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.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Accepted: October 27, 1999.
Rights and permissions
About this article
Cite this article
Lani-Wayda, B. Equivalence of the Euler Equation with a Variational Problem. J. math. fluid mech. 1, 388–408 (1999). https://doi.org/10.1007/s000210050016
Issue Date:
DOI: https://doi.org/10.1007/s000210050016