Large-friction and incompressible limits for pressureless Euler-Navier-Stokes flows
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arXiv
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2025
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| author | Li, Hai-Liang Shou, Ling-Yun Zhang, Yue |
| author_facet | Li, Hai-Liang Shou, Ling-Yun Zhang, Yue |
| contents | We study the global macroscopic limits associated with kinetic-fluid interaction models for sprays. Motivated by the Vlasov-Navier-Stokes system under the monokinetic ansatz, we consider the pressureless Euler-Navier-Stokes (Euler-NS) system in $\mathbb{R}^{d}$ ($d\geq2$) coupled through the singular drag force $\frac{1}τ ρ(u-v)$, where $τ$ is the Stokes relaxation time. For initial data uniformly close to equilibrium in critical Besov spaces, we establish global-in-time regularity estimates of solutions to the Cauchy problem for the Euler-NS system, uniformly with respect to $τ$. These estimates yield the global strong convergence of the Euler-NS system toward a one-velocity two-phase drift-flux (DF) model as $τ\to0$, with an explicit convergence rate of order $\sqrtτ$. A key point in the analysis is the introduction of an effective mixed velocity, which allows us to handle the singular relative-velocity relaxation and obtain global error estimates for ill-prepared data. We also derive large-time asymptotic estimates for the Euler-NS system, uniformly in $τ$, including the improved decay of the relative velocity and the convergence of the non-dissipative density toward an asymptotic profile. Furthermore, after introducing the Mach number $\varepsilon>0$, we justify the incompressible limit of the DF model toward the Transport-Navier-Stokes (TNS) system as $\varepsilon\to0$, and prove the combined large-friction and incompressible limit from the Euler-NS system to the TNS system in the regime $τ=\varepsilon\to0$ in an ill-prepared setting. These results provide a unified and quantitative macroscopic picture connecting the Euler-NS, DF, and TNS systems through the large-friction and incompressible regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-friction and incompressible limits for pressureless Euler-Navier-Stokes flows Li, Hai-Liang Shou, Ling-Yun Zhang, Yue Analysis of PDEs 35Q35, 76N10, 76T17 We study the global macroscopic limits associated with kinetic-fluid interaction models for sprays. Motivated by the Vlasov-Navier-Stokes system under the monokinetic ansatz, we consider the pressureless Euler-Navier-Stokes (Euler-NS) system in $\mathbb{R}^{d}$ ($d\geq2$) coupled through the singular drag force $\frac{1}τ ρ(u-v)$, where $τ$ is the Stokes relaxation time. For initial data uniformly close to equilibrium in critical Besov spaces, we establish global-in-time regularity estimates of solutions to the Cauchy problem for the Euler-NS system, uniformly with respect to $τ$. These estimates yield the global strong convergence of the Euler-NS system toward a one-velocity two-phase drift-flux (DF) model as $τ\to0$, with an explicit convergence rate of order $\sqrtτ$. A key point in the analysis is the introduction of an effective mixed velocity, which allows us to handle the singular relative-velocity relaxation and obtain global error estimates for ill-prepared data. We also derive large-time asymptotic estimates for the Euler-NS system, uniformly in $τ$, including the improved decay of the relative velocity and the convergence of the non-dissipative density toward an asymptotic profile. Furthermore, after introducing the Mach number $\varepsilon>0$, we justify the incompressible limit of the DF model toward the Transport-Navier-Stokes (TNS) system as $\varepsilon\to0$, and prove the combined large-friction and incompressible limit from the Euler-NS system to the TNS system in the regime $τ=\varepsilon\to0$ in an ill-prepared setting. These results provide a unified and quantitative macroscopic picture connecting the Euler-NS, DF, and TNS systems through the large-friction and incompressible regimes. |
| title | Large-friction and incompressible limits for pressureless Euler-Navier-Stokes flows |
| topic | Analysis of PDEs 35Q35, 76N10, 76T17 |
| url | https://arxiv.org/abs/2508.20730 |