Evolution Equations

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Surface Evolution Equations

Author : Yoshikazu Giga
Publisher : Springer Science & Business Media
ISBN 13 : 3764373911
Total Pages : 264 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis Surface Evolution Equations by : Yoshikazu Giga

Download or read book Surface Evolution Equations written by Yoshikazu Giga and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach. Although most of the results in this book are more or less known, they are scattered in several references, sometimes without proofs. This book presents these results in a synthetic way with full proofs. The intended audience are graduate students and researchers in various disciplines who would like to know the applicability and detail of the theory as well as its flavour. No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some basic knowledge about semicontinuous functions.

Abstract Parabolic Evolution Equations and their Applications

Author : Atsushi Yagi
Publisher : Springer Science & Business Media
ISBN 13 : 3642046312
Total Pages : 594 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Abstract Parabolic Evolution Equations and their Applications by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and their Applications written by Atsushi Yagi and published by Springer Science & Business Media. This book was released on 2009-11-03 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0

Evolutionary Equations

Author : Christian Seifert
Publisher : Springer Nature
ISBN 13 : 3030893979
Total Pages : 321 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Evolutionary Equations by : Christian Seifert

Download or read book Evolutionary Equations written by Christian Seifert and published by Springer Nature. This book was released on 2022 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book provides a solution theory for time-dependent partial differential equations, which classically have not been accessible by a unified method. Instead of using sophisticated techniques and methods, the approach is elementary in the sense that only Hilbert space methods and some basic theory of complex analysis are required. Nevertheless, key properties of solutions can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated by a large variety of examples, showing the applicability of the approach of evolutionary equations in various fields. Additionally, a quantitative theory for evolutionary equations is developed. The text is self-contained, providing an excellent source for a first study on evolutionary equations and a decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical research.

Invariant Manifolds and Dispersive Hamiltonian Evolution Equations

Author : Kenji Nakanishi
Publisher : European Mathematical Society
ISBN 13 : 9783037190951
Total Pages : 264 pages
Book Rating : 4.1/5 (99 download)

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Book Synopsis Invariant Manifolds and Dispersive Hamiltonian Evolution Equations by : Kenji Nakanishi

Download or read book Invariant Manifolds and Dispersive Hamiltonian Evolution Equations written by Kenji Nakanishi and published by European Mathematical Society. This book was released on 2011 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein-Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. The authors' entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors. The proofs rely on an interplay between the variational structure of the ground states and the nonlinear hyperbolic dynamics near these states. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein-Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.

Dynamical Systems and Evolution Equations

Author : John A. Walker
Publisher : Springer Science & Business Media
ISBN 13 : 1468410369
Total Pages : 244 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis Dynamical Systems and Evolution Equations by : John A. Walker

Download or read book Dynamical Systems and Evolution Equations written by John A. Walker and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of a nine-month course first given during 1976-77 in the Division of Engineering Mechanics, University of Texas (Austin), and repeated during 1977-78 in the Department of Engineering Sciences and Applied Mathematics, Northwestern University. Most of the students were in their second year of graduate study, and all were familiar with Fourier series, Lebesgue integration, Hilbert space, and ordinary differential equa tions in finite-dimensional space. This book is primarily an exposition of certain methods of topological dynamics that have been found to be very useful in the analysis of physical systems but appear to be well known only to specialists. The purpose of the book is twofold: to present the material in such a way that the applications-oriented reader will be encouraged to apply these methods in the study of those physical systems of personal interest, and to make the coverage sufficient to render the current research literature intelligible, preparing the more mathematically inclined reader for research in this particular area of applied mathematics. We present only that portion of the theory which seems most useful in applications to physical systems. Adopting the view that the world is deterministic, we consider our basic problem to be predicting the future for a given physical system. This prediction is to be based on a known equation of evolution, describing the forward-time behavior of the system, but it is to be made without explicitly solving the equation.

