• Mathematics

Blow-up Theories for Semilinear Parabolic Equations


Author: Bei Hu
Publisher: Springer Science & Business Media
ISBN: 3642184596
Category: Mathematics
Page: 127
View: 5983
There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.


    • Mathematics

Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics


Author: Tian Ma,Shouhong Wang
Publisher: American Mathematical Soc.
ISBN: 0821836935
Category: Mathematics
Page: 234
View: 1723
This book presents a geometric theory for incompressible flow and its applications to fluid dynamics. The main objective is to study the stability and transitions of the structure of incompressible flows, and applications to fluid dynamics and geophysical fluid dynamics. The development of the theory and its applications has gone well beyond the original motivation, which was the study of oceanic dynamics. One such development is a rigorous theory for boundary layer separation of incompressible fluid flows. This study of incompressible flows has two major parts, which are interconnected. The first is the development of a global geometric theory of divergence-free fields on general two-dimensional compact manifolds. The second is the study of the structure of velocity fields for two-dimensional incompressible fluid flows governed by the Navier-Stokes equations or the Euler equations. Motivated by the study of problems in geophysical fluid dynamics, the program of research in this book seeks to develop a new mathematical theory, maintaining close links to physics along the way. In return, the theory is applied to physical problems, with more problems yet to be explored.

    • Mathematics

Mathematical Aspects of Reacting and Diffusing Systems


Author: P. C. Fife
Publisher: Springer Science & Business Media
ISBN: 3642931111
Category: Mathematics
Page: 185
View: 9633
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.

Equadiff 95 - Proceedings Of The International Conference On Differential Equations


Author: Magalhaes L,Rocha Carlos,Sanchez L
Publisher: World Scientific
ISBN: 9814545074
Category:
Page: 576
View: 5694
In this volume, leading experts on differential equations address recent advances in the fields of ordinary differential equations and dynamical systems, partial differential equations and calculus of variations, and their related applications.

    • Mathematics

Spectral and Dynamical Stability of Nonlinear Waves


Author: Todd Kapitula,Keith Promislow
Publisher: Springer Science & Business Media
ISBN: 1461469953
Category: Mathematics
Page: 361
View: 9283
This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles. Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The structure of the linear eigenvalue problem for Hamiltonian systems is carefully developed, including the Krein signature and related stability indices. The Evans function for the detection of point spectra is carefully developed through a series of frameworks of increasing complexity. Applications of the Evans function to the Orientation index, edge bifurcations, and large domain limits are developed through illustrative examples. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability.

    • Mathematics

Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications


Author: Janusz Mierczynski,Wenxian Shen
Publisher: CRC Press
ISBN: 9781584888963
Category: Mathematics
Page: 336
View: 3716
Providing a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective. Taking a clear, unified, and self-contained approach, the authors first develop the abstract general theory in the framework of weak solutions, before turning to cases of random and nonautonomous equations. They prove that time dependence and randomness do not reduce the principal spectrum and Lyapunov exponents of nonautonomous and random parabolic equations. The book also addresses classical Faber–Krahn inequalities for elliptic and time-periodic problems and extends the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems. The final chapter presents applications to Kolmogorov systems of parabolic equations. By thoroughly explaining the spectral theory for nonautonomous and random linear parabolic equations, this resource reveals the importance of the theory in examining nonlinear problems.

    • Mathematics

Delay Equations

Functional-, Complex-, and Nonlinear Analysis
Author: Odo Diekmann,Stephan A.van Gils,Sjoerd M.V. Lunel,Hans-Otto Walther
Publisher: Springer Science & Business Media
ISBN: 1461242061
Category: Mathematics
Page: 536
View: 9242
The aim here is to provide an introduction to the mathematical theory of infinite dimensional dynamical systems by focusing on a relatively simple - yet rich - class of examples, delay differential equations. This textbook contains detailed proofs and many exercises, intended both for self-study and for courses at graduate level, as well as a reference for basic results. As the subtitle indicates, this book is about concepts, ideas, results and methods from linear functional analysis, complex function theory, the qualitative theory of dynamical systems and nonlinear analysis. The book provides the reader with a working knowledge of applied functional analysis and dynamical systems.

