Asymptotic analysis of Vlasov-type equations under strong local alignment regime

TL;DR

This paper rigorously analyzes the hydrodynamic limit of a collisionless kinetic equation under strong local alignment, showing convergence to a pressureless Euler system using entropy methods.

Contribution

It provides a rigorous justification of the hydrodynamic limit for Vlasov-type equations with strong local alignment, extending previous singular limit analyses.

Findings

Weak solutions converge to local equilibrium distributions.

The kinetic equation converges to the pressureless Euler system.

The relative entropy method is effective for this analysis.

Abstract

We consider the hydrodynamic limit of a collisionless and non-diffusive kinetic equation under strong local alignment regime. The local alignment is first considered by Karper, Mellet and Trivisa in [24], as a singular limit of an alignment force proposed by Motsch and Tadmor in [32]. As the local alignment strongly dominate, a weak solution to the kinetic equation under consideration converges to the local equilibrium, which has the form of mono-kinetic distribution. We use the relative entropy method and weak compactness to rigorously justify the weak convergence of our kinetic equation to the pressureless Euler system.