Equivalence of the Euler Equation with a Variational Problem

Abstract.

We show in detail in which sense the following two properties of a time dependent, C ²-smooth, divergence-free vector field v are equivalent:¶a) v satisfies the Euler equation of hydrodynamics (with some pressure function p)¶b) v is a stationary point of a suitable Lagrange functional.¶Important steps are the study of surjectivity properties of the derivative of the action functional, and the identification of vector fields orthogonal to the divergence-free fields as gradients, in the sense of classical differentiability. Thus, a foundation of the Euler equation from a variational principle is provided in a form which, to the author's knowledge, was not available so far.