Global Weak Solutions to Compressible Navier–Stokes Equations for Quantum Fluids

Abstract

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier–Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [Derivation of viscous correction terms for the isothermal quantum Euler model, 2009, submitted] from a Wigner equation using a moment method and a Chapman–Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier–Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo–Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier–Stokes system if the viscosity constant is smaller than the scaled Planck constant.