Some applications of generalized moments of the density in an inviscid compressible flow

A generalized moment of an inviscid compressible fluid contained in a domain ΩΩ is a function of the form I=∫ΩρΦdV, where ρρ is the density and ΦΦ is an arbitrary, time-independent function. The moment II satisfies an evolution equation which relates the energy of the flow, the distribution of matter and the pressure at the boundary. By considering convex domains and choosing appropriately ΦΦ we show that any flow which does not blow-up must satisfy a balance between these magnitudes which preclude e.g. a vacuum at the boundary. Another possibility is to take Φ=rαΦ=rα, where rr is the distance to the center of a ball or to an axis; the classical moment of inertia corresponds to α=2α=2. For this case we find estimates on the evolution of the growth of II which show that in general this growth rate decreases with αα.