UNIFORM STABILITY AND MEAN-FIELD LIMIT OF A THERMODYNAMIC CUCKER-SMALE MODEL

We present a uniform-in-time stability and uniform mean-field limit of a thermodynamic Cucker-Smale model with small diffusion velocityf (for short, the SDV-TCS model). The original Cucker-Smale model deals with flocking dynamics of mechanical particles, in which the position and momentum are only macroscopic observables. Thus, the original Cucker-Smale model cannot describe some thermodynamic phenomena resulting from the temperature variations among particles and internal variables not taken into account. In [SIAM J. Math. Anal. 50 (2018), pp. 3092-3121] and [Arch. Rational. Mech. Anal. 223 (2017), pp. 1397-1425], a new thermodynamically consistent particle model was proposed from the system of gas mixtures in a rational way. In this paper, we discuss two issues for the SDV-TCS model. First we present a uniform stability of the SDV-TCS model with respect to initial data in the sense that the distance between two solutions is uniformly bounded by that of initial data in a mixed Lebesgue norm. Second, we derive a uniform mean-field limit from the SDV-TCS model to the Vlasov-type kinetic equation for some class of initial data whose empirical measure approximation guarantees exponential flocking in the SDV-TCS model.