This work is devoted to the study of the existence of positive solutions and the hydrodynamic limit of the steady Boltzmann equation with in-flow boundary condition. The proof is based on a $L^6-L^{\mathrm{\infty }}$ framework developed by Esposito et al. [Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann PDE. 2018;4(1):1–119] and a refined positivity-preserving scheme in deriving positivity of solutions with in-flow boundary condition and external force. The incompressible Navier–Stokes–Fourier limit with a Dirichlet boundary condition is justified for in-flow boundary data as a small perturbation of a global Maxwellian.

这项工作致力于研究具有流入边界条件的稳态玻尔兹曼方程的正解的存在性和流体动力学极限。证明是基于一个${\mathrm{大号}}^6-{\mathrm{大号}}^{\mathrm{\infty }}$Esposito 等人开发的框架。[流体动力学极限中玻尔兹曼方程的平稳解。安 PDE。2018;4(1):1-119] 和一种改进的正性保持方案，用于推导具有流入边界条件和外力的解的正性。具有狄利克雷边界条件的不可压缩的 Navier-Stokes-Fourier 极限对于流入边界数据作为全局 Maxwellian 的小扰动是合理的。