Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions

ISBN-13: 9780817641733 / Angielski / Twarda / 2010 / 314 str.

Giga; Yoshikazu Giga; Mi-Ho Giga

Opis

The purpose of this book is to present typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial di?erential equations of di?usion type. For instance, we examine such eq- tions by analyzing special so-called self-similar solutions. We are in particular interested in equations describing various phenomena such as the Navier- Stokesequations.Therescalingmethod describedherecanalsobeinterpreted as a renormalization group method, which represents a strong tool in the asymptotic analysis of solutions of nonlinear partial di?erential equations. Although such asymptotic analysis is used formally in various disciplines, not seldom there is a lack of a rigorous mathematical treatment. The intention of this monograph is to ?ll this gap. We intend to develop a rigorous mat- matical foundation of such a formalasymptotic analysis related to self-similar solutions. A self-similar solution is, roughly speaking, a solution invariant under a scaling transformationthat does not change the equation. For several typical equations we shall give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions. Since nonlinear partial di?erential equations are used not only in mat- matics but also in various ?elds of science and technology, there is a huge variety of approaches. Moreover, even the attempt to cover only a few typical ?elds and methods requires many pages of explanations and collateral tools so that the approaches are self-contained and accessible to a large audience.

Recenzje

From the reviews:

"This book studies the asymptotic behavior of solutions to some nonlinear evolution problems by using rescaling ... methods with self-similar solutions ... . not only are there exercises but also answers to these exercises. In any case this book is a very welcome and useful addition to the literature." (Jesús Hernández, Mathematical Reviews, Issue 2011 f)

"The book presents typical methods ... for the examination of the behavior of solutions of nonlinear partial differential equations of diffusion type. ... The aim of the authors was to teach the readers to deal with such tools during the study of PDEs and to give them a strong motivation for their study. ... The book is written in a very pedagogical way. Each chapter contains plenty of exercises whose detailed solutions can be found at the end of the book." (Pavol Quittner, Zentralblatt MATH, Vol. 1215, 2011)

Spis treści

Preface.- Part I Asymptotic Behavior of Solutions of Partial Differential Equations.- 1 Behavior Near Time Infinity of Solutions of the Heat Equation.- 2 Behavior Near Time Infinity of Solutions of the Vorticity Equations.- 3 Self-Similar Solutions for Various Equations.- Part II Useful Analytic Tools.- 4 Various Properties of Solutions of the Heat Equation.- 5 Compactness Theorems.- 6 Calculus Inequalities.- 7 Convergence Theorems in the Theory of Integration.- Answers to Exercises.- Comments on Further References.- References.- Glossary.- Index.

Blurb

The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts with recent developments and several open problems.  With challenging exercises, examples, and illustrations to help explain the rigorous analytic basis for the Navier–-Stokes equations, mean curvature flow equations, and other important equations describing real phenomena, this book is written for graduate students and researchers, not only in mathematics but also in other disciplines.

Nonlinear Partial Differential Equations will serve as an excellent textbook for a first course in modern analysis or as a useful self-study guide. Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented.  The only prerequisite required is a basic course in calculus.