Asymptotic profile for diffusion wave terms of the compressible Navier-Stokes-Korteweg system

Asymptotic profile for diffusion wave terms of solutions to the compressible Navier-Stokes-Korteweg system is studied on $R^2$. The diffusion wave with time decay estimate is studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002) and Kobayashi and Tsuda (2018) for the compressible Navier-Stokes system and the compressible Navier-Stokes-Korteweg system. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space-time $L^2$ of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by $L^2$ on space, a decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz's energy estimate, and the Fefferman-Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.