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
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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+v⋅∇xf−∇xU(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 t⩾0. 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(|x−y|)f(t,y,v∗)(v∗−v)dydv∗. |
The interaction kernel function φ(|x−y|) 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=1N∑Nj=1φ(|xi−xj|)(vj−vi)dt−∇xU(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:Rd→R+ 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(|x−y|)f(t,y,v∗i)(v∗i−vi)dydv∗. |
Then, we construct three special weighted Sobolev spaces with power ω(x,v)=(1+x2+v2)12:
L1ω(Ω):={f(t,x,v):‖f‖L1ω(Ω)<∞},H1ω(Ω):={f(t,x,v):‖f‖H1ω(Ω)<∞},X(Ω):={f(t,x,v):‖f‖X(Ω)<∞},‖f‖H1ω:=‖f‖L2ω+‖∇xf‖L2ω+‖∇vf‖L2ω,‖f‖X:=‖ω∇vf‖L2ω+‖ω∇xf‖L2ω+‖ω2f‖L2ω, |
where
‖f‖L1ω:=‖f‖L1ω(Ω)=∫Ωωf(t,x,v)dxdv,‖f‖L2ω:=‖f‖L2ω(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 0⩽f(t,x,v)∈C([0,∞);L1ω(Ω)). The function f(t,x,v) is a weak solution to system (1.1) if
∂tf+v⋅∇xf−∇xU(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|2⩽U(x)⩽A2|x|2,a|x|⩽|∇U(x)|⩽A|x|,∀x∈R3,0<a⩽A. | (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 ‖ω∇vfn(t)‖2L2ω and ‖ω∇xfn(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 0⩽f0∈X(Ω)∩L1ω(Ω) and ∫Ωf0(x,v)dxdv=1. Then, for ∀ t⩾0, we have
(1) ∫Ωf(t)dxdv=1, ‖f(t)‖L1=1, and f(t)⩾0;
(2) ‖f(t)‖L1ω⩽C(t);
(3) ‖f(t)‖2L2ω+2σ∫t0‖∇vf(τ)‖2L2ωdτ⩽Cexp(∫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
ddt‖f(t)‖L1=0, |
which implies ‖f(t)‖L1=‖f0‖L1=1,∀t⩾0. 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]dxdv−∫Ωf(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,∀t⩾0.
(2) Multiplying system (1.1) by ω and integrating by parts over Ω leads to
ddt‖ωf(t)‖L1=−∫Ωωv⋅∇xf(t,x,v)dxdv+∫Ωωx⋅∇vf(t,x,v)dxdv−∫Ωω∇v⋅(L[f]f(t,x,v))dxdv+σ∫ΩωΔvf(t,x,v)dxdv=∫Ωf(t,x,v)∇x⋅(vω)dxdv−∫Ωf(t,x,v)∇v⋅(xω)dxdv+∫Ωf(t,x,v)L[f]⋅∇vωdxdv+σ∫Ωf(t,x,v)Δvωdxdv=∫Ωf(t,x,v)L[f]⋅vωdxdv+σ∫Ωf(t,x,v)Δvωdxdv⩽Cd,σ‖ωf(t)‖L1. |
By applying Grönwall's lemma, we obtain
‖ωf(t)‖L1⩽‖ωf0‖L1eCd,σt=:C(t), |
where C(t) is the function of t depending on d, σ, and ‖f0‖L1ω.
