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

Abstract

A generalized moment of an inviscid compressible fluid contained in a domain Ω is a function of the form I = ∫ Ω ρ Φ d V , where ρ is the density and Φ is an arbitrary, time-independent function. The moment I 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 Φ 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 = r α , where r is the distance to the center of a ball or to an axis; the classical moment of inertia corresponds to α = 2 . For this case we find estimates on the evolution of the growth of I which show that in general this growth rate decreases with α .