucker-Smale model has been extended to various complexities, including the presence of different network structures [11], communication mechanism [12], self-propulsion and friction forces [13], and external fields such as fluid field [14], temperature field [15,16], potential fields [1], etc. These extensions significantly influence the dynamics of the system, leading to behaviors that are markedly different from the original model. Moreover, the connection between the kinetic Cucker-Smale model and the Euler-alignment system has been rigorously explored in recent literature. For the Euler-alignment system with pressure effects, Karper et al. [17] rigorously justified the hydrodynamic limit of the kinetic Cucker-Smale flocking model. Furthermore, Poyato and Soler [18] provided detailed analysis of a compressible Euler-type equation with singular commutator, which is derived from a hyperbolic limit of the kinetic Cucker-Smale model. In the pressure-less case, the derivation from the kinetic Cucker-Smale model to the nonlocal Euler-alignment system was established by [19]. Recently, Fabisiak and Peszek [20] rigorously derived the macroscopic fractional Euler-alignment system from the kinetic Cucker-Smale equation without performing any hydrodynamic limit.

    The well-posedness of solution is a fundamental concept in the theory of partial differential equations. Previous works [21,22,23] have established the well-posedness of weak and strong solution to the kinetic Cucker-Smale model without external potentials. Recently, Jin [24] developed a unified framework to establish the well-posedness of the model with or without noise. In this paper, the global well-posedness of the noisy version of kinetic Cucker-Smale model with harmonic potential field is studied. We prove the global nonnegativity, existence, and uniqueness of the strong solution for the system (1.1). Our approach is based on a combination of weighted Sobolev spaces and approximation schemes, which have been shown to be effective in dealing with inherent nonlinear and nonlocal problems in [25,26,27].

    Notation: We denote the usual Lp norms on Ω by f(t)Lp:=f(t)Lp(Ω), and the ith element L[f]i of the vector L[f] by

    L[f]i=Ωφ(|xy|)f(t,y,vi)(vivi)dydv.

    Then, we construct three special weighted Sobolev spaces with power ω(x,v)=(1+x2+v2)12:

    L1ω(Ω):={f(t,x,v):fL1ω(Ω)<},H1ω(Ω):={f(t,x,v):fH1ω(Ω)<},X(Ω):={f(t,x,v):fX(Ω)<},fH1ω:=fL2ω+xfL2ω+vfL2ω,fX:=ωvfL2ω+ωxfL2ω+ω2fL2ω,

    where

    fL1ω:=fL1ω(Ω)=Ωωf(t,x,v)dxdv,fL2ω:=fL2ω(Ω)=(Ωω2f2(t,x,v)dxdv)12.

    In the rest of the paper, we denote Cd and Cd,σ as positive constants while subscripts are used to indicate specific dependencies of such constants.

    Definition 1.1. Let 0f(t,x,v)C([0,);L1ω(Ω)). The function f(t,x,v) is a weak solution to system (1.1) if

    tf+vxfxU(x)vf+v(L[f]f)=σΔvf,inD([0,+)×Ω).

    We say f(t,x,v) is a strong solution if f(t,x,v) is a weak solution and f(t,x,v)C([0,);H1ω(Ω)).

    Now, our main results are stated as follows.

    Theorem 1.1. Assume initial datum f0(x,v)X(Ω)L1ω(Ω). Then, the system (1.1) admits a unique strong solution in sense of Definition 1.1.

    Remark 1.1. In this paper, we consider a special potential function U=12|x|2. In fact, similar to the study in [1], the potential U can be extended to a more general case: assume that the potential function U satisfies the following conditions:

    a2|x|2U(x)A2|x|2,a|x||U(x)|A|x|,xR3,0<aA. (1.3)

    One can obtain the existence and uniqueness of a strong solution to the system (1.1) with (1.3).

    The existence of a strong solution is constructed in the weighted Sobolev space H1ω(Ω). However, to find the Cauchy sequences in this space, we need to give extra estimates for terms such as ωvfn(t)2L2ω and ωxfn(t)2L2ω. For this purpose, we construct the weighted Sobolev space X(Ω) and establish the a priori estimate for the preparation; see Proposition 3.1.

    The rest of this paper is organized as follows. In Section 2, we establish a priori estimates of the system (1.1) by taking advantage of three special weighted Sobolev spaces. In Section 3, we first prove the local existence and uniqueness of the strong solution to system (1.1) by an iteration scheme and extend the local existence to the global one.

