Modulated energy estimates for singular kernels and their applications to asymptotic analyses for kinetic equations

TL;DR

This paper develops modulated energy estimates for singular kernels and applies them to analyze the asymptotic behavior of kinetic equations, including limits to hydrodynamic models with nonlocal interactions.

Contribution

It introduces a novel method for modulated energy estimates for singular kernels and applies it to derive rigorous asymptotic limits in kinetic equations.

Findings

Quantified small inertia limit for kinetic equations with singular interactions

Rigorous derivation of aggregation equations with singular kernels

Hydrodynamic limits leading to Euler systems with nonlocal forces

Abstract

In this paper, we provide modulated interaction energy estimates for the kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified small inertia limit of kinetic equation with singular nonlocal interactions. The aggregation equations with singular interaction kernels are rigorously derived. We also study the rigorous quantified hydrodynamic limit of the kinetic equation to derive the isothermal Euler or pressureless Euler system with the nonlocal singular interactions forces.