全空间可压缩Euler--Vlasov--Fokker--Planck系统的低马赫数极限
Low Mach number limit of the compressible Euler--Vlasov--Fokker--Planck system in the whole space
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中文总结 AI 辅助
本文通过精细能量方法和修正声学变量，证明了全空间可压缩Euler--Vlasov--Fokker--Planck系统在低马赫数极限下以$\mathcal O(\varepsilon)$速率收敛到不可压缩系统，解决了流体-粒子模型间的连接问题。
中文摘要 AI 辅助
尽管关于可压缩和不可压缩流体-粒子相互作用模型分别已有许多重要贡献，但如何通过低马赫数极限将两类流体-粒子模型联系起来仍然是一个具有挑战性的开放问题。本文在全空间$\mathbb{R}^3$中解决了可压缩等熵流体-粒子模型（Euler--Vlasov--Fokker--Planck（Euler--VFP）系统）的这一难题。首先，我们在全局Maxwellian附近建立了关于马赫数$\varepsilon$一致的光滑解的全局时间先验估计。证明依赖于一种精细的能量方法，该方法结合了由流体-粒子相互作用引起的松弛结构$b^\varepsilon-u^\varepsilon$以及模型中可压缩Euler部分的对称化声学结构。在良好准备初始数据的假设下，我们在$H^2$框架下推导了可压缩Euler--VFP系统的解与极限不可压缩Euler--VFP系统的解之间的全局时间一致误差估计。一个关键点是引入修正声学变量$q^\varepsilon-\varepsilon [P'(1)]^{-1}\pi$，该变量捕捉了低马赫数极限中的压力修正项。这也使我们能够利用奇异声学项的精确相消，并闭合全局时间误差估计。阻尼项$b^\varepsilon-u^\varepsilon$在纯Euler方程中不存在，它在恢复相对速度耗散和控制耦合流体-粒子动力学中起着至关重要的作用。因此，我们证明了可压缩Euler--VFP系统的低马赫数极限，其收敛速度为$\mathcal O(\varepsilon)$，在时间连续的$H^2$拓扑下成立。
英文摘要
Although there are many important contributions on compressible and incompressible fluid-particle interaction models respectively, how to connect the two-type fluid-particle models via the low Mach number limit remains a challenging open problem. In this paper, we resolve it for the compressible isentropic fluid-particle model (Euler--Vlasov--Fokker--Planck (Euler--VFP) system) in the whole space $\mathbb{R}^3$. First, we establish the global-in-time {\it a priori estimates} of strong solutions that are uniform with respect to the Mach number $\varepsilon$ near the global Maxwellian. The proof relies on a refined energy method that combines the relaxation structure $b^\varepsilon-u^\varepsilon$ induced by the fluid-particle interaction and the symmetrized acoustic structure of the compressible Euler part in the model. Under the assumption of well-prepared initial data, we derive a {\it global-in-time} uniform error estimate in the $H^2$ framework between the solution of the compressible Euler--VFP system and that of the limiting incompressible Euler--VFP system. A key point is to introduce the corrected acoustic variable $ q^\varepsilon-\varepsilon [P'(1)]^{-1}π$, which captures the pressure corrector in the low Mach number limit. This also allows us to exploit the exact cancellation of the singular acoustic terms and to close the {\it global-in-time} error estimate. The damping term $b^\varepsilon-u^\varepsilon$, which is absent in the pure Euler equations, plays an essential role in recovering the relative velocity dissipation and in controlling the coupled fluid-particle dynamics. As a consequence, we prove the low Mach number limit of the compressible Euler--VFP system with the convergence rate $\mathcal O(\varepsilon)$ in the time-continuous $H^2$ topology.
发表机构
- Nanjing University（南京大学）
- Ocean University of China（中国海洋大学）
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