Yong Lu, Milan Pokorný
We study the homogenization of stationary compressible Navier--Stokes--Fourier system in a bounded three dimensional domain perforated with a large number of very tiny holes. Under suitable assumptions imposed on the smallness and distribution of the holes, we show that the homogenized limit system remains the same in the domain without holes.
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This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of the same type as the problem with oscillating coefficients.
The paper deals with homogenization of Navier-Stokes-type system describing electrorheologial fluid with random characteristics. Under non-standard growth conditions we construct the homogenized model and prove the convergence result. The structure of the limit equations is also studied
In this paper we study the deterministic homogenization problems for unsteady Navier-Stokes type equations, on one hand in an open set Ω of R^{N}, on the other hand in porous media Ω^{ε}. In the second case, the equations are classical unsteady Navier-Stokes one, and the porous media are periodic
Stochastic homogenization for two dimensional Navier--Stokes equations with random coefficients
本文利用弱收敛与Stratonovich--Khasminskii平均原理，证明二维随机系数Navier--Stokes方程的解在分布上收敛到常系数方程的解，实现随机均匀化。
In this paper, we study the homogenization problems for the stationary compressible Navier-Stokes system in a bounded 2D domain, where the domain is perforated with very tiny holes (or obstacles) whose diameters are much smaller than their mutual distances. We obtain that the process of homogenization doesn't change the motion of the fluids. From another point of view, we obtain the same system of equations in the asymptotic limit. It is the first result of homogenization problem in the compressible case in 2 dimensions.
We consider the homogenization problem of the compressible Navier-Stokes equations in a bounded three dimensional domain perforated with very tiny holes. As the number of holes increases to infinity, we show that, if the size of the holes is small enough, the homogenized equations are the same as the compressible Navier-Stokes equations in the homogeneous domain---domain without holes. This coincides with the previous studies for the Stokes equations and the stationary Navier-Stokes equations. It is the first result of this kind in the instationary barotropic compressible setting. The main tec…
Homogenization of the two-dimensional evolutionary compressible Navier-Stokes equations
研究二维穿孔区域内极小孔洞亚临界情形下演化可压缩Navier-Stokes方程的均匀化，证明密度与速度收敛至非穿孔区域的解。
Homogenization of Inhomogeneous Incompressible Navier-Stokes Equations in Domains with Very Tiny Holes
针对孔径远小于孔间距的三维非均匀不可压缩 Navier-Stokes 系统，利用弱解零延拓与 Aubin-Lions 紧致性论证，证明齐次化极限方程保持不变。
Homogenization of Non-homogeneous Incompressible Navier-Stokes System in Critically Perforated Domains
本文研究了3D非齐次不可压缩Navier-Stokes系统在临界尺寸穿孔区域中的均匀化问题，证明了当孔距参数ε趋于0时，速度与密度收敛于带Brinkman型摩擦项的Navier-Stokes系统解。
In this paper, we consider the homogenization problems for evolutionary incompressible Navier-Stokes system in three dimensional domains perforated with a large number of small holes which are periodically located. We first establish certain uniform estimates for the weak solutions. To overcome the extra difficulties coming from the time derivative, we use the idea of Temam and consider the equations by integrating in time variable. After suitably extending the weak solutions to the whole domain, we employ the generalized cell problem to study the limit process.
We consider the homogenization of the compressible Navier-Stokes-Fourier equations in a randomly perforated domain in $\mathbb{R}^3$. Assuming that the particle size scales like $\varepsilon^α$, where $\varepsilon>0$ is their mutual distance and $α>3$, we show that in the limit $\varepsilon\to 0$, the velocity, density, and temperature converge to a solution of the same system. We follow the methods of Lu and Pokorný [https://doi.org/10.1016/j.jde.2020.10.032], where they considered the full system in periodically perforated domains.
In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in $\mathbb{R}^3$. Assuming that the particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit $\varepsilon\to 0$, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Höfer, Kowalczik and Schwarzacher [arXiv:2007.09031], where they proved convergence to Darcy's law for th…
Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems
本文研究了具有非单调压力的可压缩 Navier-Stokes 系统中由初始数据诱导的持续振荡的精确解，并利用守恒律的动理学公式导出了描述粘弹性材料相变的一维模型中振荡传播的有效均匀化方程。
We study the inverse of the divergence operator on a domain $Ω\subset R^3$ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space $L^p(Ω)$ for any $1< p < 3$, with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.
We analyze behavior of weak solutions to compressible fluid flows in a bounded domain in $\mathbb{R}^3$, randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like $\varepsilon^α$, $α> 3$, with $\varepsilon$ denoting the average distance between the balls, the problem homogenize with the same limiting equation. Our main contribution is a construction of the Bogovski\uı operator, uniformly in $\varepsilon$, without any assumptions on the minimal distance between the balls.
