On the porous medium equations, fast diffusion equations and compressible Navier-Stokes equations, new results on the quasi-solutions and on the scaling of the equations

Boris Haspot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In \cite{arxiv,cras1,cras2}, we have developed a new tool called the \textit{quasi solution} 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 \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 large initial data in the energy space in dimension $N=2$). In this paper we are interesting in studying in details this notion of \textit{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. We are also going to discuss the notion of scaling of the solution for compressible Navier-Stokes equations which justifies the notion of quasi solution.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. On the porous medium equations, fast diffusion equations and compressible Navier-Stokes equations, new results on the quasi-solutions and on the scaling of the equations. 2012. ⟨hal-00809574⟩