From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations. Existence of Global Weak Solution for the Quasi-Solutions
Haspot, Boris (2016), From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations. Existence of Global Weak Solution for the Quasi-Solutions, Journal of Mathematical Fluid Mechanics, 18, 2, p. 243-291. 10.1007/s00021-015-0226-5
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-00770248
Journal nameJournal of Mathematical Fluid Mechanics
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) 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 it exists classical quasi-solutions in the sense that they are C∞C∞ on (0,T)×RN(0,T)×RN for any T>0T>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 process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to μ(ρ)=ραμ(ρ)=ρα with α>1α>1 . Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density ρ0ρ0 .
Subjects / KeywordsGlobal weak solutions; porous media solutions; compactness arguments; quasi-solutions; high compressible limit
Showing items related by title and author.
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) Communication / Conférence
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 Haspot, Boris (2013) Document de travail / Working paper
New effective pressure and existence of global strong solution for compressible Navier-Stokes equations with general viscosity coefficient in one dimension Burtea, Cosmin; Haspot, Boris (2020) Article accepté pour publication ou publié