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 $[0,T_{\mathrm{m}\mathrm{a}\mathrm{x}})$. Moreover, $T_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{\infty }$ in the two dimensional setting.