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 Grönwall-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.