    This section is devoted to the a priori estimates for the system (1.1). Directly integrating system (1.1) over [0,t]×Ω gives that any smooth solution of it satisfies the following conservation law: Ωf0(x,v)dxdv=Ωf(t,x,v)dxdv=:Ωf(t)dxdv. Therefore, w.o.l.g., we assume Ωf0(x,v)dxdv=1 in the rest of paper. We first give the following a priori estimates for f(t)L1ω and f(t)L2ω, which are better integrabilities of f with large v and x.

    Lemma 2.1. Assume the function f(t,x,v) is a smooth solution to system (1.1) with initial datum 0f0X(Ω)L1ω(Ω) and Ωf0(x,v)dxdv=1. Then, for  t0, we have

    (1) Ωf(t)dxdv=1, f(t)L1=1, and f(t)0;

    (2) f(t)L1ωC(t);

    (3) f(t)2L2ω+2σt0vf(τ)2L2ωdτCexp(t0C(τ)dτ);

    where C and C(t) are positive constant and positive continuous functions of t both depending on σ,d and the weighted norms of the initial datum f0.

    Proof. (1) We multiply system (1.1) by sgn(f) and integrate it over Ω to obtain

    ddtf(t)L1=0,

    which implies f(t)L1=f0L1=1,t0. Note that

    Ωf(t)dxdv=Ωf(t)1[f(t)0]dxdv+Ωf(t)1[f(t)<0]dxdv=1, (2.1)
    f(t)L1=Ωf(t)1[f(t)0]dxdvΩf(t)1[f(t)<0]dxdv=1. (2.2)

    We subtract system (2.1) from (2.2) to obtain Ωf(t)1[f(t)<0]dxdv=0, which gives f(t)0,t0.

    (2) Multiplying system (1.1) by ω and integrating by parts over Ω leads to

    ddtωf(t)L1=Ωωvxf(t,x,v)dxdv+Ωωxvf(t,x,v)dxdvΩωv(L[f]f(t,x,v))dxdv+σΩωΔvf(t,x,v)dxdv=Ωf(t,x,v)x(vω)dxdvΩf(t,x,v)v(xω)dxdv+Ωf(t,x,v)L[f]vωdxdv+σΩf(t,x,v)Δvωdxdv=Ωf(t,x,v)L[f]vωdxdv+σΩf(t,x,v)ΔvωdxdvCd,σωf(t)L1.

    By applying Grönwall's lemma, we obtain

    ωf(t)L1ωf0L1eCd,σt=:C(t),

    where C(t) is the function of t depending on d, σ, and f0L1ω.

    (3) We multiply system (1.1) by 2ω2f to obtain

    ddt(ω2f2)=ω2vx(f2)+ω2xv(f2)2(vL[f])ω2f2ω2L[f]v(f2)+2σω2fΔvf. (2.3)

    Then, we integrate system (2.3) by parts over Ω to obtain

    ddtf(t)2L2ω=Ωx(ω2v)f2(t,x,v)dxdvΩv(ω2x)f2(t,x,v)dxdv2Ω(vL[f])ω2f2(t,x,v)dxdv+Ωv(ω2L[f])f2(t,x,v)dxdv+2σΩω2f(t,x,v)Δvf(t,x,v)dxdv=2Ωxvf2(t,x,v)dxdv2Ωxvf2(t,x,v)dxdvdΩ(Ωφ(|xy|)f(t,y,v)dydv)ω2f2(t,x,v)dxdv+2ΩvL[f]f2(t,x,v)dxdv2σΩvf(t,x,v)v(ω2f(t,x,v))dxdvCd,σ(1+ωf(t)L1)f(t)2L2ω2σvf(t)2L2ω, (2.4)

    where

    2σΩvf(t,x,v)v(ω2f(t,x,v))dxdv=2σΩvf(t,x,v)(vω2f(t,x,v))dxdv2σΩvf(t,x,v)(ω2vf(t,x,v))dxdv=2σΩvf(t,x,v)(vω2f(t,x,v))dxdv2σvf(t)2L2ω=σΩv|f(t,x,v)|2vω2dxdv2σvf(t)2L2ω=σΩ|f(t,x,v)|2Δvω2dxdv2σvf(t)2L2ω.

    Applying Grönwall's lemma to system (2.4), we obtain

    f(t)2L2ω+2σt0vf(τ)2L2ωdτf02L2ωexp(t0C(τ)dτ),

    where C(t) is the function of t depending on d, σ, and f0L1ω.

    Now we derive the estimates in the weighted Sobolev spaces. The first weighted space H1ω(Ω) is prepared for the strong solution, while the second space X(Ω) is constructed to estimate the approximate solutions.

