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 R 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 torus T 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 . KEYWORDS compressible Navier-Stokes-Korteweg system, existence, uniqueness, Van der Waals gas MSC CLASSIFICATION 35Q30; 35B35; 35L65; 76D33; 74J40 and the stress tensor S is given by S = 2 D(u) + (divu)I − pI.Here p = p( , ) and D(u) = 1 2 (∇u+(∇u) T ) denote pressure and strain tensor, respectively. E = e+ u 2 2 is total energy where e = e( , ) is the inertial energy, and the viscosity coefficients , satisfy the usual physical conditions > 0, + 2 d ≥ 0. Also, and represent capillary coefficient and heat conduction, respectively. I denotes the unit matrix. In addition, f(t, x) is a given external force.

Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type

“…Chikami and Kobayashi [11] established global existence and decay of strong solution in the L 2 critical Besov spaces. Huang, Hong and Shi [31] studied more general pressure law including Van der Waals equation of state and showed the local-in-time existence of smooth solutions to the Cauchy problem of (1.1)-(1.2). Furthermore, the global-in-time existence of smooth solutions for the periodic problem is also established.…”