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 L 2 and L p , p ⩾ 2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L 2 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.