    Proposition 2.1. Assume the function f(t,x,v) is a smooth solution to system (1.1) with initial datum satisfying the condition of Lemma 2.1. Then, for  t0, we have

    (1) f(t)2H1ω+ωf(t)2L2ωCexp(t0C(τ)dτ);

    (2) f(t)2X+σt0vf(τ)2XdτCexp(t0C(τ)dτ);

    where C and C(t) are positive constant and positive continuous functions of t both depending on σ,d and the weighted norms of the initial datum f0.

    Proof. (1) Applying v to system (1.1) gives

    tvf=xfvx(vf)+xv(vf)v(vf)L[f]+(d+1)(Ωφ(|xy|)f(t,y,v)dydv)vf+σΔv(vf). (2.5)

    Multiplying system (2.5) by 2ω2vf leads to

    ddt(ω2|vf|2)=2ω2vfxfω2vx(|vf|2)+ω2xv(|vf|2)ω2v(|vf|2)L[f]+2(d+1)(Ωφ(|xy|)f(t,y,v)dydv)ω2|vf|2+2σω2v(Δvf)vf. (2.6)

    Then we integrate system (2.6) by parts over Ω to obtain

    ddtvf(t)2L2ω=2Ωω2vf(t,x,v)xf(t,x,v)dxdv+Ω2xv|vf(t,x,v)|2dxdvΩ2xv|vf(t,x,v)|2dxdv+2(d+1)Ω(Ωφ(|xy|)f(t,y,v)dydv)ω2|vf(t,x,v)|2dxdvΩω2v(|vf(t,x,v)|2)L[f]dxdv+2σΩω2v(Δvf(t,x,v))vf(t,x,v)dxdv2Ωω2vf(t,x,v)xf(t,x,v)dxdv+Cd,σ(1+ωf(t)L1)vf(t)2L2ω2σ2vf(t)2L2ω. (2.7)

    By applying x to system (1.1), we obtain

    xft=vfvx(xf)+xv(xf)x(vL[f])fvL[f]xfx(vfL[f])+σx(Δvf). (2.8)

    Multiplying system (2.8) by 2ω2xf gives

    ddt(ω2|xf|2)=2ω2xfvfω2vx(|xf|2)+ω2xv(|xf|2)ω2x(vL[f])x(f2)2ω2(vL[f])|xf|2ω2v(|xf|2)L[f]2ω2xf(xL[f]vf)+2σω2xfx(Δvf). (2.9)

    Then, we integrate system (2.9) by parts over Ω to obtain

    ddtxf(t)2L2ω=2Ωω2vf(t,x,v)xf(t,x,v)dxdv+2Ω|xf(t,x,v)|2x(ω2v)dxdv2Ω|xf(t,x,v)|2v(ω2x)dxdv+Ωf2(t,x,v)x(ω2x(vL[f]))dxdv2Ωω2(vL[f])|xf|2dxdv+Ω|xf(t,x,v)|2v(ω2L[f])dxdv2σdi,j=1Ωvi(ω2xjf)(2vixjf)dxdv2Ωω2xf(t,x,v)(xL[f]vf(t,x,v))dxdv2Ωω2xf(t,x,v)vf(t,x,v)dxdv+Cd,σ(1+ωf(t)L1)xf(t)2L2ω+Cdf(t)2L2ω2σvxf(t)2L2ω2Ωω2xf(t,x,v)(xL[f]vf(t,x,v))dxdvA. (2.10)

    Now we estimate the last term A on the right hand side of system (2.10).

    A=2di,j=1Ωω2vif(t,x,v)xjf(t,x,v)xjL[f]idxdv=2di,j=1Ωf(t,x,v)vi(ω2xjf(t,x,v)xjL[f]i)dxdv=4di,j=1Ωf(t,x,v)vixjf(t,x,v)xjL[f]idxdv+2di,j=1Ωf(t,x,v)ω22vixjf(t,x,v)xjL[f]idxdv+2di,j=1Ωf(t,x,v)ω2xjf(t,x,v)2vixjL[f]idxdvCd,σ(1+ωf(t)L1+ωf(t)2L1)(f(t)2L2ω+ωf(t)2L2ω)+Cdxf(t)2L2ω+σvxf(t)2L2ω, (2.11)

    where we have used the σ-Young's inequality and the following facts:

    xjL[f]i=Ωxjφ(|xy|)f(t,y,vi)(vivi)dydvC(1+v2)12Ωf(t,y,vi)(1+|v|2)12dydvCωωfL1

    and

    2vixjL[f]i=Ωxjφ(|xy|)f(t,y,vi)vi(vivi)dydv=Ωxjφ(|xy|)f(t,y,vi)dydvC.

