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Nonlinear Diffusion Problems The Large Time Behavior
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Book Synopsis Nonlinear Diffusion Equations by : Zhuoqun Wu
Download or read book Nonlinear Diffusion Equations written by Zhuoqun Wu and published by World Scientific. This book was released on 2001 with total page 521 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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 Clarendon Press. This book was released on 2006-10-26 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.
Book Synopsis The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise by : Arnaud Debussche
Download or read book The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise written by Arnaud Debussche and published by Springer. This book was released on 2013-10-01 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.
Book Synopsis Nonlinear Diffusion Equations and Their Equilibrium States I by : W.-M. Ni
Download or read book Nonlinear Diffusion Equations and Their Equilibrium States I written by W.-M. Ni and published by Springer. This book was released on 1988-06-24 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations by : P.L. Sachdev
Download or read book Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations written by P.L. Sachdev and published by Springer Science & Business Media. This book was released on 2009-10-29 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/ boundary conditions; these equations, in general, do not admit exact solution. The present monograph gives constructive mathematical techniques which bring out large time behavior of solutions of these model equations. These approaches, in conjunction with modern computational methods, help solve physical problems in a satisfactory manner. The asymptotic methods dealt with here include self-similarity, balancing argument, and matched asymptotic expansions. The physical models discussed in some detail here relate to porous media equation, heat equation with absorption, generalized Fisher's equation, Burgers equation and its generalizations. A chapter each is devoted to nonlinear diffusion and fluid mechanics. The present book will be found useful by applied mathematicians, physicists, engineers and biologists, and would considerably help understand diverse natural phenomena.
Book Synopsis Micropolar Fluids by : Grzegorz Lukaszewicz
Download or read book Micropolar Fluids written by Grzegorz Lukaszewicz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Micropolar fluids are fluids with microstructure. They belong to a class of fluids with nonsymmetric stress tensor that we shall call polar fluids, and include, as a special case, the well-established Navier-Stokes model of classical fluids that we shall call ordinary fluids. Physically, micropolar fluids may represent fluids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. The model of micropolar fluids introduced in [65] by C. A. Eringen is worth studying as a very well balanced one. First, it is a well-founded and significant generalization of the classical Navier-Stokes model, covering, both in theory and applications, many more phenomena than the classical one. Moreover, it is elegant and not too complicated, in other words, man ageable to both mathematicians who study its theory and physicists and engineers who apply it. The main aim of this book is to present the theory of micropolar fluids, in particular its mathematical theory, to a wide range of readers. The book also presents two applications of micropolar fluids, one in the theory of lubrication and the other in the theory of porous media, as well as several exact solutions of particular problems and a numerical method. We took pains to make the presentation both clear and uniform.
Book Synopsis Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation by : Alfredo Berm?dez
Download or read book Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation written by Alfredo Berm?dez and published by SIAM. This book was released on 2000-01-01 with total page 1062 pages. Available in PDF, EPUB and Kindle. Book excerpt: This conference was held in Santiago de Compostela, Spain, July 10-14, 2000. This volume contains papers presented at the conference covering a broad range of topics in theoretical and applied wave propagation in the general areas of acoustics, electromagnetism, and elasticity. Both direct and inverse problems are well represented. This volume, along with the three previous ones, presents a state-of-the-art primer for research in wave propagation. The conference is conducted by the Institut National de Recherche en Informatique et en Automatique with the cooperation of SIAM.
Book Synopsis Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws by : Tai-Ping Liu
Download or read book Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws written by Tai-Ping Liu and published by American Mathematical Soc.. This book was released on 1997 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: We are interested in the time-asymptotic behavior of solutions to viscous conservation laws. Through the pointwise estimates for the Green's function of the linearized system and the analysis of coupling of nonlinear diffusion waves, we obtain explicit expressions of the time-asymptotic behavior of the solutions. This yields optimal estimates in the integral norms. For most physical models, the viscosity matrix is not positive definite and the system is hyperbolic-parabolic, and not uniformly parabolic. This implies that the Green's function may contain Dirac [lowercase Greek]Delta-functions. When the corresponding inviscid system is non-strictly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. These are illustrated by applying our general theory to the compressible Navier-Stokes equations and the equations of magnetohydrodynamics.
Book Synopsis Nonlocal Diffusion Problems by : Fuensanta Andreu-Vaillo
Download or read book Nonlocal Diffusion Problems written by Fuensanta Andreu-Vaillo and published by American Mathematical Soc.. This book was released on 2010 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the $p$-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.
Book Synopsis Smoothing and Decay Estimates for Nonlinear Diffusion Equations by : Juan Luis Vázquez
Download or read book Smoothing and Decay Estimates for Nonlinear Diffusion Equations written by Juan Luis Vázquez and published by Oxford University Press, USA. This book was released on 2006-08-03 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is concerned with quantitative aspects of the theory of nonlinear diffusion equations, whichappear as mathematical models in different branches of Physics, Chemistry, Biology and Engineering.
