Premium
Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law
Author(s) -
Huang Feimin,
Hong Hakho,
Shi Xiaoding
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6252
Subject(s) - barotropic fluid , mathematics , uniqueness , sobolev space , mathematical analysis , initial value problem , conservation law , weak solution , mechanics , physics
This paper is concerned with a barotropic model of capillary compressible fluids describing the dynamics of a liquid‐vapor mixture with diffuse interphase, in the case of general pressure law including Van der Waals gas. In the Sobolev spaces as close as possible to the physical energy spaces, we first prove local in time existence and uniqueness of the smooth solutions to the Cauchy problem inR d ( d = 2 , 3 ) , based on the estimates of a linearized system and the contraction mapping principle. Next, we show that there exists a global unique solution for the initial boundary value problem with periodic conditions in torusT d ( d = 2 , 3 ) , by using a continuation argument of local solution. Notice that it is one of the main difficulties that pressure p is not increasing function of density ρ .