Anomalous smoothing effect on the incompressible Navier-Stokes-Fourier limit from Boltzmann with periodic velocity

TL;DR

This paper proves the existence of global weak solutions to an incompressible 3D Navier-Stokes-Fourier system derived from the Boltzmann equation with periodic velocity, highlighting an anomalous smoothing effect due to microstructure terms.

Contribution

It introduces a novel approach using hydrodynamic limits from Boltzmann equations with periodic velocity to establish solutions for a complex fluid system.

Findings

Existence of global-in-time weak solutions with bounded enstrophy.

Development of a new method employing hydrodynamic limits.

Identification of an anomalous smoothing effect in the system.

Abstract

Adding some nontrivial terms composed from a microstructure, we prove the existence of a global-in-time weak solution, whose enstrophy is bounded for all the time, to an incompressible 3D Navier-Stokes-Fourier system for arbitrary initial data. It cannot be expected to directly derive the energy inequality for this new system of equations. The main idea is to employ the hydrodynamic limit from the Boltzmann equation with periodic velocity and a specially designed collision operator.