A LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC CUCKER-SMALE EQUATION WITH RANDOM INPUTS

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1 A LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC CUCKER-SMALE EQUATION WITH RANDOM INPUTS SEUNG-YEAL HA, SHI JIN, AND JINWOOK JUNG Abstract. We present a local sensitivity analysis for the kinetic Cucker-Smale C-S equation with random inputs. This is a companion work to our previous local sensitivity analysis for the particle C-S model. Random imputs in the coefficients of the kinetic C-S equation can be caused by diverse sources such as the incomplete measurement and interactions with unknown environments, and will enter the problem through the communication function or initial data. For the proposed random kinetic C-S equation, we present sufficient conditions for the pathwise well-posedness and flocking estimates. For the local sensitivity analysis, we study the propagation of regularity of the kinetic density function in random space. 1. Introduction The purpose of this paper is to extend the local sensitivity analysis [21] for the particle C- S model to the corresponding mesoscopic model. The jargon flocking denotes a collective behavior in which particles in a many-body system organize into an ordered motion using the environmental information based on simple rules [4, 5, 7, 37, 45, 46], e.g., flocking of birds, swarming of fish and herding of sheep, etc. Recently, due to emerging applications [34, 39, 40] in sensor networks, robot systems and unmanned aerial vehicles, research on the collective dynamics has received lots of attention from diverse scientific disciplines. After Vicsek s seminal work [48] on the collective dynamics, several physical and mathematical models were proposed in literature [11, 12, 37, 38, 47]. Among them, our main interest lies on the kinetic C-S equation which can be derived from the mean-field limit from the particle C-S model with random inputs. In literature, the particle and kinetic C-S models have been extensively studied from diverse perspectives, to name a few, collision avoidance [1, 9], effects of white noises [2, 10, 17, 18], kinetic limit [16, 19], dynamics of kinetic model [13, 19, 20], uncertainty quantificationuq problems [3, 6], general networks [8, 42], variants [30, 31, 32, 33] of the C-S model, etc. In this paper we will focus on the kinetic C-S equation with a random comunication weight and random initial input. Let f = ft, x, v, z be a one-particle distribution function of the C-S ensemble at position x with velocity v R d, random vector z, here z = z 1,, z m is a random vector defined on the sample space Ω. The random vector z registers the random effect on the communication Date: January 7, Mathematics Subject Classification. 15B48, 92D25. Key words and phrases. Cucker-Smale model, flocking, local sensitivity analysis, random communication, uncertainty quantification. Acknowledgment. The work of S.-Y. Ha was supported by National Research Foundation of KoreaNRF-2017R1A2B , and the work of S. Jin was supported by NSF grants DMS and DMS : RNMS KI-Net, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin. 1

2 2 HA, JIN, AND JUNG weight. We also assume that each component random variables z i are i.i.d., and let π = πz be a probability density function for the random vector z. Note that the communication between particles, denoted by ψ = ψx, z, is usually determined a priori by empirical data, thus inevitably contains uncertainty, modeled by z. In this situation, the dynamics of the kinetic density f is governed by the mean-field kinetic equation with random inputs: 1.1 t f + v x f + v F a [f]f = 0, x, v R d, z Ω, t > 0, F a [f]t, x, v, z := ψx x, zv v ft, x, v, zdv dx, R 2d where ψx, z =: ψ x, z satisfies several structural properties such as the positivity, boundedness, monotonicity and Lipschitz continuity in the first argument: there exists a positive random variable ψ M z > 0 such that < ψx, z ψ M z <, ψ x, z = ψx, z, x, z R d Ω, ψ x 2, z ψ x 1, z x 2 x 1 0, ψ, z LipR; R +. For each sample of z, i.e., the randomness is quenched, then 1.1 becomes a deterministic kinetic C-S equation which has been extensively studied in literature [4, 7, 16, 19, 20, 31, 32, 33]. In this paper, we are mainly interested in the effect of randomness of 1.1 on flocking dynamics and regularity of the solution in the random kinetic equation 1.1, with also random initial data f 0 x, v, z, via the local sensitivity analysis [41, 44]. Note that the kinetic density function ft, x, v, z + dz can be expanded in z-variable via Taylor s expnasion: ft, x, v, z + dz = ft, x, v, z + m i=1 f z i dz i + m i,j=1 Then, we can define the sensitivity matrices consisting of coefficients: S 1 := f, S 2 2 f :=,. z k z i z j 2 f z i z j dz i dz j +. Thus, the local sensitivity analysis deals with the regularity and stability estimates for the sensitivity matrices S i, i = 1, 2,. Such analysis for a wide class of random kinetic equations have been extensively investigated by the group of the second author in [21, 22, 24, 25, 28, 29, 35, 36, 43], where the regularity and sensitivity were studied using weighted Sobolve energy estimates and coercivity or hypocoercivity for perturnative solution near the global equilibrium of the kinetic operators. For our kinetic flocking model, we use a mixed norm HπL k x,v to be defined in 1.3, based on method of characteristics. This is the standard tool in Vlasov theory for perturbative solutions near vacuum, where energy methods do not work well [14]. There are two main results in this paper. First, we derive pathwise Wx,v m, -estimate and asymptotic flocking estimates for 1.1 along the sample paths. More precisely, for a fixed random vector z, we provide Wx,v m, -estimate and flocking estimate using the Lyapunov functional approach as in the deterministic case. Under the suitable regularity and compact support assumptions on the initial data, we show that there exists a unique regular solution

