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

Abstract

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.