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

TL;DR

This paper proves the global existence and low Mach number limit of solutions to the compressible Navier-Stokes-Transport system in critical Besov spaces, revealing unique decay properties due to the transport nature of temperature.

Contribution

It establishes the global well-posedness and optimal decay rates for the NST system in critical Besov spaces, and rigorously justifies the low Mach number limit even with ill-prepared initial data.

Findings

Solutions exist globally in time in critical Besov spaces.

Solutions converge to incompressible inhomogeneous Navier-Stokes solutions as Mach number tends to zero.

Density remains uniformly bounded and exhibits distinct asymptotic behavior from NSF systems.

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.