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The time-relaxation limit for weak solutions to the quantum hydrodynamics system

Paolo Antonelli, Pierangelo Marcati, Hao Zheng

Abstract

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. \newline 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. \newline 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. \newline 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. \newline As a by-product of our analysis, we also establish the existence of global in time $H^2$ solutions to a nonlinear Schrödinger-Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.

The time-relaxation limit for weak solutions to the quantum hydrodynamics system

Abstract

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. \newline 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. \newline 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. \newline 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. \newline As a by-product of our analysis, we also establish the existence of global in time solutions to a nonlinear Schrödinger-Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.

Paper Structure

This paper contains 12 sections, 21 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Physical framework and weak solutions
  3. Main result: global well-posedness
  4. Main result: relaxation-time limit
  5. Outline of the paper
  6. Definitions and preliminaries
  7. Global well-posedness of GCP solutions
  8. Rescaling of QHD system with damping and uniform estimates in $\tau$
  9. Entropy estimates and Dissipation
  10. Dissipation for small $\tau$ and internal energy $f(s)=(s-M_0)^{2n}$
  11. Relaxation-time limit and convergence rate
  12. Hydrodynamical computation of identities \ref{['eq:en_disp']} and \ref{['eq:rl_I']}

Key Result

Proposition 1

Let $(\rho_0, J_0)$ be a finite energy initial datum such that, and let us further assume that $J_0/\sqrt{\rho_0}=0$ a.e. on the set $\{\rho_0=0\}$. Then, there exists a global in time finite energy weak solution to the Cauchy problem that satisfies

Theorems & Definitions (43)

  • Proposition 1: Global Existence of finite energy weak solutions
  • Theorem 2: Global well-posedness of GCP weak solutions
  • Remark 3
  • Theorem 4
  • Remark 5
  • Theorem 6: Relaxation-time limit
  • Theorem 7: Dissipation for small $\tau$
  • Remark 8
  • Definition 9: Finite energy weak solutions
  • Remark 10
  • ...and 33 more