A rigorous derivation of the stationary compressible Reynolds equation via the Navier-Stokes equations

TL;DR

This paper rigorously derives the stationary compressible Reynolds equation from the Navier-Stokes equations on thin domains, establishing solution existence and uniqueness in certain cases, and introduces new bounds for pressure laws.

Contribution

It provides a rigorous derivation of the Reynolds system as a limit of Navier-Stokes equations, including existence results and new a priori bounds for pressure laws.

Findings

Existence of solutions to Navier-Stokes with non-homogeneous boundary conditions.

Rigorous derivation of the Reynolds equation as a singular limit.

Uniqueness of the limit problem in 1D case.

Abstract

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier-Stokes system on a thin domain. In particular, the existence of solutions to the Navier-Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the 1D case.