Analysis of a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model

Abstract

In this paper, existence of generalized solutions to a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model introduced in Eleuteri et al. (2015) is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak–strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.