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(g). 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(g) = kappa/g. (C) 2021 Elsevier Masson SAS. All rights reserved.

Automated summary

This paper investigates the dissipation effect of Korteweg tensor with density-dependent capillarity in the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effect. The linear third-order capillarity behaves like heat diffusion of density fluctuation, allowing the development of L-p energy methods independent of spectral analysis. Time-decay estimates of L-q-L-r type can be established for this system, with a focus on nonlinear capillarity treatment using Besov product estimates and Sobolev embeddings and interpolations. The results can be applied to the quantum Navier-Stokes system through a special choice of capillarity.