(3) We multiply system (1.1) by 2ω2f to obtain
ddt(ω2f2)=−ω2v⋅∇x(f2)+ω2x⋅∇v(f2)−2(∇v⋅L[f])ω2f2−ω2L[f]⋅∇v(f2)+2σω2fΔvf. | (2.3) |
Then, we integrate system (2.3) by parts over Ω to obtain
ddt‖f(t)‖2L2ω=∫Ω∇x⋅(ω2v)f2(t,x,v)dxdv−∫Ω∇v⋅(ω2x)f2(t,x,v)dxdv−2∫Ω(∇v⋅L[f])ω2f2(t,x,v)dxdv+∫Ω∇v⋅(ω2L[f])f2(t,x,v)dxdv+2σ∫Ωω2f(t,x,v)Δvf(t,x,v)dxdv=2∫Ωx⋅vf2(t,x,v)dxdv−2∫Ωx⋅vf2(t,x,v)dxdv−d∫Ω(∫Ωφ(|x−y|)f(t,y,v∗)dydv∗)ω2f2(t,x,v)dxdv+2∫Ωv⋅L[f]f2(t,x,v)dxdv−2σ∫Ω∇vf(t,x,v)⋅∇v(ω2f(t,x,v))dxdv⩽Cd,σ(1+‖ωf(t)‖L1)‖f(t)‖2L2ω−2σ‖∇vf(t)‖2L2ω, | (2.4) |
where
−2σ∫Ω∇vf(t,x,v)⋅∇v(ω2f(t,x,v))dxdv=−2σ∫Ω∇vf(t,x,v)⋅(∇vω2f(t,x,v))dxdv−2σ∫Ω∇vf(t,x,v)⋅(ω2∇vf(t,x,v))dxdv=−2σ∫Ω∇vf(t,x,v)⋅(∇vω2f(t,x,v))dxdv−2σ‖∇vf(t)‖2L2ω=−σ∫Ω∇v|f(t,x,v)|2⋅∇vω2dxdv−2σ‖∇vf(t)‖2L2ω=σ∫Ω|f(t,x,v)|2Δvω2dxdv−2σ‖∇vf(t)‖2L2ω. |
Applying Grönwall's lemma to system (2.4), we obtain
‖f(t)‖2L2ω+2σ∫t0‖∇vf(τ)‖2L2ωdτ⩽‖f0‖2L2ωexp(∫t0C(τ)dτ), |
where C(t) is the function of t depending on d, σ, and ‖f0‖L1ω.
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 ∀ t⩾0, we have
(1) ‖f(t)‖2H1ω+‖ωf(t)‖2L2ω⩽Cexp(∫t0C(τ)dτ);
(2) ‖f(t)‖2X+σ∫t0‖∇vf(τ)‖2Xdτ⩽Cexp(∫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
∂t∇vf=−∇xf−v⋅∇x(∇vf)+x⋅∇v(∇vf)−∇v(∇vf)⋅L[f]+(d+1)(∫Ωφ(|x−y|)f(t,y,v∗)dydv∗)∇vf+σΔv(∇vf). | (2.5) |
Multiplying system (2.5) by 2ω2∇vf leads to
ddt(ω2|∇vf|2)=−2ω2∇vf⋅∇xf−ω2v⋅∇x(|∇vf|2)+ω2x⋅∇v(|∇vf|2)−ω2∇v(|∇vf|2)⋅L[f]+2(d+1)(∫Ωφ(|x−y|)f(t,y,v∗)dydv∗)ω2|∇vf|2+2σω2∇v(Δvf)⋅∇vf. | (2.6) |
Then we integrate system (2.6) by parts over Ω to obtain
ddt‖∇vf(t)‖2L2ω=−2∫Ωω2∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+∫Ω2x⋅v|∇vf(t,x,v)|2dxdv−∫Ω2x⋅v|∇vf(t,x,v)|2dxdv+2(d+1)∫Ω(∫Ωφ(|x−y|)f(t,y,v∗)dydv∗)ω2|∇vf(t,x,v)|2dxdv−∫Ωω2∇v(|∇vf(t,x,v)|2)⋅L[f]dxdv+2σ∫Ωω2∇v(Δvf(t,x,v))⋅∇vf(t,x,v)dxdv⩽−2∫Ωω2∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+Cd,σ(1+‖ωf(t)‖L1)‖∇vf(t)‖2L2ω−2σ‖∇2vf(t)‖2L2ω. | (2.7) |
By applying ∇x to system (1.1), we obtain
∇xft=∇vf−v⋅∇x(∇xf)+x⋅∇v(∇xf)−∇x(∇v⋅L[f])f−∇v⋅L[f]∇xf−∇x(∇vf⋅L[f])+σ∇x(Δvf). | (2.8) |
Multiplying system (2.8) by 2ω2∇xf gives
ddt(ω2|∇xf|2)=2ω2∇xf⋅∇vf−ω2v⋅∇x(|∇xf|2)+ω2x⋅∇v(|∇xf|2)−ω2∇x(∇v⋅L[f])⋅∇x(f2)−2ω2(∇v⋅L[f])|∇xf|2−ω2∇v(|∇xf|2)⋅L[f]−2ω2∇xf⋅(∇xL[f]⋅∇vf)+2σω2∇xf⋅∇x(Δvf). | (2.9) |
Then, we integrate system (2.9) by parts over Ω to obtain
ddt‖∇xf(t)‖2L2ω=2∫Ωω2∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+2∫Ω|∇xf(t,x,v)|2∇x⋅(ω2v)dxdv−2∫Ω|∇xf(t,x,v)|2∇v⋅(ω2x)dxdv+∫Ωf2(t,x,v)∇x⋅(ω2∇x(∇v⋅L[f]))dxdv−2∫Ωω2(∇v⋅L[f])|∇xf|2dxdv+∫Ω|∇xf(t,x,v)|2∇v⋅(ω2L[f])dxdv−2σd∑i,j=1∫Ω∂vi(ω2∂xjf)(∂2vixjf)dxdv−2∫Ωω2∇xf(t,x,v)⋅(∇xL[f]⋅∇vf(t,x,v))dxdv⩽2∫Ωω2∇xf(t,x,v)⋅∇vf(t,x,v)dxdv+Cd,σ(1+‖ωf(t)‖L1)‖∇xf(t)‖2L2ω+Cd‖f(t)‖2L2ω−2σ‖∇v∇xf(t)‖2L2ω−2∫Ωω2∇xf(t,x,v)⋅(∇xL[f]⋅∇vf(t,x,v))dxdv⏟A. | (2.10) |
Now we estimate the last term A on the right hand side of system (2.10).
