Quantum Hydrodynamic equation for semiconductor devices in 2-dimensional space and the relaxation time limit

This paper extends the author's previous analysis in \cite{AMZ3} on weak solutions with large norms for the collisional quantum hydrodynamic (QHD) equations in semiconductor modeling to 2-dimensional tori. We first establish the global existence of weak solutions with strictly positive density within the functional framework of GCP solutions introduced in \cite{AMZ1}. A logarithmic Sobolev-type inequality is employed to control oscillations in the mass density. Furthermore, by constructing a combined functional that incorporates both the GCP energy and the physical entropy, we derive the exponential decay of solutions. 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 that remain bounded away from vacuum, we justify the time-relaxation limit and provide an explicit convergence rate. Our analysis relies on compactness techniques that does not require the existence or smoothness of solutions to the limiting equations. Moreover, our results impose no well-preparedness conditions on the initial data, thereby accommodating the possible formation of an initial layer.