Fluid models with phase transition for kinetic equations in swarming

Abstract

We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.

Local well‐posedness of classical solutions for the kinetic self‐organized model of Cucker–Smale type

“…In this paper, we consider the following kinetic model of Cucker-Smale type for self-organized system introduced by [1,2]:…”

“…In this paper, we consider the following kinetic model of Cucker–Smale type for self‐organized system introduced by [1, 2]: $$$$ {\partial}_tf+v\cdotp {\nabla}_xf=\sigma {\Delta}_vf+{\nabla}_v\cdotp \left\{f\left(v-u\left[f\right]\right)\right\}+{\nabla}_v\cdotp \left\{f{\nabla}_vV\right\}. $$$$ …”

“…Moreover, the Brownian motion is represented by the velocity diffusion $$$ \sigma {\Delta}_vf $$$, with the constant diffusion coefficient $$$ \sigma \ge 0 $$$. More details can be found in [1, 2, 4].…”