The Oberbeck-Boussinesq system with non-local boundary conditions

We consider the Oberbeck--Boussinesq system with non--local boundary conditions arising as a singular limit of the full Navier--Stokes--Fourier system in the regime of low Mach and low Froude number. The existence of strong solutions is shown on a maximal time interval $[0, T_{\rm max})$. Moreover, $T_{\rm max} = \infty$ in the two dimensional setting.