Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

Given constant data of density $\rho_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(\rho_0, u_0, p_0)$ and the length of the nozzle

Scaling of Coulomb pseudo-potential in s-wave narrow-band superconductors

The Coulomb pseudo-potential $\mu^*$ is extracted by fitting the numerically calculated transition temperature $T_c$ of the Eliashberg-Nambu equation which is extended to incorporate the narrow-band effects, that is, the vertex correction and the frequency dependence of the screened Coulomb interaction. It is shown that even for narrow-band superconductors, where the fermi energy $\epsilon_F$ is comparable with the phonon frequency $\omega_{ph}$, the Coulomb pseudo-potential is a pertinent parameter, and is still given by $\mu^* = \mu/[1+\mu \ln(\epsilon_F/\omega_{ph})]$, provided $\omega_{ph}$ is appropriately scaled.Comment: 5 pages, 3 figures, accepted for publication by Phys. Rev.