Operator approach in linear problems of hydrodynamics:volume 1:self-adjoint problems for an ideal fluid液体动力学线性问题的算子方法 第1卷：理想流体的自伴问题

作者： Nikolay D. Kopachevskii等著

出 版 社：

出版时间： 2001-10-1字数：版次： 1页数： 384印刷时间： 2001/10/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783764354060包装： 精装内容简介

This is the first volume of a set of two devoted to the operator approach to linear problems in hydrodynamics. It presents functional analytical methods applied to the study of small movements and normal oscillations of hydromechanical systems having cavities filled with either ideal or viscous fluids. The work is a sequel to and at the same time substantially extends the volume "Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems" by N.D. Kopachevsky, S.G. Krein and Ngo Zuy Kan, published in 1989 by Nauka in Moscow. It includes several new problems on the oscillations of partially dissipative hydrosystems and the oscillations of visco-elastic or relaxing fluids. The work relies on the authors' and their students' works of the last 30-40 years. The readers are not supposed to be familiar with the methods of functional analysis. In the first part of the present volume, the main facts of linear operator theory relevant to linearized problems of hydrodynamics are summarized, including elements of the theories of distributions, self-adjoint operators in Hilbert spaces and in spaces with an indefinite metric, evolution equations and asymptotic methods for their solutions, the spectral theory of operator pencils. The book is particularly useful for researchers, engineers and students in fluid mechanics and mathematics interested in operator theoretical methods for the analysis of hydrodynamical problems.

目录

Volume I

Preface

Table of Contents

Volume I

Volume II

Introduction

Part 1 Mathematical Foundations of Linear Hydrodynamics . .

Chapter 1: Operators on Hilbert Spaces

1.1 General Facts

1.1.1 The Concept of a Hilbert Space

1.1.2 The Space L2

1.1.3 Orthogonality. Projection onto a Subspace

1.1.4 Equivalent Norms

1.1.5 Linear Functionals. Riesz Theorem

1.1.6 Embeddings of Spaces. Riesz Theorem for Equipments

1.1.7 Orthonormal Systems and Bases

1.1.8 Bounded Linear Operators

1.1.9 Adjoint Operators

1.1.10 Self-Adjoint Operators

1.1.11 Self-Adjoint Compact Operators

1.1.12 Compact operators, s-numbers

1.1.13 Riesz Bases and p-Bases

1.1.14 Direct Sum of Subspaces. Invariant Subspaces

1.1.15 Eigen- and Associated (Root) Elements. Root Subspaces .

1.1.16 Unbounded Linear Operators

1.1.17 Resolvent and Spectrum of a Linear Operator

1.1.18 Classification of Points in the Spectrum of a Linear Operator

1.1.19 Spectrum of a Self-Adjoint Operator. Weyl Theorem . . .

1.1.20 Riesz Projections

1.1.21 Symmetric and Self-Adjoint Operators

1.1.22 Spectral Decomposition of Self-Adjoint Operators. Functions of Operators

1.1.23 Spaces with Degenerate Scalar Products. Seminorms . . .

1.1.24 Equivalent Corrections of Seminorms

1.2 Sobolev Spaces

1.2.1 Finite Functions

1.2.2 Generalized Derivatives

1.2.3 The Definition of Sobolev Spaces

1.2.4 The Space L2. Regions of the First Type .

1.2.5 The Subspace H01

1.2.6 Embedding H1 into L2 Regions of the Second Type

1.2.7 The Trace Operator. Regions of the Third Type

1.3 Spaces With Indefinite Metrics

1.3.1 J-Spaces

1.3.2 Uniformly Definite Subspaces

1.3.3 J-Orthonormal Systems and Bases

1.3.4 Linear Operators on J-Spaces

1.3.5 Invariant Subspaces of J-Self-Adjoint Operators

1.3.6 Pontryagin Spaces

1.3.7 On Completeness and Basicity for the System of Root Elements of a J-Self-Adjoint Operator

1.4 Eigenvalue Problems

1.4.1 Operator B is the Identity Operator

1.4.2 Operator B is Positive Definite

1.4.3 Positivity Condition for a Matrix Operator

1.4.4 Simplifying Equations with an Alternating Operator

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Chapter 2:Fundamental Spaces and Operators of Linear Hydrodynamics

Part 2: Motion of Bodies With Cavities Containing Ideal Fluids

Chapter 3:Oscillations of a Heavy Ideal Fluid in Stationary and Nonstationay Contatiners

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