# Time decay estimate with diffusion wave property and smoothing effect for solutions to the compressible Navier-Stokes-Korteweg system

## TL;DR

This paper establishes time decay estimates with diffusion wave properties for solutions to the compressible Navier-Stokes-Korteweg system, highlighting the smoothing effects and lower regularity requirements compared to classical systems.

## Contribution

It introduces decay estimates with diffusion wave behavior for both linearized and nonlinear compressible Navier-Stokes-Korteweg systems, emphasizing lower initial data regularity and smoothing effects.

## Key findings

- Decay estimates with diffusion wave property for linearized system
- Lower regularity initial data suffices due to smoothing effect
- Diffusion wave property for nonlinear system with less regular initial data

## Abstract

Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial data is derived. As an application, the time decay estimate of solutions to the nonlinear problem is given. In contrast to the compressible Navier-Stokes system, for linear system regularities of the initial data are lower and independent of the order of derivative of solutions owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusion wave property is obtained with an initial data having lower regularity than that of study of the compressible Navier-Stokes system.