Vanishing Capillarity Limit of the Compressible Fluid Models of Korteweg Type to the Navier--Stokes Equations

Abstract

In this paper, we consider the three-dimensional isentropic compressible fluid models of Korteweg type, called the compressible Navier--Stokes--Korteweg system. We mainly present the vanishing capillarity limit of the smooth solution to the initial value problem. Precisely, we first establish the uniform estimates of the global smooth solution with respect to the capillary coefficient $\kappa$. Then by the Lions--Aubin lemma, we show that the unique smooth solution of the three-dimensional Navier--Stokes--Korteweg system converges globally in time to the smooth solution of the three-dimensional Navier--Stokes system as $\kappa$ tends to zero. Also, we give the convergence rate estimates for any given positive time.