A priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regime

We show a priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regime using the framework of Hairer's regularity structures theory. Assuming the model bounds our estimates imply global existence of solutions and existence of an invariant measure. We extend the method developed for the usual heat operator by Chandra, Moinat and Weber [CMW23] to the fractional heat operator, thereby treating a more physically relevant model. A key ingredient in this work is the development of localised multilevel Schauder estimates for the fractional heat operator which is not covered by Hairer's original work. Furthermore, the algebraic arguments from [CMW23] are streamlined significantly.