Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction.A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is continuous as the volume fraction tends to zero. Our approach has its own advantages, as we will see. In particular, it provides another method to deduce optimal estimates on polarization tensors in any dimension from those on homogenized tensors, along with the information on underlying microstructures.
This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusiv…
We consider the homogenization for time-fractional diffusion equations in a periodic structure and we derive the homogenized time-fractional diffusion equation. Then we discuss the determination of the constant diffusion coefficient by minimum data. Combining the results obtained above, we investigate the inverse problems of determining the diffusion coefficient for the homogenized equations from the data in the periodic structure and vice versa, that is, we investigate the inverse problem of determining the diffusion coefficient for the periodic equations from the data in the homogenized stru…
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 stochastic heat equation $\partial_{s}u =\frac{1}{2}Δu +(βV(s,y)-λ)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition $u(0,x)=u_0(\varepsilon x)$ and a suitably chosen $λ\in{\mathbb R}$. It is known that, for $β$ small enough, the diffusively rescaled solution $u^{\varepsilon}(t,x)=u(\varepsilon^{-2}t,\varepsilon^{-1}x)$ converges weakly to a scalar multiple of the solution $\bar u(t,x)$ of the heat equation with an effective diffusivity $a$, and that fluctuations converge, also in a weak sense, to the solu…
Sediment Concentration Estimation via Multiscale Inverse Problem and Stochastic Homogenization
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；Ningxia University（宁夏大学）；Xiangtan University（湘潭大学）；City University of Hong Kong（香港城市大学）
We present examples of Padé approximation of the $α$-effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow, where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore application of Padé approximants for computation of tensors of ma…
A parameter identification problem in stochastic homogenization
INRIA Rocquencourt（法国国家信息与自动化研究所）；MATHERIALS research-team（MATHERIALS 研究团队）；CNRS, UMR 8234 PHENIX（法国国家科学研究中心）；ANDRA（法国放射性废物管理局）；Laboratoire Navier, École Nationale des Ponts et Chaussées, Université Paris-Est（国立路桥学院拉维埃实验室，巴黎东大学）
Stable determination of coefficients for the nonlinear third-order acoustic equations
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics, Zhejiang University（浙江大学数学系）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
