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  • 7 - Euler–Arnold Theory: PDEs via Geometry

  • Alexander Schmeding, Nord Universitet, Norway
  • Book:
    An Introduction to Infinite-Dimensional Differential Geometry
    Published online:
    08 December 2022
    Print publication:
    22 December 2022, pp 138-156
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  • 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.