Nonlinear Partial Differential Equations (eBook)

Asymptotic Behavior of Solutions and Self-Similar Solutions

Mi-Ho Giga, Yoshikazu Giga, Jurgen Saal (Autoren)

eBook Download: PDF

2010 | 1. Auflage
294 Seiten
Birkhauser Boston (Verlag)
978-0-8176-4651-6 (ISBN)

Infos

This work will serve as an excellent first course in modern analysis. The main focus 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. This textbook will be an excellent resource for self-study or classroom use.

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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.

Produkt-Details

• Erscheint lt. Verlag: 19.6.2010
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 0-8176-4651-5 / 0817646515
• ISBN-13: 978-0-8176-4651-6 / 9780817646516