3 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 3 process f = ft, x, v, z to such that for some nonnegative process Φt, z, v v c z 2 ft, x, v, zdvdx Φt, z v v c z 2 f 0 x, v, zdvdx, z Ω, t 0, R 2d R 2d see Theorem 3.1 and Theorem 3.2. Second, we provide the pathwise local stability estimates. For the regular solutions processes f and f with compact supports, there exists a nonnegative process Ct, z such that m zft, l z z l ft, m z Ct, z L zf l 0 z l f z 0 z L x,v, z Ω, t 0, T, x,v see Theorem 4.3 and Theorem 4.4 for detailed description. In Proposition 4.1, we also show that the solution process f is bounded in any finite time interval in terms of mixed norm H m π L x,v: for T 0,, ft H m π L x,v CT f 0 H m π L x,v, t [0, T, where the mixed norm H m π L x,v is defined at the end of this section. The rest of this paper is organized as follows. In Section 2, we briefly introduce the particle and kinetic C-S models with random inputs, and present several elementary estimates along the sample paths. In Section 3, we study a pathwise well-posedness and asymptotic flocking estimate using the Lyapunov functional approach. In Section 4, we study the propagation of H k z -regularity and local sensitivity on the random input parameter for in a mixed norm. Finally, Section 5 is devoted to the brief summary of our main results and future direction. Gallery of notation: Let π : Ω R + {0} be a nonnegative p.d.f. function, and let y = yz be a scalar-valued random function defined on Ω. Then, we define the expected value as E[ϕ] := ϕzπzdz, a weighted L 2 -space: Ω L 2 πω := {y : Ω R Ω yz 2 πzdz < }, with an inner product and norm: y 1, y 2 L 2 π Ω := y 1 zy 2 zπzdz, y L 2 π Ω := Ω For k Z + {0}, set Ω 1 yz 2 2 πzdz k 1 y H k π Ω := zy l 2 2 L, k 1, y 2 πω H 0 π Ω := y L 2 π Ω. = E[ y 2 ]. Let h = hx, v, z be scalar-valued random function defined on the extended phase space R 2d Ω. For such h, we define a mixed norm HπL k x,v as follows. 1.3 h 2 HπL k x,v := z α hx, v 2 L 2 πω;l R 2d. α k

4 4 HA, JIN, AND JUNG Moreover, as long as there is no confusion, we suppress π and Ω dependence in L 2 πω- norm and H k πω-norm: y L 2 z := y L 2 π Ω, y H k z := y H k π Ω. For a vector-valued function yz = y 1 z,, y d z R d, we set d yz := y i z 2 1 d 1 2, y L 2 z := y i 2 2 L. 2 z i=1 i=1 2. Preliminaries In this section, we discuss particle and kinetic C-S models with random inputs, and study their basic properties on the propagation of velocity moments The C-S model with random inputs. Consider an ensemble consisting of N identical C-S particles in a random communication registered by ψ = ψx, z. Let x i t, z, v i t, z R 2d be the position-velocity processes of the i-th C-S particle. Then, their dynamics is governed by Cauchy problem for the random C-S model: t x i t, z = v i t, z, t > 0, z Ω, i = 1,, N, 2.1 t v i t, z = 1 N ψx j t, z x i t, z, zv j t, z v i t, z, N j=1 x i 0, z, v i 0, z = x 0 i z, v0 i z. Note that for each fixed z Ω, system 2.1 becomes a deterministic C-S model which has been extensively studied in literature see [7] for detailed survey and references therein. Next, we present definition of pathwise mono-cluster flocking for the C-S ensemble. Set the first and second velocity moments: m 1 t, z := N v i t, z, m 2 t, z := i=1 N v i t, z 2, t 0, z Ω. i=1 Proposition 2.1. [21] Let {x i t, z, v i t, z} N i=1 be a solution process to the C-S model 2.1 with zero total momentum: Then, we have m 1 0, z = 0. i m 1 t, z = 0, z k m 1 t, z = 0, t > 0, z Ω, k 1. ii t m 2 t, z = 1 N ψx j t, z x i t, z, z v j t, z v i t, z 2. N i,j=1 Remark 2.1. It follows from Proposition 2.1 that the total momentum is conserved and total energy is non-increasing along the C-S flow 2.1.

5 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS The kinetic C-S model with random inputs. Consider an system of N C-S particles with random inputs on the phase space R 2d, with N very large. In this case, it becomes computationally expensive to integrate the infinite number of ODE system 2.1. Thus, we introduce a one-particle distribution function f = ft, x, v, z for the infinite ensemble. Via the mean-field limit N in 2.1, the kinetic density f satisfies the Vlasov equation see [16, 19] for rigorous justification: 2.2 t f + v x f + v F a [f]f = 0, x, v R d, z Ω, t > 0, F a [f]t, x, v, z = ψx x, zv v ft, x, v, z, tdv dx, R 2d f0, x, v, z = f 0 x, v, z. For notational simplicity, we introduce a simplified notation: σ := x, v, and suppress t-dependance in f as well: dσ := dxdv, fσ, z := ft, σ, z, σ R 2d, z Ω, t 0. Next, we define the k-th velocity momentum of f as follows. M k t, z := v k fσ, zdσ. R 2d We now list a few basic propertieis of the moments of f. Lemma 2.1. Let f = fσ, z be a regular solution process to 2.2 decaying fast enough at infinity in the phase space R 2d. Then, for z Ω and t 0, i fσ, zdσ = f 0 σ, zdσ, vfσ, zdσ = vf 0 σ, zdσ. R 2d R 2d R 2d R 2d ii v 2 fσ, zdσ = ψx x, z v v 2 fσ, zfσ, zdσdσ. t R 2d R 4d iii fσ, z p dσ = dp 1 ψx x, zfσ, zf p σ, zdσ dσ, p [1,. t R 2d R 4d Proof. The proof is elementary, as in its deterministic counterpart [20]. Thus, we omit the proof. 3. Local sensity analysis: Zeroth-order estimates In this section, we briefly discuss the pathwise well-posedness and asymptotic flocking estimates of smooth solution to the kinetic C-S equation 1.1 and Pathwise well-posedness. As mentioned in Introduction, the deterministic kinetic C-S equation admits a global C 1 -solution in any finite-time interval, as long as the initial datum is C 1 -regular and compactly supported in x and v see [20] for details. More precisely, we can derive a W k, -estimate using the method of characteristics in any finite time interval see [20] for detailed estimates. In the standard Vlasov theory [14] in L 1 -framework, W k, - estimate is standard. For this reason, we use the mixed norm H m π L x,v throughout the paper. In contrast, when initial datum is a Radon measure, global measure-valued solutions to 1.1 was also studied in [19]. Now, we present the pathwise well-posedness of the kinetic C-S equation 2.2. Since the local existence can be obtained in a standard manner, we obtain a priori W 1, -estimates

