Global weak solutions to one-dimensional compressible viscous hydrodynamic equations

In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ2 ρ((φ(ρ))xxφ′(ρ)x with φ(ρ)=ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term ɛuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1](α=12)to0<α≤1. In addition, we perform the limit ɛ→ 0 with respect to 0 < α ≤ ½.