Existence of renormalized weak solutions to the steady equations describing compressible fluids in barotropic regime

We investigate the steady compressible Navier-Stokes system describing compressible fluids in barotropic regime in a bounded three-dimensional domain with slip boundary conditions. Considering a general pressure law of the form p = p(Q), where p(Q) is increasing at infinity with the rate Q(gamma), we show existence of renormalized weak solutions for gamma > 1. This improves considerably the recent result of Frehse et al. (2012) [9] which requires (gamma) > 4/3, and thus allows to treat various multi-atomic gases as well as the relativistic gas where (gamma) = 4/3. This result is obtained by using an original bootstrapping argument suggested in Jiang and Zhou (2011) [13] to estimate the density combined with a new potential type estimates up to the boundary obtained via a testing of the momentum equation by a specially constructed test function. (C) 2012 Elsevier Masson SAS. All rights reserved.