Partially dissipative 2D Boussinesq equations with Navier type boundary conditions

This paper concerns itself with two systems of the 2D Boussinesq equations with partial dissipation in bounded domains with the Navier type boundary conditions. We attempt to achieve two main goals: first, to prove the global existence and uniqueness under minimal regularity assumptions on the initial data; and second, to provide a direct and transparent approach that explicitly reveals the impacts of the Navier boundary conditions. The 2D Boussinesq equations with partial dissipation have attracted considerable interests in the last few years, although most of the results are aimed at sufficiently regular solutions in the whole space or periodic domains. Larios et al. (2013) made serious efforts to minimize the regularity assumptions necessary for the uniqueness of solutions in the spatially periodic setting. In contrast to the whole space and the periodic domains, the Navier boundary conditions generate boundary terms and require compatibility conditions. In addition, due to the lack of boundary conditions for the pressure, we resort to the existence and regularity result on the associated Stokes problem with Navier boundary conditions. The uniqueness relies on the Yudovich techniques and the introduction of a lower regularity counterpart of the temperature.