Quantitative fluid approximation in fractional regimes of transport equations with more invariants

TL;DR

This paper extends the analysis of macroscopic limits of linear kinetic equations to fractional regimes with heavy-tailed equilibria, revealing new spectral behaviors and scaling differences in the eigenvalues.

Contribution

It develops a unified spectral framework for operators with multiple invariants and heavy tails, generalizing previous results and addressing spectral gap challenges.

Findings

Spectral analysis uncovers faster convergence of transversal wave eigenvalues in fractional regimes.

The framework handles operators with mass, momentum, and energy invariants and heavy-tailed equilibria.

Distinct scaling behaviors are identified between fractional and classical regimes.

Abstract

We present an extension of results in a previous paper by the first and the last author [PMP, 2022] about macroscopic limits of linear kinetic equations in (potentially) fractional regimes. More precisely, we develop a unified framework inspired by Ellis and Pinsky [J. Math. Pures Appl., 1975] for operators that preserve mass, momentum and energy, and have microscopic equilibrium with heavy tails (typically polynomial). This paper also generalizes one of Hittmeir and Merino [KRM, 2016] in a related framework. The main difficulty, that leads to our main contribution, is the understanding of the spectrum of the generator in the Fourier space, which is significantly complicated by the lack of spectral gap and the fat tails of the equilibrium. Indeed, the scaling of the eigenelements in the suitable macroscopic rescaling is subtle to handle. In particular, our study uncovered an interesting difference in scaling in the fractional regime, where the transversal wave eigenvalues converge faster to zero than the Boussinesq and acoustic wave eigenvalues.