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Stochastic Porous Media Equations
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Book Synopsis Stochastic Porous Media Equations by : Viorel Barbu
Download or read book Stochastic Porous Media Equations written by Viorel Barbu and published by Springer. This book was released on 2016-09-30 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Book Synopsis Stochastic Methods for Flow in Porous Media by : Dongxiao Zhang
Download or read book Stochastic Methods for Flow in Porous Media written by Dongxiao Zhang and published by Elsevier. This book was released on 2001-10-11 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties explores fluid flow in complex geologic environments. The parameterization of uncertainty into flow models is important for managing water resources, preserving subsurface water quality, storing energy and wastes, and improving the safety and economics of extracting subsurface mineral and energy resources. This volume systematically introduces a number of stochastic methods used by researchers in the community in a tutorial way and presents methodologies for spatially and temporally stationary as well as nonstationary flows. The author compiles a number of well-known results and useful formulae and includes exercises at the end of each chapter. - Balanced viewpoint of several stochastic methods, including Greens' function, perturbative expansion, spectral, Feynman diagram, adjoint state, Monte Carlo simulation, and renormalization group methods - Tutorial style of presentation will facilitate use by readers without a prior in-depth knowledge of Stochastic processes - Practical examples throughout the text - Exercises at the end of each chapter reinforce specific concepts and techniques - For the reader who is interested in hands-on experience, a number of computer codes are included and discussed
Book Synopsis Weak Solutions to Stochastic Porous Media Equations by : Giuseppe Da Prato
Download or read book Weak Solutions to Stochastic Porous Media Equations written by Giuseppe Da Prato and published by . This book was released on 2003 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Stochastic Dynamics. Modeling Solute Transport in Porous Media by : Don Kulasiri
Download or read book Stochastic Dynamics. Modeling Solute Transport in Porous Media written by Don Kulasiri and published by Elsevier. This book was released on 2002-11-22 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most of the natural and biological phenomena such as solute transport in porous media exhibit variability which can not be modeled by using deterministic approaches. There is evidence in natural phenomena to suggest that some of the observations can not be explained by using the models which give deterministic solutions. Stochastic processes have a rich repository of objects which can be used to express the randomness inherent in the system and the evolution of the system over time. The attractiveness of the stochastic differential equations (SDE) and stochastic partial differential equations (SPDE) come from the fact that we can integrate the variability of the system along with the scientific knowledge pertaining to the system. One of the aims of this book is to explaim some useufl concepts in stochastic dynamics so that the scientists and engineers with a background in undergraduate differential calculus could appreciate the applicability and appropriateness of these developments in mathematics. The ideas are explained in an intuitive manner wherever possible with out compromising rigor.The solute transport problem in porous media saturated with water had been used as a natural setting to discuss the approaches based on stochastic dynamics. The work is also motivated by the need to have more sophisticated mathematical and computational frameworks to model the variability one encounters in natural and industrial systems. This book presents the ideas, models and computational solutions pertaining to a single problem: stochastic flow of contaminant transport in the saturated porous media such as that we find in underground aquifers. In attempting to solve this problem using stochastic concepts, different ideas and new concepts have been explored, and mathematical and computational frameworks have been developed in the process. Some of these concepts, arguments and mathematical and computational constructs are discussed in an intuititve manner in this book.
Book Synopsis Exact Averaging of Stochastic Equations for Flow in Porous Media by :
Download or read book Exact Averaging of Stochastic Equations for Flow in Porous Media written by and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: It is well known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods--for example, the convergence behavior and the accuracy of truncated perturbation series--are not well studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green's functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic nonlocal equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged nonlocal equations with a unique kernel-tensor. When global symmetry does not exist, the nonlocal equation with a kernel-tensor involves complications and leads to an ill-posed problem.
