Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement

In this paper, we present quantitative local sensitivity estimates for the kinetic chemotaxis Cucker-Smale(CCS) equation with random inputs. In the absence of random inputs, the kinetic CCS model exhibits velocity alignment under suitable structural assumptions on the turning kernel and reaction term despite of the random effect due to a turning operator. We provide a global existence of a regular solution with slow velocity alignment for the random kinetic CCS model within the proposed framework. Moreover, we investigate the propagation of regularity and stability of infinitesimal variations in random space.


1.
Introduction. The purpose of this paper is to continue systematic studies begun in a series of works [1,4,10,13,14,15,16,17,18,20] for the interplay of uncertainty quantification (UQ) and collective behaviors such as flocking and synchronization. In particular, we are interested in the velocity alignment in the ensemble of bacteria exhibiting an abrupt change of velocities in their motions. As reported in [22,28,35], collective behaviors in biological groups often occur through mutual communications. In 2007, Cucker and Smale introduced a simple analytical model for the flocking behavior of biological complex systems that generalizes Vicsek's model [34]. After Cucker-Smale's seminal work, most works on the modeling of flocking adapted their simple velocity alignment mechanism, which results in the smooth change of individual velocities. However, as we often see in the school of fish in aquarium, fish change their velocity abruptly, i.e., they have several components such as free swimming, tumbling and swarming etc. Recently, to model such an abrupt change of velocity in mesoscopic level, the kinetic chemotaxis Cucker-Smale(CCS) model was introduced in [8] by adopting a turning operator commonly used in the kinetic Keller-Segel type models [5,6,7,24,25,33,32]. We also refer to [19] for the application of flocking mechanism in phototaxis and [3,29] for the collective modeling of chemotactic bacteria via a suitable modeling of the kernel in turning operator.
To incorporate uncertain effects to the kinetic CCS model, we introduce random vector z defined on the sample space Ω ⊂ R n . We assume that each component of z is i.i.d., and let π = π(z) be a probability density function for z. For the modeling of abrupt velocity change with random effects, we adopt the turning operator [8] involved with the randomness coefficients α = α(z), κ = κ(z): for a given t > 0, x ∈ R d , z ∈ Ω, we set the turning kernel T [S](t, x, v, v , z) to denote the rate of jumps from v to v , and define T * [S](t, x, v, v , z) := T [S](x, t, v , v, z) to denote the rate of jumps from v to v. Let S = S(t, x, z) be the concentration of the chemical attractant whose dynamics is governed by the reaction-diffusion equation: where κ(z)ρS is a reaction term representing random chemical interactions between the Cucker-Smale particle and chemical substance with random reaction rate κ. Note that this special ansatz for the reaction term is designed to decay to zero, as time goes on so that the effect of turning operator decays. In this way, the flocking mechanism registered by the nonlinear transport operator will be a dominant component for large-time behavior of the proposed kinetic model. Then, the contribution of the rate of change in f along the particle trajectory due to chemotactic movements will be registered by the random turning operator T [S](f ) (see [2,11,12,26,27]): where α is the desire of change in a favorable direction with randomness: where the quantity λ[S] denotes the turning frequency. Under this setting, we obtain the the random kinetic CCS model by combining (1), (2) and the kinetic Cucker-Smale model: for d ≥ 2, Here the alignment force F a [f ] is given as follows: Finally, the dynamics of (f, S) is governed by (3) with the initial data: Note that the random variable z registers the uncertain effects in the communication weight function, parameters α, κ and the initial data. If all randomness in the model are quenched, system (3)-(5) becomes the deterministic kinetic CCS model, which has been proposed to describe the dynamics of Cucker-Smale particles with chemotactic movements with velocity jumps and attraction toward chemotactic substances in [8].
In this paper, we are mainly interested in random effects on the flocking dynamics and regularity of the random kinetic equation via the local sensitivity analysis. Note that the kinetic density function f (t, x, v, z + dz) and chemotatic density function S(t, x, z + dz) can be expanded in z-variable via Taylor's expansion: Then, we can define the sensitivity matrices consisting of coefficients in the above expansion. The local sensitivity analysis deals with regularity and stability estimates for the sensitivity matrices [30,31]. Such estimates are needed not only for analytical interest, but also for numerical methods such as stochastic Galerkin or collocation methods [23]. Note that for n random effects (i.e., z ∈ Ω ⊂ R n ), we can also get the same result which is presented in this paper, but it is rather langthy and technical. So, we will assume that z ∈ Ω ⊂ R 1 for the simplicity of presentation. Next, we briefly discuss our three main results.
First, we present a framework on the initial data, the turning kernel and the communication weight leading to a regular global solution. Within our proposed framework in Section 3, our first result is concerned with pathwise W k,∞ x,v ×W k+1,∞ x estimate and flocking estimate using the Lyapunov functional approach. More precisely, we find a global unique regular solution (f, S) to (3)-(5) satisfying (see Thereom 4.7 and Theorem 4.8).
The rest of this paper is organized as follows. In Section 2, we propose structual assumptions on the turning kernel and the communication weight leading to the pathwise well-posedness estimates and flocking estimates, and provide a priori estimates for the random kinetic CCS equation. In Section 3, we provide pathwise x -estimates and flocking estimates within the proposed framework. We also investigate the propagation of pathwise regularity and the pathwise stability for the z-variations of solutions in Section 4. Finally, Section 5 is devoted to a brief summary of our main results.
Gallery of notation: For simplicity, we use the following handy notation.
Let π : Ω → R + ∪ {0} be a nonnegative probability density function, and we define a weighted L 2 -space: with an inner product and the corresponding L 2 -norm: For m ∈ N ∩ {0}, we set 2. Preliminaries. In this section, we discuss structural assumptions on the truning kernel and communication weight function, and study a priori estimate for the random kinetic CCS model.

