  In \cite{arxiv,arxiv1,Kor,cras1,cras2}, we have developed a new tool called
\textit{quasi solutions} which approximate in some sense the compressible
Navier-Stokes equation. In particular it allows us to obtain global strong
solution for the compressible Navier-Stokes equations with \textit{large}
initial data on the irrotational part of the velocity (\textit{large} in the
sense that the smallness assumption is subcritical in terms of scaling, it
turns out that in this framework we are able to obtain a family of large
initial data in the energy space in dimension N=2). In this paper we are
interested in proving the result anounced in \cite{cras3} concerning the
existence of global weak solution for the quasi-solutions, we also observe that
for some choice of initial data (irrotationnal) the quasi solutions verify the
porous media, the heat equation or the fast diffusion equations in function of
the structure of the viscosity coefficients. In particular it implies that it
exists classical quasi-solutions in the sense that they are $C^{\infty}$ on
$(0,T)\times\R^{N}$ for any $T>0$. Finally we show the convergence of the
global weak solution of compressible Navier-Stokes equations to the quasi
solutions in the case of a vanishing pressure limit process. In particular we
show that for highly compressible equations the speed of propagation of the
density is quasi finite when the viscosity corresponds to
$\mu(\rho)=\rho^{\alpha}$ with $\alpha>1$. Furthermore the density is not far
from converging asymptotically in time to the Barrenblatt solution of mass the
initial density $\rho_{0}$.
