Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density
Burtea, Cosmin; Haspot, Boris (2021), Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density. https://basepub.dauphine.psl.eu/handle/123456789/22341
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-03158400
Series titleCahier de recherche du CEREMADE
MetadataShow full item record
Institut de Mathématiques de Jussieu - Paris Rive Gauche [IMJ-PRG (UMR_7586)]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients µ = µ (ρ) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective velocity is bounded uniformly w.r.t. the capillary coefficient. This turns out to be crucial in providing a lower bound for the density. The second main result states the existene of unique finite-energy global strong solutions for the 1d Navier-Stokes system assuming only that ρ0, 1/ρ0 ∈ L ∞. This last result finds itself a natural application in the context of the mathematical modeling of multiphase flows.
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
Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system Haspot, Boris; Charve, Frédéric (2013) Article accepté pour publication ou publié