Global strong solutions to the compressible Navier--Stokes--Coriolis system for large data

TL;DR

This work addresses the global well-posedness of the 3D compressible Navier–Stokes system with Coriolis force on $\mathbb{R}^3$ for large initial data. By leveraging a scaling-critical Besov/Chemin–Lerner framework and exploiting the dispersive interaction between rotation and acoustic waves, the authors prove the existence of a unique global strong solution in the transformed variables $(a,u)$ under the regime $\Omega_0\le |\Omega|\le c_0/\varepsilon$, with $a$ remaining strictly above $-1/\varepsilon$ and initial data in critical Besov spaces. The analysis combines a low-frequency $4^{\text{th}}$-order dissipation behavior, momentum formulation to tame nonlinearities, and frequency-separated a priori estimates (low/mid/high) to close the global bounds, enabling large-data results in the fast-rotation, low-Mach limit $1\ll|\Omega|\ll 1/\varepsilon$. This extends the Matsumura–Nishida approach to rotating compressible flows on $\mathbb{R}^3$ and provides new insight into how rotational dispersion can stabilize nonlinear dynamics, with potential implications for low Mach number limits and geophysical fluid dynamics.

Abstract

We consider the compressible Navier--Stokes system with the Coriolis force on the $3$D whole space. In this model, the Coriolis force causes the linearized solution to behave like a $4$th order dissipative semigroup $\{ e^{-tΔ^2} \}_{t>0}$ with slower time decay rates than the heat kernel, which creates difficulties in nonlinear estimates in the low-frequency part and prevents us from constructing the global strong solutions by following the classical method. On account of this circumstance, the existence of unique global strong solutions has been open even in the classical Matsumura--Nishida framework. In this paper, we overcome the aforementioned difficulties and succeed in constructing a unique global strong solution in the framework of scaling critical Besov spaces. Furthermore, our result also shows that the global solution is constructed for arbitrarily large initial data provided that the speed of the rotation is high and the Mach number is low enough by focusing on the dispersive effect due to the mixture of the Coriolis force and acoustic wave.

Theorems & Definitions (33)

• Theorem 2.1
• Remark 2.2
• Lemma 3.1: Fuj-Wat-25*Lemma 2.1
• Lemma 3.2
• Remark 3.3
• proof
• Lemma 3.4: Fuj-Wat-25*Lemma 2.5
• Lemma 3.5: Fuj-Wat-25*Proposition 3.1
• Lemma 4.1
• proof