    • Mathematics

Radially Symmetric Patterns of Reaction-diffusion Systems


Author: Arnd Scheel
Publisher: American Mathematical Soc.
ISBN: 0821833731
Category: Mathematics
Page: 86
View: 9410
In this paper, bifurcations of stationary and time-periodic solutions to reaction-diffusion systems are studied. We develop a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns. In particular, we show the existence of localized pulses near saddle-nodes, critical Gibbs kernels in the cusp, focus patterns in Turing instabilities, and active or passive target patterns in oscillatory instabilities.

    • Mathematics

Topological Methods in Differential Equations and Inclusions


Author: Andrzej Granas,Marlène Frigon
Publisher: Springer Science & Business Media
ISBN: 9401103399
Category: Mathematics
Page: 522
View: 6223
The papers collected in this volume are contributions to the 33rd session of the Seminaire de Mathematiques Superieures (SMS) on "Topological Methods in Differential Equations and Inclusions". This session of the SMS took place at the Universite de Montreal in July 1994 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together a considerable group of young researchers from various parts of the world and to present to them coherent surveys of some of the most recent advances in this area of Nonlinear Analysis. During the meeting 89 mathematicians from 20 countries have had the opportunity to get acquainted with various aspects of the subjects treated in the lectures as well as the chance to exchange ideas and learn about new problems arising in the field. The main topics teated in this ASI were the following: Fixed point theory for single- and multi-valued mappings including topological degree and its generalizations, and topological transversality theory; existence and multiplicity results for ordinary differential equations and inclusions; bifurcation and stability problems; ordinary differential equations in Banach spaces; second order differential equations on manifolds; the topological structure of the solution set of differential inclusions; effects of delay perturbations on dynamics of retarded delay differential equations; dynamics of reaction diffusion equations; non smooth critical point theory and applications to boundary value problems for quasilinear elliptic equations.

    • Mathematics

Controllability of Partial Differential Equations Governed by Multiplicative Controls


Author: Alexander Y. Khapalov
Publisher: Springer
ISBN: 3642124135
Category: Mathematics
Page: 284
View: 6539
This monograph addresses the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The methodology is illustrated with a variety of model equations.

    • Mathematics

Hamiltonian Dynamical Systems and Applications


Author: Walter Craig
Publisher: Springer Science & Business Media
ISBN: 1402069642
Category: Mathematics
Page: 441
View: 1273
This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.

    • Mathematics

Abstract Parabolic Evolution Equations and their Applications


Author: Atsushi Yagi
Publisher: Springer Science & Business Media
ISBN: 3642046312
Category: Mathematics
Page: 581
View: 4293
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


    • Computers

Infinite Dimensional Optimization and Control Theory


Author: Hector O. Fattorini
Publisher: Cambridge University Press
ISBN: 9780521451253
Category: Computers
Page: 798
View: 6297
Treats optimal problems for systems described by ODEs and PDEs, using an approach that unifies finite and infinite dimensional nonlinear programming.

    • Mathematics

Shadowing in Dynamical Systems

Theory and Applications
Author: K.J. Palmer
Publisher: Springer Science & Business Media
ISBN: 9780792361794
Category: Mathematics
Page: 300
View: 1860
In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic. It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic. Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

    • Mathematics

Introduction to Differentiable Manifolds


Author: Serge Lang
Publisher: Springer Science & Business Media
ISBN: 038721772X
Category: Mathematics
Page: 250
View: 5191
Author is well-known and established book author (all Serge Lang books are now published by Springer); Presents a brief introduction to the subject; All manifolds are assumed finite dimensional in order not to frighten some readers; Complete proofs are given; Use of manifolds cuts across disciplines and includes physics, engineering and economics