Porous Media Equations, Fast Diffusions Equations and the Existence of Global Weak Solution for the Quasi-solutions of Compressible Navier-stokes Equations

In [3, 4, 5], we have developed a new tool called tquasi solutions which approximate in some sense the compressible Navier-Stokes equation. In particular it allows to obtain global strong solution for the compressible Navier-Stokes equations with large initial data on the irrotational part of the velocity (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 large initial data in the energy space in dimension N = 2). In this paper we are interesting in studying in details this notion of quasi solution and in particular proving global weak solution, we also observe that for some choice of initial data (irrotationnal) we obtain some quasi solutions verifying the porous medium equation, the heat equation or the fast diffusion equation in function of the structure of the viscosity coefficients. Finally we show the convergence of the global weak solution of compressible Navier-Stokes equations to the quasi solutions when the pressure vanishing.