Quantitative Closure Analysis toward Ideal Fluids

TL;DR

This paper proves the low-Mach/high-Reynolds limit of the Boltzmann equation to incompressible fluids without asymptotic expansion, providing quantitative estimates and convergence results in two dimensions.

Contribution

It introduces a new quasi-linear analysis using macro-micro decomposition to establish rigorous convergence to incompressible Euler equations.

Findings

Quantitative estimates for microscopic fluctuations derived.

Bounds established for kinetic vorticity and entropic fluctuations.

Convergence to incompressible Euler equations in 2D proven.

Abstract

We establish the incompressible low--Mach/high--Reynolds limit for the Boltzmann equation for a broad class of initial data, without recourse to any asymptotic expansion. Exploiting the local Maxwellian manifold and the macro--micro decomposition in a new quasi-linear analysis, we derive quantitative estimates for the purely microscopic fluctuation, as well as bounds for the kinetic vorticity and the entropic fluctuation in terms of the initial data. As a consequence, in two space dimensions, the rescaled velocity and temperature converge to a global solution of the incompressible Euler equations coupled to a transported temperature, within the frameworks of DiPerna--Lions--Majda and Delort.