AnL ^p-theory for then-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions

Abstract

In this paper we study the system (1.1), (1.3), which describes the stationary motion of a given amount of a compressible heat conducting, viscous fluid in a bounded domain Ω ofR ⁿ, n≧2. Hereu(x) is the velocity field, σ(x) is the density of the fluid, ς(x) is the absolute temperature,f(x) andh(x) are the assigned external force field and heat sources per unit mass, andp(σ, ς) is the pressure. In the physically significant case one hasg=0. We prove that for small data (f, g, h) there exists a unique solution (u, σ, ς) of problem (1.1), (1.3)₁, in a neighborhood of (0,m, ς₀); for arbitrarily large data the stationary solution does not exist in general (see Sect. 5). Moreover, we prove that (for barotropic flows) the stationary solution of the Navier-Stokes equations (1.8) is the incompressible limit of the stationary solutions of the compressible Navier-Stokes equations (1.7), as the Mach number becomes small. Finally, in Sect. 5 we will study the equilibrium solutions for system (4.1). For a more detailed explanation see the introduction.