Stationary Motions and Incompressible Limit for Compressible Viscous Fluids

摘要

This paper considers the non-linear system of partial differential equation, describing the barotropic stationary motion of a compressible fluid, in a bounded region Omega. Assume that the total mass of fluid inside Omega is fixed, and equal to (m) abs. vol. Omega, where the mean density m is given. For small f and g, there exists a unique solution u(x), rho(x) in a neighborhood of (0, m). Here, u(x) is the field of velocities, rho(x) the density of the fluid, p(rho(x)) the pressure field, and f(x) the external force field (in the physical interesting case one has g = 0). Moreover, the solutions of system converge to the solution of the Navier-Stokes equation as lambda approaches + infinity, i.e. when the Mach number becomes small. The solution of the Navier-Stokes equations are the incompressible limit of the solutions of the compressible Navier-Stokes equations. The proofs given here, apply, without supplementary difficulties, in the context of Sobolev spaces H superscript k,p, and other functional spaces. The results can be extended to the heat depending case, too. Keywords: Non-linear partical differential equations; Viscous compressible fluid; Incompressible limit; Stationary solutions.