Context. The anomalously large radii of hot Jupiters has long been a mystery. However, by combining both theoretical arguments and 2D models, a recent study has suggested that the vertical advection of potential temperature leads to a hotter adiabatic temperature profile in the deep atmosphere than the profile obtained with standard 1D models. Aims. In order to confirm the viability of that scenario, we extend this investigation to 3D, time-dependent models. Methods. We use a 3D general circulation model DYNAMICO to perform a series of calculations designed to explore the formation and structure of the driving atmospheric circulations, and detail how it responds to changes in both the upper and deep atmospheric forcing. Results. In agreement with the previous, 2D study, we find that a hot adiabat is the natural outcome of the long-term evolution of the deep atmosphere. Integration times of the order of 1500 yr are needed for that adiabat to emerge from an isothermal atmosphere, explaining why it has not been found in previous hot Jupiter studies. Models initialised from a hotter deep atmosphere tend to evolve faster toward the same final state. We also find that the deep adiabat is stable against low-levels of deep heating and cooling, as long as the Newtonian cooling timescale is longer than ~3000 yr at 200 bar. Conclusions. We conclude that steady-state vertical advection of potential temperature by deep atmospheric circulations constitutes a robust mechanism to explain the inflated radii of hot Jupiters. We suggest that future models of hot Jupiters be evolved for a longer time than currently done, and when possible that models initialised with a hot deep adiabat be included. We stress that this mechanism stems from the advection of entropy by irradiation-induced mass flows and does not require a (finely tuned) dissipative process, in contrast with most previously suggested scenarios.