Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

Abstract

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 in , 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 torus , 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.