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We derive the incompressible Euler equations with heat convection with theno-penetration boundary condition from the Boltzmann equation with the diffuseboundary in the hydrodynamic limit for the scale of large Reynold number.Inspired by the recent framework in [30], we consider the Navier-Stokes-Fouriersystem with no-slip boundary conditions as an intermediary approximation anddevelop a Hilbert-type expansion of the Boltzmann equation around the globalMaxwellian that allows the nontrivial heat transfer by convection in the limit.To justify our expansion and the limit, a new direct estimate of the heat fluxand its derivatives in the Navier-Stokes-Fourier system is established adoptinga recent Green's function approach in the study of the inviscid limit.

Passage from the Boltzmann equation with Diffuse Boundary to the Incompressible Euler equation with Heat Convection