具有记忆项的Darcy定律：演化微结构的均匀化

Homogenisation of the Stokes equations for evolving microstructure

We consider the homogenisation of the Stokes equations in a porous medium which is evolving in time. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables to model a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit with the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method and construct a family of $\varepsilon$-scaled operators $\operatorname{div}_\varepsilon^{-1}$, whi…

高对比度卷积型算子的均匀化与谱收敛

Homogenisation and spectral convergence of high-contrast convolution type operators

本文将双尺度收敛方法推广到高对比度非局部卷积型算子，得到全空间及有界域中的均匀化结果，并刻画了极限双尺度算子的谱与原问题谱的极限行为之间的关系。

Darcy's Law for Porous Media with Multiple Microstructures

In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar's method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pres…

扩散方程中由振荡微结构诱导的有效平流

An effective advection induced by oscillating microstructure in a diffusion equation

本研究针对多孔介质扩散方程的均匀化问题，发现小时间尺度振荡的时变微结构会在均匀化方程中诱导新型平流，且该微结构可产生任意局部变化的平流速度场。

Analysis of the Stokes-Darcy problem with generalised interface conditions

Fluid flows in coupled systems consisting of a free-flow region and the adjacent porous medium appear in a variety of environmental settings and industrial applications. In many applications, fluid flow is non-parallel to the fluid-porous interface that requires a generalisation of the Beavers-Joseph coupling condition typically used for the Stokes-Darcy problem. Generalised coupling conditions valid for arbitrary flow directions to the interface are recently derived using the theory of homogenisation and boundary layers. The aim of this work is the mathematical analysis of the Stokes-Darcy pr…

薄多孔介质中流体流动的达西定律理论推导

Theoretical derivation of Darcy's law for fluid flow in thin porous media

本文推导了薄多孔介质中流体流动的二维达西定律，通过引入依赖于两个小参数的展开方法，实现了对流体流动的理论建模。