We study the homogenization of an obstacle problem in a perforated domain. The holes are periodically distributed but have random size and shape. The capacity of the holes is assumed to be stationary ergodic. As in the periodic case, we show that the asymptotic behavior of the solutions is described by an elliptic equation involving an additional term that takes into account the effects of the obstacle.
In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.
We study homogenization of a boundary obstacle problem on $ C^{1,α} $ domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $γ$. For any $ ε\in\mathbb{R}_+$, $\partial D=Γ\cup Σ$, $Γ\cap Σ=\emptyset $ and $ S_ε\subset Σ$ with suitable assumptions,\ we prove that as $ε$ tends to zero, the energy minimizer $ u^ε $ of $ \int_{D} |γ\nabla u|^{2} dx $, subject to $ u\geq φ$ on $ S_{\varepsilon} $, up to a subsequence, converges weakly in $ H^{1}(D) $ to $ \widetilde{u} $ which minimizes the energy functional $\int_{D}|γ\nabla u|^{2}+\int_Σ (u-φ)^{2}_{-}μ(x) dS_{x}$, wh…
We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable…
This paper concerns a space-time homogenization limit of nonnegative weak solutions to porous medium equations. In particular, the so-called homogenized matrix will be characterized in terms of solutions to cell problems, which drastically vary in a scaling parameter $r > 0$. A similar problem has already been studied in [1], where the growth of the power nonlinearity is strictly restricted due to some substantial obstacles. In the present paper, such obstacles will be overcome by developing local uniform estimates for the gradients of nonnegative weak solutions.
Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes
Faculty of Mathematics and Physics of the Charles University（查理大学数学物理学院）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）
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.
Homogenization of the compressible Navier-Stokes equations via two-scale convergence in perforated domains
Augsburg University（奥格斯堡大学）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity we prove that the $Γ$-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, that can be extended to this `asymptotically-local…
We obtain the first quantitative stochastic homogenization result for reaction-diffusion equations, for ignition reactions in dimensions $d\le 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton-Jacobi equations.
In this paper we extend the homogenization results obtained in (G. Allaire, A. Mikelić, A. Piatnitski, J. Math. Phys. 51 (2010), 123103) for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid periodic porous medium, to the case of random disperse porous media. We present a study of the nonlinear Poisson-Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scal…
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.
We study a general class of discrete $p$-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional $p$-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.
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.
We consider the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. This problem has been studied by the same authors in [A. Giunti and R.M. Höfer, Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes] where convergence of the fluid velocity field towards the solution of the Brinkman equations has been established. In the present we consider the pressure associated to the solution of the Stokes equation…
In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations the homogenized stored energy function $W_{\rm hom}$ is $C^3$ and that $W_{\rm hom}$, the stress-tensor $DW_{\rm hom}$, and the tangent-moduli $D^2W_{\rm hom}$ can be represented with help of stochastic correctors. Furthermore, we study the error of an approximation of these quantities via representative volume elements. More precisely, we consider periodic RVEs obtained by periodizing the distribu…
In the present work we study how the standard homogenization commutator, a random field that plays a central role in the theory of fluctuations, quantitatively decorrelates on large scales.
In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(Φ, \mathcal{R})$. The point process $Φ$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+β)$-moment, for $β> 0$. We study the rate of converge…
In this paper homogenization of a mathematical model for biomechanics of a plant tissue with randomly distributed cells is considered. Mechanical properties of a plant tissue are modelled by a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells and cell walls, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and the Stokes equations for fluid flow inside the cells. The nonlinear coupling between the mechanics and chemistry is given by the dependence of elastic properties of plant tissue on dens…
We prove a quantitative rate of homogenization for the G equation in a random environment with finite range of dependence. Using ideas from percolation theory, the proof bootstraps a result of Cardaliaguet and Souganidis, who proved qualitative homogenization in a more general ergodic environment.
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $Λ$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1,\infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random co…
We prove stochastic homogenization for reaction-advection-diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP r…