A=−2d∑i,j=1∫Ωω2∂vif(t,x,v)∂xjf(t,x,v)∂xjL[f]idxdv=2d∑i,j=1∫Ωf(t,x,v)∂vi(ω2∂xjf(t,x,v)∂xjL[f]i)dxdv=4d∑i,j=1∫Ωf(t,x,v)vi∂xjf(t,x,v)∂xjL[f]idxdv+2d∑i,j=1∫Ωf(t,x,v)ω2∂2vixjf(t,x,v)∂xjL[f]idxdv+2d∑i,j=1∫Ωf(t,x,v)ω2∂xjf(t,x,v)∂2vixjL[f]idxdv⩽Cd,σ(1+‖ωf(t)‖L1+‖ωf(t)‖2L1)(‖f(t)‖2L2ω+‖ωf(t)‖2L2ω)+Cd‖∇xf(t)‖2L2ω+σ‖∇v∇xf(t)‖2L2ω, | (2.11) |
where we have used the σ-Young's inequality and the following facts:
∂xjL[f]i=∫Ω∂xjφ(|x−y|)f(t,y,v∗i)(v∗i−vi)dydv∗⩽C(1+v2)12∫Ωf(t,y,v∗i)(1+|v∗|2)12dydv∗⩽Cω‖ωf‖L1 |
and
∂2vixjL[f]i=∫Ω∂xjφ(|x−y|)f(t,y,v∗i)∂vi(v∗i−vi)dydv∗=−∫Ω∂xjφ(|x−y|)f(t,y,v∗i)dydv∗⩽C. |
Combining system (2.11) with (2.10) yields
ddt‖∇xf(t)‖2L2ω⩽2∫Ωω2∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+Cd,σ(1+‖ωf(t)‖L1)‖∇xf(t)‖2L2ω−σ‖∇v∇xf(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
ddt‖ωf(t)‖2L2ω+2σ‖ω∇vf(t)‖2L2ω⩽Cd,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+‖ω∇vf(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+σ∫t0(‖∇vf(τ)‖2H1ω+‖ω∇vf(τ)‖2L2ω)dτ⩽(‖f0‖H1ω+‖ωf0‖2L2ω)exp(∫t0C(τ)dτ), |
where C(t) depends on d, σ, and ‖f0‖L1ω.
