Dimension reduction for the compressible Navier-Stokes system with density dependent viscosity

TL;DR

This paper demonstrates that weak solutions of a 3D compressible Navier-Stokes system with density-dependent viscosity converge to the strong solution of a 1D system as the domain's cross-sectional size shrinks to zero, establishing a dimension reduction result.

Contribution

It provides a rigorous mathematical proof of the dimension reduction from 3D to 1D for the compressible Navier-Stokes system with density-dependent viscosity.

Findings

Weak solutions converge to 1D strong solutions as domain collapses.

Dimension reduction holds for density-dependent viscosity models.

Mathematical framework for analyzing asymptotic behavior of solutions.

Abstract

We consider a compressible Navier-Stokes system for a barotropic fluid with density dependent viscosity in a three-dimensional time-space domain $(0,T)\times \Omega_\varepsilon$ where $\Omega_\varepsilon = (0,\varepsilon)^2\times (0,1)$. We show that the weak solutions of the 3D system converges to the strong solution of the respective 1D system as $\varepsilon\rightarrow0.$