Download or read book Hypocoercivity written by Cdric Villani and published by American Mathematical Soc.. This book was released on 2009-10-08 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, taking the general form ${\frac{\partial f}{\partial t}}+ L f =0$. The question is whether and how one can overcome the degeneracy by exploiting commutators.
Book Synopsis Asymptotics for Dissipative Nonlinear Equations by : Nakao Hayashi
Download or read book Asymptotics for Dissipative Nonlinear Equations written by Nakao Hayashi and published by Springer Science & Business Media. This book was released on 2006-04-21 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
Book Synopsis Large-Time Behavior of Solutions of Linear Dispersive Equations by : Daniel B. Dix
Download or read book Large-Time Behavior of Solutions of Linear Dispersive Equations written by Daniel B. Dix and published by Springer. This book was released on 2006-11-13 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book studies the large-time asymptotic behavior of solutions of the pure initial value problem for linear dispersive equations with constant coefficients and homogeneous symbols in one space dimension. Complete matched and uniformly-valid asymptotic expansions are obtained and sharp error estimates are proved. Using the method of steepest descent much new information on the regularity and spatial asymptotics of the solutions are also obtained. Applications to nonlinear dispersive equations are discussed. This monograph is intended for researchers and graduate students of partial differential equations. Familiarity with basic asymptotic, complex and Fourier analysis is assumed.
Book Synopsis Nonlinear Variational Problems and Partial Differential Equations by : A Marino
Download or read book Nonlinear Variational Problems and Partial Differential Equations written by A Marino and published by CRC Press. This book was released on 1995-02-27 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains proceedings of a conference held in Italy in late 1990 dedicated to discussing problems and recent progress in different aspects of nonlinear analysis such as critical point theory, global analysis, nonlinear evolution equations, hyperbolic problems, conservation laws, fluid mechanics, gamma-convergence, homogenization and relaxation methods, Hamilton-Jacobi equations, and nonlinear elliptic and parabolic systems. Also discussed are applications to some questions in differential geometry, and nonlinear partial differential equations.
Book Synopsis Mathematical Aspects of Reacting and Diffusing Systems by : P. C. Fife
Download or read book Mathematical Aspects of Reacting and Diffusing Systems written by P. C. Fife and published by Springer Science & Business Media. This book was released on 2013-03-08 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modeling and analyzing the dynamics of chemical mixtures by means of differ- tial equations is one of the prime concerns of chemical engineering theorists. These equations often take the form of systems of nonlinear parabolic partial d- ferential equations, or reaction-diffusion equations, when there is diffusion of chemical substances involved. A good overview of this endeavor can be had by re- ing the two volumes by R. Aris (1975), who himself was one of the main contributors to the theory. Enthusiasm for the models developed has been shared by parts of the mathematical community, and these models have, in fact, provided motivation for some beautiful mathematical results. There are analogies between chemical reactors and certain biological systems. One such analogy is rather obvious: a single living organism is a dynamic structure built of molecules and ions, many of which react and diffuse. Other analogies are less obvious; for example, the electric potential of a membrane can diffuse like a chemical, and of course can interact with real chemical species (ions) which are transported through the membrane. These facts gave rise to Hodgkin's and Huxley's celebrated model for the propagation of nerve signals. On the level of populations, individuals interact and move about, and so it is not surprising that here, again, the simplest continuous space-time interaction-migration models have the same g- eral appearance as those for diffusing and reacting chemical systems.
Book Synopsis Far-from-equilibrium Dynamics by : Yasumasa Nishiura
Download or read book Far-from-equilibrium Dynamics written by Yasumasa Nishiura and published by American Mathematical Soc.. This book was released on 2002 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of evolution of nonequilibrium systems. Such a system usually consists of regions with different dominant scales, which coexist in the space-time where the system lives. In the case of high nonuniformity in special direction, one can see patterns separated by clearly distinguishable boundaries or interfaces. The author considers several examples of nonequilibrium systems. One of the examples describes the invasion of the solid phase into the liquidphase during the crystallization process. Another example is the transition from oxidized to reduced states in certain chemical reactions. An easily understandable example of the transition in the temporal direction is a sound beat, and the author describes typical patterns associated with thisphenomenon. The main goal of the book is to present a mathematical approach to the study of highly nonuniform systems and to illustrate it with examples from physics and chemistry. The two main theories discussed are the theory of singular perturbations and the theory of dissipative systems. A set of carefully selected examples of physical and chemical systems nicely illustrates the general methods described in the book.
Download or read book Inequalities written by Everitt and published by CRC Press. This book was released on 1990-11-30 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of an international conference organized by the London Mathematical Society, held July 1987 at the U. of Birmingham, and dominated by the ghosts of Hardy, Littlewood and Polya, whose Inequalities (still the primary reference in the field) appeared in 1934. Thirteen essays summarize subse
