WIAS Preprint No. 2739, (2020)

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


  • Lasarzik, Robert
    ORCID: 0000-0002-1677-6925

2010 Mathematics Subject Classification

  • 35Q35 35D99 76D05


  • Weak-strong uniqueness, phase transition, Navier--Stokes, Cahn--Hilliard, existence, thermodynamical consistent, dissipative solutions, relative energy




In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] 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.

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