Steady compressible Navier-Stokes equations with large potential forces via a method of decomposition

Abstract

We investigate the steady compressible Navier–Stokes equations near the equilibrium state v = 0, ρ = ρ0 (v the velocity, ρ the density) corresponding to a large potential force. We introduce a method of decomposition for such equations: the velocity field v is split into a non-homogeneous incompressible part u (div (ρ0u) = (0) and a compressible (irrotational) part ∇ϕ. In such a way, the original complicated mixed elliptic–hyperbolic system is split into several ‘standard’ equations: a Stokes-type system for u, a Poisson-type equation for ϕ and a transport equation for the perturbation of the density σ = ρ − ρ0. For ρ0 = const. (zero potential forces), the method coincides with the decomposition of Novotny and Padula [21]. To underline the advantages of the present approach, we give, as an example, a ‘simple’ proof of the existence of isothermal flows in bounded domains with no-slip boundary conditions. The approach is applicable, with some modifications, to more complicated geometries and to more complicated boundary conditions as we will show in forthcoming papers. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.