The Lp energy methods and decay for the compressible Navier-Stokes equations with capillarity

We consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effect. Referring to those studies in the non-capillary case, the purpose of this paper is to investigate the dissipation effect of Korteweg tensor with the density-dependent capillarity $\kappa(\varrho)$. It is observed by the pointwise estimate that the linear third-order capillarity behaves like the heat diffusion of density fluctuation, which allows to develop the $L^p$ energy methods (independent of spectral analysis). As a result, the time-decay estimates of $L^q$-$L^r$ type regarding this system can be established. The treatment of nonlinear capillarity depends mainly on new Besov product estimates and the elaborate use of Sobolev embeddings and interpolations. Our results can be also applied to the quantum Navier-Stokes system, since it is a special choice of capillarity $\kappa(\varrho)=\kappa/\varrho$.