Analysis of semidiscretization of the compressible Navier–Stokes equations

The objective of this work is to investigate the time discretization of two-dimensional Navier–Stokes system with the slip boundary conditions. First, the existence of weak solutions for fixed time step △ t > 0 is presented and then the limit passage as △ t → 0 + is carried out. The proof is based on a new technique established for the steady Navier–Stokes equations by P.B. Mucha and M. Pokorný [P.B. Mucha, M. Pokorný, On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations, Nonlinearity 19 (8) (2006) 1747–1768] which enables to estimate the growth of L ∞ norm of the density when △t goes to 0.