Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

TL;DR

This paper proves the global existence of smooth solutions for quantum hydrodynamic equations with viscosity and heat conduction, and demonstrates their convergence to classical solutions, addressing challenges in energy estimates and Sobolev space selection.

Contribution

It establishes the first rigorous proof of global solutions and semiclassical limits for quantum hydrodynamic equations with energy considerations.

Findings

Global existence of smooth solutions for small initial data.

Convergence of quantum solutions to classical hydrodynamics.

Addressed new difficulties in energy estimates with heat conduction.

Abstract

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.