Nonlinear Diffusion
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The Nonlinear Diffusion Equation
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Author : J.M. Burgers
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-11
The Nonlinear Diffusion Equation written by J.M. Burgers and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-11 with Mathematics categories.
Since the 'Introduction' to the main text gives an account of the way in which the problems treated in the following pages originated, this 'Preface' may be limited to an acknowledgement of the support the work has received. It started during the pe riod when I was professor of aero- and hydrodynamics at the Technical University in Delft, Netherlands, and many discussions with colleagues ha ve in:fluenced its devel opment. Oftheir names I mention here only that ofH. A. Kramers. Papers No. 1-13 ofthe list given at the end ofthe text were written during that period. Severa! ofthese were attempts to explore ideas which later had to be abandoned, but gradually a line of thought emerged which promised more definite results. This line began to come to the foreground in pa per No. 3 (1939}, while a preliminary formulation ofthe results was given in paper No. 12 (1954}. At that time, however, there still was missing a practica! method for manipulating a certain distribution function of central interest. A six months stay at the Hydrodynamics Laboratories ofthe California Institute of Technology, Pasadena, California (1950-1951}, was supported by a Contract with the Department of the Air F orce, N o. AF 33(038}-17207. A course of lectures was given during this period, which were published in typescript under the title 'On Turbulent Fluid Motion', as Report No. E-34. 1, July 1951, of the Hydrodynamics Laboratory.
Nonlinear Diffusion Equations
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Author : Huilai Li
language : en
Publisher: World Scientific
Release Date : 2001-11-12
Nonlinear Diffusion Equations written by Huilai Li and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-11-12 with Mathematics categories.
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations.This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.
Nonlinear Diffusion
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Author : Homer Franklin Walker
language : en
Publisher: Pitman Publishing
Release Date : 1977
Nonlinear Diffusion written by Homer Franklin Walker and has been published by Pitman Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 1977 with Mathematics categories.
Degenerate Nonlinear Diffusion Equations
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Author : Angelo Favini
language : en
Publisher: Springer
Release Date : 2012-05-08
Degenerate Nonlinear Diffusion Equations written by Angelo Favini and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-08 with Mathematics categories.
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asymptotic behaviour, discretization schemes, coefficient identification, and to introduce relevant solving methods for each of them.
Semigroup Approach To Nonlinear Diffusion Equations
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Author : Viorel Barbu
language : en
Publisher: World Scientific
Release Date : 2021-09-23
Semigroup Approach To Nonlinear Diffusion Equations written by Viorel Barbu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-23 with Mathematics categories.
This book is concerned with functional methods (nonlinear semigroups of contractions, nonlinear m-accretive operators and variational techniques) in the theory of nonlinear partial differential equations of elliptic and parabolic type. In particular, applications to the existence theory of nonlinear parabolic equations, nonlinear Fokker-Planck equations, phase transition and free boundary problems are presented in details. Emphasis is put on functional methods in partial differential equations (PDE) and less on specific results.
Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces
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Author : Luigi Ambrosio
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-02-13
Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces written by Luigi Ambrosio and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-13 with Education categories.
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.
Travelling Waves In Nonlinear Diffusion Convection Reaction
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Author : Brian H. Gilding
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Travelling Waves In Nonlinear Diffusion Convection Reaction written by Brian H. Gilding and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion. Brian worked under the guidance of Bert Peletier at the University of Sussex in Brighton, England, and, later at Delft University of Technology in the Netherlands on extending the earlier mathematics to include nonlinear convection; while Robert worked at Lomonosov State Univer sity in Moscow under the supervision of Anatolii Kalashnikov on generalizing the earlier mathematics to include nonlinear absorption. We first met at a conference held in Rome in 1985. In 1987 we met again in Madrid at the invitation of Ildefonso Diaz, where we were both staying at 'La Residencia'. As providence would have it, the University 'Complutense' closed down during this visit in response to student demonstra tions, and, we were very much left to our own devices. It was natural that we should gravitate to a research topic of common interest. This turned out to be the characterization of the phenomenon of finite speed of propagation for nonlin ear reaction-convection-diffusion equations. Brian had just completed some work on this topic for nonlinear diffusion-convection, while Robert had earlier done the same for nonlinear diffusion-absorption. There was no question but that we bundle our efforts on the general situation.
Smoothing And Decay Estimates For Nonlinear Diffusion Equations
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Author : Juan Luis Vázquez
language : en
Publisher: OUP Oxford
Release Date : 2006-08-03
Smoothing And Decay Estimates For Nonlinear Diffusion Equations written by Juan Luis Vázquez and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-08-03 with Mathematics categories.
This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porous medium type"), the aim of this text is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.
Nonlinear Diffusion Equations And Their Equilibrium States I
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Author : W.-M. Ni
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Nonlinear Diffusion Equations And Their Equilibrium States I written by W.-M. Ni and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = D.u + f(u). Here D. denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium D.u + f(u) = o. Particular cases arise, for example, in population genetics, the physics of nu clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified com bustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In the latter direction, for example, the problem of finding conformal metrics with prescribed curvature leads to a ground state problem involving critical exponents. Thus not only analysts, but geome ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the struc ture of the nonlinear function f(u) influences the behavior of the solution.
Nonlinear Diffusion Equations And Their Equilibrium States 3
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Author : N.G Lloyd
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Nonlinear Diffusion Equations And Their Equilibrium States 3 written by N.G Lloyd and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn damental questions concern the existence, uniqueness and regularity of so lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.