Compressible Navier--Stokes--Coriolis system in critical Besov spaces

TL;DR

This paper proves the long-time existence and uniqueness of solutions for the three-dimensional compressible Navier--Stokes system with Coriolis force in critical Besov spaces, highlighting the effects of high rotation and low Mach numbers.

Contribution

It is the first to establish well-posedness of the compressible Navier--Stokes system with Coriolis force in the whole space using dispersive linear estimates.

Findings

Solutions exist uniquely for any finite time with high rotation and low Mach numbers.

Dispersive linear estimates are established despite anisotropic linearized equations.

The results apply to initial data in critical Besov spaces.

Abstract

We consider the three-dimensional compressible Navier--Stokes system with the Coriolis force and prove the long-time existence of a unique strong solution. More precisely, we show that for any $0<T<\infty$ and arbitrary large initial data in the scaling critical Besov spaces, the solution uniquely exists on $[0,T]$ provided that the speed of rotation is high and the Mach numbers are low enough. To the best of our knowledge, this paper is the first contribution to the well-posedness of the \textit{compressible} Navier--Stokes system with the Coriolis force in the whole space $\mathbb R^3$. The key ingredient of our analysis is to establish the dispersive linear estimates despite a quite complicated structure of the linearized equation due to the anisotropy of the Coriolis force.