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

Haspot, Boris (2014), Porous media equations, fast diffusion equations and the existence of global weak solution for the quasi-solution of compressible Navier-Stokes equations, in Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea Marson, Hyperbolic Problems: Theory, Numerics, Applications, AIMS, p. 667-674

Type
Communication / Conférence
Date
2014
Book title
Hyperbolic Problems: Theory, Numerics, Applications
Book author
Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea Marson
Publisher
AIMS
Published in
Paris
ISBN
978-1-60133-017-8
Pages
667-674
Metadata
Show full item record
Author(s)
Haspot, Boris
Abstract (EN)
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 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 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. 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 $\alpha>1$ and that the density is not far from converging asymptoticaly to the Barrenblatt solution of mass the initial density $\rho_{0}$.
Subjects / Keywords
Weak solutions; Navier–Stokes equations

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