Abstract
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 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.
Acknowledgements
The research is supported by NSFC under the grant number 11371147, Fundamental Research Funds for the Central Universities under the grant number 2019MS112, and Foundation for Basic and Applied Basic Research of Guangdong under the grant number 2020A1515011562.
Disclosure statement
No potential conflict of interest was reported by the author(s).