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

## TL;DR

This paper investigates the long-term behavior of diffusion wave components in solutions to the compressible Navier-Stokes-Korteweg system on R^2, revealing differences in decay rates between density and potential flow parts under Hardy space initial conditions.

## Contribution

It demonstrates that, in the Hardy space setting, the asymptotic decay rates of diffusion wave parts differ between density and potential flow, with potential flow decaying more slowly.

## Key findings

- Decay rates differ between density and potential flow parts.
- Potential flow decay is slower than Stokes flow decay.
- Asymptotic behaviors are characterized in space-time L^2 using advanced energy estimates.

## Abstract

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.