A framework.
In this subsection, we discuss structural assumptions on the turning kernel T and the communication weight ψ leading to the pathwise wellposedness theory and the flocking estimate. Our proposed framework (A) is formulated in terms of the turning kernel and communication weight function as follows.
(A 1 ) (Compactness and regularities of the turning kernel): (i) The nonnegative function T [S](t, x, v, v , z) has compact supports in v and v : x, v, v , z) satisfies regularity conditions: x, v, v , z) satisfies the regularity conditions: Remark 1. In the sequel, we provide several comments on the above framework.
1. The assumptions (i) and (ii) in (A 1 ) are the natural extensions of [8] for a global well-posedness, and additional assumption (iii) in (A 1 ) is needed for local sensitivity analysis. We also note that the difference T [S] − T [S] in (ii) and (iii) in (A 1 ) requires one-less regularity compared to the regularity of each turning kernel.
2. As in [21], we can admit a relaxed communication weight ψ satisfying to derive a well-posedness of strong solution in W k,∞ -space and asymptotic flocking analysis pathwise. In fact, following the method of bi-characteristics, we only need to assume the uniform boundedness of ψ in spatial variable for the boundedness of velocity support and second velocity moment (see Lemma 3.1 of [8]): for T ∈ (0, ∞), In [8], the existence of a positive lower bound for ψ as in (A 2 ) was used for the simplicity of presentation and we also adopted the same condition for the same reason. Thus, one can use the assumption (6) to derive the well-posedness of strong solution and asymptotic flocking estimate pathwise with more technical estimates.

2.2.
A priori estimates. In this subsection, we present a priori estimates for the random kinetic CCS model. First, we set the heat kernel on R d × R d as and we set T ∈ (0, ∞).
Lemma 2.1. Suppose that the framework (A 1 )-(A 2 ) holds, and random parameters and the initial data (f 0 , S 0 ) satisfy , and let (f, S) be a regular solution process to (3). Then, for d ≥ 2, z ∈ Ω and t ∈ [0, T ), we have where C and C(T ) are positive constants.
Proof. The estimates in (7) can be derived from the integral equation: which comes from Duhamel's principle for the inhomogeneous heat equation. We refer to [8] for details.
Next, we discuss the propagation of velocity moments in f . We define the first three velocity moments, M 0 , M 1 and M 2 which represent the total mass, momentum, and twice the value of energy, respectively: for (t, z) ∈ R + × R 2d , For notational simplicity, we drop f from the moments of (3): In the following two propositions, we study the temporal evolution of velocity moments.
x, v, z), S(t, x, z)) be a regular solution process to (3), and f decays fast enough at infinity in phase space R 2d . Then, for d ≥ 2, z ∈ Ω and t ≥ 0, Proof. The proof is essentially the same as the determinstic counterpart. See [8] for details.
Proposition 2. Suppose that the framework (A 1 )-(A 2 ) holds, and random parameters and the initial data (f 0 , S 0 ) satisfy and let (f, S) be a regular solution process to (3). Then, for d ≥ 2, z ∈ Ω and t ≥ 0, Proof. (i) We use Proposition 1 to derive where Thus, we have (ii) Again, we use Proposition 1 to obtain Then Grönwall's lemma yields , t ≥ 0 and z ∈ Ω. Now, we present a global well-posedness of a strong solution to (3).
Theorem 2.2. Suppose that the framework (A 1 )-(A 2 ) holds, and random parameters, the initial data satisfy Then, there is a unique global solution process (f, S) to (3) such that for any T > 0, where V 0 is a compact set containing B R (0) and V is another compact set containing V 0 .
Proof. The proof is basically the same as that of Theorem 3.1 in [8].

