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 $\hat{I}\pm \sim {\epsilon }^{1/2}$ [Jiang-Masmoudi2017]. We establish global in time strong diffusive limit for all accommodation coefficients $\hat{I}\pm \in [0,1]$ within strong solutions framework. The main novelties of our proof include: (1) a $\epsilon$-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 $\hat{I}\pm \ll \epsilon$.

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