Scaling laws for Rayleigh-B\'enard convection between Navier-slip boundaries

We consider the two-dimensional Rayeigh-B\'enard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the slip-length and on the Prandtl number, our upper bounds indicate five possible scaling laws: Nu ~ (Lₛ⁻¹Ra)1/3, Nu ~ (Lₛ-2/5Ra)5/13, Nu ~ Ra5/12, Nu ~ Pr-1/6 (Lₛ-4/3Ra)1/2 and Nu ~ Pr-1/6 (Lₛ-1/3Ra)1/2