Global Existence and Vanishing Dispersion Limit of Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

TL;DR

This paper proves the global existence and large-time behavior of strong solutions to the one-dimensional compressible quantum Navier-Stokes equations with large initial data, and establishes the vanishing dispersion limit with convergence rates.

Contribution

It extends previous results by handling the case where viscosity and Planck constant are not equal, and introduces a new effective velocity to analyze the equations.

Findings

Proved global existence of strong solutions for large initial data.

Established the vanishing dispersion limit with convergence rates.

Derived uniform bounds on the specific volume.

Abstract

We are concerned with the global existence and vanishing dispersion limit of strong/classical solutions to the Cauchy problem of the one-dimensional barotropic compressible quantum Navier-Stokes equations, which consists of the compressible Navier-Stokes equations with a linearly density-dependent viscosity and a nonlinear third-order differential operator known as the quantum Bohm potential. The pressure $p(\rho)=\rho^\gamma$ is considered with $\gamma\geq1$ being a constant. We focus on the case when the viscosity constant $\nu$ and the Planck constant $\varepsilon$ are not equal. Under some suitable assumptions on $\nu,\varepsilon, \gamma$, and the initial data, we proved the global existence and large-time behavior of strong and classical solutions away from vacuum to the compressible quantum Navier-Stokes equations with arbitrarily large initial data. This result extends the previous ones on the construction of global strong large-amplitude solutions of the compressible quantum Navier-Stokes equations to the case $\nu\neq\varepsilon$. Moreover, the vanishing dispersion limit for the classical solutions of the quantum Navier-Stokes equations is also established with certain convergence rates. The proof is based on a new effective velocity which converts the quantum Navier-Stokes equations into a parabolic system, and some elaborate estimates to derive the uniform-in-time positive lower and upper bounds on the specific volume.