Existence of Weak Solutions to Kinetic Flocking Model with Cut-off Interaction Function

TL;DR

This paper proves the existence of weak solutions for a kinetic flocking model with a cut-off interaction function, using fixed point theorems and velocity averaging, and removes previous integrability constraints.

Contribution

It establishes the existence of weak solutions under natural boundedness assumptions, extending prior results by relaxing integrability conditions.

Findings

Velocity support remains uniformly bounded over time.

Existence of weak solutions proven using Schauder fixed point theorem.

Removed the need for strong integrability conditions on initial data.

Abstract

We prove the existence of weak solutions to kinetic flocking model with cut-off interaction function by using Schauder fixed pointed theorem and velocity averaging lemma. Under the natural assumption that the velocity support of the initial distribution function is bounded, we show that the velocity support of the distribution function is uniformly bounded in time. Employing this property, we remove the constraint in the paper of Karper, Mellet and Trivisa[SIAM. J. Math. Anal., (45)2013, pp.215-243] that the initial distribution function should have better integrability for large $|\bx|$.