Harnack inequalities for nonlocal operators with supercritical drifts and their applications

In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}^{\alpha \mathrm{/}2}u+b\cdot \mathrm{\nabla }u+f,$$ where ${\mathrm{\Delta }}^{\alpha \mathrm{/}2}$ denotes the fractional Laplacian, $b$ is a divergence-free vector field in a critical or supercritical regularity regime, and $f$ is a distribution in a fractional Sobolev space with negative indices. As applications of the analytical results obtained in this paper, we establish the well-posedness of critical stochastic quasi-geostrophic equations driven by additive Brownian noise, prove the existence of weak solutions to the two-dimensional fractional Navier–Stokes equations with measure-valued initial vorticity, and demonstrate the well-posedness of generalized martingale problems associated with critical stochastic differential equations.