Remark 2.
(1) The local well-posedness of a solution in Theorem 2.2 is shown in [8] by constructing of approximate solutions (f n , S n ) using the standard successive approximation. In other words, the proof is based on the method of characteristics for (3) 1 : For x, v ∈ R d and z ∈ Ω, we set (x n (t, z), v n (t, x), f n (t, z)) := (x n (t; 0, σ, z), v n (t; 0, σ, z), f n (t, x n (t; 0, σ, z), v n (t; 0, σ, z), z)), as a solution to the following ODE system: for t > 0 and z ∈ Ω, and Duhamel's principle for (3) 2 : For a global well-posedness, the authors in [8] extended a local solution to a global one by showing the corresponding norm of solution does not blow-up in finite time.
(2) If we denote the diameter of the velocity support and the position support by V (t, z) and X(t, z), respectively, it follows from Theorem 2.2 and (8) that for some constant C independent of t, z.
3. Pathwise well-posedness and flocking estimate. In this section, we present the pathwise well-posedness and velocity alignment estimate for the smooth solution to (3). For the sake of simplicity, we briefly denote the random turning operator by

3.1.
Pathwise well-posedness. In this subsection, we present a global existence of unique regular solution (f, S) ∈ W k,∞ × W k+1,∞ to (3). Although we considered the global well-posedness of a strong solution in W 1,∞ × W 2,∞ -space in Section 2.2, we need to extend its analysis to regular solutions with high regularity for a later use.
Theorem 3.1. Suppose that the framework (A 1 )-(A 2 ) holds, and in addition, the following assumptions hold: for d ≥ 2, (1) Initial datum f 0 is compactly supported with respect to v variable, and has finite moments: there exists a positive bounded function V (z) such that (3) ψ(|x|, z), α(z) and κ(z) are sufficiently regular: Then, there exists a unique global solution process (f, S) to (3) with the following regularity: for z ∈ Ω and T ∈ (0, ∞), there exists a random variable C(T, z) such that Proof. The proof is basically the same as that of Theorem 2.2. So, we will provide a priori estimate of (10) using induction argument on k ≥ k ≥ 1. To do this, we first introduce two functionals: • Initial step (k = 1): It is obviously true due to Theorem 2.2.
• Inductive step (k ≥ 2): In this step, suppose that for z ∈ Ω, Next, we will show that for z ∈ Ω, respectively. As a result we can conclude Step A. (finiteness of B k +1 (t, z)): For this, we consider the cases: ν = 0 and ν ≥ 1, separately.
We integrate the equation (11) along the particle trajectory to obtain For the term F a [f ] in I 21 and I 22 , we observe that For the terms T, T * in I 22 , we use the assumption (A 1 ) to have Thus, we substitute these estimates into (12) to obtain This yields (ii) (|ν| ≥ 1 case): For µ and ν with |µ| We use the assumption (A 1 ) in Section 2.1 for the terms T and T * : Similary, the estimate for T * can be made as follows.
Next, we choose j such that the j-th-component of µ is not zero, and e j denotes the unit vector with jth-component 1. Then, we apply ∂ µ x to S(t, x, z) to obtain We insert (18) into (17) to see This concludes sup 0≤t<T C k +1 (t, z) ≤ c 3 (T, z).

