Browsing by Author "Yeh, Li-Ming"

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A Holder estimate for non-uniform elliptic equations in a random medium
(2017-01)
Wang, Shiah-Sen
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Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary-ergodic assumption, a uniform Holder estimate in omega, epsilon, lambda for the elliptic solutions is derived, where w is a realization of the random ensemble, epsilon is an element of (0,1] is the length scale of the interfaces, and lambda(2) is an element of (0,1] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Holder estimate in omega, is an element of, lambda holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset. (C) 2016 Elsevier Ltd. All rights reserved.
A priori estimate for non-uniform elliptic equations
(2011-02-15)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists. of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let c denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Holder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in epsilon. But Holder gradient estimate and L(P) estimate of the second order derivatives of the solutions in general are not bounded uniformly in epsilon. (c) 2010 Elsevier Inc. All rights reserved.
Convergence for elliptic equations in periodic perforated domains
(2013-10-01)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let epsilon denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in epsilon in L-2 space. Also, when epsilon approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W-1,W-p space for p > 1. More precisely, a uniform W-1,W-p estimate in epsilon for p is an element of (1, infinity] and a W-1,W-p convergence result for p is an element of (n/n-2, infinity] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the L-p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in epsilon. (c) 2013 Elsevier Inc. All rights reserved.
Elliptic and parabolic equations in fractured media
(2015-12-15)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
The elliptic and the parabolic equations with Dirichlet boundary conditions in fractured media are considered. The fractured media consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let epsilon is an element of (0, 1] denote the size ratio of the matrix blocks to the whole domain and let omega(2) is an element of (0, 1] denote the permeability ratio of the disconnected subset to the connected sub-region. It is proved that the W-1,W-P norm of the elliptic and the parabolic solutions in the high permeability sub-region are bounded uniformly in omega, epsilon. However, the W-1,W-P norm of the solutions in the low permeability subset may not be bounded uniformly in omega, epsilon. For the elliptic and the parabolic equations in periodic perforated domains, it is also shown that the W-1,W-P norm of their solutions are bounded uniformly in epsilon. (C) 2015 Elsevier Inc. All rights reserved.
Elliptic equations in highly heterogeneous porous media
(2010-01-30)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Uniform estimate and convergence for highly heterogeneous elliptic equations are concerned. The domain considered consists of a connected fractured subregion (with high permeability) and a disconnected matrix block subregion (with low permeability). Let epsilon denote the size ratio of one matrix block to the whole domain and let the permeability ratio of the matrix block region to the fractured region be of the order epsilon(2). In the fractured region, uniform Holder and uniform Lipschitz estimates in epsilon of the elliptic solutions are derived; the convergence of the solutions in L(infinity) norm is obtained as well. Copyright (C) 2009 John Wiley & Sons, Ltd.
Holder continuity for two-phase flows in porous media
(2006-07-25)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
This work is to prove the Holder continuity of the solutions of the degenerate differential equations describing two-phase, incompressible, immiscible flows in porous media. The differential equations allow degeneracy at two end points and the assumption on mild degeneracy is not required in this study. The regularity result is proved by an alternative argument. Uniqueness of the weak solutions of the differential equations is a direct consequence from this Holder continuity. Copyright (C) 2006 John Wiley & Sons, Ltd.
Holder estimate for non-uniform parabolic equations in highly heterogeneous media
(2012-06-01)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Uniform bound for the solutions of non-uniform parabolic equations in highly heterogeneous media is concerned. The media considered are periodic and they consist of a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is associated with a positive number is an element of, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the connected high permeability sub-region be of the order is an element of(2 tau) for tau is an element of (0, 1]. It is proved that the Holder norm of the solutions of the above non-uniform parabolic equations in the connected high permeability sub-region are bounded uniformly in.. One example also shows that the Holder norm of the solutions in the disconnected subset may not be bounded uniformly in is an element of. (C) 2012 Elsevier Ltd. All rights reserved.
Homogenization of two-phase flow in fractured media
(2006-10-01)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
In a fractured medium, there is an interconnected system of fracture planes dividing the porous rock into a collection of matrix blocks. The fracture planes, while very thin, form paths of high permeability. Most of the fluids reside in matrix blocks, where they move very slow. Let epsilon denote the size ratio of the matrix blocks to the whole medium and let the width of the fracture planes and the porous block diameter be in the same order. If permeability ratio of matrix blocks to fracture planes is of order epsilon(2), microscopic models for two-phase, incompressible, immiscible flow in fractured media converge to a dual-porosity model as epsilon goes to 0. If the ratio is smaller than order epsilon(2), the microscopic models approach a single-porosity model for fracture flow. If the ratio is greater than order epsilon(2), then microscopic models tend to another type of single-porosity model. In this work, these results will be proved by a two-scale method.
L-p gradient estimate for elliptic equations with high-contrast conductivities in R-n
(2016-07-15)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Uniform estimate for the solutions of elliptic equations with high-contrast conductivities R-n is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose epsilon is an element of (0, 1] is the diameter of each matrix block and omega(2) is an element of a (0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W-1,W-p norm of the elliptic solutions in the connected sub-region is bounded uniformly in epsilon, is an element of, on when epsilon <= is an element of, the L-p norm of the elliptic solutions in the whole space is bounded uniformly in epsilon, omega; the W-1,W-p norm of the elliptic solutions in perforated domains is bounded uniformly in epsilon. However, the L-p norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in epsilon, omega. (C) 2016 Elsevier Inc. All rights reserved.