6 6 HA, JIN, AND JUNG and use the continuation principle to yield the unique global solution of 1.1. First, we define the projections of the support in phase space: for t, z R + Ω, Rt, z := {x R d ft, x, v, z 0 for some v R d }, Pt, z := {v R d ft, x, v, z 0 for some x R d }. Throughout the paper, for the simplicity of presentation, we assume that Ω is one-dimensional, i.e., z R. First, we study the uniform boundedness of the velocity support Pt, z in any finite time interval. For this, we consider forward characteristics curves: For x, v R d and z Ω, set xt, z, vt, z = xt; 0, x, v, z, vt; 0, x, v, z, as a solution process to the following ODE system: txt, z = vt, z, t > 0, 3.1 t vt, z = F a[f]t, x, v, z, x0, z, v0, z = x, v. Now, we introduce two functionals Rt, z and P t, z measuring the sizes of projected supports of f as follows: 3.2 Rt, z := sup x and P t, z := sup v. x Rt,z v Pt,z Lemma 3.1. Let f = ft, x, v, z be a regular solution process to whose initial process f 0 has compact support in x and v for each z Ω. Then, the following estimates hold. 1 There exists a positive random variable D 0 z such that P t, z D 0 z1 + t, Rt, z R0, z D0 z2t + t 2, t 0, z Ω. 2 If ψ satisfies 1.1 and the extra positive lower bound condition 3.3: there exists a positive random variable ψ m z such that 3.3 inf ψx, z ψ m z > 0. x,z R d Ω Then, we have P t, z D 1 z, Rt, z R0, z + D 1 zt, t 0, z Ω, where D 1 z is a nonnegative random variable such that { D 1 ψ M z } M 2 z z := 2 max P 0, z, ψ m z. M 0 z Proof. The proof is essentially the same as the determinstic counterpart. See [20]. Remark 3.1. The results of Lemma 3.1 imply E[P t] E[P 0] + te[d 0 ], E[Rt] E[R0] + t + 12 t2 E[D 0 ]. If ψ has the extra lower bound condition 3.3, then E[P t] E[D 1 ], E[Rt] E[R0] + te[d 1 ].

7 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 7 Now, we are ready to present the global existence of the unique C k solution process to 1.1 for each z Ω. Although the global existence of C 1 solution is given in [20], we extend its analysis to the higher regular solution for later analysis. Theorem 3.1. Suppose that the following assumptions hold. 1 The communication weight is sufficiently regular in the sense that ψ L Ω; W k, R d. 2 Initial process f 0 is compactly supported and C k -regular in the sense that there exists C i z, i = 1, 2 such that R0, z + P 0, z C 1 z, v α x β f 0 z L x,v C 2 z, for each z Ω. 0 α + β k Then for any T 0,, there exists a unique C k -regular solution process f = ft, z C k R 2d [0, T to 1.1 for each z Ω. Proof. The proof is basically the same as that of Theorem 3.1 in [20] for k = 1. Consider the nonlinear transport operator T : T := t + v x + F a [f] v. Then, one has T f = d ψx x, zfσ, zfσ, zdσ d ψ L x,z M 0 z fz L x,v. R 2d For higher order estimate, we calculate T α v β x fz for α, β 1 as follows: T v α x β fz = α µ µ =1 µ =1 0 γ β v µ v x v α µ x β f α µ β Note that the following estimates hold: γ µ v γ v γ xf a [f] d ψ M 0 z, γ =1 β γ x F a [f] v v α µ v xf γ a [f] v α x β γ f β γ x f. µ v γ xf a [f] ψ M 0 z, where ψ := ψ L Ω;W k, R d. From these estimates, one can easily get T α v β x fz Bd ψ M 0 zft, z, for 0 α + β k with α, β 1, where B and Ft, z are defined by the following relations: B := α β, Ft, z := µ γ µ =1 0 γ β 0 α + β k Similarly, we calculate T α v fz and T β x fz as follows: α v β x fz L x,v.

8 8 HA, JIN, AND JUNG T v α fz = v F a [f] v α f α µ v v x v α µ f + v µ F a [f] v v α µ f, µ µ =1 T x β fz = v F a [f] x β f β v γ γ xf a [f] x β γ f + xf γ a [f] v x β γ f. Since one can obtain 1 γ β γ xf a [f] 2 ψ P t, zm 0 z, T α v fz Bd ψ M 0 zft, z, T β x fz 2Bd ψ P t, zm 0 zft, z. We combine all above results to get a Gronwall s inequality: Ft, z CzP t, zft, z. t Therefore, by using Lemma 3.1 and Grönwall s lemma, one can complete the proof Pathwise flocking estimate. For the pathwise flocking estimate, we use the Lyapunov functional approach in [20]. First we define the mean bulk velocity v c as follows: 1 v c t, z := vft, σ, zdσ, z Ω, t 0. M 0 z R 2d Then, it follows from Lemma 2.1 that v c t, z = v c 0, z =: v c z, t 0. Next, we define the Lyapunov functional given in [20] which measures the velocity variance around the mean value v c z: for t, z R + Ω, 3.4 L[f]t, z := v v c z 2 ft, σ, zdσ. R 2d As discussed in [17], the zero convergence of L as t implies the formation of velocity alignment in probability sense. This can be seen easily from the Chebyshev inequality. More precisely, let f be a probability density function over R 2d. Then, for any ε > 0 and z Ω, L[f]t, z = v v c z 2 fdvdx v v c z 2 fdvdx R 2d v v cz >ε ε 2 fdvdx = ε 2 P[ v v c 0 > ε]. This implies v v cz >ε lim P[ v v cz > ε] 1 t ε 2 lim L[f]t, z = 0. t Lemma 3.2. Let f be a global solution process to 1.1 with initial process f 0 which is compactly supported in x and v for each z Ω. Then for each z Ω, we have L[f]t, z = ψx x, z v v 2 fσ, zfσ, zdσ dσ. t R 4d