(2) We estimate the first term of ‖f(t)‖2X. Multiplying system (2.5) by 2ω4∇vf gives
ddt(ω4|∇vf|2)=−2ω4∇vf⋅∇xf−ω4v⋅∇x(|∇vf|2)+ω4x⋅∇v(|∇vf|2)−ω4∇v(|∇vf|2)⋅L[f]+2(d+1)(∫Ωφ(|x−y|)f(t,y,v∗)dydv∗)ω4|∇vf|2+2σω4∇v(Δvf)⋅∇vf=:6∑k=1Ik. |
Integrating it over Ω, we can get
∫ΩI1dxdv=−2∫Ωω4∇vf(t,x,v)⋅∇xf(t,x,v)dxdv; |
∫ΩI2dxdv=∫Ω|∇vf(t,x,v)|2∇x⋅(ω4v)dxdv=4∫Ω|∇vf(t,x,v)|2ω2x⋅vdxdv; |
∫ΩI3dxdv=−∫Ω|∇vf(t,x,v)|2∇v⋅(ω4x)dxdv=−4∫Ω|∇vf(t,x,v)|2ω2x⋅vdxdv; |
∫ΩI4dxdv=∫Ω|∇vf(t,x,v)|2∇v⋅(ω4L[f])dxdv=4∫Ω|∇vf(t,x,v)|2ω2v⋅L[f]dxdv+∫Ω|∇vf(t,x,v)|2ω4(∇v⋅L[f])dxdv⩽Cd(1+‖ωf(t)‖L1)‖ω∇vf(t)‖2L2ω; |
∫ΩI5dxdv⩽Cd‖ω∇vf(t)‖2L2ω; |
∫ΩI6dxdv=−2σd∑i,j=1∫Ω∂2vivjf∂vi(ω4∂vjf)dxdv=−2σd∑i,j=1∫Ω∂2vivjf∂viω4∂vjfdxdv−2σd∑i,j=1∫Ω∂2vivjfω4∂2vivjfdxdv=−σd∑i,j=1∫Ω∂vi|∂vjf|2∂viω4dxdv−2σ‖ω∇2vf(t)‖2L2ω=σd∑i,j=1∫Ω|∂vjf|2(∂2viviω4)dxdv−2σ‖ω∇2vf(t)‖2L2ω⩽−2σ‖ω∇2vf(t)‖2L2ω+Cd,σ‖ω∇vf(t)‖2L2ω. |
Adding up the above estimates leads to
ddt‖ω∇vf(t)‖2L2ω+2σ‖ω∇2vf(t)‖2L2ω⩽−2∫Ωω4∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+Cd,σ(1+‖ωf(t)‖L1)‖ω∇vf(t)‖2L2ω. | (2.14) |
Then, we estimate the second term of ‖f(t)‖2X. Multiplying system (2.8) by 2ω4∇xf leads to
ddt(ω4|∇xf|2)=2ω4∇vf⋅∇xf−ω4v⋅∇x(|∇xf|2)+ω4x⋅∇v(|∇xf|2)−ω4∇v(|∇xf|2)⋅L[f]−ω4∇x(∇v⋅L[f])⋅∇x(f2)−2ω4(∇v⋅L[f])|∇xf|2+2σω4∇xf⋅∇x(Δvf)−2ω4∇xf⋅(∇xL[f]⋅∇vf)=:8∑k=1Jk. |
Similarly, we integrate it over Ω as follows:
∫ΩJ1dxdv=2∫Ωω4∇vf(t,x,v)⋅∇xf(t,x,v)dxdv; |
∫ΩJ2dxdv=4∫Ω|∇xf(t,x,v)|2ω2x⋅vdxdv; |
∫ΩJ3dxdv=−4∫Ω|∇xf(t,x,v)|2ω2x⋅vdxdv; |
∫ΩJ4dxdv=∫Ω|∇xf(t,x,v)|2∇v⋅(ω4L[f])dxdv=4∫Ω|∇xf(t,x,v)|2ω2v⋅L[f]dxdv+∫Ω(∇v⋅L[f])|∇xf(t,x,v)|2ω4dxdv⩽Cd(1+‖ωf(t)‖L1)‖ω∇xf(t)‖2L2ω; |
∫ΩJ5dxdv=∫Ωf2(t,x,v)∇x⋅(ω4∇x(∇v⋅L[f]))dxdv=4∫Ωf2(t,x,v)ω2x⋅∇x(∇v⋅L[f])dxdv+∫Ωf2(t,x,v)ω4Δx(∇v⋅L[f])dxdv⩽Cd‖ωf(t)‖2L2ω, |
where we use the fact max{|φ|,|φ′|,|φ″|}⩽1;
∫ΩJ6dxdv⩽Cd‖ω∇xf(t)‖2L2ω; |
∫ΩJ7dxdv=−2σd∑i,j=1∫Ω∂vi(ω4∂xjf)(∂2vixjf)dxdv=−2σd∑i,j=1∫Ω(∂viω4∂xjf)(∂2vixjf)dxdv−2σd∑i,j=1∫Ω(ω4∂2vixjf)(∂2vixjf)dxdv=−σd∑i,j=1∫Ω(∂viω4)∂vi|∂xjf|2dxdv−2σ‖ω∇v∇xf(t)‖2L2ω=σd∑i,j=1∫Ω(∂2viviω4)|∂xjf|2dxdv−2σ‖ω∇v∇xf(t)‖2L2ω⩽−2σ‖ω∇v∇xf(t)‖2L2ω+Cd,σ‖ω∇xf(t)‖2L2ω. |
Similar to the way of estimating the term A, we have
∫ΩJ8dxdv⩽Cd,σ(1+‖ωf(t)‖L1+‖ωf(t)‖2L1)(‖ω2f(t)‖2L2ω+‖ω∇xf(t)‖2L2ω)+σ‖ω∇v∇xf(t)‖2L2ω. |
Adding up the above estimates for the integrals ∫ΩJidxdv (i=1,2,⋯,8), we obtain
ddt‖ω∇xf(t)‖2L2ω+σ‖ω∇x∇vf(t)‖2L2ω⩽2∫Ωω4∇vf(t,x,v)⋅∇xf(t,x,v)dxdv+Cd,σ(1+‖ωf(t)‖L1+‖ωf(t)‖2L1)(‖ω∇xf(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
