Global Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

Abstract

We are concerned with the global existence and singular 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 densitydependent viscosity and a nonlinear third-order differential operator known as the quantum Bohm potential. The pressure p(ρ) = ρ γ is considered with γ ≥ 1 being a constant. We focus on the case when the viscosity constant ν and the Planck constant ε are not equal. Under some suitable assumptions on ν, ε, γ, 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 large initial data. This extends the previous results on the construction of global strong large-amplitude solutions of the compressible quantum Navier-Stokes equations to the case ν = ε. Moreover, the vanishing dispersion limit for the classical solutions of the quantum Navier-Stokes equations is also established with certain convergence rates. Some new mathematical techniques and useful estimates are developed to deduce the uniform-in-time positive lower and upper bounds on the specific volume.