9 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 9 Proof. The proof is almost the same as that of Lemma 4.2 in [20], so we skip the details. Now, Lemma 3.2 yields the exponential decay of the functional L[f]. Theorem 3.2. Let f be a global solution process to 1.1 with initial process f 0 whose support is compactly supported in x and v for each z Ω. Then, t 3.5 L[f]t, z L[f 0 ]z exp 2M 0 z ϕs, zds, z Ω, t 0, where ϕt, z is defiend by ϕt, z := ψ 2R0, z + D 0 z2t + t 2, z. 0 Proof. It follows from Lemma 3.1 that 3.6 ψx x, z ϕt, z for any x, x Rt, z. On the other hand, we use 3.6 and Lemma 3.2 to obtain L[f]t, z = ψx x, z v v 2 fσ, zfσ, zdσ dσ t R 4d ϕt, z v v 2 fσ, zfσ, zdσ dσ R 4d where we used the identity: = 2ϕt, zm 0 zl[f]t, z, v v 2 = v v 0 c z 2 + v v 0 c z 2 2v v 0 c z v v 0 c z. Then, Grönwall s lemma completes the proof. Remark 3.2. Suppose that initial datum f 0 is deterministic: Hence, L[f 0 ] is also deterministic: f 0 σ, z = f 0 σ, for all z Ω. E[L[f 0 ] ] = L[f 0 ]. The relation 3.5 yields [ E[L[f]t, ] L[f 0 ]E exp 2M 0 z t 0 ϕs, zds ]. 4. Local sensitivity analysis: Higher-order estimates In this section, we present two local sensitivity analysis for regular solution processes such as the propagation of H k z -regularity and L x,v-stability. Recall that for the simplicity of notation, we assume that the random space Ω is one-dimensional, i.e. z R. For a given m Z +, applying the operator z m 4.1 t z m f + v x z m f + m l 0 l m to 1.1 gives v l zf a [f] m l z f = 0, As in previous section, we introduce the following notation: for m Z +,

10 10 HA, JIN, AND JUNG R m t, z := {x R d m z fx, v, z, t 0 for some v R d }, P m t, z := {v R d m z fx, v, z, t 0 for some x R d }. and P 0 t, z := Pt, z, R 0 t, z := Rt, z. For a fixed x, v, z R d R d Ω, we introduce the forward characteristics x m t, z, v m t, z associated with 4.1: 4.2 Recall that t xm t, z = v m t, z, t vm t, z = F a [f]t, x m t, z, v m t, z, z. σ := x, v, dσ := dvdx and dσ := dv dx Propagation of pathwise-regularity. In this subsection, we study propagation of regularity in random space for f in terms of L x,v-norm. Note that for a fixed z Ω, z m f satisfies the deterministic equation: t z m f + v x z m f + m v l l zf a [f] z m l f = 0 x, v R d, z Ω, t > 0, 0 l m and we can use all estimates available for the deterministic kinetic C-S equation in [19, 20]. Lemma 4.1. Let f = ft, x, v, z be a regular solution process to 1.1 decaying fast enough at infinity in phase space for each z Ω and t > 0. Then, we have i m z fσ, zdσ = 0, v z m fσ, zdσ = 0. t R 2d t R 2d ii v 2 z m fσ, zdσ t R 2d m l m l = fσ, z z m l fσ, zdσ dσ. l =0 0 l m v v 2 z ψ z l R 4d Proof. i The first estimate follows from the divergence structure of 4.1, and the second estimate can be treated as follows. v z m fσ, zdσ t R 2d = d m l zf l a [f] z m l fdσ R 2d = d m l =0 m l l v v z ψz z l R 4d fσ, z z m l fσ, zdσ dσ.

11 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 11 Note that the following relation holds: 4.3 m l =0 m l l = m z ψz z l fσ, z z m l fσ, z l =0 m l l z ψz l z fσ, z z m l fσ, z. We use 4.3 and change of variable x, v x, v to derive the desired result. ii One can easily follow the calculation in ii. We use the same argument in Lemma 2.1 and change of variable to yield the desired estimate. Next, define two functionals that measure the size of projected supports of m z ft,, z, as we did in Section 3: For m Z +, t 0, z Ω, set 4.4 R m t, z := sup x, P m t, z := sup v. x R mt,z v P mft,z For m = 0, it follows from 3.1 and 3.2 that R 0 t, z = Rt, z, P 0 t, z = P t, z, t 0, z Ω. Below, we estimate the sizes of R m and P m in the following lemma. Lemma 4.2. Let f = ft, x, v, z be a regular solution process to with initial process f 0 whose z-derivative m z f 0 has compact support in x and v for each z Ω. Then, the functionals R m and P m defined in 4.4 satisfy the following estimates: 1 For m Z +, there exists a positive random variable D 0 mz such that P m t, z D 0 mz1 + t, R m t, z R m 0, z D0 mz2t + t 2, t 0, z Ω. 2 If ψ satisfies 1.2 and extra positive lower bound condition below: there exists a positive random variable ψ m z such that Then, inf ψx, z ψ m z > 0. x,z R d Ω P m t, z D 1 mz, R m t, z R m 0, z + D 1 mzt, t 0, z Ω, where Dmz 1 is a nonnegative random variable such that { Dmz 1 ψ M z } M 2 z := 2 max P m 0, z, ψ m z, M 0 z Proof. 1 For z Ω, let x m t, z, v m t, z be the characteristic curve to 4.2 such that P m t, z = v m t, z.

12 12 HA, JIN, AND JUNG Then, t P mt, z 2 = ψx m t, z x, zv v m t, z v m t, zfσ, zdσ R 2d ψx m t, z x, zv v m t, zfσ, zdσ R 2d ψ M zp m t, z M 2 t, z M 0 t, z ψ M zp m t, z M 2 z M 0 z, where we used Lemma 2.1 and ψ M z := max x ψx, z. Next, we divide 4.5 by P m t, z and integrate with respect to t to obtain P m t, z P m 0, z + tψ M z M 2 z M 0 z. This yields the desired estimate 3.4 with { Dmz 0 := max P m 0, z, ψ M z M 2 z } M 0 z. For R m t, z, we choose the particle trajectory x m t, z, v m t, z that yields R m t, z: R m t, z = x m t, z. Then, by using and estimate of P m t, z, one has R m t, z R m 0, z + t 0 R m 0, z D mz2t + t 2. v m s, z ds x m 0, z + D 0 mz t sds ii Again it follows from the first equality in 3.6 that 1 2 t P mt, z 2 = ψx m t, z x, zv v m t, z v m t, zfσ, zdσ R 2d = ψx m t, z x, zv v m t, zfσ, zdσ 4.6 R 2d P m t, z 2 R 2d ψx m t, z x, zfσ, zdσ ψ m zp m t, z 2 M 0 z + ψ M zp t, z M 2 z M 0 z. Dividing 4.6 by P m t, z and use Gronwall s lemma, one finds where D 1 mz is defined by P m t, z P m 0, ze ψmzm 0zt + ψ Mz M 2 z ψ m z M 0 z D1 mz, D 1 mz := 2 max { P m 0, z, ψ M z } M 2 z ψ m z. M 0 z Thus, we have the desired uniform boundedness. On the other hand, by the same analysis in??, one has R m t, z R m 0, z + D 1 zt.

