Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces

TL;DR

This work analyzes the compressible NST system in critical Besov spaces on $\mathbb{R}^d$, proving global well-posedness and a low Mach number limit to the incompressible inhomogeneous Navier–Stokes equations under ill-prepared data. The authors introduce a novel mode $\omega=\gamma a-z$ to compensate for the lack of dissipation in density and temperature, and develop a frequency-localized Lyapunov framework to obtain uniform estimates for $(a,u,z)$. They prove that as $\varepsilon\to0$, the solutions converge to the incompressible limit with precise convergence of the divergence-free velocity and the derived density mode, even without well-prepared data. Additionally, the paper establishes optimal time decay rates for the velocity and pressure-like variables, highlighting distinct decay behavior from the Navier–Stokes–Fourier system due to the non-dissipative density. The results advance understanding of non-isentropic flows in critical regularity and provide rigorous justification of singular limits in Besov spaces.

Abstract

In this paper, we investigate the Navier-Stokes-Transport (NST) system in the framework of Besov spaces. This system contains of a compressible Navier-Stokes system for the density and momentum of a fluid, and a transport equation for the potential temperature of the fluid. In stark contrast to the well-known Navier-Stokes-Fourier (NSF) system where the temperature satisfies a parabolic type equation providing dissipative effect for the temperature and the density, the temperature in our NST system enjoys a transport equation which precludes a dissipative mechanism for the density, leading to significant different effects to the whole system. We first establish the global well-posedness of strong solutions to the compressible NST system in critical Besov spaces over $\mathbb{R}^d$ with $d \geq 2$. Furthermore, by introducing the Mach number $\varepsilon > 0$, we rigorously prove the low Mach number limit as $\varepsilon \to 0$, showing that the solutions converge to that of the incompressible inhomogeneous Navier-Stokes system. This singular limit holds globally in time, even for {\it ill-prepared} initial data. To address the challenge posed by the lack of dissipation on the density and temperature, we develop a refined energy analysis and establish optimal time decay rates for strong solutions in $\mathbb{R}^d$ with $d \geq 3$. Notably, the density remains uniformly bounded in time, displaying asymptotic behavior fundamentally distinct from that in the NSF system, where the density possesses a dissipative structure via the momentum and temperature equations and exhibits temporal decay.

Theorems & Definitions (63)

• Theorem 1.1: Local well-posedness
• Remark 1.1
• Theorem 1.2: Global well-posedness
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.3: Low Mach number limit
• Remark 1.5
• Remark 1.6
• Theorem 1.4: Upper-bound: the bounded condition