Rigorous derivation of the Kinetic/Fluid coupling involving a Kinetic layer on a toy problem

TL;DR

This paper rigorously derives the kinetic/fluid coupling from a hydrodynamical limit on a toy model, revealing the role of a kinetic layer like the Milne problem in complex shock scenarios.

Contribution

It provides a rigorous derivation of the kinetic/fluid coupling involving a kinetic layer, especially in the presence of shocks, using blow-up techniques for strong convergence.

Findings

Kinetic/fluid coupling can be derived from hydrodynamical limits.

The Milne problem appears as a kinetic layer in the coupling.

Strong convergence is achieved despite the presence of shocks.

Abstract

In this article, we investigate the kinetic/fluid coupling on a toy model. We obtain it rigorously from a hydrodynamical limit. The idea is that at the level of the full kinetic model, the coupling is obvious. We then investigate the coupling obtained when passing into the limit. We show that, especially in presence of a shock stuck on the interface, the coupling involves a kinetic layer known as the Milne problem. Due to this layer, the limit process is quite delicate and some blow-up techniques are needed to ensure its strong convergence.