13 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 13 Before we present our main theorem in this subsection, we provide a preliminary lemma. Lemma 4.3. For each z Ω and m N, let f = ft, x, v, z be a regular solution process to where every z-derivative of initial process zf l 0 and f 0 itself are compactly supported in x and v. Then, there exist C i t, z, i = 1, 2 such that i zf l a [f] + x µ zf l a [f] C 1 t, z z f L x,v, ii vi l zf a [f] + vi µ x l zf a [f] C 2 t, z where µ k, l m and ψ := ψ W m, Ω;W k, R d. Proof. i It follows from that zf l a [f]t, x, v, z = l 2 ψ l 2 ψ l 2 ψ where we used and C 1 t, z is given by l P t,z R t,z C 1 t, z P t, z P t, z l z z f L x,v, ψx x, zv v z fσ, zdσ P t,z R t,z P t,z R t,z z fσ, z dσ z fz L x,v dσ P t, z P t, zr t, z d z f L x,v z f L x,v, P t, z R t, z P t, zr t, z d, C 1 t, z := 2 ψ For x µ zf l a [f], one can use similar estimate: x µ zf l a [f] l 2 ψ P t,z R t,z l C 1 t, z z f L x,v. l P t, z P t, zr t, z d. x µ z l ψx x, z v v z fσ, z dσ P t, z P t, zt, zr t, z d z f L x,v

14 14 HA, JIN, AND JUNG ii For vi zf l a [f], by direct estimate, one has vi zf l a [f] l ψ where C 2 t, z is defined by C 2 t, z P t,z R t,z l C 2 t, z := ψ z l ψx x, z z fσ, z dσ P t, zr t, z d z f L x,v z f L x,v, l P t, zr t, z d. For vi x µ zf l a [f], similarly one has vi x µ zf l a [f] l ψ C 2 t, z P t,z R t,z l x µ z l ψx x, z z fσ, z dσ P t, zr t, z d z f L x,v z f L x,v. Remark 4.1. From Lemma 4.2, one can deduce that P, z := Ot, R, z := Ot 2, for each z Ω. Thus, this yields the following estimates: for each z Ω, we have C 1, z = Ot 3d+1, C 2, z = Ot 3d, where C 1 and C 2 are constants given in Lemma 4.3. Now we present the global existence of a unique regular solution process to 4.1 for each z Ω. First we would like to consider the case m = 1. Theorem 4.1. For each z Ω and k 2, suppose that the initial process f 0 satisfies the following conditions: 1 Initial processes f 0 z and z f 0 z are compactly supported in phase space for each z Ω: R l t, z, P l t, z <, for each l = 0, 1, and each z Ω. 2 Initial processes f 0 z and zf l 0 z are C k -regular and C k 1 -regular, respectively and bounded for each z Ω: v α x β zf l 0,, z L x,v <. 0 l 1 0 α + β k l

15 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 15 3 The communication weight ψ belongs to W 1, Ω; W k, R d. Then, for any T 0,, there exists a unique C k 1 -regular solution process z fz C k 1 R 2d [0, T to 4.1 for each z Ω. Proof. The proof is almost the same as Theorem 3.1. So we provide a priori W k 1, - estimate for z f. Again consider the nonlinear transport operator T := t + v x + F a [f] v. Case A zeroth-order estimate: It follows from 4.1 that T z f = v F a [f] z f v z F a [f]f z F a [f] v f. By Remark 3.1, f is C k -regular and bounded. Set F 0 t, z := v α x β ft,,, z L x,v, 0 α + β k Then, F 0 t, z is bounded for each t T and z Ω, and T z f d ψ M 0 z z f + d ψ M 0 z + P 1 t, zr 1 t, z d z f L x,v f + 2 ψ P 0 t, zm 0 z + P 1 t, z P 1 t, zr 1 t, z d z f L x,v v f 4.7 Ct, z1 + F 0 t, z Ct, z z f L x,v + 1 z f L x,v + 1, where ψ := ψ W k, R d W 1, Ω and Ct, z is a constant that depends on d, f 0, P l t, z, R l t, z, but independent of z f, since v z F a [f] = d z ψzf z + ψz z fσ, zdσ R 2d d ψ M 0 z + P 1 t, zr 1 t, z d z f L x,v, z F a [f] v v z ψzfσ, z + φz z fσ, z dσ R 2d 2 ψ P 0 t, zm 0 z + P 1 t, z P 1 t, zr 1 t, z d z f L x,v, v F a [f] d ψ M 0 z. Case B Higher-order estimate: Now we estimate higher-order terms. First, consider the term: α v β x z f, where 1 α, β k 1 and α + β k 1. Then, it follows from 1.1 that we have T v α x β z f = v z F a [f] v α x β f z F a [f] v α x β v f v F a [f] v α x β z f G 1 α, β, µf H 1 α, β, µ 2, µ 3 f, µ 1 =1 0 µ µ 3 β µ 2 + µ 3 =0

16 16 HA, JIN, AND JUNG where G 1 and H 1 are given by the following relations: α G 1 α, β, µ 1 f = µ 1 µ 1 v v α µ 1 v x β x z f, H 1 α, β, µ 2, µ 3 f α β = µ 2 v µ 3 x v z F a [f] α µ 2 x f + µ 2 v µ 3 x z F a [f] α µ 2 x v f µ 2 µ 3 + µ 2 v µ 3 x v F a [f] α µ 2 x z f + µ 2 v µ 3 x F a [f] α µ 2 x v z f. One can easily see that Here, by using Theorem 3.1, one finds α G 1 α, β, µ 1 f α µ 1 v x β x z f L x,v. α µ 2 x f, α µ 2 Combining this with Lemma 4.3 gives µ 1 x v f F 0 t, z Ct, z, H 1 α, β, µ 2, µ 3 f 1 Ct, z zf l L x,v + α µ 2 x z f L x,v + α µ 2 x v z f L x,v. Then, we again use Lemma 4.3 and combine all the estimates above to get 4.8 T v α x β z f 1 Ct, z zf l L x,v + v α x β z f L x,v µ µ 3 β µ 2 + µ 3 =0 Ct, z F 1 t, z + 1, µ 1 =1 α µ 1 v x β x z f L x,v α µ 2 x z f L x,v + α µ 2 x v z f L x,v where Ct, z depends on d, F 0 t, z, ψ, P l t, z, R l t, z for l = 0, 1, and F 1 t, z is given by F 1 t, z := 0 α + β k 1 α v β x z ft, z L x,v. Similarly, for β 1 and α 1, we compute T β x z f and T α v z f as follows:

