On some compressible fluid models: Korteweg, lubrication, and shallow water systems

In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in (Dunn 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 κ with 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 κ(ρ) = κ̃ = cte 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.