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

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

2009-02-19

arxiv.org/abs/0902.3405v3

Proc. Roy. Soc. A 465 (2009), 2461-2488

Physics

Fluid Dynamics

24 pages; published version with misprints corrected

Scientific paper

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.

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