    Combining system (2.11) with (2.10) yields

    ddtxf(t)2L2ω2Ωω2vf(t,x,v)xf(t,x,v)dxdv+Cd,σ(1+ωf(t)L1)xf(t)2L2ωσvxf(t)2L2ω+Cd,σ(1+ωf(t)L1+ωf(t)2L1)(f(t)2L2ω+ωf(t)2L2ω). (2.12)

    Due to the term ωf(t)2L2ω which appeared in system (2.12), we need to analyze it in detail to close the a priori estimate in H1ω. Similarly to the estimate of system (2.4), we multiply system (1.1) by 2ω4f and integrate it over Ω to have

    ddtωf(t)2L2ω+2σωvf(t)2L2ωCd,σ(1+ωf(t)L1)ωf(t)2L2ω. (2.13)

    Adding up systems (2.4), (2.7), (2.12), and (2.13), we can get

    ddt(f(t)2H1ω+ωf(t)2L2ω)+σ(vf(t)2H1ω+ωvf(t)2L2ω)Cd,σ(1+ωf(t)L1+ωf(t)2L1)(f(t)2H1ω+ωf(t)2L2ω).

    By Grönwall's lemma and the estimate in Lemma 2.1, we obtain

    f(t)2H1ω+ωf(t)2L2ω+σt0(vf(τ)2H1ω+ωvf(τ)2L2ω)dτ(f0H1ω+ωf02L2ω)exp(t0C(τ)dτ),

    where C(t) depends on d, σ, and f0L1ω.

    (2) We estimate the first term of f(t)2X. Multiplying system (2.5) by 2ω4vf gives

    ddt(ω4|vf|2)=2ω4vfxfω4vx(|vf|2)+ω4xv(|vf|2)ω4v(|vf|2)L[f]+2(d+1)(Ωφ(|xy|)f(t,y,v)dydv)ω4|vf|2+2σω4v(Δvf)vf=:6k=1Ik.

    Integrating it over Ω, we can get

    ΩI1dxdv=2Ωω4vf(t,x,v)xf(t,x,v)dxdv;
    ΩI2dxdv=Ω|vf(t,x,v)|2x(ω4v)dxdv=4Ω|vf(t,x,v)|2ω2xvdxdv;
    ΩI3dxdv=Ω|vf(t,x,v)|2v(ω4x)dxdv=4Ω|vf(t,x,v)|2ω2xvdxdv;
    ΩI4dxdv=Ω|vf(t,x,v)|2v(ω4L[f])dxdv=4Ω|vf(t,x,v)|2ω2vL[f]dxdv+Ω|vf(t,x,v)|2ω4(vL[f])dxdvCd(1+ωf(t)L1)ωvf(t)2L2ω;
    ΩI5dxdvCdωvf(t)2L2ω;
    ΩI6dxdv=2σdi,j=1Ω2vivjfvi(ω4vjf)dxdv=2σdi,j=1Ω2vivjfviω4vjfdxdv2σdi,j=1Ω2vivjfω42vivjfdxdv=σdi,j=1Ωvi|vjf|2viω4dxdv2σω2vf(t)2L2ω=σdi,j=1Ω|vjf|2(2viviω4)dxdv2σω2vf(t)2L2ω2σω2vf(t)2L2ω+Cd,σωvf(t)2L2ω.

    Adding up the above estimates leads to

    ddtωvf(t)2L2ω+2σω2vf(t)2L2ω2Ωω4vf(t,x,v)xf(t,x,v)dxdv+Cd,σ(1+ωf(t)L1)ωvf(t)2L2ω. (2.14)

    Then, we estimate the second term of f(t)2X. Multiplying system (2.8) by 2ω4xf leads to

    ddt(ω4|xf|2)=2ω4vfxfω4vx(|xf|2)+ω4xv(|xf|2)ω4v(|xf|2)L[f]ω4x(vL[f])x(f2)2ω4(vL[f])|xf|2+2σω4xfx(Δvf)2ω4xf(xL[f]vf)=:8k=1Jk.

