Book contents
- Frontmatter
- Contents
- Preface
- 1 Calculus in Locally Convex Spaces
- 2 Spaces and Manifolds of Smooth Maps
- 3 Lifting Geometry to Mapping Spaces I: Lie Groups
- 4 Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
- 5 Weak Riemannian Metrics with Applications in Shape Analysis
- 6 Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
- 7 Euler–Arnold Theory: PDEs via Geometry
- 8 The Geometry of Rough Paths
- Appendix A A Primer on Topological Vector Spaces and Locally Convex Spaces
- Appendix B Basic Ideas from Topology
- Appendix C Canonical Manifold of Mappings
- Appendix D Vector Fields and Their Lie Bracket
- Appendix E Differential Forms on Infinite-Dimensional Manifolds
- Appendix F Solutions to Selected Exercises
- References
- Index
7 - Euler–Arnold Theory: PDEs via Geometry
Published online by Cambridge University Press: 08 December 2022
- Frontmatter
- Contents
- Preface
- 1 Calculus in Locally Convex Spaces
- 2 Spaces and Manifolds of Smooth Maps
- 3 Lifting Geometry to Mapping Spaces I: Lie Groups
- 4 Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
- 5 Weak Riemannian Metrics with Applications in Shape Analysis
- 6 Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
- 7 Euler–Arnold Theory: PDEs via Geometry
- 8 The Geometry of Rough Paths
- Appendix A A Primer on Topological Vector Spaces and Locally Convex Spaces
- Appendix B Basic Ideas from Topology
- Appendix C Canonical Manifold of Mappings
- Appendix D Vector Fields and Their Lie Bracket
- Appendix E Differential Forms on Infinite-Dimensional Manifolds
- Appendix F Solutions to Selected Exercises
- References
- Index
Summary
In this chapter, we shall give an introduction to Euler–Arnold theory for partial differential equations (PDEs). The main idea of this theory is to reinterpret certain PDEs as smooth ordinary differential equations (ODEs) on infinite-dimensional manifolds. One advantage of this idea is that the usual solution theory for ODEs can be used to establish properties for the PDE under consideration. This principle has been successfully applied to a variety of PDE arising for example in hydrodynamics. Among these are the Euler equations for an ideal fluid, the Camassa–Holm equation, the Hunter–Saxton and the inviscid Burgers equation. Indeed there is a much longer list of physically relevant PDE which fit into this setting. We shall mainly orient ourselves along the classical exposition by Arnold and Ebin and Marsden and study the Euler equation of an incompressible ideal fluid.
Keywords
- Type
- Chapter
- Information
- An Introduction to Infinite-Dimensional Differential Geometry , pp. 138 - 156Publisher: Cambridge University PressPrint publication year: 2022
- Creative Commons
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC-ND 4.0 https://creativecommons.org/cclicenses/
You have
Access
Open access