AI 中文总结
该研究从双组分玻尔兹曼系统出发，在特定标度假设下严格推导不可压缩弗拉索夫-纳维-斯托克斯系统，为该模型的论证提供了关键理论支撑。
AI 中文摘要
弗拉索夫-纳维-斯托克斯系统的严格论证仍是一个未解决的重要问题，既作为N粒子流体相互作用系统的平均场极限，也作为从多相玻尔兹曼方程导出的流体动力学极限。受Bernard、Desvillettes、Golse和Ricci（2017年发表于《Commun. Math. Sci.》第15卷第6期，1703-1741页）工作的启发，我们从用于弹性硬球碰撞的双组分玻尔兹曼系统出发，对不可压缩弗拉索夫-纳维-斯托克斯系统进行了严格推导。为此，我们针对小初始数据建立了重标度多组分玻尔兹曼系统的适定性，具体得到了解的估计，该估计对热速度比ε和质量比η均一致成立。此外，在弗拉索夫-纳维-斯托克斯标度假设下，即当ε→0时满足O(1)ε³≤η≤o(1)ε²，我们建立了解到流体-动力学耦合极限系统的弱收敛性。
英文摘要
The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an $N$-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [{\it Commun. Math. Sci.}, {\bf 15}(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio $\varepsilon$ and the mass ratio $η$. Furthermore, under the Vlasov-Navier-Stokes scaling assumption $O(1)\varepsilon^3 \leqslant η\leqslant o(1)\varepsilon^2$ as $\varepsilon \rightarrow 0$, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.
Comments54 pages