17 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS i T x β z f Ct, z zf l L x,v + x β z f L x,v + 1 µ 3 β β µ 3 x z f L x,v + β µ 3 x v z f L x,v Ct, z F 1 t, z + 1, 1 ii T v α z f Ct, z zf l L x,v + v α z f L x,v µ 2 1 Ct, z F 1 t, z + 1. µ 1 =1 α µ 2 v z f L x,v + α µ 2 v v z f L x,v µ 2 =1 Finally, we combine all estimates 4.7, 4.8 and 4.9 to obtain t F 1 t, z Ct, z F 1 t, z + 1. Then, Grönwall s lemma yields the desired estimate. Now we generalize the idea in Theorem 4.1 to the case m 1. α µ 1 v x z f L x,v Theorem 4.2. For each z Ω and m 2, assume that the initial process f 0 z := f0, z satisfies the followings: 1 Every z-derivative of initial process l zf 0 z is compactly supported in the phase space for each l = 0, 1,, m and each z Ω, i.e. R l 0, z <, P l 0, z <, for each l = 0, 1,, m and each z Ω. 2 Every z-derivative of initial process zf l 0 zis C m l+1 -regular and bounded for each l = 0, 1,, m and each z Ω, i.e. v α x β zf l 0 z L x,v <. 0 l m 0 α + β m l+1 3 The communication weight ψ belongs to W m+1, R d Ω. Then, for any T 0,, there exists a unique C m l+1 -regular solution process l zfz C m l+1 R 2d [0, T to 4.1 for each l = 1,, m and each z Ω. Proof. We proceed the proof by induction on m. Note that we have already proved the initial step m = 2 case in Theorem 4.1. So one just needs to verify the induction step for m. However, to show this, one needs to use induction on l, i.e. to show the following step: For fixed m 2 and l m, under the assumptions in this theorem, if there exists a unique C m +1 -regular solution process z fz for all < l, then there also exists a unique C m l+1 -regular solution process l zfz.

18 18 HA, JIN, AND JUNG Note that we have also proved the initial step l = 1 in Theorem 4.1. Since it suffices to show the above statement, again we consider the nonlinear transport operator T := t + v x + F a [f] v and define F t, z as follows: 4.10 F t, z := v α x β z fz L x,v, for = 0, 1,, l. 0 α + β m +1 Case A Zeroth-order estimate: From 4.1, T zf l = v F a [f] zf l v zf l a [f]f zf l a [f] v f l 1 l v z l F a [f] z f + z l F a [f] v z f. =1 Then by using Lemma 4.3, one can estimate T l zf as follows: 4.11 T zf l Ct, z zf l + l 1 z f L x,v + Ct, z F l t, z + F t, z + Ct, z F l t, z + 1, l =1 r=0 l 1 l 1 =1 f zf r L x,v z f + v z F r t, z F t, z where Ct, z depends on F t, z, d, ψ, P t, z and R t, z with 0 l 1. Since they can be bounded by induction hypothesis, we get the last inequality. Case B Higher-order estimate: It follows from tedious, direct computation that one can obtain from 4.1 that for 1 α, β with α + β m l + 1, t v α x β zf+ α l µ 1 v v x α µ 1 v x β zf+ l Hα, β, l, µ 2, µ 3, f = 0, 0 µ 1 1 where H is given by µ 1 Hα, β, l, µ 2, µ 3, f = It follows from Lemma 4.3 that µ 2 v µ 3 x v z F a [f] α µ 2 µ 2 v µ 3 x z F a [f] v α µ 2 x x l α µ 2 β µ 2 µ 3 r=0 0 µ µ 3 β v µ 3 x v z F a [f] α µ 2 + µ 2 v µ 3 x z F a [f] v α µ 2 l z f Ct, z l z f Ct, z x x l z f zf r L x,v α µ 2 r=0 zf r L x,v α µ 2 r=0. x x z l f l z f L x,v, v l z f L x,v.

19 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 19 Now let us define a set Il, β given by Il, β := {, µ 2, µ 3 N {0} 2d+1 l, µ 2 1, µ 3 β, + µ 2 + µ 3 > 0}. Then using the estimates above, one can get for, µ 2, µ 3 Il, β, Hα, β, l, µ 2, µ 3, f Ct, z zf r L x,v α µ 2 r=0 Ct, z F r t, z F l t, z. r=0 This yields the following estimates: 4.12 µ 1 =1 µ 1 x l z f L x,v + α µ 2 x v l z f L x,v T v α x β zf l = α µ 1 v v x α µ 1 v x β zf l v F a [f] v α x β zf l Ct, z + Ct, z,µ 2,µ 3 Il,β µ 1 =1,µ 2,µ 3 Il,β Hα, β, l, µ 2, µ 3, f x α µ 1 v x β zf l L x,v + v α x β zf l L x,v F r t, z F l t, z r=0 1 + F 0 t, zf l t, z + l 1 + F r t, z F l t, z =1 r=0 Ct, z F l t, z + 1, l 1 F r t, z F 0 t, z where Ct, z depends on d, ψ, F t, z, P t, z and R t, z with 0 l 1, but independent of l zf. From the assumptions and induction hypothesis, for each z Ω, one has that Ct, z can be bounded by a constant. We use 4.12 to get estimates for T v α zf l and T x β zf: l T v α zf, l T x β zf l Ct, z F l t, z + 1. Therefore, combination of 4.10, 4.11 and 4.12 yields t F l t, z Ct, z F l t, z + 1, 0 < t T, z Ω. The Grönwall s lemma gives F l t, z Ct, z F l 0, z + 1, 0 t T. r=0

