流体动力学中的拓扑方法

章节摘录

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前言

Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of math-metical science. Many important achievements in this field are based on profound theories rather than on experiments. In ram, those hydro dynamical theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology, stability theory, bifurcation theory, and completely integral dynamical systems. In spite of all this acknowledged success, hydrodynamics with its spec-tabular empirical laws remains a challenge for mathematicians. For instance, the phenomenon of turbulence has not yet acquired a rigorous mathematical theory. Furthermore, the existence problems for the smooth solutions of hydrodynamic equations of a three-dimensional fluid are still open.  The simplest but already very substantial mathematical model for fluid dynamics is the hydrodynamics of an ideal （i.e., of an incompressible and in viscid）homogeneous fluid. From the mathematical point of view.

内容概要

作者：(法国)阿诺德

书籍目录

PrefaceAcknowledgmentsI.Group and Hamiltonian Structures of Fluid Dynamics  1.Symmetry groups for a rigid body and an ideal fluid  2.Lie groups, Lie algebras, and adjoint representation  3.Coadjoint representation of a Lie group    3.A.Definition of the coadjoint representation    3.B.Dual of the space of plane divergence-free vector fields    3.C.The Lie algebra of divergence-free vector fields and its    dual in arbitrary dimension  4.Left-invariant metrics and a rigid body for an arbitrary group  5.Applications to hydrodynamics  6.Hamiltonian structure for the Euler equations  7.Ideal hydrodynamics on Riemannian manifolds    7.A.The Euler hydrodynamic equation on manifolds    7.B.Dual space to the Lie algebra of divergence-free fields    7.C.Inertia operator of an n-dimensional fluid  8.Proofs of theorems about the Lie algebra of divergence-free    fields and its dual  9.Conservation laws in higher-dimensional hydrodynamics  10.The group setting of ideal magnetohydrodyuamics    10.A.Equations of magnetohydrodynamics and the    Kirchhoff equations    10.B.Magnetic extension of any Lie group    10.C.Hamiltonian formulation of the Kirchhoff and    magnetohydrodynamics equations  11.Finite-dimensional approximations of the Euler equation    11.A.Approximations by vortex systems in the plane    11.B.Nonintegrability of four or more point vortices    11.C.Hamiltonian vortex approximations in threedimensions    11.D.Finite-dimensional approximations of diffeomorphismgroups     12.The Navier-Stokes equation from the group viewpointII.Topology of Steady Fluid Flows  1.Classification of three-dimensional steady flows    1.A.Stationary Euler solutions and Bernoulli functions    1.B.Structural theorems  2.Variational principles for steady solutions and applications to two-dimensional flows    2.A.Minimization of the energy    2.B.The Dirichlet problem and steady flows    2.C.Relation of two variational principles    2.D.Semigroup variational principle for two-dimensional    steady flows  3.Stability of stationary points on Lie algebras  4.Stability of planar fluid flows    4.A.Stability criteria for steady flows    4.B.Wandering solutions of the Euler equation  5.Linear and exponential stretching of particles and rapidly    oscillating perturbations    5.A.The linearized and shortened Euler equations    5.B.The action-angle variables    5.C.Spectrum of the shortened equation    5.D.The Squire theorem for shear flows    5.E.Steady flows with exponential stretching of particles    5.E  Analysis of the linearized Euler equation    5.G.Inconclusiveness of the stability test for space steady flows  6.Features of higher-dimensional steady flows    6.A.Generalized Beltrami flows    6.B.Structure of four-dimensional steady flows    6.C.Topology of the vorticity function    6.D.Nonexistence of smooth steady flows and sharpness of    the restrictionsIII.Topological Properties of Magnetic and Vorticity Fields  1.Minimal energy and helicity of a frozen-in field    1.A.Variational problem for magnetic energy    I.B.Extremal fields and their topology    1.C.Helicity bounds the energy    1.D.Helicity of fields on manifolds  2.Topological obstructions to energy relaxation    2.A.Model example: Two linked flux tubes    2.B.Energy lower bound for nontrivial linking  3.Salcharov-Zeldovich minimization problem  ……IV.Differential Geometry of diffeomorphism GroupsV.Kinematic Fast Dynarno ProblemsVI.Dynamical Systems With Hydrodynamical BackgroudReferencesIndex

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作者简介

《流体动力学中的拓扑方法(英文版)》讲述了：Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of math-metical science. Many important achievements in this field are based on profound theories rather than on experiments. In ram, those hydro dynamical theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology, stability theory, bifurcation theory, and completely integral dynamical systems. In spite of all this acknowledged success, hydrodynamics with its spec-tabular empirical laws remains a challenge for mathematicians.

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