Nonrelativistic limits of the relativistic Cucker-Smale model and its kinetic counterpart

We present sufficient frameworks for the uniform-in-time nonrelativistic limits for the relativistic Cucker-Smale (RCS) model and the relativistic kinetic Cucker-Smale (RKCS) equation. For the RCS model, one can easily show that the difference between the solutions to the RCS model and the CS model can be bounded by a quantity proportional to the exponential of time and inversely proportional to some power of the speed of light via a standard Gronwall-type differential inequality. However, this finite-in-time nonrelativistic limit result cannot be used in a uniform-in-time estimate due to the exponential factor of lifespan of solution as it is. For the uniform-in-time nonrelativistic limit, we split the deviation functional between the relativistic solution and the nonrelativistic solution into two parts (finite-time interval and infinite-time interval). In the finite-time interval, the deviation functional is bounded by a finite-in-time nonrelativistic limit result, and then, after a finite time, we use asymptotic flocking estimates with the same asymptotic momentum-like quantity for the RCS model and the CS model to show that the deviation functional can be made as small as possible. In this manner, we can derive a uniform-in-time nonrelativistic limit for the RCS model. For the RKCS equation, we use a uniform-in-time mean-field limit in a measure theoretic framework and a uniform-in-time nonrelativistic limit result for the RCS model to derive a uniform-in-time nonrelativistic limit for the RKCS equation. Published under an exclusive license by AIP Publishing.N

KINETIC AND HYDRODYNAMIC MODELS FOR THE RELATIVISTIC CUCKER-SMALE ENSEMBLE AND EMERGENT BEHAVIORS

We study emergent behaviors of relativistic kinetic and hydrodynamic models which can be derived from the relativistic Cucker-Smale (RCS) model. The RCS model was introduced to describe the flocking dynamics of relativistic particles in authors' recent work. The proposed relativistic kinetic and hydrodynamic models correspond to relativistic counterparts of the Vlasov-type CS model and pressureless gas dynamics, respectively, and their classical limits can be made in any finite-time interval as the speed of light tends to infinity. For the proposed relativistic models, we provide a global well-posedness of classical solutions and asymptotic flocking estimates using Lyapunov functional approach.N

Asymptotic flocking dynamics of a relativistic Cucker-Smale flock under singular communications

We study collision avoidance and flocking dynamics for the relativistic Cucker-Smale (RCS) model with a singular communication weight. For a bounded and regular communication weight, RCS particles can exhibit collisions in finite time depending on the geometry of the initial configuration. In contrast, for a singular communication weight, when particles collide, the associated Cucker-Smale vector field becomes unbounded and the standard Cauchy-Lipschitz theory cannot be applied so that existence theory after collisions is problematic. Thus, the collision avoidance problem is directly linked to the global solvability of the singular RCS model and asymptotic flocking dynamics. In this paper, we present sufficient frameworks leading to the nonexistence of finite-time collisions and asymptotic flocking in terms of initial configuration and blow-up rate at the singular point of communication weight.N

Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit

ABSTRACT. We present a uniform(-in-time) stability of the relativistic CuckerSmale (RCS) model in a suitable framework and study its application to a uniform mean-field limit which lifts earlier classical results for the CS model in a relativistic setting. For this, we first provide a sufficient framework for an exponential flocking for the RCS model in terms of the diameters of state observables, coupling strength and communication weight function, and then we use the obtained exponential flocking estimate to derive a uniform lq,pstability of the RCS model under appropriate conditions on initial data and system parameters. As an application of the derived uniform lq,p-stability estimate, we show that a uniform mean-field limit of the RCS model can be made for some admissible class of solutions uniformly in time. This justifies a formal derivation of the kinetic RCS equation [18] in a rigorous setting.N

Octave bandwidth photonic fishnet-achromatic-metalens

Here the authors demonstrate all-dielectric fishnet-achromatic-metalenses from the visible to the near-infrared region. This metalens performs efficiently independent of polarization over about an octave from 640 nm to 1200 nm