Homogenization of compressible Navier-Stokes equations under a hard sphere pressure law
研究二维和三维有界穿孔区域中可压缩Navier-Stokes方程，在硬球压力律下，当穿孔尺寸趋于零且数量趋于无穷时，证明极限方程形式不变。
Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes
该研究针对带极小孔洞区域的可压缩Navier-Stokes-Korteweg近似系统，证明当障碍物尺寸远快于间距趋于零时极限系统不变，扩展了单相可压缩Navier-Stokes方程的均质化结果至两相流体情形。
We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: 1) dimensional reduction of the layer and 2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter $ε>0,$ which is related to the thickness of the layer as well as the size of the periodic structure. By letting $ε$ tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit v…
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
On the non-stationary Navier-Stokes type flows and reiterated homogenization
本文研究了在固定域中具有周期性快速变化系数的非平稳纳维-斯托克斯类型方程的确定性重复均质化，证明了一个收敛定理和一个修正结果，并推导出宏观均质模型。
Homogenization of Leray's flux problem for the steady-state Navier-Stokes equations in a multiply-connected planar domain
本文研究带小孔多连通平面域内粘性不可压缩流体稳态运动的Navier-Stokes方程均匀化问题，通过能量法和基于伯努利定律的反证法得到一致界，证明孔洞趋于零时等效方程不变。
Stochastic-periodic Homogenization of Non-stationary Incompressible Navier-Stokes Type Equations
本文利用随机双尺度收敛方法，研究了各向异性非均匀介质上非平稳Navier-Stokes型方程变分公式的随机-周期均匀化，证明了极限方程与原变分形式一致。
Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent $γ>3$
本文研究了穿孔三维域中非定常可压缩 Navier-Stokes 方程的均匀化，通过两种限制算子获得压力估计，将绝热指数下界从 $γ>6$ 降至 $γ>3$，并推广至 Navier-Stokes-Fourier 系统及讨论了多维结果的最优性。
Homogenization of the Navier-Stokes-Cahn-Hilliard system in the small-hole regime
研究含大量小孔区域中三维Navier-Stokes-Cahn-Hilliard系统均匀化，考虑粘度和迁移率依赖相场变量，通过建立渐近区域，在不同毛细强度条件下得出极限系统，首次对演化NSCH流在亚临界孔缩放时进行严格均匀化分析。
Homogenization of the compressible Navier-Stokes equations via two-scale convergence in perforated domains
通过双尺度收敛方法，研究周期穿孔区域中可压缩等熵Navier-Stokes方程的均匀化，导出Darcy定律和孔隙介质方程，并建立密度强双尺度收敛，将绝热常数扩展到γ>9/5。
Quantitative homogenization of the compressible Navier-Stokes equations towards Darcy's law
该研究针对周期穿孔区域内的可压缩Navier-Stokes方程，在孔洞直径与孔间距同阶的设定下，利用相对能量不等式证明了极限解足够正则时方程解收敛到多孔介质方程的定量版本，拓展了已有收敛性结论。
Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit
本文研究三维空间中带周期小颗粒穿孔区域上Navier-Stokes方程在无粘极限下的均匀化问题，在颗粒局部Reynolds数可忽略时证明定量收敛，并根据集体摩擦量级得到Euler-Brinkman方程、Euler方程和Darcy定律三种有效宏观方程。
The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local …
In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is suitable for the passage to the limit.
Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit
研究随机孔隙域中黏度趋于零的纳维-斯托克斯方程解的行为，证明在亚临界和临界 regime 下，解收敛到欧拉方程或欧拉-布林克曼方程。
Ad hoc test functions for homogenization of compressible viscous fluid with application to the obstacle problem in dimension two
本文提出特设测试函数以改进二维和三维空间中带微小孔洞的可压缩粘性流体均匀化结果，并证明二维中小障碍不影响流体动力学。
On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization
本文研究了可压缩纳维-斯托克斯方程弱解的Lipschitz连续依赖性及其正则化误差，得出了在弱范数下数据的依赖性和双尺度正则化版本的误差估计。
For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in $L^2$ or $H^1_{\text{loc}}$. Our results rely on the recent progress on the homogenization of boundary layer problems.
In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in [L. Desvillettes, F. Golse and V. Ricci. The mean field limit for solid particles in a Navier-Stokes flow. J. Stat. Phys. 131: 941-967, 2008], under suitable dilution assumptions, that the solution is well-approximated asymptotically by solving a Stokes-Brinkman equation. We provide here quantitative estimates in L p-norms of this convergence.
Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains
研究穿孔域中NSCH系统的临界均匀化，在Stokes容量尺度下得到带Brinkman阻力的极限方程，并处理相依赖粘度与校正子的相互作用，同时分析毛细系数消失的极限。
Navier-Stokes-Cahn-Hilliard system in a $3$D perforated domain with free slip and source term: Existence and homogenization
研究周期性穿孔多孔介质中二元不可压缩混合物的扩散界面模型，证明微观NSCH系统弱解的存在性，并通过均匀化得到两种宏观模型：无毛细力时解耦为线性Stokes和Cahn-Hilliard系统；平衡时得到耦合的Navier-Stokes-Cahn-Hilliard系统。
On the steady motion of a Navier-Stokes flow across a sieve with prescribed pressure drop in a finite pipe
本文研究有限管道中带规定压降的Navier-Stokes流穿过筛网的稳态运动，通过均匀化理论证明孔径趋于零时筛网渐变为墙，有效方程为两个独立无滑移Navier-Stokes系统，且无外力时流体趋于静止。
We define the Ladyzhenskaya-Lions exponent $α_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Δ)^α$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $α\geq α_{\rm {\tiny \sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1\over 2} \|\nabla u \|^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $α<α_{\rm {\tiny \sc l}} (n)$. The ener…
The two dimensional Navier-Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling ($\e^2$ for the viscosity and 1 for the density) that will lead to a time dependent limit problem. However, the noncritical scaling ($\e^β,\quad β>1$) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy's law with memory with two permeabil…
This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions $(G_\varepsilon,Π_\varepsilon)$ (and fundamental solutions $(Γ_\varepsilon, Q_\varepsilon)$) for the Stokes system with periodically oscillating coefficients (including a system of linear incompressible elasticity). Particular emphasis will be placed on the new oscillation estimates for the pressure component $Π_\varepsilon$. Also, for the first time we prove the \textit{adjustable} uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by makin…
Homogenization of an optimal control problem for a Navier-Stokes-Cahn-Hilliard system in a 3D perforated domain
该研究针对三维穿孔域中Navier-Stokes-Cahn-Hilliard系统的最优控制问题，证明其代价泛函序列Γ-收敛到均匀化问题的代价泛函，得到最优值与控制的收敛性及聚点的最优性。
We consider a non--homogeneous incompressible and heat conducting fluid confined to a 3D domain perforated by tiny holes. The ratio of the diameter of the holes and their mutual distance is critical, the former being equal to $ε^3$, the latter proportional to $ε$, where $ε$ is a small parameter. We identify the asymptotic limit for $ε\to 0$, in which the momentum equation contains a friction term of Brinkman type determined uniquely by the viscosity and geometric properties of the perforation. Besides the inhomogeneity of the fluid, we allow the viscosity and the heat conductivity coefficient …
We consider the homogenization limit of the compressible barotropic Navier-Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is increasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy's law. The range of sizes of particles we consider are exact…
Low Mach number limit on perforated domains for the evolutionary Navier-Stokes-Fourier system
本文研究穿孔域上可压缩粘性热传导流体的低马赫数极限，证明其收敛到Oberbeck-Boussinesq近似，并利用弱-强唯一性原理获得强收敛。
Brinkman's law as $Γ$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains
该研究证明低马赫数下可压缩Navier-Stokes方程的极限为不可压缩Navier-Stokes-Brinkman方程，并补充了随机穿孔区域临界情形的随机均匀化结果。
Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes
本文研究了粘弹性与可压缩纳维-斯托克斯方程中持续振荡的解，通过动能公式推导有效方程，结合宏观流动分析振荡机制。
We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average distance between the balls is of size $\eps$ and their average radius is $\varepsilon^α$, $α\in (1; 3)$. We prove that, as in the periodic case [G. Allaire, ``Homogenization of the Navier-Stokes equations in domains perforated with tiny holes. II''], the solutions converge to the solution of Darcy's law (or its scalar analogue in the case of Poisson). In the …
We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can …
Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes
该研究针对含小颗粒的变形管道内黏性不可压缩流体稳态运动，建立带混合边界条件的Navier-Stokes方程模型，用均匀化能量方法分析小参数趋于零时的渐近行为，证明有效方程含额外Brinkman项，并通过反证法得到规定通量问题的一致界。
Lagrangian controllability in perforated domains
本文研究小体积占比穿孔区域内黏性不可压 Navier-Stokes 流体的 Lagrangian 可控性，结合均匀化、弱-强稳定性与 Runge 型逼近，按孔洞和初始数据参数给出定量结果。
We perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \; Ω^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon = F(x,u^\varepsilon) & \mbox{in} \; Ω^\varepsilon,\\ u^\varepsilon = 0 & \mbox{on} \; \partial Ω^\varepsilon.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carathéodory function such that $0 \leq F(x,s) \leq h(x)/Γ(s)$ a.e. $x\inΩ$ for every $s > 0$, with $h$ in some $L^r(Ω)$ and $Γ$ a $C^1([0, +\infty[)$ function such that $Γ(0) = 0$ and $Γ'(s) > 0$ for every $s > 0$. On the other hand the…
Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries
本文研究了高度振荡粗糙John域上平稳Navier-Stokes方程的大尺度正则性，通过构造一阶和二阶边界层证明了包括大尺度Calderón-Zygmund、Lipschitz及高阶$C^{1,\gamma}$和$C^{2,\gamma}$估计在内的正则性结果。
Influence of the Reynolds number on non-Newtonian flow in thin porous media
本文研究了雷诺数对薄多孔介质中非牛顿流体流动的影响，推导了达西定律并提出数值方法验证其应用。
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