Conserved integrals for inviscid compressible fluid flow in Riemannian manifolds

TL;DR

This paper explicitly derives all local conservation laws, constants of motion, and symmetries for inviscid compressible fluid flow on curved Riemannian manifolds, enhancing understanding of fluid dynamics in geometric settings.

Contribution

It provides a comprehensive determination of all kinematic conservation laws, symmetries, and Casimirs for Euler equations on Riemannian manifolds, which was previously uncharted.

Findings

All local conservation laws of kinematic type are explicitly characterized.

Kinematic constants of motion and Hamiltonian symmetries are fully identified.

The Hamiltonian structure of the equations yields all associated Casimirs.

Abstract

An explicit determination of all local conservation laws of kinematic type on moving domains and moving surfaces is presented for the Euler equations of inviscid compressible fluid flow on curved Riemannian manifolds in n>1 dimensions. All corresponding kinematic constants of motion are also determined, along with all Hamiltonian kinematic symmetries and kinematic Casimirs which arise from the Hamiltonian structure of the inviscid compressible fluid equations.