Environmental averaging

Abstract

We introduce a new analytical framework designed to systematically study models of collective behavior based on smooth communication. Many examples including the classical Cucker-Smale, Motsch-Tadmor, multi-species, and several new models fall under the framework and can be viewed as special cases of 'environmental averaging'. By identifying special properties of the averaging protocols we reveal new alignment results that are otherwise obscured by specificity of the model if analyzed individually. In particular, it is shown that the kinetic Cucker-Smale model based on a Bochner-positive communication kernel φ = ψ * ψ aligns provided the density of the flock obeys a lower bound ρ 1 t 1/4 , and no bound from above is required. Similar results are proved for other positive models.A series of studies are presented based on the introduced framework -mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, global hypocoercivity and relaxation for dissipative kinetic models, several general alignment results based on chain connectivity of the system. In each study we identify a set of natural regularity conditions on the averaging protocol sufficient for the end-result in question. Numerous applications to existing models and new examples are revealed. A detailed well-posedness theory for kinetic Vlasov-alignment and Fokker-Planck-alignment models is developed. It is shown that solutions to the Fokker-Planck model based on a smooth environmental averaging gain Gaussian tails uniformly in time and relax unconditionally. This applies, in particular, to the classical Cucker-Smale model and presents a substantial improvement over previously obtained perturbative results.