Vorticity convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class

TL;DR

This paper rigorously analyzes the hydrodynamic limit of Boltzmann equations to 2D Euler equations with unbounded vorticity, establishing convergence and rates even in singular vorticity regimes.

Contribution

It provides the first rigorous proof of hydrodynamic limits toward unbounded vorticity solutions of 2D Euler equations, including convergence rates in the Yudovich class.

Findings

Established convergence of kinetic vorticity to Euler vorticity.

Proved hydrodynamic limit for solutions with unbounded vorticity.

Derived convergence rates as vorticity becomes unbounded.

Abstract

It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) $$ toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded $$\partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\text{div }u =0.$$ We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of the hydrodynamic limit toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ($\omega \in L^\mathfrak{p}$ for any fixed $1 \leq \mathfrak{p} < \infty$). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in $L^\mathfrak{p}$ as $\mathfrak{p} \rightarrow \infty$ (localized Yudovich class).