Harnack inequalities for nonlocal operators with supercritical drifts and their applications

TL;DR

This paper establishes Harnack inequalities for nonlocal equations with supercritical drift terms and applies these results to prove well-posedness and existence results for several critical stochastic PDEs and fluid dynamics models.

Contribution

It introduces new Harnack estimates for nonlocal operators with supercritical drifts and applies them to solve open problems in stochastic PDEs and fluid mechanics.

Findings

Proved Harnack inequalities for nonlocal equations with supercritical drifts.

Established well-posedness for critical stochastic quasi-geostrophic equations.

Proved existence of weak solutions for fractional Navier-Stokes with measure-valued vorticity.

Abstract

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.