Moving Interfaces and Quasilinear Parabolic Evolution Equations

Author : Jan Prüss
Publisher : Birkhäuser
ISBN 13 : 3319276980
Total Pages : 609 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Moving Interfaces and Quasilinear Parabolic Evolution Equations by : Jan Prüss

Download or read book Moving Interfaces and Quasilinear Parabolic Evolution Equations written by Jan Prüss and published by Birkhäuser. This book was released on 2016-07-25 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasilinear parabolic evolution equations, elliptic and parabolic boundary value problems, transmission problems, one- and two-phase Stokes problems, and the equations of incompressible viscous one- and two-phase fluid flows. The theory of maximal regularity, an essential element, is also fully developed. The authors present a modern approach based on powerful tools in classical analysis, functional analysis, and vector-valued harmonic analysis. The theory is applied to problems in two-phase fluid dynamics and phase transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and a variety of geometric evolution equations. The book also includes a discussion of the underlying physical and thermodynamic principles governing the equations of fluid flows and phase transitions, and an exposition of the geometry of moving hypersurfaces.

Nonlinear Evolution Equations - Global Behavior of Solutions

Author : Alain Haraux
Publisher : Springer
ISBN 13 : 3540385347
Total Pages : 324 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Nonlinear Evolution Equations - Global Behavior of Solutions by : Alain Haraux

Download or read book Nonlinear Evolution Equations - Global Behavior of Solutions written by Alain Haraux and published by Springer. This book was released on 2006-11-15 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Evolution Equations and Approximations

Author : Kazufumi Ito
Publisher : World Scientific
ISBN 13 : 9789812380265
Total Pages : 524 pages
Book Rating : 4.3/5 (82 download)

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Book Synopsis Evolution Equations and Approximations by : Kazufumi Ito

Download or read book Evolution Equations and Approximations written by Kazufumi Ito and published by World Scientific. This book was released on 2002 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation Ito (North Carolina State U.) and Kappel (U. of Graz, Austria) offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K. and Y. Kobayashi and S. Oharu. They also explore abstract approximation results for evolution equations. Their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses. Annotation copyrighted by Book News, Inc., Portland, OR

Semigroup Methods for Evolution Equations on Networks

Author : Delio Mugnolo
Publisher : Springer
ISBN 13 : 3319046217
Total Pages : 294 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Semigroup Methods for Evolution Equations on Networks by : Delio Mugnolo

Download or read book Semigroup Methods for Evolution Equations on Networks written by Delio Mugnolo and published by Springer. This book was released on 2014-05-21 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: This concise text is based on a series of lectures held only a few years ago and originally intended as an introduction to known results on linear hyperbolic and parabolic equations. Yet the topic of differential equations on graphs, ramified spaces, and more general network-like objects has recently gained significant momentum and, well beyond the confines of mathematics, there is a lively interdisciplinary discourse on all aspects of so-called complex networks. Such network-like structures can be found in virtually all branches of science, engineering and the humanities, and future research thus calls for solid theoretical foundations. This book is specifically devoted to the study of evolution equations – i.e., of time-dependent differential equations such as the heat equation, the wave equation, or the Schrödinger equation (quantum graphs) – bearing in mind that the majority of the literature in the last ten years on the subject of differential equations of graphs has been devoted to elliptic equations and related spectral problems. Moreover, for tackling the most general settings - e.g. encoded in the transmission conditions in the network nodes - one classical and elegant tool is that of operator semigroups. This book is simultaneously a very concise introduction to this theory and a handbook on its applications to differential equations on networks. With a more interdisciplinary readership in mind, full proofs of mathematical statements have been frequently omitted in favor of keeping the text as concise, fluid and self-contained as possible. In addition, a brief chapter devoted to the field of neurodynamics of the brain cortex provides a concrete link to ongoing applied research.

