Young Researchers Workshop: Current trends in kinetic theory

Energy conservation for the Euler-Korteweg equations

Tomasz Debiec

University of Warsaw

The isothermal Euler-Korteweg system, consisting of the compressible Euler equations augmented by dispersive terms that model capillarity, is a well-known model for describing liquid-vapor flows. It is easy to show that strong solutions satisfy an energy balance equation. However irregular weak solutions are known to exist, which violate the principle of energy conservation. We ask the question: what is the minimal regularity of a weak solution so that the energy is conserved? We prove a sufficient regularity condition, using commutator estimates similar to those used by Constantin et al. for the homogeneous incompressible Euler equations. The related Quantum Hydrodynamics system is also analyzed. Joint work with P.Gwiazda, A.?wierczewska-Gwiazda and A.Tzavaras.