    Similarly, we integrate it over Ω as follows:

    ΩJ1dxdv=2Ωω4vf(t,x,v)xf(t,x,v)dxdv;
    ΩJ2dxdv=4Ω|xf(t,x,v)|2ω2xvdxdv;
    ΩJ3dxdv=4Ω|xf(t,x,v)|2ω2xvdxdv;
    ΩJ4dxdv=Ω|xf(t,x,v)|2v(ω4L[f])dxdv=4Ω|xf(t,x,v)|2ω2vL[f]dxdv+Ω(vL[f])|xf(t,x,v)|2ω4dxdvCd(1+ωf(t)L1)ωxf(t)2L2ω;
    ΩJ5dxdv=Ωf2(t,x,v)x(ω4x(vL[f]))dxdv=4Ωf2(t,x,v)ω2xx(vL[f])dxdv+Ωf2(t,x,v)ω4Δx(vL[f])dxdvCdωf(t)2L2ω,

    where we use the fact max{|φ|,|φ|,|φ|}1;

    ΩJ6dxdvCdωxf(t)2L2ω;
    ΩJ7dxdv=2σdi,j=1Ωvi(ω4xjf)(2vixjf)dxdv=2σdi,j=1Ω(viω4xjf)(2vixjf)dxdv2σdi,j=1Ω(ω42vixjf)(2vixjf)dxdv=σdi,j=1Ω(viω4)vi|xjf|2dxdv2σωvxf(t)2L2ω=σdi,j=1Ω(2viviω4)|xjf|2dxdv2σωvxf(t)2L2ω2σωvxf(t)2L2ω+Cd,σωxf(t)2L2ω.

    Similar to the way of estimating the term A, we have

    ΩJ8dxdvCd,σ(1+ωf(t)L1+ωf(t)2L1)(ω2f(t)2L2ω+ωxf(t)2L2ω)+σωvxf(t)2L2ω.

    Adding up the above estimates for the integrals ΩJidxdv (i=1,2,,8), we obtain

    ddtωxf(t)2L2ω+σωxvf(t)2L2ω2Ωω4vf(t,x,v)xf(t,x,v)dxdv+Cd,σ(1+ωf(t)L1+ωf(t)2L1)(ωxf(t)2L2ω+ω2f(t)2L2ω). (2.15)

    Finally, we consider the last term of f(t)2X. Similar to the estimate of system (2.4), we multiply system (2.5) by 2ω6f and integrate over Ω to have

    ddtω2f(t)2L2ω+2σω2vf(t)2L2ωCd,σ(1+ωf(t)L1)ω2f(t)2L2ω. (2.16)

    Adding up systems (2.14)–(2.16), one has

    ddtω2f(t)2X+σvf(t)2XCd,σ(1+ωf(t)L1+ωf(t)2L1)ω2f(t)2X.

    Applying Grönwall's lemma and the estimate f(t)L1ω in Lemma 2.1, we obtain

    f(t)2X+σt0vf(τ)2Xdτf02Xexp(t0C(τ)dτ),

    where C(t) depends on d, σ, and f0L1ω. Therefore, we finish the proof.

    In this section, we prove the local existence and uniqueness of the strong solution to system (1.1) by constructing a sequence of approximate solutions and extend the local existence to the global. First, let us recall a Grönwall-type lemma in [28].

    Lemma 3.1. Let T>0 and (an)nN be the sequence of the nonnegative continuous functions on [0,T]. Assume that (an) satisfies

    an+1(t)A+Bt0an(s)ds+Ct0an+1(s)ds,0tT,

    where A,B, and C are nonnegative constants.

    If A=0, there exists a constant K0 depending on B,C such that

    an(t)Kntnn!,0tT,nN.

    If A>0, there exists a constant K0 depending on A,B,C such that

    an(t)Kexp(Kt),0tT,nN.

    To begin with, we construct the following iteration scheme on finite time [0,T].

    {tfn+vxfnxvfn+v(L[fn1]fn)=σΔvfn,in[0,T]×Ω,fn(t,x,v)|t=0=f0(x,v),in{t=0}×Ω, (3.1)

    for n1. We define a sequence of approximate solutions {fn} as the solution to the above iterative system (3.1) by induction.

    Initial step (n=1): we set

    f0(t,x,v):=f0(x,v).

    With this, we solve the initial value problem for the Cucker-Smale model with external potential field and noise:

    tf1+vxf1xvf1+v[L(f0)f1]=σΔvf1,(x,v)Ω

    subject to

    f1(0,x,v)=f0(x,v).

    Inductive step: Suppose we have the sequence of smooth approximate solutions {fk}nk=1. Then, we can solve the following model:

    tfn+1+vxfn+1xvfn+1+v[L(fn)fn+1]=σΔvfn+1,in [0,T]×Ω, (3.2)

    subject to initial datum:

    fn+1(0,x,v)=f0(x,v).

    Thus, we can construct the smooth function {fn+1} from {fn}. The solvability of system (3.2) is similar to that in the appendix of reference [29]. Therefore, the sequence {fn} is well-defined.