ddt‖ω2f(t)‖2L2ω+2σ‖ω2∇vf(t)‖2L2ω⩽Cd,σ(1+‖ωf(t)‖L1)‖ω2f(t)‖2L2ω. | (2.16) |
Adding up systems (2.14)–(2.16), one has
ddt‖ω2f(t)‖2X+σ‖∇vf(t)‖2X⩽Cd,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+σ∫t0‖∇vf(τ)‖2Xdτ⩽‖f0‖2Xexp(∫t0C(τ)dτ), |
where C(t) depends on d, σ, and ‖f0‖L1ω. 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)n∈N be the sequence of the nonnegative continuous functions on [0,T]. Assume that (an) satisfies
an+1(t)⩽A+B∫t0an(s)ds+C∫t0an+1(s)ds,0⩽t⩽T, |
where A,B, and C are nonnegative constants.
If A=0, there exists a constant K⩾0 depending on B,C such that
an(t)⩽Kntnn!,0⩽t⩽T,n∈N. |
If A>0, there exists a constant K⩾0 depending on A,B,C such that
an(t)⩽Kexp(Kt),0⩽t⩽T,n∈N. |
To begin with, we construct the following iteration scheme on finite time [0,T].
{∂tfn+v⋅∇xfn−x⋅∇vfn+∇v⋅(L[fn−1]fn)=σΔvfn,in[0,T]×Ω,fn(t,x,v)|t=0=f0(x,v),in{t=0}×Ω, | (3.1) |
for n⩾1. 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+v⋅∇xf1−x⋅∇vf1+∇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+v⋅∇xfn+1−x⋅∇vfn+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 f0∈C∞(Ω)∩X(Ω)∩L1ω(Ω) and ∫Ωf0(x,v)dxdv=1. Then, for ∀ t⩽T1, we have
(1) ∫Ωfn(t)dxdv=1, ‖fn(t)‖L1=1, and fn(t)⩾0;
(2) ‖ωfn(t)‖L1⩽C;
(3) ‖fn(t)‖L2ω⩽C;
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
ddt‖ωfn(t)‖L1=−∫Ωωv⋅∇xfn(t,x,v)dxdv+∫Ωωx⋅∇vfn(t,x,v)dxdv−∫Ωω∇v⋅(L[fn−1]fn(t,x,v))dxdv+σ∫ΩωΔvfn(t,x,v)dxdv=∫Ωfn(t,x,v)L[fn−1]⋅∇vωdxdv+σ∫Ωfn(t,x,v)Δvωdxdv⩽Cd‖ωfn−1(t)‖L1+Cd,σ‖ωfn(t)‖L1. | (3.3) |
Then we integrate system (3.3) over [0,t] to obtain
‖ωfn(t)‖L1⩽‖ωf0‖L1+Cd∫t0‖ωfn−1(τ)‖L1dτ+Cd,σ∫t0‖ωfn(τ)‖L1dτ. | (3.4) |
Applying Lemma 3.1 to system (3.4), one gets
‖ωfn(t)‖L1⩽Cexp(Ct),n∈N, |
where the positive constant C depends on ‖f0‖L1ω, 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 n⩾1,
‖ωfn(t)‖L1(Ω)⩽C,∀ 0⩽t⩽T1. |
(3) Following the way of the computation for equations (2.3) and (2.4), we can conclude that ‖fn(t)‖L2ω⩽C.
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 ∀ t⩽T1, we have
(1) (‖fn(t)‖2H1ω(Ω)+‖ωfn(t)‖2L2ω)⩽C;
(2) ‖fn(t)‖2X⩽C;
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+1−fn, it follows from system (3.1) that
∂thn=−v⋅∇xhn+x⋅∇vhn−∇v⋅(L[fn]hn)−∇v⋅(L[hn−1]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.
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