Evolutionary Equations with Applications in Natural Sciences

Author : Jacek Banasiak
Publisher : Springer
ISBN 13 : 3319113224
Total Pages : 493 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Evolutionary Equations with Applications in Natural Sciences by : Jacek Banasiak

Download or read book Evolutionary Equations with Applications in Natural Sciences written by Jacek Banasiak and published by Springer. This book was released on 2014-11-07 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the unifying theme of abstract evolutionary equations, both linear and nonlinear, in a complex environment, the book presents a multidisciplinary blend of topics, spanning the fields of theoretical and applied functional analysis, partial differential equations, probability theory and numerical analysis applied to various models coming from theoretical physics, biology, engineering and complexity theory. Truly unique features of the book are: the first simultaneous presentation of two complementary approaches to fragmentation and coagulation problems, by weak compactness methods and by using semigroup techniques, comprehensive exposition of probabilistic methods of analysis of long term dynamics of dynamical systems, semigroup analysis of biological problems and cutting edge pattern formation theory. The book will appeal to postgraduate students and researchers specializing in applications of mathematics to problems arising in natural sciences and engineering.

Evolution Equations, Semigroups and Functional Analysis

Author : Alfredo Lorenzi
Publisher : Birkhäuser
ISBN 13 : 3034882211
Total Pages : 404 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Evolution Equations, Semigroups and Functional Analysis by : Alfredo Lorenzi

Download or read book Evolution Equations, Semigroups and Functional Analysis written by Alfredo Lorenzi and published by Birkhäuser. This book was released on 2012-12-06 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Brunello Terreni (1953-2000) was a researcher and teacher with vision and dedication. The present volume is dedicated to the memory of Brunello Terreni. His mathematical interests are reflected in 20 expository articles written by distinguished mathematicians. The unifying theme of the articles is "evolution equations and functional analysis", which is presented in various and diverse forms: parabolic equations, semigroups, stochastic evolution, optimal control, existence, uniqueness and regularity of solutions, inverse problems as well as applications. Contributors: P. Acquistapace, V. Barbu, A. Briani, L. Boccardo, P. Colli Franzone, G. Da Prato, D. Donatelli, A. Favini, M. Fuhrmann, M. Grasselli, R. Illner, H. Koch, R. Labbas, H. Lange, I. Lasiecka, A. Lorenzi, A. Lunardi, P. Marcati, R. Nagel, G. Nickel, V. Pata, M. M. Porzio, B. Ruf, G. Savaré, R. Schnaubelt, E. Sinestrari, H. Tanabe, H. Teismann, E. Terraneo, R. Triggiani, A. Yagi.

Harmonic Analysis Method for Nonlinear Evolution Equations, I

Author : Baoxiang Wang
Publisher : World Scientific
ISBN 13 : 9814458392
Total Pages : 300 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Harmonic Analysis Method for Nonlinear Evolution Equations, I by : Baoxiang Wang

Download or read book Harmonic Analysis Method for Nonlinear Evolution Equations, I written by Baoxiang Wang and published by World Scientific. This book was released on 2011-08-10 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods. This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students. Contents:Fourier Multiplier, Function Spaces Xsp,qNavier–Stokes EquationStrichartz Estimates for Linear Dispersive EquationsLocal and Global Wellposedness for Nonlinear Dispersive EquationsThe Low Regularity Theory for the Nonlinear Dispersive EquationsFrequency-Uniform Decomposition TechniquesConservations, Morawetz' Estimates of Nonlinear Schrödinger EquationsBoltzmann Equation without Angular Cutoff Readership: Graduate students and researchers interested in analysis and PDE. Keywords:Nonlinear Dispersive Equation;Harmonic Analysis MethodKey Features:From PDE point of view, this book gives a self-contained introduction to the theory of function spaces including Besov, modulation and Triebel–Lizorkin spacesThe main topics are concentrated in four kinds of important equations, nonlinear Schrödinger, Navier–Stokes, KdV and Boltzmann equationsThis monograph is a unique treatment of the frequency-uniform localization techniques for nonlinear evolution equationsReviews: "The book under review is well and clearly written and pleasant to read. It is aimed at advanced graduate students; hence, familiarity with basic topics in measure theory, real analysis, complex analysis, functional analysis, etc., is assumed on the part of the reader. Those mathematicians who wish to learn harmonic analysis methods used in PDEs, and who wish to enter into this active area of research, will surely find this book interesting. The book also contains a reasonably large bibliography." Mathematical Reviews

