Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917241905217536 |
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| author | Li, Fucai Ni, Jinkai Wang, Yuzhu |
| author_facet | Li, Fucai Ni, Jinkai Wang, Yuzhu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_02019 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces Li, Fucai Ni, Jinkai Wang, Yuzhu Analysis of PDEs 76N10, 35Q30, 35B25, 35B40 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. |
| title | Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces |
| topic | Analysis of PDEs 76N10, 35Q30, 35B25, 35B40 |
| url | https://arxiv.org/abs/2602.02019 |