Relaxation Limit from the Quantum Navier–Stokes Equations to the Quantum Drift–Diffusion Equation

The relaxation time limit from the quantum Navier–Stokes–Poisson system to the quantum drift–diffusion equation is performed in the framework of finite energy weak solutions. No assumptions on the limiting solution are made. The proof exploits the suitably scaled a priori bounds inferred by the energy and BD entropy estimates. Moreover, it is shown how from those estimates the Fisher entropy and free energy estimates associated to the diffusive evolution are recovered in the limit. As a byproduct, our main result also provides an alternative proof for the existence of finite energy weak solutions to the quantum drift–diffusion equation.