Book Synopsis Homogenization of Stochastic Partial Differential Equations in Perforated Porous Media by : Chigoziem A. Emereuwa
Download or read book Homogenization of Stochastic Partial Differential Equations in Perforated Porous Media written by Chigoziem A. Emereuwa and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study the homogenization of a stochastic model of groundwater pollution in periodic porous media and the homogenization of a stochastic model of a single-phase uid ow in partially ssured media. In the rst study, we investigated the ow of a uid carrying reacting substances through a porous medium. We modeled this ow using a coupled system of equations; the velocity of the uid is modeled using steady Stokes equations, the concentration of the solute while being moved by the uid under the action of random forces is modeled by a stochastic convection-di usion equation driven by a Wiener type random force and the concentration of the solute on the surface of the pore skeleton is modeled using reaction-di usion equations. The homogenization process was carried out using the multiple scale expansion, Tartar's method of oscillating test functions and stochastic calculus together with deep probability compactness results due to Prokhorov and Skorokhod. This part of the thesis is the rst in the scienti c literature dealing with the important problem of groundwater pollution using stochastic partial di erential equations. Our results in this regard are original. Also as a by-product of our work, we establish the rst homogenization result for stochastic convection-di usion equation The second study is devoted to a single-phase ow under the in uence of external random forces through partially ssured media arising in reservoir engineering (oil and gas industries). We undertake to model this ow using a system of nonlinear stochastic di usion equations with monotone operators in the pore system and the ssure system; on the interface of the pores and ssures, we prescribe transmission boundary conditions. We carried out the homogenization process using the two-scale convergence method, Prokhorov- Skorokhod compactness process and Minty's monotonicity method. While some works have been undertaken in the deterministic case and in the case of nonlinear di usion equations with randomly oscillating coe cients, our work is novel in the sense that it uses the more advanced tool of stochastic partial di erential equations driven by random forces to investigate the in uence of random uctuations on the ow. To the best of our knowledge, our work also initiates the study of stochastic evolution transmission problems by means of homogenization.
Book Synopsis Mathematical and Numerical Modeling in Porous Media by : Martin A. Diaz Viera
Download or read book Mathematical and Numerical Modeling in Porous Media written by Martin A. Diaz Viera and published by CRC Press. This book was released on 2012-07-24 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Porous media are broadly found in nature and their study is of high relevance in our present lives. In geosciences porous media research is fundamental in applications to aquifers, mineral mines, contaminant transport, soil remediation, waste storage, oil recovery and geothermal energy deposits. Despite their importance, there is as yet no complete
Book Synopsis Stochastic Analysis: A Series of Lectures by : Robert C. Dalang
Download or read book Stochastic Analysis: A Series of Lectures written by Robert C. Dalang and published by Birkhäuser. This book was released on 2015-07-28 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in thirteen refereed survey articles an overview of modern activity in stochastic analysis, written by leading international experts. The topics addressed include stochastic fluid dynamics and regularization by noise of deterministic dynamical systems; stochastic partial differential equations driven by Gaussian or Lévy noise, including the relationship between parabolic equations and particle systems, and wave equations in a geometric framework; Malliavin calculus and applications to stochastic numerics; stochastic integration in Banach spaces; porous media-type equations; stochastic deformations of classical mechanics and Feynman integrals and stochastic differential equations with reflection. The articles are based on short courses given at the Centre Interfacultaire Bernoulli of the Ecole Polytechnique Fédérale de Lausanne, Switzerland, from January to June 2012. They offer a valuable resource not only for specialists, but also for other researchers and Ph.D. students in the fields of stochastic analysis and mathematical physics. Contributors: S. Albeverio M. Arnaudon V. Bally V. Barbu H. Bessaih Z. Brzeźniak K. Burdzy A.B. Cruzeiro F. Flandoli A. Kohatsu-Higa S. Mazzucchi C. Mueller J. van Neerven M. Ondreját S. Peszat M. Veraar L. Weis J.-C. Zambrini
Book Synopsis Computational Modelling of Multi-scale Solute Dispersion in Porous Media by : Don Kulasiri
Download or read book Computational Modelling of Multi-scale Solute Dispersion in Porous Media written by Don Kulasiri and published by BoD – Books on Demand. This book was released on 2011-11-04 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research monograph presents a mathematical approach based on stochastic calculus which tackles the "cutting edge" in porous media science and engineering - prediction of dispersivity from covariance of hydraulic conductivity (velocity). The problem is of extreme importance for tracer analysis, for enhanced recovery by injection of miscible gases, etc. This book explains a generalised mathematical model and effective numerical methods that may highly impact the stochastic porous media hydrodynamics. The book starts with a general overview of the problem of scale dependence of the dispersion coefficient in porous media. Then a review of pertinent topics of stochastic calculus that would be useful in the modeling in the subsequent chapters is succinctly presented. The development of a generalised stochastic solute transport model for any given velocity covariance without resorting to Fickian assumptions from laboratory scale to field scale is discussed in detail. The mathematical approaches presented here may be useful for many other problems related to chemical dispersion in porous media.