20 20 HA, JIN, AND JUNG Again, this yields the boundedness of F l t, z for any t T and each z Ω Pathwise stability analysis. In this subsection, we present L x,v-stability of 4.1. First we present pathwise L - stability in the following theorem. Theorem 4.3. For each z Ω, suppose that two initial processes f 0 z := f0, z and f 0 z := f0, z satisfy the following conditions: 1 Initial processes are compactly supported in the phase space for each z Ω: R0, z + R0, z <, P 0, z + P 0, z <, for each z Ω. 2 Initial processes f 0 z and f 0 z are C 1 -regular and bounded for each z Ω: v α x β f 0 z L x,v, v α x β f 0 z L x,v <. 0 α + β 1 0 α + β 1 3 The communication weight ψ, z belongs to W 1, R d. Then, for any T 0,, there exists a nonnegative random variable C, z L 0, T for each z Ω such that ft, z ft, z L x,v Ct, z f 0 z f 0 z L x,v, t 0, T. Proof. From Theorem 3.1 in [20], for each z Ω, there uniquely exist C 1 -regular processes ft, z and ft, z in C 1 R 2d [0, T whose initial processes are f 0 z and f 0 z respectively, and F 0 t, z := F 0 t, z := Now, by using 1.1 one gets 0 α + β 1 0 α + β 1 α v β x ft, z L x,v <, α v β x ft, z L x,v < t f f + v x f f + v F a [f] F [ f] f + v F a [f]f f = 0. Then 4.14 v F a [f] F [ f] d ψx x, z fσ, z fσ, z dσ R 2d C 1 t, z ft, z ft, z L x,v, F a [f] F [ f] ψx x, z v v fσ, z fσ, z dσ R 2d C 2 t, z ft, z ft, z L x,v, where C i, z = Ot 3d+1 for each z Ω and i = 1, 2 by Lemma 3.1. Now by integrating 4.13 using the relations 4.14 one gets f ft, z L x,v f 0 f 0 z L x,v + t 0 Cs, z f fs, z L x,v ds.

21 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 21 Note that Cs, z L 0, T for each z Ω and its estimate can be followed from Lemma 3.1, Remark 3.1 and the estimate of F 0 and F 0, which has exponential order in t for each z Ω. Hence by using Grönwall s lemma one obtains the desired result. Similarly, we present the local sensitivity analysis. Theorem 4.4. For each z Ω and m 2, assume that two initial processes f 0 z := f0, z and f 0 z := f0, z satisfy the following conditions: 1 z-derivatives of initial processes zf l 0 z and z l f 0 z are compactly supported in the phase space for each l = 0, 1,, m and each z Ω, i.e. P l 0, z + P l 0, z + R l 0, z + R l 0, z <, for each z Ω. 0 l m 2 z-derivatives of initial processes zf l 0 z and z l f 0 z are C m l+1 -regular and bounded for each l = 0, 1,, m and each z Ω, i.e. v α x β zf l 0 z L x,v + v α x β z l f 0 z L x,v <. 0 l m 0 α + β m l+1 3 The communication weight ψ belongs to W m, R d Ω. Then, for any T 0,, there exists a nonnegative random variable CT, z such that m zft, l z z l ft, m z CT, z L zf l 0 z l f z 0 z L x,v, t 0, T. x,v Proof. It follows from Theorem 4.2 that, for each l = 0, 1,, m, there exist unique C m l+1 regular solution processes l zft, z and l z ft, z to 4.1 corresponding to initial processes l zf 0 z and l z fz respectively. Then, it follows from 4.1 that 4.15 t zf l f + v x zf l f l l + v z F a [f] F [ f] z l f + z F a [f] z l f f =0 Note that the following estimates hold for each ; z F a [f] F [ f] v v r zψx r x, z z r fσ, z fσ, z dσ r=0 R 2d Ct, z zf r ft, z L x,v, r=0 v z F a [f] F [ f] d r r=0 Ct, z r=0 = 0. zψx r x, z z r fσ, z fσ, z dσ R 2d r zf ft, z L x,v,

22 22 HA, JIN, AND JUNG where C, z = Ot 3d+1 and non-decreasing in t for each z Ω by Lemma 4.2. Define F t, z and F t, z as Thus, by integrating 4.15 over the particle trajectory of zf l one obtains t zf l ft, z L x,v zf l 0 f l 0 z L x,v + Cs, z z f fs, z L x,v ds, 0 where t 0, T and C, z = Os 3d+1 and non-decreasing in s for each z Ω by Lemma 4.1, Remark 4.1 and Lemma 4.2. Since this holds for each l = 0, 1,, m, one can get m zf l ft, m t z L x,v zf l 0 f m 0 z L x,v +m Cs, z zf l fs, z L x,v ds. Then, by using Grönwall s lemma one obtains m zf l ft, z L x,v e m t m Cs,zds 0 zf l 0 f 0 z L x,v. We set CT, z := e m T 0 Cs,zds to obtain the desired estimate Uniform bound in H m π L x,v-norm. In this subsection, we provide L 2 -estimates by using L -estimates for m z f. To be precise, we estimate m z f in a space L 2 πω; L x,vr 2d. Proposition 4.1. For T 0, and m 2, suppose the following conditions: 1 For every l =0, 1,, m, z-derivatives of initial process zf l 0 have compact velocity and position support which are uniform in z, i.e. sup P l 0, z + R l 0, z <. 0 l m z Ω 2 For every l =0, 1,, m, z-derivatives of initial process zf l 0 are C m l+1 -regular for each z Ω and belongs to L 2 π L Ω; W m l+1, R 2d, i.e. zf l L 2 π L Ω;W m l+1, R 2d <. 0 l m 0 =0 3 The communication weight ψ belongs to W m, R d Ω. Then one can obtain ft 2 Hπ ml x,v CT f 0 2 Hπ ml x,v, t 0, T. Proof. Multiplying 4.1 by z m f and πz, and then integrating the resulting relation over Ω, give 1 2 t m z f 2 L 2 π Ω + v x z m f 2 L 2 π Ω + F a[f] v z m f 2 L 2 π Ω 4.16 = 1 l m Ω m l v Ω l zf a [f] m l z v F a [f] m z f 2 πzdz. f m z fσ, zπzdz

