Classical solutions to one-dimensional stationary quantum Navier–Stokes equations

Abstract

The existence of classical solutions to the stationary quantum Navier–Stokes equations in one space dimension is studied. The main idea of the proof is to reformulate the quantum Navier–Stokes equations as the viscous quantum Euler system. The existence of classical solutions to the stationary viscous quantum Euler system is shown by using an exponential variable transformation and the Leray–Schauder fixed-point theorem. As a consequence, the existence of classical solutions to the stationary quantum Navier–Stokes equations can be deduced.