Effective velocity in compressible Navier–Stokes equations with third-order derivatives

A formulation of certain barotropic compressible Navier–Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier–Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch–Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier–Stokes equations with strictly positive particle densities is proved.