Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime

Abstract

We present a hydrodynamic limit from the kinetic thermomechanical Cucker-Smale (TCS) model to the hydrodynamic Cucker-Smale (CS) model in a strong local alignment regime. For this, we first provide a global existence of weak solution, and flocking dynamics for classical solution to the kinetic TCS model with local alignment force. Then we consider one-parameter family of well-prepared initial data to the kinetic TCS model in which the temperature tends to common constant value determined by initial datum, as singular parameter ε tends to zero. In a strong local alignment regime, the limit model is the hydrodynamic CS model in [8]. To verify this hydrodynamic limit rigorously, we adopt the technique introduced in [5] which combines the relative entropy method together with the 2-Wasserstein metric.

Propagation of the mono-kinetic solution in the Cucker-Smale-type kinetic equations

“…For a rigorous study on the hydrodynamic limit of the kinetic equation (1.3), we refer to [11]. There, they proved a hydrodynamic limit of (1.3) with a strong local alignment towards (1.2), by considering the temperature support of the initial data f 0 degenerating to a single value as the scaling parameter tends to 0.…”