Global existence and optimal decay rate for the strong solutions to the compressible fluid models of Korteweg type

In this paper, we consider the compressible Navier–Stokes–Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the global existence of a strong solution to the compressible Navier–Stokes–Korteweg system when the initial perturbation ‖ ρ 0 − ρ ¯ ‖ H 2 + ‖ u 0 ‖ H 1 is small. Furthermore, if the L 1 norm of the initial perturbation is finite, we can obtain the optimal L 2 decay rates.