Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition

Abstract

While weak diffusive limit from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system was established for the Maxwell boundary condition within renormalized solutions framework [Saint.Raymond2009][Jiang-Masmoudi2017], the corresponding strong diffusive limit has remained outstanding except when the accommodation coefficient $\alpha \sim \varepsilon^{1/2}$ [Jiang-Masmoudi2017]. We establish global in time strong diffusive limit for all accommodation coefficients $\alpha \in [0, 1]$ within strong solutions framework. The main novelties of our proof include: (1) a $\varepsilon$-stretching method for reduction to a single-bounce $L^\infty$ estimate; (2) a dissipation estimate for a carefully constructed rotating Maxwellian in the near-specular regime $\alpha \ll \varepsilon$.