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In \cite{arxiv,arxiv1,Kor,cras1,cras2}, we have developed a new tool called\textit{quasi solutions} which approximate in some sense the compressibleNavier-Stokes equation. In particular it allows us to obtain global strongsolution for the compressible Navier-Stokes equations with \textit{large}initial data on the irrotational part of the velocity (\textit{large} in thesense that the smallness assumption is subcritical in terms of scaling, itturns out that in this framework we are able to obtain a family of largeinitial data in the energy space in dimension N=2). In this paper we areinterested in proving the result anounced in \cite{cras3} concerning theexistence of global weak solution for the quasi-solutions, we also observe thatfor some choice of initial data (irrotationnal) the quasi solutions verify theporous media, the heat equation or the fast diffusion equations in function ofthe structure of the viscosity coefficients. In particular it implies that itexists 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 theglobal weak solution of compressible Navier-Stokes equations to the quasisolutions in the case of a vanishing pressure limit process. In particular weshow that for highly compressible equations the speed of propagation of thedensity is quasi finite when the viscosity corresponds to$\mu(\rho)=\rho^{\alpha}$ with $\alpha>1$. Furthermore the density is not farfrom converging asymptotically in time to the Barrenblatt solution of mass theinitial density $\rho_{0}$.

Porous media, Fast diffusion equations and the existence of global weak solution for the quasi-solution of compressible Navier-Stokes equations