Linear elliptic equations in composite media with anisotropic fibres
(2019-05-05)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the Holder and the gradient L-P estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions. (C) 2019 Elsevier Inc. All rights reserved.
Modeling contaminant transport in a two-aquifer system with an intervening aquitard
(2013-08-30)
Liu, Chih-Tien
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Yeh, Hund-Der
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Yeh, Li-Ming
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環境工程研究所
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應用數學系
This study deals with the issue of one-dimensional solute transport in a two-aquifer system, where an aquitard lies between two aquifers. Different from previous studies on analysis of the contaminant transport affected by the presence of an aquitard, we developed a mathematical transport model in an aquifer-aquitard-aquifer system with considering transport of solutes in the aquitard governed by both advection and diffusion. The Laplace-domain solution of the model for concentration distributions is obtained by the Laplace transform technique and its corresponding time-domain results are computed numerically by using Laplace numerical inversion. An explicit finite difference model is also developed to simulate two-dimensional contaminant transport process in the system. The simulated depth-averaged concentrations in the lower and upper aquifers slightly differ from those predicted by the present solution. The results show that the movement of contaminant in the upper aquifer is slowed down considerably due to the advective transport in aquitard. When neglecting the aquitard advection (a zero Pedet number), the concentration level in the lower aquifer will be underestimated, especially at late times. In addition, the contaminant concentration in the lower aquifer increases significantly with aquitard's Peclet number. (C) 2013 Elsevier B.V. All rights reserved.
Modeling volatilization of residual VOCs in unsaturated zones: A moving boundary problem
(2012-06-15)
Abbas, Talib R.
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Yu, Jung-Hau
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Fen, Chiu-Shia
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Yeh, Hund-Der
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Yeh, Li-Ming
It is of practical interest to investigate the natural evaporation of volatile organic compounds (VOCs) after the removal of a leaking tank situated on the top of the soil. This study aims to develop a mathematical model to predict mole fraction distributions and migration of evaporation front for two VOCs emanating from residual non-aqueous phase liquid (NAPL) due to the leak from the tank in a homogeneous soil. Considering the location of the front and the regions above and below the front, a numerical model for the diffusive transport of VOCs in unsaturated soils was developed using the finite difference method with a moving grid approach. The model was further simplified to the case of single VOC and solved analytically by Boltzmann's transformation with a moving boundary. Analytical expressions for the depth and moving speed of the front for a single VOC were then obtained for practical use. Finally, the developed model was used to predict the concentration distributions of VOCs below the land surface and examine the factors affecting the location and moving speed of the evaporation front. (C) 2012 Elsevier B.V. All rights reserved.
Non-uniform elliptic equations in convex Lipschitz domains
(2015-05-01)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Non-uniform elliptic equations in convex Lipschitz domains are concerned. The non-smooth domains consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let epsilon is an element of (0, 1] denote the size ratio of the matrix blocks to the whole domain and let omega(2) is an element of (0, 1] denote the permeability ratio of the disconnected matrix block subset to the connected sub-region. The W-1,W-p norm for p is an element of (1, infinity) of the elliptic solutions in the high permeability sub-region is shown to be bounded uniformly in omega, epsilon. However, the W-1,W-p norm of the solutions in the low permeability subset may not be bounded uniformly in omega, epsilon. Roughly speaking, if the sources in the low permeability subset are small enough, the solutions in that subset are bounded uniformly in omega, epsilon. Otherwise the solutions cannot be bounded uniformly in omega, epsilon. Relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work. (C) 2015 Elsevier Ltd. All rights reserved.
POINTWISE ESTIMATE FOR ELLIPTIC EQUATIONS IN PERIODIC PERFORATED DOMAINS
(2015-09-01)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Pointwise estimate for the solutions of elliptic equations in periodic perforated domains is concerned. Let epsilon denote the size ratio of the period of a periodic perforated domain to the whole domain. It is known that even if the given functions of the elliptic equations are bounded uniformly in epsilon, the C-1,C- alpha norm and the W-2,W- p norm of the elliptic solutions may not be bounded uniformly in epsilon. It is also known that when epsilon closes to 0, the elliptic solutions in the periodic perforated domains approach a solution of some homogenized elliptic equation. In this work, the Holder uniform bound in epsilon and the Lipschitz uniform bound in epsilon for the elliptic solutions in perforated domains are proved. The L-infinity and the Lipschitz convergence estimates for the difference between the elliptic solutions in the perforated domains and the solution of the homogenized elliptic equation are derived.
Uniform bound and convergence for elliptic homogenization problems
(2016-12)
Yeh, Li-Ming
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應用數學系
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Department of Applied Mathematics
Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let denote the size ratio of the period to the whole domain, and let denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in in Holder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as converge to 0. In this work, the convergence rate for is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors for the former and related to the factors for the latter.