Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh-Bénard convection (RBC) exhibits the classical [Formula: see text] scaling as the Rayleigh number [Formula: see text] with Prandtl number unity, consistent with Malkus-Howard's marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin-Doering-Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same [Formula: see text] scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the [Formula: see text] scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.
