Homogenization of Inhomogeneous Incompressible Navier-Stokes Equations in Domains with Very Tiny Holes
Yong Lu, Jiaojiao Pan, Peikang Yang
TL;DR
This work analyzes the homogenization of the 3D inhomogeneous incompressible Navier–Stokes equations in a perforated domain with very tiny holes, where hole diameters are $\varepsilon^{\alpha}$ with $\alpha>3$ and mutual spacing is $O(\varepsilon)$. The authors reformulate the problem on the hole-free domain via zero extensions, obtain uniform energy-type estimates, and employ compactness (Aubin–Lions), Bogovskii operators in perforated domains, and Leray–Helmholtz projections to pass to the limit. They prove strong convergence of the density $\widetilde{\rho}_{\varepsilon}\to\rho$ in $C([0,T];L^{p}(\Omega))$ and of the momentum, leading to a limit system that is the same in the homogeneous domain: $\partial_t\rho+\mathrm{div}(\rho\mathbf{u})=0$, $\partial_t(\rho\mathbf{u})+\mathrm{div}(\rho\mathbf{u}\otimes\mathbf{u})-\mu\Delta\mathbf{u}+\nabla p=\rho\mathbf{f}$, $\mathrm{div}\mathbf{u}=0$, with $\mathbf{u}=0$ on $\partial\Omega$ and corresponding initial data. In particular, for very small holes the homogenized limit remains unchanged, reinforcing that perforations of vanishingly small size do not alter the macroscopic NS dynamics in this regime. The results provide a rigorous foundation for effective models of multi-phase incompressible flows in highly perforated media.
Abstract
In this paper, we study the homogenization problems of $3D$ inhomogeneous incompressible Navier-Stokes system perforated with very tiny holes whose diameters are much smaller than their mutual distances. The key is to establish the equations in the homogeneous domain without holes for the zero extensions of the weak solutions. This allows us to derive time derivative estimates and show the strong convergence of the density and the momentum by Aubin-Lions type argument. For the case of small holes, we finally show the limit equations remain unchanged in the homogenization limit.