Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions
type: Item
instance of: scholarly article
title: Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions
zbMATH Open document ID: 1215.35001
DOI: 10.1007/978-0-8176-4651-6
author: Mi-Ho Giga
author: Yoshikazu Giga
author: Jürgen Saal
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 2009-10-28T00:00:00Z
review text: This very nice and useful book is an extended and revised version of a Japanese book written by the first two authors in 1999. The book presents typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial differential equations of diffusion type. In particular, the authors examine such equations by analyzing special so-called self-similar solutions which are, roughly speaking, solutions invariant under a scaling transformation that does not change the equation. For several typical equations they give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions. Instead of explaining fundamental universal theory for PDE analysis, such as functional analysis, and discussing general issues for PDEs (like solvability) in that framework, the authors study directly the behavior of solutions of particular equations without preparing the fundamental theory. The fundamental tools used in the study (calculus inequalities, for example) are discussed in the second part of the book. 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. Part~I of the book consists of three chapters. In Chapter~1 the authors use two different methods to show that the large-time behavior of solutions of the heat equation is asymptotically self-similar. The first method is based on a representation formula of the solution. The second method is based on the study of convergence of suitable rescaled solutions and applies to a wide class of problems. In particular, it is also used in Chapter~2 in order to analyze the two-dimensional vorticity equations (obtained from the Navier-Stokes equations). This detailed analysis also includes an application concerning the formation of the Burgers vortex in three dimensions and the proof of uniqueness of weak solutions. Chapter~3 deals mainly…
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35-01
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35C06
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K93
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 5622734
zbMATH Keywords: vorticity equations
zbMATH Keywords: porous medium equation
zbMATH Keywords: mean curvature flow
zbMATH Keywords: Burgers vortex
zbMATH Keywords: rescaling methods
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-4651-6
OpenAlex ID: W1003373857
description: scientific article; zbMATH DE number 5622734