Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

Abstract

We consider an interacting N -particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuitytype of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles N goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman [J. Funct. Anal., 56, (1984), 311-336] to that one with discontinuous kernels. 14 4. Propagation of chaos 15 4.1. Law of large numbers like estimates 15 4.2. Proof of Theorem 1.3 21 Appendix A. Gronwall-type inequalities 21 Appendix B. A representation for solutions to the PDE (1.4) 22 Acknowledgments 24 References 24 1. IntroductionMathematical modelling of collective behaviors, such as flocks of birds, schools of fish, or aggregation of bacteria, etc, has received a bulk of attention because of its possible applications in the field of engineering, biology, industry, and sociology [2,19,23,30,31]. These models are usually based on incorporating different mechanisms of interactions between individuals, for instance, a short-range repulsion, a longrange attraction, and an alignment in certain spatial regions. We refer the reader to [5,7,9] and the references therein for recent surveys of collective behavior models. In this current work, we consider Date: August 27, 2018.

Collective behavior models with vision geometrical constraints: Truncated noises and propagation of chaos

“…We first show law of large numbers-like estimates whose proofs rely on the nice property of our communication weights in Lemma 2.1. Even though similar results observed in [14,25], for the sake of completeness we give the details of it in Appendix A.…”

“…In the current work, it seems convenient to consider p-Wasserstein distance with p ∈ 2N due to the multiplicative noises. However it has already been pointed out in [7,Remark 3.1] by the authors and their collaborators that our strategy does not work in Wasserstein distance of order p with p ∈ (1, ∞), thus the make use of W 1 or W ∞ is essential in the framework because of the form of force fields, see [14]. For those reasons, we introduce a modified Wasserstein 1 distance W γ 1 as…”

“…Then we can find a probability space (Ω, P, (F t ) t≥0 , F ), a Brownian motion (B t ) t≥0 on that basis and a process (X t , V t ) t≥0 solution to (4.1), which has the time marginalf t at any time t ≥ 0. We refer to [14] for more details on that.…”

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

“…The existence of weak and strong solutions, and the rigorous derivation of the kinetic equation (1.5) are well studied in [2,24,25]. We also refer to [1,4,5,6,17] for the local/glocal-in-time existence theories and the mean-field limit for the swarming models with singular interaction kernels. At the formal level, in order to derive the hydrodynamic equations of the form (1.1), we can take into account the moments on the kinetic equations: ρ = f dv and ρu = vf dv.…”

Propagation of chaos for the Keller–Segel equation over bounded domains

“…The key observation is that due to the convexity of the domain, this extra term possesses a good sign (see in (4.13) and (4.14)), and thus it does not add essential difficulties to the stability estimates. Such observation has been used in earlier works [15,16,20].…”