Book Synopsis Non-fickian Solute Transport in Porous Media by : Don Kulasiri
Download or read book Non-fickian Solute Transport in Porous Media written by Don Kulasiri and published by Springer. This book was released on 2013-04-23 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt: The advection-dispersion equation that is used to model the solute transport in a porous medium is based on the premise that the fluctuating components of the flow velocity, hence the fluxes, due to a porous matrix can be assumed to obey a relationship similar to Fick’s law. This introduces phenomenological coefficients which are dependent on the scale of the experiments. This book presents an approach, based on sound theories of stochastic calculus and differential equations, which removes this basic premise. This leads to a multiscale theory with scale independent coefficients. This book illustrates this outcome with available data at different scales, from experimental laboratory scales to regional scales.
Book Synopsis Optimal Regularity in Time and Space for Stochastic Porous Medium Equations by : Stefano Bruno
Download or read book Optimal Regularity in Time and Space for Stochastic Porous Medium Equations written by Stefano Bruno and published by . This book was released on 2022 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Porous Medium Equation by : Juan Luis Vazquez
Download or read book The Porous Medium Equation written by Juan Luis Vazquez and published by Oxford University Press. This book was released on 2007 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at research students and academics in mathematics and engineering, as well as engineering specialists, this book provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation.
Book Synopsis Invariant Measures for a Stochastic Porous Medium Equation by : Giuseppe Da Prato
Download or read book Invariant Measures for a Stochastic Porous Medium Equation written by Giuseppe Da Prato and published by . This book was released on 2003 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonlinear Stochastic Differential Equations and the Viscous Porous Medium Equation by : Robert Philipowski
Download or read book Nonlinear Stochastic Differential Equations and the Viscous Porous Medium Equation written by Robert Philipowski and published by . This book was released on 2006 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematical Modelling Of Flow Through Porous Media - Proceedings Of The Conference by : Alain P Bourgeat
Download or read book Mathematical Modelling Of Flow Through Porous Media - Proceedings Of The Conference written by Alain P Bourgeat and published by World Scientific. This book was released on 1995-11-30 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: This proceedings volume contains contributions from leading scientists working on modelling and numerical simulation of flows through porous media and on mathematical analysis of the equations associated to the modelling. There is a number of contributions on rigorous results for stochastic media and for applications to numerical simulations. Modelling and simulation of environment and pollution are also subject of several papers. The published material herein gives an insight to the state of the art in the field with special attention for rigorous discussions and results.
Book Synopsis Stochastic Partial Differential Equations by : Helge Holden
Download or read book Stochastic Partial Differential Equations written by Helge Holden and published by Springer Science & Business Media. This book was released on 2009-12-01 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time Lévy process noise. Applications of the theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance. Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.
Book Synopsis Geometry and Invariance in Stochastic Dynamics by : Stefania Ugolini
Download or read book Geometry and Invariance in Stochastic Dynamics written by Stefania Ugolini and published by Springer Nature. This book was released on 2022-02-09 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of the Random Transformations and Invariance in Stochastic Dynamics conference held in Verona from the 25th to the 28th of March 2019 in honour of Sergio Albeverio. It presents the new area of studies concerning invariance and symmetry properties of finite and infinite dimensional stochastic differential equations.This area constitutes a natural, much needed, extension of the theory of classical ordinary and partial differential equations, where the reduction theory based on symmetry and invariance of such classical equations has historically proved to be very important both for theoretical and numerical studies and has given rise to important applications. The purpose of the present book is to present the state of the art of the studies on stochastic systems from this point of view, present some of the underlying fundamental ideas and methods involved, and to outline the main lines for future developments. The main focus is on bridging the gap between deterministic and stochastic approaches, with the goal of contributing to the elaboration of a unified theory that will have a great impact both from the theoretical point of view and the point of view of applications. The reader is a mathematician or a theoretical physicist. The main discipline is stochastic analysis with profound ideas coming from Mathematical Physics and Lie’s Group Geometry. While the audience consists essentially of academicians, the reader can also be a practitioner with Ph.D., who is interested in efficient stochastic modelling.