Semilinear Evolution Equations and Their Applications

Author : Toka Diagana
Publisher : Springer
ISBN 13 : 303000449X
Total Pages : 189 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Semilinear Evolution Equations and Their Applications by : Toka Diagana

Download or read book Semilinear Evolution Equations and Their Applications written by Toka Diagana and published by Springer. This book was released on 2018-10-23 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, which is a continuation of Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, presents recent trends and developments upon fractional, first, and second order semilinear difference and differential equations, including degenerate ones. Various stability, uniqueness, and existence results are established using various tools from nonlinear functional analysis and operator theory (such as semigroup methods). Various applications to partial differential equations and the dynamic of populations are amply discussed. This self-contained volume is primarily intended for advanced undergraduate and graduate students, post-graduates and researchers, but may also be of interest to non-mathematicians such as physicists and theoretically oriented engineers. It can also be used as a graduate text on evolution equations and difference equations and their applications to partial differential equations and practical problems arising in population dynamics. For completeness, detailed preliminary background on Banach and Hilbert spaces, operator theory, semigroups of operators, and almost periodic functions and their spectral theory are included as well.

Strong and Weak Approximation of Semilinear Stochastic Evolution Equations

Author : Raphael Kruse
Publisher : Springer
ISBN 13 : 3319022318
Total Pages : 177 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Strong and Weak Approximation of Semilinear Stochastic Evolution Equations by : Raphael Kruse

Download or read book Strong and Weak Approximation of Semilinear Stochastic Evolution Equations written by Raphael Kruse and published by Springer. This book was released on 2013-11-18 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book we analyze the error caused by numerical schemes for the approximation of semilinear stochastic evolution equations (SEEq) in a Hilbert space-valued setting. The numerical schemes considered combine Galerkin finite element methods with Euler-type temporal approximations. Starting from a precise analysis of the spatio-temporal regularity of the mild solution to the SEEq, we derive and prove optimal error estimates of the strong error of convergence in the first part of the book. The second part deals with a new approach to the so-called weak error of convergence, which measures the distance between the law of the numerical solution and the law of the exact solution. This approach is based on Bismut’s integration by parts formula and the Malliavin calculus for infinite dimensional stochastic processes. These techniques are developed and explained in a separate chapter, before the weak convergence is proven for linear SEEq.

Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I

Author : Atsushi Yagi
Publisher : Springer Nature
ISBN 13 : 9811618968
Total Pages : 68 pages
Book Rating : 4.8/5 (116 download)

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Book Synopsis Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I written by Atsushi Yagi and published by Springer Nature. This book was released on 2021-05-31 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical Łojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Łojasiewicz–Simon gradient inequality. This book presents a unified method to show asymptotic convergence of solutions to a stationary solution for abstract parabolic evolution equations of the gradient form by utilizing this Łojasiewicz–Simon gradient inequality. In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Łojasiewicz–Simon inequality is extended, which is published here for the first time. In the second version, these abstract results are applied to reaction–diffusion equations with discontinuous coefficients, reaction–diffusion systems, and epitaxial growth equations. The results are also applied to the famous chemotaxis model, i.e., the Keller–Segel equations even for higher-dimensional ones.

Dynamics of Evolutionary Equations

Author : George R. Sell
Publisher : Springer Science & Business Media
ISBN 13 : 1475750374
Total Pages : 680 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Dynamics of Evolutionary Equations by : George R. Sell

Download or read book Dynamics of Evolutionary Equations written by George R. Sell and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. This book serves as an entrée for scholars beginning their journey into the world of dynamical systems, especially infinite dimensional spaces. The main approach involves the theory of evolutionary equations.

Beyond Partial Differential Equations

Author : Horst Reinhard Beyer
Publisher : Springer
ISBN 13 : 3540711295
Total Pages : 291 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Beyond Partial Differential Equations by : Horst Reinhard Beyer

Download or read book Beyond Partial Differential Equations written by Horst Reinhard Beyer and published by Springer. This book was released on 2007-04-10 with total page 291 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.