Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

TL;DR

This paper proves that solutions to the Boltzmann equation converge to the Euler equations with contact discontinuities in the hydrodynamic limit, using energy methods and micro-macro decomposition, for suitable initial data.

Contribution

It establishes the global-in-time convergence of Boltzmann solutions to contact discontinuities in the Euler system without initial layer effects.

Findings

Convergence of Boltzmann solutions to Euler contact discontinuities as Knudsen number approaches zero.

Existence and uniqueness of Boltzmann solutions for all time under specified initial conditions.

Uniform convergence away from contact discontinuities in the hydrodynamic limit.

Abstract

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exists a unique solution to the Boltzmann equation globally in time for any given Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number $\varepsilon$ tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.