The Mean Field Kinetic Equation for Interacting Particle Systems with Non-Lipschitz Force

TL;DR

This paper establishes the global existence of weak solutions for a mean field kinetic equation describing interacting particle systems with non-Lipschitz forces, using advanced mathematical techniques to handle non-local nonlinearities.

Contribution

It proves the existence of solutions for the mean field kinetic equation with non-Lipschitz forces, extending previous results to more general interaction forces.

Findings

Proved global existence of weak solutions for the kinetic equation.

Handled non-local nonlinear interactions with compactness methods.

Applied Aubin-Lions theorem to overcome analytical challenges.

Abstract

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the $N$-particle Newtonian system. For $L^1\cap L^\infty$ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the non-local non-linear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.