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)=ρ^α. 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 ?u_xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in $1]\left(\alpha =\frac{1}{2}\right)\text{to}0<\alpha \le 1.$ In addition, we perform the limit ?→ 0 with respect to 0 < α ≤ ½.