Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density
Haspot, Boris (2018), Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density. https://basepub.dauphine.fr/handle/123456789/17968
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01716150
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum ρ 0 u 0 belongs to the set of the finite measure M(R) and when the initial density ρ 0 is in the set of bounded variation functions BV (R). In particular it includes at the same time the case of initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong then the initial density which admits initially shocks is instantaneously regularizing and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity v = u + µ(ρ) ρ 2 ∂ x ρ (see [20, 18, 3]) with µ(ρ) the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is to weak (it means ρ 0 v 0 ∈ M(R)) then we prove the existence of weak energy in finite time but the density remains a prirori discontinuous on the time interval of existence.
Subjects / KeywordsNavier-Stokes equations
Showing items related by title and author.
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
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