  This paper concerns the existence of global weak solutions \`a la Leray for
compressible Navier-Stokes equations with a pressure law that depends on the
density and on time and space variables $t$ and $x$. The assumptions on the
pressure contain only locally Lipschitz assumption with respect to the density
variable and some hypothesis with respect to the extra time and space
variables. It may be seen as a first step to consider heat-conducting
Navier-Stokes equations with physical laws such as the truncated virial
assumption. The paper focuses on the construction of approximate solutions
through a new regularized and fixed point procedure and on the weak stability
process taking advantage of the new method introduced by the two first authors
with a careful study of an appropriate regularized quantity linked to the
pressure.