Velocity alignment.
In this subsection, we present the velocity alignment estimate by using the Lyapunov functional approach in [8]. First we define Lyapunov functional L as follows.
where σ = (x, v) and σ * = (x * , v * ). Then, it is easy to check that the functional can be rewritten as The following theorem provides that the velocity alignment emerge algebraically fast under the same assumptions as in Theorem 3.1.
Proof. We use Proposigion 1 and 2 to find the time-variation of L[f ](t, z) as follows: The last inequality is due to the decay estimate of S L ∞ x in Lemma 2.1. Then, Grönwall type lemma yields

4.
Local sensitivity analysis: Higher-order estimates. In this section, we study two quantitative local sensitivity estimates such as the propagation of wellposedness and stability for the infinitesimal in a random space. For the simplicity of presentation, we assume that the random space Ω is one-dimensional, i.e. z ∈ R. For m ∈ N, we first apply the operator ∂ m z to equation (3) to get to see that f and its z-variations ∂ m z f have the same transport structure as in (8) 3 . Thus the global unique solvability of (∂ m z f, ∂ m z S) is fundamentally based on the same argument as in the proof of Theorem 2.2 and Theorem 3.1. Moreover, it is easy to check that the velocity and spatial supports of z-variations ∂ m z f are subsets of those of f . To be precise, for m ∈ N, we denote the diameter of the velocity support and the position support for ∂ m z f by V m (t, z) and X m (t, z), respectively. If the initial datum f 0 is compactly supported in x, v, then it follows from (9) that for z ∈ Ω, t > 0.

Propagation of pathwise regularities. In this subsection, we study the global existence of z-variations
In the sequel, we will consider the two cases: m = 1 and m ≥ 2.
Before we present our main results, we provide a series of a priori estimates.
Lemma 4.1. Let (f, S) = (f (t, x, v, z), S(t, x, z)) be a regular solution process to (3) whose initial datum f 0 is compactly supported in x, v. Then, for d ≥ 2, there exist C 1 (t, z) and C 2 (t, z) such that Proof. (i) It follows from (4) that where the random function C 1 is given as follows Similarly, we can derive (ii) By direct calculations, one has where the random function C 2 is given as follows Similarly, we have Remark 3. Note that the relations (20) imply that the constants C 1 and C 2 in Lemma 4.1 under the assumption (A 2 ) satisfy the following estimates: for each z ∈ Ω, x, v, z), S(t, x, z)) be a regular solution process to (3) whose initial datum f 0 is compactly supported in x, v. Then, for d ≥ 2, z ∈ Ω and t > 0, where 0 < |µ| ≤ k, 0 < l ≤ m, |µ| + l < k + 1.
Proof. (i) We use Duhamel's principle to rewrite (19) 2 as follows. Since Here, we used the property: R d e t∆ (x, y) dy = 1.
(ii) Again we apply Duhamel's principle to rewrite (19) 2 as where G 2 (s, y, z) Note that we select the index j such that the j th-component of µ is not zero and denote the unit vector with j th-component 1 by e j . Again, it follows from (21) that This yields The following lemma deals with a priori estimate for the turning operator x, v, z), S(t, x, z)) be a regular solution process to (3) whose initial datum f 0 is compactly supported in x, v. Then, for d ≥ 2, z ∈ Ω and t > 0, Proof. (i) We apply ∂ l z for the turning operator to get Note that This leads to (ii) Similarly, we take ∂ ν v ∂ µ x ∂ m z to the turning operator to obtain Note that Then, we have  For k ≥ 1 and for each z ∈ Ω, suppose that the initial data (f 0 , S 0 ) = (f 0 (x, v, z), S 0 (x, z)) satisfy the following conditions: for d ≥ 2, 1. Initial datum f 0 is compactly supported with respect to x, v, and has finite moments: there exist positive bounded functions X(z), V (z) such that 4. ψ(|x|, z), α(z) and κ(z) are sufficiently regular: Then, for any T ∈ (0, ∞), there exists a global unique solution process satisfying for each z ∈ Ω.
Proof. The proof is almost the same as that of Theorem 3.1. So, we provide a priori W k−1,∞ x,v -estimate for ∂ z f and W k,∞ x -estimate for ∂ z S. Throughout this proof, we set Then, by Theorem 3.1, sup 0≤t<T F(t, z) + S(t, z) ≤ C(T, z), for z ∈ Ω.
• Case A (k = 1): Consider the equation (19) 1 with m = 1 to get We integrate the above equation along the particle trajectory to obtain By Lemma 4.1 and 4.3, we have Thus, we have On the other hand, one has This yields Finally, we apply Gronwall's lemma for (23) and (24) to get Moreover, the estimate (ii) in Lemma 4.2 gives sup 0≤t<T |µ|=1 • Case B (k ≥ 2): We will verify (22) by using an induction argument on k ≥ k ≥ 2. Suppose that the following estimate holds.
Then, we have Then, we integrate the above relation along the particle trajectory and apply Lemma 4.1, the estimate (ii) in Lemma 4.3 and the induction hypothesis to derive This leads We collect (25), (26) and use the induction hypothesis to find We apply Grönwall's lemma to the above integral estimate to obtain Finally, we substitute this estimate into the estimate (ii) in Lemma 4.2 to conclude Now, we extend the result of Theorem 4.4 to the case m ≥ 2.
Theorem 4.5. For k ≥ 2 and z ∈ Ω, suppose that the initial data (f 0 , S 0 ) satisfy the following conditions: for d ≥ 2, (1) Initial datum f 0 is compactly supported with respect to x, v and has finite moments: there exist positive bounded functions X(z) and V (z) such that (4) ψ(|x|, z), α(z) and κ(z) are sufficiently regular: Then, for any T ∈ (0, ∞) and z ∈ Ω, there exists a global unique solution process satisfying Proof. Since the proof is rather lengthy, we leave its proof in Appendix A.

