Local conservation laws, symmetries and Hamiltonian structure of three-dimensional incompressible Euler flow

Abstract

All smooth local conservation laws, symmetries and adjoint symmetries of three-dimensional homogeneous incompressible Euler flow are classified in the primitive velocity--pressure variables. Explicit dependence on time and position and arbitrary finite orders of mixed time and spatial derivatives are allowed. Up to trivial currents, the conservation laws consist of incompressibility, linear and angular momentum, kinetic energy, helicity and generalised time-dependent momentum. Every local adjoint symmetry is represented by a multiplier of these currents. Every finite-order local symmetry is equivalent on solutions to a classical point symmetry, resolving a conjecture of Olver. An explicit local first-order Hamiltonian structure of the primitive system is also established in the geometric theory of partial differential equations. It maps the energy multiplier to time translation; its kernel on local adjoint symmetries is spanned by the helicity and incompressibility multipliers. The proof first excludes higher-order symmetries by comparing their responses to transverse and harmonic characteristic directions in compatible derivative data. The Hamiltonian map and its kernel then determine all local adjoint symmetries from the symmetry classification. Exact multiplier identities and the correspondence between multipliers and current classes complete the conservation-law classification.