    Paralleling to the a priori estimate for the solution f in Section 2, we can establish the uniform energy estimates for the approximate sequence fn in the weighted Sobolev spaces.

    Lemma 3.2. Let T1>0. Assume the function fn(t,x,v) is a smooth solution to system (3.1) with the initial datum satisfying f0C(Ω)X(Ω)L1ω(Ω) and Ωf0(x,v)dxdv=1. Then, for  tT1, we have

    (1) Ωfn(t)dxdv=1, fn(t)L1=1, and fn(t)0;

    (2) ωfn(t)L1C;

    (3) fn(t)L2ωC;

    where C denotes the positive constant only depending on σ,d,T1, and the weighted norms of the initial datum f0.

    Proof. (1) The results are obvious.

    (2) We multiply system (3.1) by ω and integrate it over Ω to obtain

    ddtωfn(t)L1=Ωωvxfn(t,x,v)dxdv+Ωωxvfn(t,x,v)dxdvΩωv(L[fn1]fn(t,x,v))dxdv+σΩωΔvfn(t,x,v)dxdv=Ωfn(t,x,v)L[fn1]vωdxdv+σΩfn(t,x,v)ΔvωdxdvCdωfn1(t)L1+Cd,σωfn(t)L1. (3.3)

    Then we integrate system (3.3) over [0,t] to obtain

    ωfn(t)L1ωf0L1+Cdt0ωfn1(τ)L1dτ+Cd,σt0ωfn(τ)L1dτ. (3.4)

    Applying Lemma 3.1 to system (3.4), one gets

    ωfn(t)L1Cexp(Ct),nN,

    where the positive constant C depends on f0L1ω, d, and σ. Then, for any given T1>0, there exists a positive C depending σ,d,T1, and the weighted norms of the initial datum f0 such that for all n1,

    ωfn(t)L1(Ω)C, 0tT1.

    (3) Following the way of the computation for equations (2.3) and (2.4), we can conclude that fn(t)L2ωC.

    Proposition 3.1. Let T1>0. Assume the function fn(t,x,v) is a smooth solution to system (3.1) with initial datum satisfying the condition of Lemma 3.2. Then, for  tT1, we have

    (1) (fn(t)2H1ω(Ω)+ωfn(t)2L2ω)C;

    (2) fn(t)2XC;

    where C denotes positive constant only depending on σ,d,T1, and the weighted norms of the initial datum f0.

    Proof. Following the similar proof of Proposition 2.1 in Section 2, we can obtain the parallel results.

    Next, we show that the approximate solution fn is the Cauchy sequence in C([0,T];L1ω(Ω)H1ω(Ω)), where 0<T<1. Setting hn:=fn+1fn, it follows from system (3.1) that

    thn=vxhn+xvhnv(L[fn]hn)v(L[hn1]fn)+σΔvhn. (3.5)

    It is obvious to see that

    hn(0,x,v)=fn+1(0,x,v)fn(0,x,v)=f0(x,v)f0(x,v)0.

    Proposition 3.2. Assume that initial datum f0(x,v)C(Ω)X(Ω)L1ω(Ω), , and . For any given positive small time , we have

    where denotes the positive constant only depending on , and the weighted norms of the initial datum .

    Furthermore, there exists a limit function such that

    Proof. For arbitrary and , multiplying system (3.5) by and integrating it over by parts lead to

    (3.6)

    where depends on , and the weighed norms of the initial datum and , since we used the conclusion of Lemma 3.2. Then, we integrate system (3.6) to obtain

    Using Lemma 3.1, we can derive that

    Then, for any given small time , we have

    (3.7)

    where denotes the positive constant only depending on , and the weighted norms of the initial datum . This means that is the Cauchy sequence in . Moreover, there exists a limit function such that

    Proposition 3.3. Assume that the initial datum satisfies the condition of Proposition 3.2. For any given positive small time , we have

    (3.8)
    (3.9)
    (3.10)

    where denotes the positive constant only depending on , and the weighted norms of the initial datum . Furthermore, there exists a limit function such that

    Proof. (1) Multiplying system (3.5) by leads to

    (3.11)

    Similar to the way to estimate system (2.3), we only need to estimate two extra terms and . Note that

    where , and we have used the -Young's inequality.

    Thus, by integrating system (3.11) over and using the estimate of Lemma 3.2 and Proposition 3.1, we have

    (3.12)

    Here, depends on , and weighed norms of initial datum.

    By applying Grönwall's lemma to system (3.12) on and using the conclusion of Proposition 3.2, we obtain

    (3.13)

    Then for any given small time , we have

    (3.14)

    Hence, we obtain the estimate system (3.8).

