Convergence rate of the three-dimensional compressible fluid models of Korteweg type to the Navier–Stokes system

In this paper, we focus on the three-dimensional isentropic compressible fluids models of Korteweg type, that is, the three-dimensional compressible Navier–Stokes–Korteweg system. We establish the convergence rate for the solution of three-dimensional compressible Navier–Stokes–Korteweg system to the solution of three-dimensional Navier–Stokes system as the capillary coefficient tends to zero. Once the capillary coefficient tends to zero, this convergence rate will tend to zero. This explicit convergence rate result extracted in the vanishing capillary limit improves the important work of [Bian et al. SIAM J. Math. Anal. 46(2), 1633 (2014)].