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

TL;DR

This paper proves the existence of generalized solutions for a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model using measure-valued and dissipative solution concepts, ensuring well-posedness in any space dimension.

Contribution

It introduces measure-valued and dissipative solution frameworks for the model, establishing existence, uniqueness, and regularity results in a comprehensive thermodynamic setting.

Findings

Existence of measure-valued solutions in any space dimension.

Weak-strong uniqueness of solutions.

Generalized solutions with regularity are strong solutions.

Abstract

In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [18] 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.