Regularity properties of passive scalars with rough divergence-free drifts

TL;DR

This paper establishes sharp regularity conditions for passive scalars with divergence-free drifts in Lebesgue spaces, ensuring properties like local boundedness and Harnack inequalities, and constructs counterexamples to demonstrate the sharpness of these conditions.

Contribution

It provides new precise criteria on divergence-free drifts in Lebesgue spaces for regularity properties of passive scalars, including sharpness via counterexamples.

Findings

Sharp conditions on divergence-free drifts for regularity properties.

Demonstration of these properties for specific Lebesgue space classes.

Construction of counterexamples showing the limits of these conditions.

Abstract

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $\frac{2}{q} + \frac{n}{p} < 2$, or $L^p_x L^q_t$, where $\frac{3}{q} + \frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of drifts with `bounded total speed' is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before' they can be dissipated.