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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 aframework 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.

Positive solutions and hydrodynamic limit of the steady Boltzmann equation with in-flow boundary