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

Mikihiro Fujii, Keiichi Watanabe

TL;DR

The paper studies the Cauchy problem for the 3D compressible Navier–Stokes system with the Coriolis force on $\mathbb{R}^3$ in the scaling-critical Besov framework. It combines dispersive effects from rapid rotation with acoustic waves and develops linear energy and dispersive/Strichartz estimates, including an inviscid reduction to obtain explicit spectral decay. Through a local fixed-point argument in the scaling-critical spaces and a detailed continuation analysis, it proves long-time existence and uniqueness of strong solutions for arbitrarily large data provided $|\Omega|$ is sufficiently large and the Mach number $\varepsilon$ is small, with the regime $\Omega_T \le |\Omega| \le 1/\varepsilon$. The work demonstrates how fast rotation can regularize the compressible NSC system in the whole space by inducing dispersion despite anisotropy, establishing a first well-posedness result in the critical Besov setting for this model. Overall, the results offer a rigorous framework for analyzing geophysical flows under fast rotation and low Mach number in critical regularity classes.

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.

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

TL;DR

The paper studies the Cauchy problem for the 3D compressible Navier–Stokes system with the Coriolis force on in the scaling-critical Besov framework. It combines dispersive effects from rapid rotation with acoustic waves and develops linear energy and dispersive/Strichartz estimates, including an inviscid reduction to obtain explicit spectral decay. Through a local fixed-point argument in the scaling-critical spaces and a detailed continuation analysis, it proves long-time existence and uniqueness of strong solutions for arbitrarily large data provided is sufficiently large and the Mach number is small, with the regime . The work demonstrates how fast rotation can regularize the compressible NSC system in the whole space by inducing dispersion despite anisotropy, establishing a first well-posedness result in the critical Besov setting for this model. Overall, the results offer a rigorous framework for analyzing geophysical flows under fast rotation and low Mach number in critical regularity classes.

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 and arbitrary large initial data in the scaling critical Besov spaces, the solution uniquely exists on 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 . 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.

Paper Structure

This paper contains 6 sections, 13 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Linear energy estimates
  3. Linear dispersive estimates
  4. Local well-posedness on the short time interval
  5. Proof of Theorem \ref{['thm']}
  6. Proof of Lemma \ref{['lemm:stab']}

Key Result

Theorem 1.1

Let $0 < T < \infty$ and Then, there exists a positive constant $\Omega_{T}$ depending on $T$, $\mu$, $P$, $a_0$, and $u_0$ such that if $\Omega \in \mathbb R \setminus \{0 \}$ and $0 < \varepsilon \leqslant 1$ satisfy then the equation eq:NSC-2 possesses a unique solution $(a,u)$ on $[0,T]$ in the class with $\rho = 1 +\varepsilon a >0$ on $[0,T] \times \mathbb{R}^3$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • proof : Proof of Corollary \ref{['cor:low-ene']}
  • Lemma 2.5
  • ...and 20 more