Global Weak Solutions for the High--Friction Quantum Navier--Stokes--Poisson Model

Abstract

In [1], the Authors rigorously establish the relaxation limit from the Quantum Navier Stokes Poisson (QNSP) system to the Quantum Drift Diffusion (QDD) equation, while providing only a brief outline of the global existence theory for weak solutions to QNSP in the high friction regime (see Appendix A therein). In this manuscript, we present a complete and fully self contained proof of global existence. More precisely, we prove the global existence of finite energy weak solutions to the QNSP system with high friction and large initial data on the three-dimensional torus. The model describes a compressible, viscous quantum fluid with Korteweg type capillarity effects, and allows for degenerate viscosity and vacuum regions. The construction proceeds in two main steps. First, it is introduced a Faedo Galerkin approximation endowed with suitable damping mechanisms, which yields smooth approximate solutions through compactness arguments. Then, it will be justify the convergence of the approximating sequence by combining a truncation of the momentum equation with DiPerna Lions commutator estimates, providing the required control over the nonlinear transport structure.