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Harnack inequalities for nonlocal operators with supercritical drifts and their applications

Zhen-Qing Chen, Xicheng Zhang

TL;DR

The paper develops Harnack-type estimates for weak solutions to a nonlocal parabolic equation with a divergence-free drift in critical or supercritical regimes, enabling a robust regularity theory even when the source is distributional. By embedding the problem in De Giorgi–Moser frameworks and proving local upper bounds and weak Harnack inequalities, it derives global $L^\infty$ bounds and Hölder continuity, including in the challenging critical case. These analytic results are then applied to stochastic models, establishing well-posedness for critical stochastic quasi-geostrophic equations, weak solutions for 2D fractional Navier–Stokes with measure-valued data, and the well-posedness of generalized martingale problems in the critical setting. The combination of nonlocal diffusion, rough drifts, and distributional forcing yields a versatile toolkit with implications for stochastic fluid dynamics and kinetic-type SDEs, supported by Krylov-type estimates and careful functional-analytic arguments.

Abstract

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

Harnack inequalities for nonlocal operators with supercritical drifts and their applications

TL;DR

The paper develops Harnack-type estimates for weak solutions to a nonlocal parabolic equation with a divergence-free drift in critical or supercritical regimes, enabling a robust regularity theory even when the source is distributional. By embedding the problem in De Giorgi–Moser frameworks and proving local upper bounds and weak Harnack inequalities, it derives global bounds and Hölder continuity, including in the challenging critical case. These analytic results are then applied to stochastic models, establishing well-posedness for critical stochastic quasi-geostrophic equations, weak solutions for 2D fractional Navier–Stokes with measure-valued data, and the well-posedness of generalized martingale problems in the critical setting. The combination of nonlocal diffusion, rough drifts, and distributional forcing yields a versatile toolkit with implications for stochastic fluid dynamics and kinetic-type SDEs, supported by Krylov-type estimates and careful functional-analytic arguments.

Abstract

In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: where denotes the fractional Laplacian, is a divergence-free vector field in a critical or supercritical regularity regime, and 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.

Paper Structure

This paper contains 16 sections, 33 theorems, 374 equations.

Table of Contents

  1. Introduction
  2. Stochastic quasi-geostrophic equations
  3. Fractional Navier-Stokes equation in ${\mathbb R}^2$
  4. Main results
  5. Organization of the paper
  6. Preliminary
  7. Function spaces
  8. De Giorgi and Moser classes
  9. Local upper bound of sub-solutions: Proof of Theorem \ref{['Le32']}
  10. Weak Harnack inequality for super-solutions
  11. Global $L^\infty$-estimate and Hölder regularity
  12. Global well-posedness
  13. Hölder regularity for $\alpha\in(0,1]$
  14. Critical stochastic quasi-geostrophic equations with additive noises
  15. Existence of weak solutions to DDSDE
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $\alpha \in (0, 2)$, $\beta\in[0, \alpha/2)$ and $(q_0,{\boldsymbol{p}}_0)\in{\mathbb I}_\alpha^\beta$. Under KK0 and (H$_b$), for any $p>0$, there is a constant $C=C(\Theta,\beta,q_0,{\boldsymbol{p}}_0, p)>0$ such that for any sub-solution $u$ of PDE PDE0 and $t\in{\mathbb R}$, where ${\mathcal{I}}_t:={\mathbbm 1}_{(-\infty,t]}$ and for $j=1,2$, $\chi_j\in C^\infty_c(Q_{2j})$, are nonnegati

Theorems & Definitions (73)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 63 more