Vanishing capillarity limit of the compressible non-isentropic Navier-Stokes-Korteweg system to Navier-Stokes system

In this paper, we consider the three dimensional compressible nonisentropic fluid model of Korteweg type. We mainly present the vanishing capillarity limit of smooth solution to the initial value problem. Based on the uniform estimates with respect to the capillary coefficient κ, we show the unique smooth solutions of the three dimensional Korteweg model converges globally in time to the smooth solution of the three dimensional compressible nonisentropic Navier-Stokes system, as κ tends to zero. Also, we give the convergence rate estimates in H2-norm with respect to κ.