Modeling and energy estimates for viscous compressible Korteweg type equations

Abstract

Viscous compressible Korteweg type equations are given by the balance equations for the mass, momentum and energy of a fluid, where capillarity is taken into account. Capillary tensors of Korteweg type follow from Van der Waal's concept of capillarity and depend on the gradient of the density. We motivate the Navier-Stokes equations by reformulating the balance equations for a fluid and derive them from the Boltzmann BGK model. Additionally, we motivate the dependency of the capillary tensor on the gradient of the density and derive a general form of the Korteweg tensor from thermodynamic constitutive equations. Furthermore, we will discuss the mathematical properties of viscous compressible Korteweg type equations. We will prove an additional physical energy estimate, the BD entropy estimate, if the viscosity coefficients satisfy a certain relation. This estimate yields additional a priori regularity on the gradient of the density and allows us to prove stability results for degenerated models with vacuum initial conditions. Using the a priori bounds resulting from the classical physical energy estimate and the BD entropy estimate, we prove the stability of weak solutions of the shallow water equations in two dimensions and the barotropic Navier-Stokes equations with a power pressure law in two and three dimensions.