Singular limits for non-isentropic compressible rotating fluids

In this article, we study the singular limit of non-isentropic compressible rotating fluids. We incorporate the capillary effect into both the $\alpha=1$ and $\alpha=0$ cases, and investigate the Navier-Stokes-Korteweg equations involving the terms of low Mach number, low Rossby number and high Reynolds number. When $\alpha=1$, the dispersion estimate of the acoustic wave equation is derived by Rage's theorem. When $\alpha=0$, we obtain the convergence results by error estimate. Moreover, we obtain that the three dimensions compressible Navier-Stokes-Korteweg equations converge to the two dimensions incompressible Euler equations.