These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.1 (Young-Pil Choi)

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

To investigate the effects of pipe materials on biofilm accumulation and water quality, an annular reactor with the sample coupons of four pipe materials (steel, copper, stainless steel, and polyvinyl chloride) was operated under hydraulic conditions similar to a real plumbing system for 15 months. The bacterial concentrations were substantially increased in the steel and copper reactors with progression of corrosion, whereas those in stainless steel (STS) and polyvinyl chloride (PVC) reactors were affected mainly by water temperature. The heterotrophic plate count (HPC) of biofilms was about 100 times higher on steel pipe than other pipes throughout the experiment, with the STS pipe showing the lowest bacterial number at the end of the operation. Analysis of the 16S rDNA sequences of 176 cultivated isolates revealed that 66.5% was Proteobacteria and the others included unclassified bacteria, Actinobacteria, and Bacilli. Regardless of the pipe materials, Sphingomonas was the predominant species in all biofilms. PCR-DGGE analysis showed that steel pipe exhibited the highest bacterial diversity among the metallic pipes, and the DGGE profile of biofilm on PVC showed three additional bands not detected from the profiles of the metallic materials. Environmental scanning electron microscopy showed that corrosion level and biofilm accumulation were the least in the STS coupon. These results suggest that the STS pipe is the best material for plumbing systems in terms of the microbiological aspects of water quality.

Abstract. In this paper, we study the local well-posedness of two types of generalized kinetic CuckerSmale (in short C-S) equations. We consider two different communication weights in space with nonlinear coupling of the velocities, v|v| β−2 for β >, where singularities are present either in space or in velocity. For the singular communication weight in space, ψ 1 (x) = 1/|x| α with α ∈ (0, d−1), d ≥ 1, we consider smooth velocity coupling, β ≥ 2. For the regular one, we assume ψ, 2). We also present the various dynamics of the generalized C-S particle system with the communication weights ψ i , i = 1, 2 when β ∈ (0, 3). We provide sufficient conditions of the initial data depending on the exponent β leading to finite-time alignment or to no collisions between particles in finite time.

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