Energy conservation for the Euler-Korteweg equations

The Euler-Korteweg equations consist of the compressible Euler equations together with dispersive terms that model capillary effects. It is a well-known model for describing liquid-vapour flows.
Strong solutions of the system satisfy an energy balance equation.
The possibility of both energy-conservative and energy-dissipative weak solutions raises the issue of studying the Onsager conjecture for the Euler-Korteweg system.
We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy.
The result applies to the system of Quantum Hydrodynamics.
The talk is based on joint work with Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda and Athanasios Tzavaras (KAUST).