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On local and non‐local Navier‐Stokes‐Korteweg systems for liquid‐vapour phase transitions

We consider a compressible fluid that can occur in a liquid and in a vapour phase. By means of a variational method and basic thermodynamical laws evolution equations to describe the isothermal dynamics can be derived. This method requires only the knowledge of the free energy. Starting from the Van‐der‐Waals free energy one obtains the classical Navier‐Stokes‐Korteweg system. However the Van‐der‐Waals free energy is not the only possible choice. Non‐local energies might be physically more relevant. In this paper we consider this class of energies and introduce a new non‐local variant of the classical Navier‐Stokes‐Korteweg system. To ensure the wellposedness of the non‐local system we present a short‐time existence theorem for the Cauchy‐problem.