Optimal decay rates for the compressible fluid models of Korteweg type

We consider the compressible Navier-Stokes-Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L2 and Lp, p⩾2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L2 decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.