Porous media equations, fast diffusion equations and the existence of global weak solution for the quasisolution of compressible NavierStokes equations
Haspot, Boris (2014), Porous media equations, fast diffusion equations and the existence of global weak solution for the quasisolution of compressible NavierStokes equations, in Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea Marson, Hyperbolic Problems: Theory, Numerics, Applications, AIMS, p. 667674
Type
Communication / ConférenceDate
2014Book title
Hyperbolic Problems: Theory, Numerics, ApplicationsBook author
Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea MarsonPublisher
AIMS
Published in
Paris
ISBN
9781601330178
Pages
667674
Metadata
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Haspot, BorisAbstract (EN)
We have developed a new tool called \textit{quasi solutions} which approximate in some sense the compressible NavierStokes equation. In particular it allows us to obtain global strong solution for the compressible NavierStokes 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 quasisolutions, 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 quasisolutions 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 NavierStokes 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 equationsRelated items
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