    (2) Multiplying system (3.5) by and integrating it over leads to

    which is similar in form to the result of system (3.12). Hence, we obtain estimate system (3.9).

    (3) Applying to system (3.5) gives

    (3.15)

    We multiply system (3.15) by and integrate it over . Comparing to system (2.7), we only need to estimate two extra terms and . Note that

    (3.16)

    where we have used the -Young's inequality.

    (3.17)

    Thus, similar to system (2.7), we use systems (3.16) and (3.17) to obtain

    (3.18)

    Applying to system (3.5) gives

    (3.19)

    We multiply system (3.19) by and integrate it over . Comparing to system (2.9), we only estimate the extra terms , , and .

    (3.20)

    Note that

    (3.21)

    where we have used the -Young's inequality.

    (3.22)

    Thus, similar to system (2.12), we use systems (3.20)–(3.22) to obtain

    (3.23)

    Adding up systems (3.12), (3.18), and (3.23), and applying Lemma 3.2-Proposition 3.2 and Proposition 3.3(2), we obtain

    Similar to systems (3.13) and (3.14), for any given positive small time , we have

    (3.24)

    which is system (3.10). This means that the approximate solution is the Cauchy sequence in Thus, it converges strongly to the limit function as

    With the help of estimates system (3.7) in Proposition 3.2 and (3.24) in Proposition 3.3, we conclude that there exists a constant depending on , and weighed norms of initial datum, such that for any given positive small time ,

    Therefore, the limit function of Cauchy sequence is a local strong solution to the system (1.1). The uniqueness of the solution can be derived easily. Let and be the two strong solutions above corresponding to the same initial datum . Set

    Then, by the same argument as in Lemma 3.2-Proposition 3.3, satisfies Grönwall's inequality:

    and the standard Grönwall's lemma implies that

    which gives the uniqueness of the local solution.

    When initial datum is smooth, the limit function from Proposition 3.2–Proposition 3.3 is the unique local smooth solution to system (1.1). Combining with Lemma 2.1 and Proposition 2.1, one can extend the local smooth solution to be global-in-time. Hence, we obtain the global smooth solution.

    When initial datum is not smooth, we first mollify the initial datum by convolution, i.e.,

    where is the standard mollifier. Then, we consider the following modified system

    Following the basic idea about the proof of Theorem 3.1 and Theorem 3.2 in [24], we can also prove that there exists a sequence , with , such that

    and satisfies system (1.1). Hence, the limit function is the desired unique strong solution.

    All authors contributed equally to the study and the writing of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Natural Science Foundation of China (No.12001097), Natural Science Foundation of Shanghai Municipality (No. 22ZR1402300), and AI-Enhanced Research Program of Shanghai Municipal Education Commission (No. SMEC-AI-DHUY-01).

    The authors declare there is no conflict of interest.



