In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = \Delta^{ \alpha/2} u + b \cdot \nabla u + f, $$ where $\Delta^{ \alpha/2}$ 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.
Categories: math.AP math.PR