Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions

TL;DR

This paper classifies conservation laws for the Euler equations of compressible isentropic fluid flow in multiple dimensions, identifying special equations of state that admit extra conserved quantities and extending Kelvin's circulation theorem.

Contribution

It provides a complete classification of local conservation laws for these flows, highlighting the unique role of the polytropic equation of state and generalizing Kelvin's theorem to higher even dimensions.

Findings

Polytropic equation of state with specific exponent admits extra conserved integrals.

No distinguished equations of state in the vorticity case.

Generalized Kelvin's circulation theorem in even dimensions.

Abstract

For the Euler equations governing compressible isentropic fluid flow with a barotropic equation of state (where pressure is a function only of the density), local conservation laws in $n>1$ spatial dimensions are fully classified in two primary cases of physical and analytical interest: (1) kinematic conserved densities that depend only on the fluid density and velocity, in addition to the time and space coordinates; (2) vorticity conserved densities that have an essential dependence on the curl of the fluid velocity. A main result of the classification in the kinematic case is that the only equation of state found to be distinguished by admitting extra $n$-dimensional conserved integrals, apart from mass, momentum, energy, angular momentum and Galilean momentum (which are admitted for all equations of state), is the well-known polytropic equation of state with dimension-dependent exponent $\gamma=1+2/n$. In the vorticity case, no distinguished equations of state are found to arise, and here the main result of the classification is that, in all even dimensions $n\geq 2$, a generalized version of Kelvin's two-dimensional circulation theorem is obtained for a general equation of state.