# Global well-posedness and time-decay estimates of the compressible Navier-Stokes-Korteweg system in critical Besov spaces

## TL;DR

This paper proves the global existence and decay rates of solutions to the compressible Navier-Stokes-Korteweg system in critical Besov spaces, highlighting the system's stability and parabolic properties across all frequencies.

## Contribution

It establishes the global well-posedness and optimal decay rates for the system in critical Besov spaces, extending previous results to include cases with non-negative sound speed.

## Key findings

- Global solutions exist under linear stability in critical Besov spaces.
- No derivative loss occurs due to pressure, enabling fixed point methods.
- Optimal decay rates are achieved in the L^2 framework.

## Abstract

We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(\mathbb{R}^d)$-framework.