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

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 $L^6-L^{\mathrm{\infty }}$ 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.

Keywords:

• Steady Boltzmann equation
• in-flow boundary condition
• positivity preserving scheme
• incompressible Navier–Stokes–Fourier limit

2010 Mathematics Subject Classifications:

• 35Q20
• 76P05

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).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11371147], Foundation for Basic and Applied Basic Research of Guangdong [2020A1515011562] and Fundamental Research Funds for the Central Universities [2019MS112].