    [1] R. Shu, E. Tadmor, Flocking hydrodynamics with external potentials, Arch. Ration. Mech. Anal., 238 (2020), 347–381. https://doi.org/10.1007/s00205-020-01544 doi: 10.1007/s00205-020-01544
    [2] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852–862. https://doi.org/10.1109/TAC.2007.895842 doi: 10.1109/TAC.2007.895842
    [3] F. Cucker, S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197–227. https://doi.org/10.1007/s11537-007-0647-x doi: 10.1007/s11537-007-0647-x
    [4] S. Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415–435. https://doi.org/10.3934/krm.2008.1.415 doi: 10.3934/krm.2008.1.415
    [5] D. Benedetto, E. Caglioti, M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615–641.
    [6] G. Russo, P. Smereka, Kinetic theory for bubbly flow I: Collisionless case, SIAM J. Appl. Math., 56 (1996), 327–357. https://doi.org/10.1137/S0036139993260563 doi: 10.1137/S0036139993260563
    [7] G. Russo, P. Smereka, Kinetic theory for bubbly flow II: Fluid dynamic limit, SIAM J. Appl. Math., 56 (1996), 358–371. https://doi.org/10.1137/S0036139993260575 doi: 10.1137/S0036139993260575
    [8] J. A. Carrillo, M. Fornasier, G. Toscani, F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, In: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Boston: Birkhäuser, 2010,297–336. https://doi.org/10.1007/978-0-8176-4946-3-12
    [9] S. Y. Ha, J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297–325. http://doi.org/10.4310/CMS.2009.v7.n2.a2 doi: 10.4310/CMS.2009.v7.n2.a2
    [10] J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218–236. https://doi.org/10.1137/090757290 doi: 10.1137/090757290
    [11] Y. P. Choi, D. Oh, O. Tse, Controlled pattern formation of stochastic Cucker-Smale systems with network structures, Commun. Nonlinear Sci. Numer. Simul., 111 (2022), 106474. https://doi.org/10.1016/j.cnsns.2022.106474 doi: 10.1016/j.cnsns.2022.106474
    [12] H. Dietert, R. Shvydkoy, On Cucker-Smale dynamical systems with degenerate communication, Anal. Appl., 19 (2021), 551–573. https://doi.org/10.1142/S0219530520500050 doi: 10.1142/S0219530520500050
    [13] M. Bostan, J. A. Carrillo, Reduced fluid models for self-propelled particles interacting through alignment, Math. Models Methods Appl. Sci., 27 (2017), 1255–1299. https://doi.org/10.1142/S0218202517400152 doi: 10.1142/S0218202517400152
    [14] C. Jin, Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations, Kinet. Relat. Models, 16 (2023), 69–96. https://doi.org/10.3934/krm.2022023 doi: 10.3934/krm.2022023
    [15] H. Cho, L. L. Du, S. Y. Ha, Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble, Quart. Appl. Math., 80 (2022), 1–22. http://doi.org/10.1090/qam/1602 doi: 10.1090/qam/1602
    [16] L. L. Du, S. Y. Ha, Convergence toward a periodically rotating one-point cluster in the kinetic thermodynamic Cucker-Smale model, Commun. Math. Anal. Appl., 1 (2022), 72–111. https://doi.org/10.4208/cmaa.2021-0002 doi: 10.4208/cmaa.2021-0002
    [17] T. K. Karper, A. Mellet, K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131–163. https://doi.org/10.1142/S0218202515500050 doi: 10.1142/S0218202515500050
    [18] D. Poyato, J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089–1152. https://doi.org/10.1142/S0218202517400103 doi: 10.1142/S0218202517400103
    [19] A. Figalli, M. J. Kang, A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE, 12 (2019), 843–866. https://doi.org/10.2140/apde.2019.12.843 doi: 10.2140/apde.2019.12.843
    [20] M. Fabisiak, J. Peszek, Inevitable monokineticity of strongly singular alignment, Math. Ann., 390 (2024), 589–637. https://doi.org/10.1007/s00208-023-02776-7 doi: 10.1007/s00208-023-02776-7
    [21] J. A. Cañizo, J. A. Carrillo, J. Rosado, A Well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515–539. https://doi.org/10.1142/S0218202511005131 doi: 10.1142/S0218202511005131
    [22] Y. P. Choi, S. Y. Ha, J. Jung, J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 1–34. https://doi.org/10.1007/s00021-019-0466-x doi: 10.1007/s00021-019-0466-x
    [23] Y. P. Choi, J. Jung, On weak solutions to the kinetic Cucker–Smale model with singular communication weights, Proc. Am. Math. Soc., 152 (2024), 3423–3436. https://doi.org/10.1090/proc/16837 doi: 10.1090/proc/16837
    [24] C. Jin, Well-posedness of weak and strong solutions to the kinetic Cucker-Smale model, J. Differ. Equations, 264 (2018), 1581–1612. https://doi.org/10.1016/j.jde.2017.10.001 doi: 10.1016/j.jde.2017.10.001
    [25] B. Chazelle, Q. Jiu, Q. Li, C. Wang, Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics, J. Differ. Equations, 263 (2017), 365–397. https://doi.org/10.1016/j.jde.2017.02.036 doi: 10.1016/j.jde.2017.02.036
    [26] T. K. Karper, A. Mellet, K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2023), 215–243. https://doi.org/10.1137/120866828 doi: 10.1137/120866828
    [27] Y. S. Zhou, T. F. Zhang, Q. Ma, Local well-posedness of classical solutions for the kinetic self-organized model of Cucker-Smale type, Math. Methods Appl. Sci., 47 (2024), 7329–7349. https://doi.org/10.1002/mma.9974 doi: 10.1002/mma.9974
    [28] L. Boudin, L. Desvillettes, C. Grandmont, A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differ. Integr. Equations, 22 (2009), 1247–1271. https://www.sci-hub.ru/10.57262/die/1356019415
    [29] C. Pinheiro, G. Planas, On the -Navier-Stokes-Vlasov and the -Navier-Stokes-Vlasov-Fokker-Planck equations, J. Math. Phys., 62 (2021), 031507. https://doi.org/10.1063/5.0024394 doi: 10.1063/5.0024394
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(361) PDF downloads(27) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog