Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations
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| Format: | Preprint |
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2025
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| _version_ | 1866908704311345152 |
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| author | Fei, Mingwen Shou, Ling-Yun Tang, Houzhi |
| author_facet | Fei, Mingwen Shou, Ling-Yun Tang, Houzhi |
| contents | The Euler-Navier-Stokes (E-NS) system arises as a macroscopic description of kinetic-fluid interactions, derived from the local-Maxwellian closure of the Vlasov-Fokker-Planck-Navier-Stokes flow. In this paper, we investigate the singular limit of the system in $\mathbb{R}^d$ ($d\ge2$) when the relaxation parameter $\varepsilon>0$ tends to zero. In contrast to the Euler system with velocity damping, the E-NS model features only a weaker relaxation of the relative velocity, which makes it challenging to analyze its dynamics as $\varepsilon\rightarrow 0$. We develop an energy argument to show global-in-time error estimates between the E-NS system and its limit system, the so-called Kramers-Smoluchowski-Navier-Stokes (KS-NS) system. These error estimates enable us to prove the global existence and uniform-in-$\varepsilon$ regularity of the strong solution to the E-NS system in a hybrid critical Besov space with a sharp frequency threshold of order $\mathcal{O}(\varepsilon^{-1})$ separating the low- and high-frequency regimes. Moreover, the large-time asymptotic stability of the global solution to the E-NS system is established. More precisely, we derive the optimal decay rates of the solution uniformly in $\varepsilon$, and the enhanced decay rates for the difference between the densities of the E-NS system and the KS-NS system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_09650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations Fei, Mingwen Shou, Ling-Yun Tang, Houzhi Analysis of PDEs 35B25, 35B40, 35Q99, 76N10, 76N17 The Euler-Navier-Stokes (E-NS) system arises as a macroscopic description of kinetic-fluid interactions, derived from the local-Maxwellian closure of the Vlasov-Fokker-Planck-Navier-Stokes flow. In this paper, we investigate the singular limit of the system in $\mathbb{R}^d$ ($d\ge2$) when the relaxation parameter $\varepsilon>0$ tends to zero. In contrast to the Euler system with velocity damping, the E-NS model features only a weaker relaxation of the relative velocity, which makes it challenging to analyze its dynamics as $\varepsilon\rightarrow 0$. We develop an energy argument to show global-in-time error estimates between the E-NS system and its limit system, the so-called Kramers-Smoluchowski-Navier-Stokes (KS-NS) system. These error estimates enable us to prove the global existence and uniform-in-$\varepsilon$ regularity of the strong solution to the E-NS system in a hybrid critical Besov space with a sharp frequency threshold of order $\mathcal{O}(\varepsilon^{-1})$ separating the low- and high-frequency regimes. Moreover, the large-time asymptotic stability of the global solution to the E-NS system is established. More precisely, we derive the optimal decay rates of the solution uniformly in $\varepsilon$, and the enhanced decay rates for the difference between the densities of the E-NS system and the KS-NS system. |
| title | Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations |
| topic | Analysis of PDEs 35B25, 35B40, 35Q99, 76N10, 76N17 |
| url | https://arxiv.org/abs/2512.09650 |