Quantum Navier–Stokes Equations

Abstract

Compressible Navier–Stokes models for quantum fluids are reviewed. They are derived from a collisional Wigner equation by a moment method and a Chapman–Enskog expansion around the quantum equilibrium. Introducing a new velocity variable, the barotropic quantum Navier-Stokes model can be reformulated as a viscous quantum Euler system, which possesses a new Lyapunov (energy) functional. This functional provides a priori estimates which are exploited to prove the global-in-time existence of weak solutions for general initial data. Furthermore, new numerical results for the isothermal model are presented.