Dimension Reduction of Compressible Fluid Models over Product Manifolds

In this paper we study the dimension reduction limits of the compressible Navier–Stokes equations over product Riemannian manifolds ${\mathcal{O}}_{ϵ}\cong \mathcal{M}\times ϵ\mathcal{F}$, 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 ${\mathcal{O}}_{ϵ}$ 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 $A:\mathcal{M}\to {\mathbb{R}}^{+}$ 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, $\mathit{\text{SIAM}}\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.}}$, $\mathbf{\text{48}}$ (2016), 3907–3930, and it contains as special cases the physical models of circular nozzles, thin plate limits and finite-length longitudinal nozzles.