Stability analysis.
In this subsection, we derive L ∞ -stability estimates for the z-variations of solution processes. For this, we first present a priori estimates.
x -regular solutions to (3) whose initial data f 0 andf 0 are compactly supported in x and v. Then, for d ≥ 2 and l ≤ m, 0 ≤ |µ| ≤ k − l, we have Proof. Since the proof for (ii) is similar to that of (i), we only derive the estimate (i): where V (t, z) := max{V (t, z), V (t, z)}, X(t, z) := max{X(t, x), X(t, z)}, and V (t, z), X(t, z) are the diameters of the velocity and position supports of a solutionf .
Next, we consider the L ∞ -stability estimates of global unique solution processes (f, S).
Proof. We leave its detailed proof in Appendix B.
Now, we present a local sensitivity analysis on the infinitesimal veriations of f and S.
Proof. Since the proof is rather lengthy, we leave its proof in Appendix C.
Finally, we apply Gronwall's lemma to (29) and (34) to conclude the desired result.

5.
Conclusion. In this paper, we provided several quantative local sensitivity estimates to the kinetic CCS model with random inputs. In the first author's recent work [14] on kinetic CS model with random inputs, they suggested a systematic local sensitivity analysis for the CS flocking model, and found the structural assumptions on the communication weight results in the robustness of the flocking estimates. Thus, a natural extension of the previous work is to see whether similar local sensitivity analysis can be performed for kinetic CS model combined with turning operation effect [8]. In this work, we proposed some structural assumptions on the turning operator and communication weight with the supplement of random inputs, and showed the global existence of a regular solution having the robustness of a velocity alignment. Moreover, we provided the regularity and stability estimates for a global solution in random space. Recall that we used a kinetic approach to model the abrupt change of velocities in an interacting particle system. Of course, a more natural approach is to begin with a suitable second-order particle model capturing the abrupt change of velocities. However, as far as the authors know, this is an open problem in flocking community. We leave this interesting issue as a future work.
We need to show that sup 0≤t<T F m (t, z) + S m (t, z) ≤ C(T, z).
We first apply Duhamel's principle to (38) 2 to get (S −S)(t, x, z) = We use Theorem 3.1, Lemma 4.2 and Lemma 4.6 to find We rewrite the assumption ( Thus, we have Finally, we integrate (43) and use estimates for I 4j to have (Estimate of ∂ µ x (f −f ) with |ν| = 0, k = |µ|): Note that ∂ µ x (f −f ) satisfies