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in Sobolev spaces of sufficiently high order, then we show the existence of a solution in a particular case and finally we solve the general case by using the

— We proved in Section 2 of [7] a local existence and uniqueness theorem (Theorem 2.2) for solutions of the Navier- Stokes equations with initial data in a space 7-^1 ^2,<5s ^

Tartar, Incompressible fluid flow in a porous medium: convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E. Temam,

The proposed scheme is built in such a way that the assumptions of this stability result hold (Sect. 3.3); this implies first a prediction of the density, as a non- standard first step

An essential feature of the studied scheme is that the (discrete) kinetic energy remains controlled. We show the compactness of approximate sequences of solutions thanks to a

Theorem 4.1 (Unconditional stability of the scheme) Let us suppose that the discrete heat diffusion operator satisfies the monotonicity property (3.20), that the equation of

In this section, we build a pressure correction numerical scheme for the solution of compressible barotropic Navier-Stokes equations (1), based on the low order non-conforming

The internal energy balance features a corrective source term which is needed for the scheme to compute the correct shock solutions: we are indeed able to prove a Lax consistency