From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: a quantitative error estimate

TL;DR

This paper provides a rigorous, quantitative derivation of the incompressible Navier-Stokes equations from the Boltzmann equation on the torus, using hypocoercivity to establish uniform decay and regularity bounds as the Knudsen number approaches zero.

Contribution

It introduces a new hypocoercivity-based proof of the hydrodynamic limit for the Boltzmann equation on the torus, with explicit error estimates and regularity bounds.

Findings

Uniform exponential decay of solutions near equilibrium

Explicit regularity bounds independent of the Knudsen number

Different hydrodynamic limit behavior on the torus compared to the whole space

Abstract

We investigate the Boltzmann equation, depending on the Knudsen number, in the Navier-Stokes perturbative setting on the torus. Using hypocoercivity, we derive a new proof of existence and exponential decay for solutions close to a global equilibrium, with explicit regularity bounds and rates of convergence. These results are uniform in the Knudsen number and thus allow us to obtain a strong derivation of the incompressible Navier-Stokes equations as the Knudsen number tends to $0$. Moreover, our method is also used to deal with other kinetic models. Finally, we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time, unless the initial layer conditions are satisfied.