An inequality by Gianazza, Savaré and Toscani and its applications to the viscous quantum Euler model (Q2859240)

First a simplified version of a coercivity inequality due to Gianizza, Savaré, and Toscani is given. This is then used to construct a weak solution to the initial-boundary value problem for the viscous quantum Euler model. Then these differential equations are discretised in the time variable in order to obtain an improved version of the approximation scheme in the barotropic compressible quantum Navier-Stokes equations from [\textit{A. Jüngel}, SIAM J. Math. Anal. 42, No. 3, 1025--1045 (2010; Zbl 1228.35083)]. Finally, the time step goes to 0 to get a solution of the original problem, which, by coincidence coincides with [loc. cit.]. The Grönvall inequality is used in the proof.