Global existence and optimal decay rate of the compressible Navier–Stokes–Korteweg equations with external force

In this paper, we consider the three-dimensional compressible fluid models of Korteweg type. The equation takes the form of the compressible Navier–Stokes equation, with the Cauchy stress tensor in the momentum equations. Under the smallness assumption of the external force in some Sobolev space, we first show the existence of stationary solution by standard iterative arguments. Next, we present that the global-in-time solution of the initial value problem for the three-dimensional compressible Navier–Stokes–Korteweg equations exists uniquely and approaches the stationary solution as t → ∞ , provided the prescribed initial data and the external force are sufficiently small. Finally, based on the elaborate energy estimates of the solution for the nonlinear system and L 2 -decay estimates of semigroup generated by the corresponding linearized equation, we show the optimal L 2 -convergence rates of the solution towards the stationary solution. As far as we know, this is the first result about the global existence and L 2 -decay rate of smooth solutions for the compressible Navier–Stokes–Korteweg equations with the external force.