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

Hao Zheng

Abstract

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 Schrödinger-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.

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

Abstract

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 solutions for a nonlinear Schrödinger-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.
Paper Structure (8 sections, 22 theorems, 302 equations)

This paper contains 8 sections, 22 theorems, 302 equations.

Table of Contents

  1. Introduction and main results
  2. Main result: global well-posedness of GCP solutions and stability
  3. Main result: relaxation-time limit
  4. Definitions and preliminaries
  5. Time-derivative computation of the main functionals
  6. Global well-posedness of GCP solutions on ${\mathbb T}^2$
  7. Rescaling of the QHD system and the time-relaxation limit
  8. The time-relaxation limit of the rescaled QHD system \ref{['eq:QHD_rs']} for GCP solutions

Key Result

Theorem 1

Let us consider a finite energy initial datum $(\rho_0, J_0)$ satisfying the following conditions. Then, there exists $\tau^*>0$, depending on $(\delta,M_0,E_0,I_0)$, such that for $0<\tau\leq \tau^*$, the Cauchy problem of the QHD system eq:QHD has a unique global in time GCP solution $(\rho,J)$, that for any $0\leq t<\infty$

Theorems & Definitions (45)

  • Theorem 1: Global well-posedness of GCP weak solutions
  • Remark 2
  • Theorem 3
  • Theorem 4: Dissipation for small $\tau$
  • Remark 5
  • Theorem 6: Relaxation-time limit
  • Definition 7: GCP solutions
  • Definition 8
  • Definition 9: Phase function
  • Remark 10
  • ...and 35 more