The Time-Relaxation Limit for Weak Solutions to the Quantum Hydrodynamics System

This paper analyzes weak solutions of the quantum hydrodynamics (QHD)
system with a collisional term posed on the one-dimensional torus. The main goal
of our analysis is to rigorously prove the time-relaxation limit towards solutions
to the quantum drift-diffusion (QDD) equation. The existence of global in-time,
finite energy weak solutions can be proved by straightforwardly exploiting the polar
factorization and wave function lifting tools previously developed by the authors.
However, the sole energy bounds are not sufficient to show compactness and then
pass to the limit. For this reason, we consider a class of more regular weak solutions
(termed GCP solutions), determined by the finiteness of a functional involving
the chemical potential associated with the system. For solutions in this class and
bounded away from vacuum, we prove the time-relaxation limit and provide an
explicit convergence rate. Our analysis exploits compactness tools and does not
require the existence (and smoothness) of solutions to the limiting equations or
the well-preparedness of the initial data. As a by-product of our analysis, we also
establish the existence of global in time H2 solutions to a nonlinear Schrödinger–
Langevin equation and construct solutions to the QDD equation as strong limits of
GCP solutions to the QHD system.