Large-Norm Solutions and the Relaxation-Time Limit for Quantum Hydrodynamics on the Two-Dimensional Torus

Abstract

This paper extends to the two-dimensional torus our previous analysis \cite{AMZ3} of weak solutions with large norms for the collisional quantum hydrodynamic (QHD) system in semiconductor modeling. We first establish the global well-posedness of weak solutions with strictly positive density within the functional framework of generalized chemical potential (GCP) solutions introduced in \cite{AMZ1}. Two key ingredients of the analysis are a logarithmic Sobolev-type inequality controlling oscillations of the density and a functional combining a higher-order energy with the physical entropy. This combined functional yields a coercive dissipation mechanism that allows us to establish stability and exponential convergence for solutions with large initial data. As a byproduct of our approach, we also prove the global existence of $H^2$ solutions for a nonlinear Schr\"odinger--Langevin equation. Finally, for GCP solutions with strictly positive density, we justify the relaxation-time limit and provide an explicit convergence rate. Our analysis relies on compactness techniques that do not require the existence or smoothness of solutions to the limiting system. Moreover, our results impose no well-preparedness assumptions on the initial data, thereby accommodating the possible formation of an initial layer.