On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems

Abstract In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in 1 Anderson, DM, McFadden, GB and Wheeler, GB. 1998. Diffuse interface methods in fluid mechanics. Annual Review of Fluid Mech, 30: 139–165. Crossref, Web of Science ® , Google ScholarDunn JE, Serrin J. On the thermodynamics of interstitial working. Arch Rational Mech Anal. 1985; 88(2):95–133), which can be used as a phase transition model. We consider a periodic domain Ω = T d (d = 2 ou 3) or a strip domain Ω = (0,1) × T d −1. We look at the dependence of the viscosity μ and the capillarity coefficient κwith respect to the density ρ. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity μ(ρ) = νρ and a surface tension the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.