Dimension Reduction of Compressible Fluid Models over Product Manifolds

TL;DR

This paper rigorously analyzes the dimension reduction of compressible Navier--Stokes equations on product manifolds, establishing convergence to lower-dimensional models and including effects of geometry and vanishing viscosity.

Contribution

It generalizes previous results by establishing convergence of weak solutions on product manifolds to classical solutions on the base manifold, incorporating geometric effects and vanishing viscosity limits.

Findings

Proves convergence of solutions as the fiber size shrinks

Identifies limiting equations with geometric weight functions

Includes special cases like nozzles and thin plates

Abstract

In this paper we study the dimension reduction limits of the compressible Navier--Stokes equations over product Riemannian manifolds $\mathcal{O}_\epsilon \cong \mathcal{M} \times \epsilon\mathcal{F}$, such that $\dim\,(\mathcal{M})=n$ and $\dim\,(\mathcal{F})=d$ are arbitrary. Using the method of relative entropies, we establish the convergence of the suitable weak solutions of the Navier--Stokes equations on $\mathcal{O}_\epsilon$ to the classical solution of the limiting equations on $\mathcal{M}$ as $\epsilon \rightarrow 0^+$, provided the latter exists. In addition, we also deduce the vanishing viscosity limit. The limiting equations identified through our analysis contain the weight function $A:\mathcal{M} \rightarrow \mathbb{R}^+$ as a parameter, where $A(x)$ = area of fibre $\mathcal{F}_x$. Our work is based on and generalises the results in P. Bella, E. Feireisl, M. Lewicka and A. Novotn\'{y}, A rigorous justification of the Euler and Navier--Stokes equations with geometric effects, \textit{SIAM J. Math. Anal.}, \textbf{48} (2016), 3907--3930, and it contains as special cases the physical models of circular nozzles, thin plate limits and finite-length longitudinal nozzles.