Bypassing H\"older super-critcality barriers in viscous, incompressible fluids

This is the second in a series of papers where we analyze the incompressible Navier-Stokes equations in H\"older spaces. We obtain, to our knowledge, the very first genuinely super-critical regularity criterion for this system of equations in any dimension $d\geq3$ and in the absence of physical boundaries. For \emph{any} $\beta\in(0,1)$, we show that $L_t^1C_x^{0,\beta}$ solutions emanating from smooth initial data do not develop any singularities. The novelty stems from obtaining new bounds on the fundamental solution associated with a one-dimensional drift-diffusion equation in the presence of destabilizing singular lower order terms. Such a bound relies heavily on the symmetry and pointwise structure of the problem, where the drift term is shown to ``enhance'' the parabolic nature of the equation, allowing us to break the criticality barrier. Coupled with a subtle regularity estimate for the pressure courtesy of Silvestre, we are able to treat the (incompressible) Navier-Stokes equation as a perturbation of the classical drift-diffusion problem. This is achieved by propagating moduli of continuity as was done in our previous work, based on the elegant ideas introduced by Kiselev, Nazarov, Volberg and Shterenberg.