23 LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC C-S EQUATION WITH RANDOM INPUTS 23 Note that the coefficients in the R.H.S. of 4.16 can be estimated using Lemma 4.3: for 0 l m, l zf l a [f] 2 ψ P t, z P t, zr t, z d pz fz L, x,v 4.17 l v zf l a [f] d ψ P t, zr t, z d pz fz L. x,v Set F l t, z := 0 α + β m l+1 α v β x l zfz L x,v. One can also deduce from Theorem 4.2 that z-dependence of F l t, z is given by F 0, z, P 0, z and R 0, z for all 0 l. Thus, by using the uniform boundedness of P 0, z, R 0, z and z f 0 in the assumption, one gets 4.18 zf l a [f] + v zf l Ct, 0 l m where Ct is a constant that is finite for every 0 t T. On the other hand, using Young s inequality gives v zf l a [f] z m l fσ, z z m fσ, zπzdz Ω Ct z m l f 2 L 2 π Ω + m z f 2 L 2 π Ω, 4.19 zf l a [f] v z m l fσ, z z m fσ, zπzdz Ω Ct z m f 2 L 2 π Ω + zf l 2 L 2 π L x,v. 1 l m Now combining estimates 4.17, 4.18 and 4.19 one gets 1 2 t m z f 2 L 2 πω + v x z m f 2 L 2 πω + F a[f] v z m f 2 L 2 πω Ct 0 l m l zf 2 L 2 πl x,v. Integration this inequality over the characteristic curve xt; 0, x, v, vt; 0, x, v of m z f 2 L 2 π Ω leads to This implies z m f 2 L 2 π Ωx, v t z m f 0 2 L 2 Ωx0, v0 + Cs π t z m f 0 2 L 2 π L x,v + Cs l m 0 0 l m l zf 2 L 2 π L x,v ds. 0 l m l zf 2 L 2 π L x,v ds t 4.20 z m f 2 L 2 π L x,v m z f 0 2 L 2 π L x,v + Cs zf l 2 L 2 π L ds. x,v

24 24 HA, JIN, AND JUNG Finally, we sum the above inequality 4.20 over m to obtain t f 2 Hπ ml x,v f 0 2 Hπ ml x,v + Cs f 2 Hπ ml ds. x,v Then, we apply Grönwall s lemma to complete the proof Conclusion In this paper, we provided a local sensitivity analysis for the kinetic C-S equation with random inputs. In earlier works, research on the continuous C-S model with random effects focus on the white noise perturbations as external stochastic forces. In the current work, we did not assume any specific dependence of randomness in the system dynamics. In the first two authors recent work [21] on the particle C-S model with random inputs, systematic local sensitivity analysis was performed for the C-S flocking model with finite size of ensemble. Thus, a natural extension of the previous work is to figure out whether similar local sensitivity analysis can be done for the infinite system. In this work, we have assumed the structural symmetric condition in the communication weight even for the presence of random inputs. In fact, this strong structural constraint results in the robustness of the flocking estimates independent of the size of the C-S ensemble. However, in general situation, uncertainty effect might screw up the symmetry of the communication. Thus, it results in the non-existence of global flocking as in the short-ranged communications even for the deterministic C-S equation. The direction toward the destabilizing random effects on the collective dynamics will be an interesting issue for a future research, at both the theoretical and numerical levels. One may also consult some recent numerical studies on uncertain flocking models in [3, 6]. There are many other related models for collective dynamics, decision making and selforganization in complex system in biological and social sciences [45], which are also of great interest to study the influence of uncertain random effects in various input parameters in the models. References [1] Ahn, S. M., Choi, H., Ha, S.-Y. and Lee, H.: On collision-avoiding initial configurations to Cucker-Smale type flocking models. Commun. Math. Sci. 10, [2] Ahn, S. and Ha, S.-Y.: Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises. J. Math. Phys. 51, [3] Albi, G., Pareschi, L. and Zanella, M.: Uncertain quantification in control problems for flocking models. Math. Probl. Eng. Art. ID, [4] Carrillo, J. A., Fornasier, M., Rosado, J. and Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42, [5] Carrillo, J. A. Fornasier, M., Toscani, G. and Vecil, F.: Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences , Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, [6] Carrillo, J. A., Pareschi, L. and Zanella, M.: Particle based gpc methods for mean-field models of swarming with uncertainty. Preprint. [7] Choi, Y.-P., Ha, S.-Y. and Li, Z.: Emergent dynamics of the Cucker-Smale flocking model and its variants. In N. Bellomo, P. Degond, and E. Tadmor Eds., Active Particles Vol.I - Theory, Models, Applicationstentative title, Series: Modeling and Simulation in Science and Technology, Birkhauser- Springer. [8] Cucker, F. and Dong, J.-G.: On flocks influenced by closest neighbors. Math. Models Methods Appl. Sci. 26,

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26 26 HA, JIN, AND JUNG [36] Liu, L. and Jin, S. : Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random Inputs. Preprint. [37] Motsch, S. and Tadmor, E.: Heterophilious dynamics: enhanced consensus. SIAM Review 56, [38] Motsch, S. and Tadmor, E.: A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144, [39] Paley, D. A., Leonard, N. E., Sepulchre, R., Grunbaum, D. and Parrish, J. K.: Oscillator models and collective motion. IEEE Control Systems Magazine 27, [40] Perea, L., Elosegui, P. and Gómez, G.: Extension of the Cucker-Smale control law to space flight formation. J. of Guidance, Control and Dynamics 32, [41] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. and Tarantola, S.: Introduction to sensivity analysis. Global sensivity analysis. The Primer, [42] Shen, J.: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, [43] Shu, R. and Jin, S.: Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime. Preprint. [44] Smith, R. C.: Uncertainty quantification: Theory, Implementation, and Applications. 12, SIAM, [45] Tadmor, E.: Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, [46] Toner, J. and Tu, Y.: Flocks, herds, and Schools: A quantitative theory of flocking. Physical Review E. 58, [47] Vicsek, T and Zefeiris, A.: Collective motion. Phys. Rep. 517, [48] Vicsek, T., Czirók, E. Ben-Jacob, I. Cohen and O. Schochet: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, Seung-Yeal Ha Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University, Seoul and Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea Republic of address: syha@snu.ac.kr Shi Jin Department of Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53706, USA address: jin@math.wisc.edu Jinwook Jung Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea address: warp100@snu.ac.kr