# Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition

## TL;DR

This paper proves the global existence and decay rates of solutions to the compressible Navier-Stokes-Korteweg system in a critical non-monotone pressure case, using a decomposition method for small initial data.

## Contribution

It establishes global solutions and decay estimates for a critical non-monotone pressure case, extending previous results to more realistic phase transition models.

## Key findings

- Global $L^2$ solutions are shown to exist for small data.
- Solutions exhibit parabolic decay rates over time.
- The decomposition method effectively handles low and high frequency components.

## Abstract

Global existence of solutions to the compressible Navier-Stokes-Korteweg system around a constant state is studied. This system describes liquid-vapor two phase flow with phase transition as diffuse interface model. In previous works they assume that the pressure is a monotone function for change of density similarly to the usual compressible Navier-Stokes system. On the other hand, due to phase transition the pressure is accurately non-monotone function and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. It is shown that in the critical case for small data whose momentum has derivative form there exist global $L^2$ solutions and the parabolic type decay rate of the solutions is obtained. The proof is based on decomposition method for solutions to a low frequency part and a high frequency part.